TERMS STARTING WITH
Axiomatic Architecture Description Language "language, architecture, parallel" (AADL) A language allowing concise modular specification of {multiprocessor} architectures from the compiler/operating-system interface level down to chip level. AADL is rich enough to specify target architectures while providing a concise model for clocked {microarchitectures}. ["AADL: A Net-Based Specification Method for Computer Architecture Design", W. Damm et al in Languages for Parallel Architectures, J.W. deBakker ed, Wiley, 1989]. (2003-06-30)
Axiomatic Architecture Description Language ::: (language, architecture, parallel) (AADL) A language allowing concise modular specification of multiprocessor architectures from the enough to specify target architectures while providing a concise model for clocked microarchitectures.[AADL: A Net-Based Specification Method for Computer Architecture Design, W. Damm et al in Languages for Parallel Architectures, J.W. deBakker ed, Wiley, 1989].(2003-06-30)
AXIOM "language" A commercially available subset of the {Scratchpad}, {symbolic mathematics} system from {IBM}. ["Axiom - The Scientific Computing System", R. Jenks et al, Springer 1992]. [Relationship with {AXIOM*}?] (1995-02-21)
AXIOM* "mathematics, tool" A {symbolic mathematics} system. {A
AXIOM* ::: (mathematics, tool) A symbolic mathematics system.A
Axiom of Choice "logic" (AC, or "Choice") An {axiom} of {set theory}: If X is a set of sets, and S is the union of all the elements of X, then there exists a function f:X -" S such that for all non-empty x in X, f(x) is an element of x. In other words, we can always choose an element from each set in a set of sets, simultaneously. Function f is a "choice function" for X - for each x in X, it chooses an element of x. Most people's reaction to AC is: "But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or whatever". Indeed, for any {finite} set of sets, we can simply consider each set in turn and pick an arbitrary element in some such way. We can also construct a choice function for most simple {infinite sets} of sets if they are generated in some regular way. However, there are some infinite sets for which the construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases. For example, if we express the {real number} line R as the union of many "copies" of the {rational numbers}, Q, namely Q, Q+a, Q+b, and infinitely (in fact uncountably) many more, where a, b, etc. are {irrational numbers} no two of which differ by a rational, and Q+a == {q+a : q in Q} we cannot pick an element of each of these "copies" without AC. An example of the use of AC is the theorem which states that the {countable} union of countable sets is countable. I.e. if X is countable and every element of X is countable (including the possibility that they're finite), then the sumset of X is countable. AC is required for this to be true in general. Even if one accepts the axiom, it doesn't tell you how to construct a choice function, only that one exists. Most mathematicians are quite happy to use AC if they need it, but those who are careful will, at least, draw attention to the fact that they have used it. There is something a little odd about Choice, and it has some alarming consequences, so results which actually "need" it are somehow a bit suspicious, e.g. the {Banach-Tarski paradox}. On the other side, consider {Russell's Attic}. AC is not a {theorem} of {Zermelo Fränkel set theory} (ZF). Gödel and Paul Cohen proved that AC is independent of ZF, i.e. if ZF is consistent, then so are ZFC (ZF with AC) and ZF(~C) (ZF with the negation of AC). This means that we cannot use ZF to prove or disprove AC. (2003-07-11)
Axiom of Choice ::: (mathematics) (AC, or Choice) An axiom of set theory:If X is a set of sets, and S is the union of all the elements of X, then there exists a function f:X -> S such that for all non-empty x in X, f(x) is an element of x.In other words, we can always choose an element from each set in a set of sets, simultaneously.Function f is a choice function for X - for each x in X, it chooses an element of x.Most people's reaction to AC is: But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases.For example, if we express the real number line R as the union of many copies of the rational numbers, Q, namely Q, Q+a, Q+b, and infinitely (in fact uncountably) many more, where a, b, etc. are irrational numbers no two of which differ by a rational, and Q+a == {q+a : q in Q} we cannot pick an element of each of these copies without AC.An example of the use of AC is the theorem which states that the countable union of countable sets is countable. I.e. if X is countable and every element of X is countable (including the possibility that they're finite), then the sumset of X is countable. AC is required for this to be true in general.Even if one accepts the axiom, it doesn't tell you how to construct a choice function, only that one exists. Most mathematicians are quite happy to use AC if somehow a bit suspicious, e.g. the Banach-Tarski paradox. On the other side, consider Russell's Attic.AC is not a theorem of Zermelo Fr�nkel set theory (ZF). G�del and Paul Cohen proved that AC is independent of ZF, i.e. if ZF is consistent, then so are ZFC (ZF with AC) and ZF(~C) (ZF with the negation of AC). This means that we cannot use ZF to prove or disprove AC.(2003-07-11)
Axiom of Comprehension "logic" An {axiom schema} of {set theory} which states: if P(x) is a {property} then {x : P} is a set. I.e. all the things with some property form a set. Acceptance of this axiom leads to {Russell's Paradox} which is why {Zermelo set theory} replaces it with a restricted form. (1995-03-31)
Axiom of Comprehension ::: (mathematics) An axiom schema of set theory which states: if P(x) is a property then {x : P} is a set. I.e. all the things with some property form a set.Acceptance of this axiom leads to Russell's Paradox which is why Zermelo set theory replaces it with a restricted form. (1995-03-31)
axiom ::: a. --> A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted; as, "The whole is greater than a part;" "A thing can not, at the same time, be and not be."
An established principle in some art or science, which, though not a necessary truth, is universally received; as, the axioms of political economy.
axiom: A statement that is neither proven, nor is it intended to be proven. Usually they are considered to be self-evident, although axioms are also constructed to explore the possibilities of hypothetical situations where we accept certain assertions to be true, and what theorems could be derived from it.
axiomatic ::: a. --> Alt. of Axiomatical
axiomatical ::: a. --> Of or pertaining to an axiom; having the nature of an axiom; self-evident; characterized by axioms.
axiomatically ::: adv. --> By the use of axioms; in the form of an axiom.
axiomatic semantics ::: (theory) A set of assertions about properties of a system and how they are effected by program execution. The axiomatic semantics of a program could axiomatic semantics is a set of invariants on the state which the state transformer satisfies.E.g. for a function with the type: sort_list :: [T] -> [T] list, and a postcondition that the return value is a list that is sorted.One interesting use of axiomatic semantics is to have a language that has a finitely computable sublanguage that is used for specifying pre and post conditions, and then have the compiler prove that the program will satisfy those conditions.See also operational semantics, denotational semantics. (1995-11-09)
axiomatic semantics "theory" A set of assertions about properties of a system and how they are effected by program execution. The axiomatic semantics of a program could include pre- and post-conditions for operations. In particular if you view the program as a state transformer (or collection of state transformers), the axiomatic semantics is a set of invariants on the state which the state transformer satisfies. E.g. for a function with the type: sort_list :: [T] -" [T] we might give the precondition that the argument of the function is a list, and a postcondition that the return value is a list that is sorted. One interesting use of axiomatic semantics is to have a language that has a {finitely computable} sublanguage that is used for specifying pre and post conditions, and then have the compiler prove that the program will satisfy those conditions. See also {operational semantics}, {denotational semantics}. (1995-11-09)
axiomatic set theory "theory" One of several approaches to {set theory}, consisting of a {formal language} for talking about sets and a collection of {axioms} describing how they behave. There are many different {axiomatisations} for set theory. Each takes a slightly different approach to the problem of finding a theory that captures as much as possible of the intuitive idea of what a set is, while avoiding the {paradoxes} that result from accepting all of it, the most famous being {Russell's paradox}. The main source of trouble in naive set theory is the idea that you can specify a set by saying whether each object in the universe is in the "set" or not. Accordingly, the most important differences between different axiomatisations of set theory concern the restrictions they place on this idea (known as "comprehension"). {Zermelo Fränkel set theory}, the most commonly used axiomatisation, gets round it by (in effect) saying that you can only use this principle to define subsets of existing sets. NBG (von Neumann-Bernays-Goedel) set theory sort of allows comprehension for all {formulae} without restriction, but distinguishes between two kinds of set, so that the sets produced by applying comprehension are only second-class sets. NBG is exactly as powerful as ZF, in the sense that any statement that can be formalised in both theories is a theorem of ZF if and only if it is a theorem of ZFC. MK (Morse-Kelley) set theory is a strengthened version of NBG, with a simpler axiom system. It is strictly stronger than NBG, and it is possible that NBG might be consistent but MK inconsistent. {NF (http://math.boisestate.edu/~holmes/holmes/nf.html)} ("New Foundations"), a theory developed by Willard Van Orman Quine, places a very different restriction on comprehension: it only works when the formula describing the membership condition for your putative set is "stratified", which means that it could be made to make sense if you worked in a system where every set had a level attached to it, so that a level-n set could only be a member of sets of level n+1. (This doesn't mean that there are actually levels attached to sets in NF). NF is very different from ZF; for instance, in NF the universe is a set (which it isn't in ZF, because the whole point of ZF is that it forbids sets that are "too large"), and it can be proved that the {Axiom of Choice} is false in NF! ML ("Modern Logic") is to NF as NBG is to ZF. (Its name derives from the title of the book in which Quine introduced an early, defective, form of it). It is stronger than ZF (it can prove things that ZF can't), but if NF is consistent then ML is too. (2003-09-21)
axiomatic set theory ::: (theory) One of several approaches to set theory, consisting of a formal language for talking about sets and a collection of axioms describing how they behave.There are many different axiomatisations for set theory. Each takes a slightly different approach to the problem of finding a theory that captures as much as possible of the intuitive idea of what a set is, while avoiding the paradoxes that result from accepting all of it, the most famous being Russell's paradox.The main source of trouble in naive set theory is the idea that you can specify a set by saying whether each object in the universe is in the set or not. set theory concern the restrictions they place on this idea (known as comprehension).Zermelo Fr�nkel set theory, the most commonly used axiomatisation, gets round it by (in effect) saying that you can only use this principle to define subsets of existing sets.NBG (von Neumann-Bernays-Goedel) set theory sort of allows comprehension for all formulae without restriction, but distinguishes between two kinds of set, so formalised in both theories is a theorem of ZF if and only if it is a theorem of ZFC.MK (Morse-Kelley) set theory is a strengthened version of NBG, with a simpler axiom system. It is strictly stronger than NBG, and it is possible that NBG might be consistent but MK inconsistent. (New Foundations), a theory developed by Willard Van Orman Quine, places a very different restriction on comprehension: it only works when the point of ZF is that it forbids sets that are too large), and it can be proved that the Axiom of Choice is false in NF!ML (Modern Logic) is to NF as NBG is to ZF. (Its name derives from the title of the book in which Quine introduced an early, defective, form of it). It is stronger than ZF (it can prove things that ZF can't), but if NF is consistent then ML is too.(2003-09-21)
axiom "logic" A {well-formed formula} which is taken to be true without proof in the construction of a {theory}. Compare: {lemma}. (1995-03-31)
axiom ::: (logic) A well-formed formula which is taken to be true without proof in the construction of a theory.Compare: lemma. (1995-03-31)
axiom schema "logic" A {formula} in the language of an {axiomatic system}, containing one or more. These {metasyntactic variables} (or "{schematic variables}") that stand for terms or subformulae. An example is the {Axiom of Comprehension}. (2009-02-10)
TERMS ANYWHERE
AADL {Axiomatic Architecture Description Language}
A cardinal number is inductive if it is a member of every class t of cardinal numbers which has the two properties, (1) 0∈ t, and (2) for all x, if x∈ t and y is the sum of x and 1, then y∈ t. In other (less exact) words, the inductive cardinal numbers are those which can be reached from 0 by successive additions of 1. A class b is infinite if there is a class a, different from b, such that a ⊂ b and a is equivalent to b. In the contrary case b is finite. The cardinal number of an infinite class is said to be infinite, and of a finite class, finite. It can be proved that every inductive cardinal number is finite, and, with the aid of the axiom of choice, that every finite cardinal number is inductive.
According to an important theorem of Gödel, the functional calculus of order omega with the axiom of infinity added, if consistent, is incomplete in the sense that there are formulas A containing no free variables, such that neither A nor ∼A is a theorem. The same thing holds of any logistic system obtained by adding new primitive formulas and primitive rules of inference, provided only that the effective (recursive) character of the formal construction of the system is retained. Thus the system is not only incomplete but, in the indicated sense, incompletable. The same thing holds also of a large variety of logistic systems which could be considered as acceptable substitutes for the functional calculus of order omega with axiom of infinity; in particular the Zermelo set theory (§ 9 below) is in the same sense incomplete and incompletable.
According to a view which is widely held by mathematicians, it is characteristic of a mathematical discipline that it begins with a set of undefined elements, properties, functions, and relations, and a set of unproved propositions (called axioms or postulates) involving them; and that from these all other propositions (called theorems) of the discipline are to be derived by the methods of formal logic. On its face, as thus stated, this view would identify mathematics with applied logic. It is usually added, however, that the undefined terms, which appear in the role of names of undefined elements, etc., are not really names of particulars at all but are variables, and that the theorems are to be regarded as proved for any values of these variables which render the postulates true. If then each theorem is replaced by the proposition embodying the implication from the conjunction of the postulates to the theorem in question, we have a reduction of mathematics to pure logic. (For a particular example of a set of postulates for a mathematical discipline see the article Arithmetic, foundations of.)
Actio in Distans (Latin) Action at a distance. Can force be transmitted across an empty space? On the automechanical theory of the universe, such action is inexplicable and yet inevitable, for if the universe consists entirely of matter made of atoms separated from each other by empty spaces, the transmission of force from one atom to another cannot be explained except by supposing some medium to intervene. If this medium is atomic, the old difficulty reappears; if it is continuous, there is no reason for supposing it, since matter might in the first place have been supposed to be continuous. Thus if we choose to represent reality as a system of points in space, we must assume actio in distans as an axiom. The difficulty that a body cannot act where it is not, may be gotten over by stating that wherever it can act, there it is. Scientific theories, carried to a logical conclusion, support the idea that all things in the universe are connected with each other, so that whatever affects one part affects every other part. Notions of physical space do not enter to the realm of mind, thought, and feeling.
aleph 0 "mathematics" The {cardinality} of the first {infinite} {ordinal}, {omega} (the number of {natural numbers}). Aleph 1 is the cardinality of the smallest {ordinal} whose cardinality is greater than aleph 0, and so on up to aleph omega and beyond. These are all kinds of {infinity}. The {Axiom of Choice} (AC) implies that every set can be {well-ordered}, so every {infinite} {cardinality} is an aleph; but in the absence of AC there may be sets that can't be well-ordered (don't posses a {bijection} with any {ordinal}) and therefore have cardinality which is not an aleph. These sets don't in some way sit between two alephs; they just float around in an annoying way, and can't be compared to the alephs at all. No {ordinal} possesses a {surjection} onto such a set, but it doesn't surject onto any sufficiently large ordinal either. (1995-03-29)
aleph 0 ::: (mathematics) The cardinality of the first infinite ordinal, omega (the number of natural numbers).Aleph 1 is the cardinality of the smallest ordinal whose cardinality is greater than aleph 0, and so on up to aleph omega and beyond. These are all kinds of infinity.The Axiom of Choice (AC) implies that every set can be well-ordered, so every infinite cardinality is an aleph; but in the absence of AC there may be sets that can't be well-ordered (don't posses a bijection with any ordinal) and therefore have cardinality which is not an aleph.These sets don't in some way sit between two alephs; they just float around in an annoying way, and can't be compared to the alephs at all. No ordinal possesses a surjection onto such a set, but it doesn't surject onto any sufficiently large ordinal either. (1995-03-29)
Also reasoning engine, rules engine, or simply reasoner. ::: A piece of software able to infer logical consequences from a set of asserted facts or axioms. The notion of a semantic reasoner generalizes that of an inference engine, by providing a richer set of mechanisms to work with. The inference rules are commonly specified by means of an ontology language, and often a description logic language. Many reasoners use first-order predicate logic to perform reasoning; inference commonly proceeds by forward chaining and backward chaining.
An equivalent assumption, called by Russell the multiplicative axiom and afterwards adopted by Zermelo as a statement of his Auswahl-prinzip, is as follows: Given a class K whose members are non-empty classes no two of which have a member in common, there exists a class A (the Auswahlmenge) all of whose members are members of members of K and which has one and only one member in common with each member of K. Proof of equivalence of the multiplicative axiom to the axiom of choice is due to Zermelo. -- A.C.
Anticipations of experience: In Kant's Crit. of pure Reason Antizipationen der Wahrnehmung) the second of two synthetic principles of the understanding (the other being "Axioms of Intuition") by which the mind is able to determine something a priori in regard to what is in itself empirical. While the mind cannot anticipate the specific qualities which are to be experienced, we can, nevertheless, Kant holds, predetermine or anticipate any sense experience that "in all appearances the real, which is an object of sensation, has intensive magnitude or degree." -- O.F.K.
Aristotle's Illusion: See Aristotle's Experiment. Arithmetic, foundations of: Arithmetic (i.e., the mathematical theory of the non-negative integers, 0, 1, 2, . . .) may be based on the five following postulates, which are due to Peano (and Dedekind, from whom Peano's ideas were partly derived): N(0) N(x) ⊃x N(S(x)). N(x) ⊃x [N(y) ⊃y [[S(x) = S(y)] ⊃x [x = y]]]. N(x) ⊃x ∼[S(x) = 0]. F(0)[N(x)F(x) ⊃x F(S(x))] ⊃F [N(x) ⊃x F(x)] The undefined terms are here 0, N, S, which may be interpreted as denoting, respectively, the non-negative integer 0, the propositional function to be a non-negative integer, and the function +1 (so that S(x) is x+l). The underlying logic may be taken to be the functional calculus of second order (Logic, formal, § 6), with the addition of notations for descriptions and for functions from individuals to individuals, and the individual constant 0, together with appropriate modifications and additions to the primitive formulas and primitive rules of inference (the axiom of infinity is not needed because the Peano postulates take its place). By adding the five postulates of Peano as primitive formulas to this underlying logic, a logistic system is obtained which is adequate to extant elementary number theory (arithmetic) and to all methods of proof which have found actual employment in elementary number theory (and are normally considered to belong to elementary number theory). But of course, the system, if consistent, is incomplete in the sense of Gödel's theorem (Logic, formal, § 6).
A
Assumption: A proposition which is taken or posed in order to draw inferences from it; or the act of so taking, posing, or assuming a proposition. The motive for an assumption may be (but need not necessarily be) a belief in the truth, or possible truth, of the proposition assumed; or the motive may be an attempt to refute the proposition by reductio ad absurdum (q.v.). The word assumption has also sometimes been used as a synonym of axiom, or postulate (see the article Mathematics). -- A.C.
Aufklärung: In general, this German word and its English equivalent Enlightenment denote the self-emancipation of man from mere authority, prejudice, convention and tradition, with an insistence on freer thinking about problems uncritically referred to these other agencies. According to Kant's famous definition "Enlightenment is the liberation of man from his self-caused state of minority, which is the incapacity of using one's understanding without the direction of another. This state of minority is caused when its source lies not in the lack of understanding, but in the lack of determination and courage to use it without the assistance of another" (Was ist Aufklärung? 1784). In its historical perspective, the Aufklärung refers to the cultural atmosphere and contrlbutions of the 18th century, especially in Germany, France and England [which affected also American thought with B. Franklin, T. Paine and the leaders of the Revolution]. It crystallized tendencies emphasized by the Renaissance, and quickened by modern scepticism and empiricism, and by the great scientific discoveries of the 17th century. This movement, which was represented by men of varying tendencies, gave an impetus to general learning, a more popular philosophy, empirical science, scriptural criticism, social and political thought. More especially, the word Aufklärung is applied to the German contributions to 18th century culture. In philosophy, its principal representatives are G. E. Lessing (1729-81) who believed in free speech and in a methodical criticism of religion, without being a free-thinker; H. S. Reimarus (1694-1768) who expounded a naturalistic philosophy and denied the supernatural origin of Christianity; Moses Mendelssohn (1729-86) who endeavoured to mitigate prejudices and developed a popular common-sense philosophy; Chr. Wolff (1679-1754), J. A. Eberhard (1739-1809) who followed the Leibnizian rationalism and criticized unsuccessfully Kant and Fichte; and J. G. Herder (1744-1803) who was best as an interpreter of others, but whose intuitional suggestions have borne fruit in the organic correlation of the sciences, and in questions of language in relation to human nature and to national character. The works of Kant and Goethe mark the culmination of the German Enlightenment. Cf. J. G. Hibben, Philosophy of the Enlightenment, 1910. --T.G. Augustinianism: The thought of St. Augustine of Hippo, and of his followers. Born in 354 at Tagaste in N. Africa, A. studied rhetoric in Carthage, taught that subject there and in Rome and Milan. Attracted successively to Manicheanism, Scepticism, and Neo-Platontsm, A. eventually found intellectual and moral peace with his conversion to Christianity in his thirty-fourth year. Returning to Africa, he established numerous monasteries, became a priest in 391, Bishop of Hippo in 395. Augustine wrote much: On Free Choice, Confessions, Literal Commentary on Genesis, On the Trinity, and City of God, are his most noted works. He died in 430. St. Augustine's characteristic method, an inward empiricism which has little in common with later variants, starts from things without, proceeds within to the self, and moves upwards to God. These three poles of the Augustinian dialectic are polarized by his doctrine of moderate illuminism. An ontological illumination is required to explain the metaphysical structure of things. The truth of judgment demands a noetic illumination. A moral illumination is necessary in the order of willing; and so, too, an lllumination of art in the aesthetic order. Other illuminations which transcend the natural order do not come within the scope of philosophy; they provide the wisdoms of theology and mysticism. Every being is illuminated ontologically by number, form, unity and its derivatives, and order. A thing is what it is, in so far as it is more or less flooded by the light of these ontological constituents. Sensation is necessary in order to know material substances. There is certainly an action of the external object on the body and a corresponding passion of the body, but, as the soul is superior to the body and can suffer nothing from its inferior, sensation must be an action, not a passion, of the soul. Sensation takes place only when the observing soul, dynamically on guard throughout the body, is vitally attentive to the changes suffered by the body. However, an adequate basis for the knowledge of intellectual truth is not found in sensation alone. In order to know, for example, that a body is multiple, the idea of unity must be present already, otherwise its multiplicity could not be recognized. If numbers are not drawn in by the bodily senses which perceive only the contingent and passing, is the mind the source of the unchanging and necessary truth of numbers? The mind of man is also contingent and mutable, and cannot give what it does not possess. As ideas are not innate, nor remembered from a previous existence of the soul, they can be accounted for only by an immutable source higher than the soul. In so far as man is endowed with an intellect, he is a being naturally illuminated by God, Who may be compared to an intelligible sun. The human intellect does not create the laws of thought; it finds them and submits to them. The immediate intuition of these normative rules does not carry any content, thus any trace of ontologism is avoided. Things have forms because they have numbers, and they have being in so far as they possess form. The sufficient explanation of all formable, and hence changeable, things is an immutable and eternal form which is unrestricted in time and space. The forms or ideas of all things actually existing in the world are in the things themselves (as rationes seminales) and in the Divine Mind (as rationes aeternae). Nothing could exist without unity, for to be is no other than to be one. There is a unity proper to each level of being, a unity of the material individual and species, of the soul, and of that union of souls in the love of the same good, which union constitutes the city. Order, also, is ontologically imbibed by all beings. To tend to being is to tend to order; order secures being, disorder leads to non-being. Order is the distribution which allots things equal and unequal each to its own place and integrates an ensemble of parts in accordance with an end. Hence, peace is defined as the tranquillity of order. Just as things have their being from their forms, the order of parts, and their numerical relations, so too their beauty is not something superadded, but the shining out of all their intelligible co-ingredients. S. Aurelii Augustini, Opera Omnia, Migne, PL 32-47; (a critical edition of some works will be found in the Corpus Scriptorum Ecclesiasticorum Latinorum, Vienna). Gilson, E., Introd. a l'etude de s. Augustin, (Paris, 1931) contains very good bibliography up to 1927, pp. 309-331. Pope, H., St. Augustine of Hippo, (London, 1937). Chapman, E., St. Augustine's Philos. of Beauty, (N. Y., 1939). Figgis, J. N., The Political Aspects of St. Augustine's "City of God", (London, 1921). --E.C. Authenticity: In a general sense, genuineness, truth according to its title. It involves sometimes a direct and personal characteristic (Whitehead speaks of "authentic feelings"). This word also refers to problems of fundamental criticism involving title, tradition, authorship and evidence. These problems are vital in theology, and basic in scholarship with regard to the interpretation of texts and doctrines. --T.G. Authoritarianism: That theory of knowledge which maintains that the truth of any proposition is determined by the fact of its having been asserted by a certain esteemed individual or group of individuals. Cf. H. Newman, Grammar of Assent; C. S. Peirce, "Fixation of Belief," in Chance, Love and Logic, ed. M. R. Cohen. --A.C.B. Autistic thinking: Absorption in fanciful or wishful thinking without proper control by objective or factual material; day dreaming; undisciplined imagination. --A.C.B. Automaton Theory: Theory that a living organism may be considered a mere machine. See Automatism. Automatism: (Gr. automatos, self-moving) (a) In metaphysics: Theory that animal and human organisms are automata, that is to say, are machines governed by the laws of physics and mechanics. Automatism, as propounded by Descartes, considered the lower animals to be pure automata (Letter to Henry More, 1649) and man a machine controlled by a rational soul (Treatise on Man). Pure automatism for man as well as animals is advocated by La Mettrie (Man, a Machine, 1748). During the Nineteenth century, automatism, combined with epiphenomenalism, was advanced by Hodgson, Huxley and Clifford. (Cf. W. James, The Principles of Psychology, Vol. I, ch. V.) Behaviorism, of the extreme sort, is the most recent version of automatism (See Behaviorism). (b) In psychology: Psychological automatism is the performance of apparently purposeful actions, like automatic writing without the superintendence of the conscious mind. L. C. Rosenfield, From Beast Machine to Man Machine, N. Y., 1941. --L.W. Automatism, Conscious: The automatism of Hodgson, Huxley, and Clifford which considers man a machine to which mind or consciousness is superadded; the mind of man is, however, causally ineffectual. See Automatism; Epiphenomenalism. --L.W. Autonomy: (Gr. autonomia, independence) Freedom consisting in self-determination and independence of all external constraint. See Freedom. Kant defines autonomy of the will as subjection of the will to its own law, the categorical imperative, in contrast to heteronomy, its subjection to a law or end outside the rational will. (Fundamental Principles of the Metaphysics of Morals, § 2.) --L.W. Autonomy of ethics: A doctrine, usually propounded by intuitionists, that ethics is not a part of, and cannot be derived from, either metaphysics or any of the natural or social sciences. See Intuitionism, Metaphysical ethics, Naturalistic ethics. --W.K.F. Autonomy of the will: (in Kant's ethics) The freedom of the rational will to legislate to itself, which constitutes the basis for the autonomy of the moral law. --P.A.S. Autonymy: In the terminology introduced by Carnap, a word (phrase, symbol, expression) is autonymous if it is used as a name for itself --for the geometric shape, sound, etc. which it exemplifies, or for the word as a historical and grammatical unit. Autonymy is thus the same as the Scholastic suppositio matertalis (q. v.), although the viewpoint is different. --A.C. Autotelic: (from Gr. autos, self, and telos, end) Said of any absorbing activity engaged in for its own sake (cf. German Selbstzweck), such as higher mathematics, chess, etc. In aesthetics, applied to creative art and play which lack any conscious reference to the accomplishment of something useful. In the view of some, it may constitute something beneficent in itself of which the person following his art impulse (q.v.) or playing is unaware, thus approaching a heterotelic (q.v.) conception. --K.F.L. Avenarius, Richard: (1843-1896) German philosopher who expressed his thought in an elaborate and novel terminology in the hope of constructing a symbolic language for philosophy, like that of mathematics --the consequence of his Spinoza studies. As the most influential apostle of pure experience, the posltivistic motive reaches in him an extreme position. Insisting on the biologic and economic function of thought, he thought the true method of science is to cure speculative excesses by a return to pure experience devoid of all assumptions. Philosophy is the scientific effort to exclude from knowledge all ideas not included in the given. Its task is to expel all extraneous elements in the given. His uncritical use of the category of the given and the nominalistic view that logical relations are created rather than discovered by thought, leads him to banish not only animism but also all of the categories, substance, causality, etc., as inventions of the mind. Explaining the evolution and devolution of the problematization and deproblematization of numerous ideas, and aiming to give the natural history of problems, Avenarius sought to show physiologically, psychologically and historically under what conditions they emerge, are challenged and are solved. He hypothesized a System C, a bodily and central nervous system upon which consciousness depends. R-values are the stimuli received from the world of objects. E-values are the statements of experience. The brain changes that continually oscillate about an ideal point of balance are termed Vitalerhaltungsmaximum. The E-values are differentiated into elements, to which the sense-perceptions or the content of experience belong, and characters, to which belongs everything which psychology describes as feelings and attitudes. Avenarius describes in symbolic form a series of states from balance to balance, termed vital series, all describing a series of changes in System C. Inequalities in the vital balance give rise to vital differences. According to his theory there are two vital series. It assumes a series of brain changes because parallel series of conscious states can be observed. The independent vital series are physical, and the dependent vital series are psychological. The two together are practically covariants. In the case of a process as a dependent vital series three stages can be noted: first, the appearance of the problem, expressed as strain, restlessness, desire, fear, doubt, pain, repentance, delusion; the second, the continued effort and struggle to solve the problem; and finally, the appearance of the solution, characterized by abating anxiety, a feeling of triumph and enjoyment. Corresponding to these three stages of the dependent series are three stages of the independent series: the appearance of the vital difference and a departure from balance in the System C, the continuance with an approximate vital difference, and lastly, the reduction of the vital difference to zero, the return to stability. By making room for dependent and independent experiences, he showed that physics regards experience as independent of the experiencing indlvidual, and psychology views experience as dependent upon the individual. He greatly influenced Mach and James (q.v.). See Avenarius, Empirio-criticism, Experience, pure. Main works: Kritik der reinen Erfahrung; Der menschliche Weltbegriff. --H.H. Averroes: (Mohammed ibn Roshd) Known to the Scholastics as The Commentator, and mentioned as the author of il gran commento by Dante (Inf. IV. 68) he was born 1126 at Cordova (Spain), studied theology, law, medicine, mathematics, and philosophy, became after having been judge in Sevilla and Cordova, physician to the khalifah Jaqub Jusuf, and charged with writing a commentary on the works of Aristotle. Al-mansur, Jusuf's successor, deprived him of his place because of accusations of unorthodoxy. He died 1198 in Morocco. Averroes is not so much an original philosopher as the author of a minute commentary on the whole works of Aristotle. His procedure was imitated later by Aquinas. In his interpretation of Aristotelian metaphysics Averroes teaches the coeternity of a universe created ex nihilo. This doctrine formed together with the notion of a numerical unity of the active intellect became one of the controversial points in the discussions between the followers of Albert-Thomas and the Latin Averroists. Averroes assumed that man possesses only a disposition for receiving the intellect coming from without; he identifies this disposition with the possible intellect which thus is not truly intellectual by nature. The notion of one intellect common to all men does away with the doctrine of personal immortality. Another doctrine which probably was emphasized more by the Latin Averroists (and by the adversaries among Averroes' contemporaries) is the famous statement about "two-fold truth", viz. that a proposition may be theologically true and philosophically false and vice versa. Averroes taught that religion expresses the (higher) philosophical truth by means of religious imagery; the "two-truth notion" came apparently into the Latin text through a misinterpretation on the part of the translators. The works of Averroes were one of the main sources of medieval Aristotelianlsm, before and even after the original texts had been translated. The interpretation the Latin Averroists found in their texts of the "Commentator" spread in spite of opposition and condemnation. See Averroism, Latin. Averroes, Opera, Venetiis, 1553. M. Horten, Die Metaphysik des Averroes, 1912. P. Mandonnet, Siger de Brabant et l'Averroisme Latin, 2d ed., Louvain, 1911. --R.A. Averroism, Latin: The commentaries on Aristotle written by Averroes (Ibn Roshd) in the 12th century became known to the Western scholars in translations by Michael Scottus, Hermannus Alemannus, and others at the beginning of the 13th century. Many works of Aristotle were also known first by such translations from Arabian texts, though there existed translations from the Greek originals at the same time (Grabmann). The Averroistic interpretation of Aristotle was held to be the true one by many; but already Albert the Great pointed out several notions which he felt to be incompatible with the principles of Christian philosophy, although he relied for the rest on the "Commentator" and apparently hardly used any other text. Aquinas, basing his studies mostly on a translation from the Greek texts, procured for him by William of Moerbecke, criticized the Averroistic interpretation in many points. But the teachings of the Commentator became the foundation for a whole school of philosophers, represented first by the Faculty of Arts at Paris. The most prominent of these scholars was Siger of Brabant. The philosophy of these men was condemned on March 7th, 1277 by Stephen Tempier, Bishop of Paris, after a first condemnation of Aristotelianism in 1210 had gradually come to be neglected. The 219 theses condemned in 1277, however, contain also some of Aquinas which later were generally recognized an orthodox. The Averroistic propositions which aroused the criticism of the ecclesiastic authorities and which had been opposed with great energy by Albert and Thomas refer mostly to the following points: The co-eternity of the created word; the numerical identity of the intellect in all men, the so-called two-fold-truth theory stating that a proposition may be philosophically true although theologically false. Regarding the first point Thomas argued that there is no philosophical proof, either for the co-eternity or against it; creation is an article of faith. The unity of intellect was rejected as incompatible with the true notion of person and with personal immortality. It is doubtful whether Averroes himself held the two-truths theory; it was, however, taught by the Latin Averroists who, notwithstanding the opposition of the Church and the Thomistic philosophers, gained a great influence and soon dominated many universities, especially in Italy. Thomas and his followers were convinced that they interpreted Aristotle correctly and that the Averroists were wrong; one has, however, to admit that certain passages in Aristotle allow for the Averroistic interpretation, especially in regard to the theory of intellect. Lit.: P. Mandonnet, Siger de Brabant et l'Averroisme Latin au XIIIe Siecle, 2d. ed. Louvain, 1911; M. Grabmann, Forschungen über die lateinischen Aristotelesübersetzungen des XIII. Jahrhunderts, Münster 1916 (Beitr. z. Gesch. Phil. d. MA. Vol. 17, H. 5-6). --R.A. Avesta: See Zendavesta. Avicehron: (or Avencebrol, Salomon ibn Gabirol) The first Jewish philosopher in Spain, born in Malaga 1020, died about 1070, poet, philosopher, and moralist. His main work, Fons vitae, became influential and was much quoted by the Scholastics. It has been preserved only in the Latin translation by Gundissalinus. His doctrine of a spiritual substance individualizing also the pure spirits or separate forms was opposed by Aquinas already in his first treatise De ente, but found favor with the medieval Augustinians also later in the 13th century. He also teaches the necessity of a mediator between God and the created world; such a mediator he finds in the Divine Will proceeding from God and creating, conserving, and moving the world. His cosmogony shows a definitely Neo-Platonic shade and assumes a series of emanations. Cl. Baeumker, Avencebrolis Fons vitae. Beitr. z. Gesch. d. Philos. d. MA. 1892-1895, Vol. I. Joh. Wittman, Die Stellung des hl. Thomas von Aquino zu Avencebrol, ibid. 1900. Vol. III. --R.A. Avicenna: (Abu Ali al Hosain ibn Abdallah ibn Sina) Born 980 in the country of Bocchara, began to write in young years, left more than 100 works, taught in Ispahan, was physician to several Persian princes, and died at Hamadan in 1037. His fame as physician survived his influence as philosopher in the Occident. His medical works were printed still in the 17th century. His philosophy is contained in 18 vols. of a comprehensive encyclopedia, following the tradition of Al Kindi and Al Farabi. Logic, Physics, Mathematics and Metaphysics form the parts of this work. His philosophy is Aristotelian with noticeable Neo-Platonic influences. His doctrine of the universal existing ante res in God, in rebus as the universal nature of the particulars, and post res in the human mind by way of abstraction became a fundamental thesis of medieval Aristotelianism. He sharply distinguished between the logical and the ontological universal, denying to the latter the true nature of form in the composite. The principle of individuation is matter, eternally existent. Latin translations attributed to Avicenna the notion that existence is an accident to essence (see e.g. Guilelmus Parisiensis, De Universo). The process adopted by Avicenna was one of paraphrasis of the Aristotelian texts with many original thoughts interspersed. His works were translated into Latin by Dominicus Gundissalinus (Gondisalvi) with the assistance of Avendeath ibn Daud. This translation started, when it became more generally known, the "revival of Aristotle" at the end of the 12th and the beginning of the 13th century. Albert the Great and Aquinas professed, notwithstanding their critical attitude, a great admiration for Avicenna whom the Arabs used to call the "third Aristotle". But in the Orient, Avicenna's influence declined soon, overcome by the opposition of the orthodox theologians. Avicenna, Opera, Venetiis, 1495; l508; 1546. M. Horten, Das Buch der Genesung der Seele, eine philosophische Enzyklopaedie Avicenna's; XIII. Teil: Die Metaphysik. Halle a. S. 1907-1909. R. de Vaux, Notes et textes sur l'Avicennisme Latin, Bibl. Thomiste XX, Paris, 1934. --R.A. Avidya: (Skr.) Nescience; ignorance; the state of mind unaware of true reality; an equivalent of maya (q.v.); also a condition of pure awareness prior to the universal process of evolution through gradual differentiation into the elements and factors of knowledge. --K.F.L. Avyakta: (Skr.) "Unmanifest", descriptive of or standing for brahman (q.v.) in one of its or "his" aspects, symbolizing the superabundance of the creative principle, or designating the condition of the universe not yet become phenomenal (aja, unborn). --K.F.L. Awareness: Consciousness considered in its aspect of act; an act of attentive awareness such as the sensing of a color patch or the feeling of pain is distinguished from the content attended to, the sensed color patch, the felt pain. The psychologlcal theory of intentional act was advanced by F. Brentano (Psychologie vom empirischen Standpunkte) and received its epistemological development by Meinong, Husserl, Moore, Laird and Broad. See Intentionalism. --L.W. Axiological: (Ger. axiologisch) In Husserl: Of or pertaining to value or theory of value (the latter term understood as including disvalue and value-indifference). --D.C. Axiological ethics: Any ethics which makes the theory of obligation entirely dependent on the theory of value, by making the determination of the rightness of an action wholly dependent on a consideration of the value or goodness of something, e.g. the action itself, its motive, or its consequences, actual or probable. Opposed to deontological ethics. See also teleological ethics. --W.K.F. Axiologic Realism: In metaphysics, theory that value as well as logic, qualities as well as relations, have their being and exist external to the mind and independently of it. Applicable to the philosophy of many though not all realists in the history of philosophy, from Plato to G. E. Moore, A. N. Whitehead, and N, Hartmann. --J.K.F. Axiology: (Gr. axios, of like value, worthy, and logos, account, reason, theory). Modern term for theory of value (the desired, preferred, good), investigation of its nature, criteria, and metaphysical status. Had its rise in Plato's theory of Forms or Ideas (Idea of the Good); was developed in Aristotle's Organon, Ethics, Poetics, and Metaphysics (Book Lambda). Stoics and Epicureans investigated the summum bonum. Christian philosophy (St. Thomas) built on Aristotle's identification of highest value with final cause in God as "a living being, eternal, most good." In modern thought, apart from scholasticism and the system of Spinoza (Ethica, 1677), in which values are metaphysically grounded, the various values were investigated in separate sciences, until Kant's Critiques, in which the relations of knowledge to moral, aesthetic, and religious values were examined. In Hegel's idealism, morality, art, religion, and philosophy were made the capstone of his dialectic. R. H. Lotze "sought in that which should be the ground of that which is" (Metaphysik, 1879). Nineteenth century evolutionary theory, anthropology, sociology, psychology, and economics subjected value experience to empirical analysis, and stress was again laid on the diversity and relativity of value phenomena rather than on their unity and metaphysical nature. F. Nietzsche's Also Sprach Zarathustra (1883-1885) and Zur Genealogie der Moral (1887) aroused new interest in the nature of value. F. Brentano, Vom Ursprung sittlicher Erkenntnis (1889), identified value with love. In the twentieth century the term axiology was apparently first applied by Paul Lapie (Logique de la volonte, 1902) and E. von Hartmann (Grundriss der Axiologie, 1908). Stimulated by Ehrenfels (System der Werttheorie, 1897), Meinong (Psychologisch-ethische Untersuchungen zur Werttheorie, 1894-1899), and Simmel (Philosophie des Geldes, 1900). W. M. Urban wrote the first systematic treatment of axiology in English (Valuation, 1909), phenomenological in method under J. M. Baldwin's influence. Meanwhile H. Münsterberg wrote a neo-Fichtean system of values (The Eternal Values, 1909). Among important recent contributions are: B. Bosanquet, The Principle of Individuality and Value (1912), a free reinterpretation of Hegelianism; W. R. Sorley, Moral Values and the Idea of God (1918, 1921), defending a metaphysical theism; S. Alexander, Space, Time, and Deity (1920), realistic and naturalistic; N. Hartmann, Ethik (1926), detailed analysis of types and laws of value; R. B. Perry's magnum opus, General Theory of Value (1926), "its meaning and basic principles construed in terms of interest"; and J. Laird, The Idea of Value (1929), noteworthy for historical exposition. A naturalistic theory has been developed by J. Dewey (Theory of Valuation, 1939), for which "not only is science itself a value . . . but it is the supreme means of the valid determination of all valuations." A. J. Ayer, Language, Truth and Logic (1936) expounds the view of logical positivism that value is "nonsense." J. Hessen, Wertphilosophie (1937), provides an account of recent German axiology from a neo-scholastic standpoint. The problems of axiology fall into four main groups, namely, those concerning (1) the nature of value, (2) the types of value, (3) the criterion of value, and (4) the metaphysical status of value. (1) The nature of value experience. Is valuation fulfillment of desire (voluntarism: Spinoza, Ehrenfels), pleasure (hedonism: Epicurus, Bentham, Meinong), interest (Perry), preference (Martineau), pure rational will (formalism: Stoics, Kant, Royce), apprehension of tertiary qualities (Santayana), synoptic experience of the unity of personality (personalism: T. H. Green, Bowne), any experience that contributes to enhanced life (evolutionism: Nietzsche), or "the relation of things as means to the end or consequence actually reached" (pragmatism, instrumentalism: Dewey). (2) The types of value. Most axiologists distinguish between intrinsic (consummatory) values (ends), prized for their own sake, and instrumental (contributory) values (means), which are causes (whether as economic goods or as natural events) of intrinsic values. Most intrinsic values are also instrumental to further value experience; some instrumental values are neutral or even disvaluable intrinsically. Commonly recognized as intrinsic values are the (morally) good, the true, the beautiful, and the holy. Values of play, of work, of association, and of bodily well-being are also acknowledged. Some (with Montague) question whether the true is properly to be regarded as a value, since some truth is disvaluable, some neutral; but love of truth, regardless of consequences, seems to establish the value of truth. There is disagreement about whether the holy (religious value) is a unique type (Schleiermacher, Otto), or an attitude toward other values (Kant, Höffding), or a combination of the two (Hocking). There is also disagreement about whether the variety of values is irreducible (pluralism) or whether all values are rationally related in a hierarchy or system (Plato, Hegel, Sorley), in which values interpenetrate or coalesce into a total experience. (3) The criterion of value. The standard for testing values is influenced by both psychological and logical theory. Hedonists find the standard in the quantity of pleasure derived by the individual (Aristippus) or society (Bentham). Intuitionists appeal to an ultimate insight into preference (Martineau, Brentano). Some idealists recognize an objective system of rational norms or ideals as criterion (Plato, Windelband), while others lay more stress on rational wholeness and coherence (Hegel, Bosanquet, Paton) or inclusiveness (T. H. Green). Naturalists find biological survival or adjustment (Dewey) to be the standard. Despite differences, there is much in common in the results of the application of these criteria. (4) The metaphysical status of value. What is the relation of values to the facts investigated by natural science (Koehler), of Sein to Sollen (Lotze, Rickert), of human experience of value to reality independent of man (Hegel, Pringle-Pattlson, Spaulding)? There are three main answers: subjectivism (value is entirely dependent on and relative to human experience of it: so most hedonists, naturalists, positivists); logical objectivism (values are logical essences or subsistences, independent of their being known, yet with no existential status or action in reality); metaphysical objectivism (values --or norms or ideals --are integral, objective, and active constituents of the metaphysically real: so theists, absolutists, and certain realists and naturalists like S. Alexander and Wieman). --E.S.B. Axiom: See Mathematics. Axiomatic method: That method of constructing a deductive system consisting of deducing by specified rules all statements of the system save a given few from those given few, which are regarded as axioms or postulates of the system. See Mathematics. --C.A.B. Ayam atma brahma: (Skr.) "This self is brahman", famous quotation from Brhadaranyaka Upanishad 2.5.19, one of many alluding to the central theme of the Upanishads, i.e., the identity of the human and divine or cosmic. --K.F.L.
Axiomatic Architecture Description Language "language, architecture, parallel" (AADL) A language allowing concise modular specification of {multiprocessor} architectures from the compiler/operating-system interface level down to chip level. AADL is rich enough to specify target architectures while providing a concise model for clocked {microarchitectures}. ["AADL: A Net-Based Specification Method for Computer Architecture Design", W. Damm et al in Languages for Parallel Architectures, J.W. deBakker ed, Wiley, 1989]. (2003-06-30)
Axiomatic Architecture Description Language ::: (language, architecture, parallel) (AADL) A language allowing concise modular specification of multiprocessor architectures from the enough to specify target architectures while providing a concise model for clocked microarchitectures.[AADL: A Net-Based Specification Method for Computer Architecture Design, W. Damm et al in Languages for Parallel Architectures, J.W. deBakker ed, Wiley, 1989].(2003-06-30)
AXIOM "language" A commercially available subset of the {Scratchpad}, {symbolic mathematics} system from {IBM}. ["Axiom - The Scientific Computing System", R. Jenks et al, Springer 1992]. [Relationship with {AXIOM*}?] (1995-02-21)
AXIOM* "mathematics, tool" A {symbolic mathematics} system. {A
AXIOM* ::: (mathematics, tool) A symbolic mathematics system.A
Axiom of Choice "logic" (AC, or "Choice") An {axiom} of {set theory}: If X is a set of sets, and S is the union of all the elements of X, then there exists a function f:X -" S such that for all non-empty x in X, f(x) is an element of x. In other words, we can always choose an element from each set in a set of sets, simultaneously. Function f is a "choice function" for X - for each x in X, it chooses an element of x. Most people's reaction to AC is: "But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or whatever". Indeed, for any {finite} set of sets, we can simply consider each set in turn and pick an arbitrary element in some such way. We can also construct a choice function for most simple {infinite sets} of sets if they are generated in some regular way. However, there are some infinite sets for which the construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases. For example, if we express the {real number} line R as the union of many "copies" of the {rational numbers}, Q, namely Q, Q+a, Q+b, and infinitely (in fact uncountably) many more, where a, b, etc. are {irrational numbers} no two of which differ by a rational, and Q+a == {q+a : q in Q} we cannot pick an element of each of these "copies" without AC. An example of the use of AC is the theorem which states that the {countable} union of countable sets is countable. I.e. if X is countable and every element of X is countable (including the possibility that they're finite), then the sumset of X is countable. AC is required for this to be true in general. Even if one accepts the axiom, it doesn't tell you how to construct a choice function, only that one exists. Most mathematicians are quite happy to use AC if they need it, but those who are careful will, at least, draw attention to the fact that they have used it. There is something a little odd about Choice, and it has some alarming consequences, so results which actually "need" it are somehow a bit suspicious, e.g. the {Banach-Tarski paradox}. On the other side, consider {Russell's Attic}. AC is not a {theorem} of {Zermelo Fränkel set theory} (ZF). Gödel and Paul Cohen proved that AC is independent of ZF, i.e. if ZF is consistent, then so are ZFC (ZF with AC) and ZF(~C) (ZF with the negation of AC). This means that we cannot use ZF to prove or disprove AC. (2003-07-11)
Axiom of Choice ::: (mathematics) (AC, or Choice) An axiom of set theory:If X is a set of sets, and S is the union of all the elements of X, then there exists a function f:X -> S such that for all non-empty x in X, f(x) is an element of x.In other words, we can always choose an element from each set in a set of sets, simultaneously.Function f is a choice function for X - for each x in X, it chooses an element of x.Most people's reaction to AC is: But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases.For example, if we express the real number line R as the union of many copies of the rational numbers, Q, namely Q, Q+a, Q+b, and infinitely (in fact uncountably) many more, where a, b, etc. are irrational numbers no two of which differ by a rational, and Q+a == {q+a : q in Q} we cannot pick an element of each of these copies without AC.An example of the use of AC is the theorem which states that the countable union of countable sets is countable. I.e. if X is countable and every element of X is countable (including the possibility that they're finite), then the sumset of X is countable. AC is required for this to be true in general.Even if one accepts the axiom, it doesn't tell you how to construct a choice function, only that one exists. Most mathematicians are quite happy to use AC if somehow a bit suspicious, e.g. the Banach-Tarski paradox. On the other side, consider Russell's Attic.AC is not a theorem of Zermelo Fr�nkel set theory (ZF). G�del and Paul Cohen proved that AC is independent of ZF, i.e. if ZF is consistent, then so are ZFC (ZF with AC) and ZF(~C) (ZF with the negation of AC). This means that we cannot use ZF to prove or disprove AC.(2003-07-11)
Axiom of Comprehension "logic" An {axiom schema} of {set theory} which states: if P(x) is a {property} then {x : P} is a set. I.e. all the things with some property form a set. Acceptance of this axiom leads to {Russell's Paradox} which is why {Zermelo set theory} replaces it with a restricted form. (1995-03-31)
Axiom of Comprehension ::: (mathematics) An axiom schema of set theory which states: if P(x) is a property then {x : P} is a set. I.e. all the things with some property form a set.Acceptance of this axiom leads to Russell's Paradox which is why Zermelo set theory replaces it with a restricted form. (1995-03-31)
AXLE ::: An early string processing language. Program consists of an assertion table which specifies patterns, and an imperative table which specifies replacements. AXLE: An Axiomatic Language for String Transformations, K. Cohen et al, CACM 8(11):657-661 (Nov 1965).
AXLE "language" An early {string processing} language in which a program consists of an "assertion table" specifying patterns and an "imperative table" specifying replacements. ["AXLE: An Axiomatic Language for String Transformations", K. Cohen et al, CACM 8(11):657-661, Nov 1965]. (2009-02-10)
Banach-Tarski paradox "mathematics" It is possible to cut a solid ball into finitely many pieces (actually about half a dozen), and then put the pieces together again to get two solid balls, each the same size as the original. This {paradox} is a consequence of the {Axiom of Choice}. (1995-03-29)
Banach-Tarski paradox ::: (mathematics) It is possible to cut a solid ball into finitely many pieces (actually about half a dozen), and then put the pieces together again to get two solid balls, each the same size as the original.This paradox is a consequence of the Axiom of Choice. (1995-03-29)
axiom ::: a. --> A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted; as, "The whole is greater than a part;" "A thing can not, at the same time, be and not be."
An established principle in some art or science, which, though not a necessary truth, is universally received; as, the axioms of political economy.
axiom: A statement that is neither proven, nor is it intended to be proven. Usually they are considered to be self-evident, although axioms are also constructed to explore the possibilities of hypothetical situations where we accept certain assertions to be true, and what theorems could be derived from it.
axiomatic ::: a. --> Alt. of Axiomatical
axiomatical ::: a. --> Of or pertaining to an axiom; having the nature of an axiom; self-evident; characterized by axioms.
axiomatically ::: adv. --> By the use of axioms; in the form of an axiom.
axiomatic semantics ::: (theory) A set of assertions about properties of a system and how they are effected by program execution. The axiomatic semantics of a program could axiomatic semantics is a set of invariants on the state which the state transformer satisfies.E.g. for a function with the type: sort_list :: [T] -> [T] list, and a postcondition that the return value is a list that is sorted.One interesting use of axiomatic semantics is to have a language that has a finitely computable sublanguage that is used for specifying pre and post conditions, and then have the compiler prove that the program will satisfy those conditions.See also operational semantics, denotational semantics. (1995-11-09)
axiomatic semantics "theory" A set of assertions about properties of a system and how they are effected by program execution. The axiomatic semantics of a program could include pre- and post-conditions for operations. In particular if you view the program as a state transformer (or collection of state transformers), the axiomatic semantics is a set of invariants on the state which the state transformer satisfies. E.g. for a function with the type: sort_list :: [T] -" [T] we might give the precondition that the argument of the function is a list, and a postcondition that the return value is a list that is sorted. One interesting use of axiomatic semantics is to have a language that has a {finitely computable} sublanguage that is used for specifying pre and post conditions, and then have the compiler prove that the program will satisfy those conditions. See also {operational semantics}, {denotational semantics}. (1995-11-09)
axiomatic set theory "theory" One of several approaches to {set theory}, consisting of a {formal language} for talking about sets and a collection of {axioms} describing how they behave. There are many different {axiomatisations} for set theory. Each takes a slightly different approach to the problem of finding a theory that captures as much as possible of the intuitive idea of what a set is, while avoiding the {paradoxes} that result from accepting all of it, the most famous being {Russell's paradox}. The main source of trouble in naive set theory is the idea that you can specify a set by saying whether each object in the universe is in the "set" or not. Accordingly, the most important differences between different axiomatisations of set theory concern the restrictions they place on this idea (known as "comprehension"). {Zermelo Fränkel set theory}, the most commonly used axiomatisation, gets round it by (in effect) saying that you can only use this principle to define subsets of existing sets. NBG (von Neumann-Bernays-Goedel) set theory sort of allows comprehension for all {formulae} without restriction, but distinguishes between two kinds of set, so that the sets produced by applying comprehension are only second-class sets. NBG is exactly as powerful as ZF, in the sense that any statement that can be formalised in both theories is a theorem of ZF if and only if it is a theorem of ZFC. MK (Morse-Kelley) set theory is a strengthened version of NBG, with a simpler axiom system. It is strictly stronger than NBG, and it is possible that NBG might be consistent but MK inconsistent. {NF (http://math.boisestate.edu/~holmes/holmes/nf.html)} ("New Foundations"), a theory developed by Willard Van Orman Quine, places a very different restriction on comprehension: it only works when the formula describing the membership condition for your putative set is "stratified", which means that it could be made to make sense if you worked in a system where every set had a level attached to it, so that a level-n set could only be a member of sets of level n+1. (This doesn't mean that there are actually levels attached to sets in NF). NF is very different from ZF; for instance, in NF the universe is a set (which it isn't in ZF, because the whole point of ZF is that it forbids sets that are "too large"), and it can be proved that the {Axiom of Choice} is false in NF! ML ("Modern Logic") is to NF as NBG is to ZF. (Its name derives from the title of the book in which Quine introduced an early, defective, form of it). It is stronger than ZF (it can prove things that ZF can't), but if NF is consistent then ML is too. (2003-09-21)
axiomatic set theory ::: (theory) One of several approaches to set theory, consisting of a formal language for talking about sets and a collection of axioms describing how they behave.There are many different axiomatisations for set theory. Each takes a slightly different approach to the problem of finding a theory that captures as much as possible of the intuitive idea of what a set is, while avoiding the paradoxes that result from accepting all of it, the most famous being Russell's paradox.The main source of trouble in naive set theory is the idea that you can specify a set by saying whether each object in the universe is in the set or not. set theory concern the restrictions they place on this idea (known as comprehension).Zermelo Fr�nkel set theory, the most commonly used axiomatisation, gets round it by (in effect) saying that you can only use this principle to define subsets of existing sets.NBG (von Neumann-Bernays-Goedel) set theory sort of allows comprehension for all formulae without restriction, but distinguishes between two kinds of set, so formalised in both theories is a theorem of ZF if and only if it is a theorem of ZFC.MK (Morse-Kelley) set theory is a strengthened version of NBG, with a simpler axiom system. It is strictly stronger than NBG, and it is possible that NBG might be consistent but MK inconsistent. (New Foundations), a theory developed by Willard Van Orman Quine, places a very different restriction on comprehension: it only works when the point of ZF is that it forbids sets that are too large), and it can be proved that the Axiom of Choice is false in NF!ML (Modern Logic) is to NF as NBG is to ZF. (Its name derives from the title of the book in which Quine introduced an early, defective, form of it). It is stronger than ZF (it can prove things that ZF can't), but if NF is consistent then ML is too.(2003-09-21)
axiom "logic" A {well-formed formula} which is taken to be true without proof in the construction of a {theory}. Compare: {lemma}. (1995-03-31)
axiom ::: (logic) A well-formed formula which is taken to be true without proof in the construction of a theory.Compare: lemma. (1995-03-31)
axiom schema "logic" A {formula} in the language of an {axiomatic system}, containing one or more. These {metasyntactic variables} (or "{schematic variables}") that stand for terms or subformulae. An example is the {Axiom of Comprehension}. (2009-02-10)
Both views require for their completion an exact account of the nature of the underlying logic, which, it would seem, can only be made by formalizing this logic as a logistic system (q. v,). Such a formalization of the underlying logic was employed from the beginning by Frege and by Russell, but has come into use in connection with the other -- postulational or axiomatic -- view only comparatively recently (with, perhaps, a partial exception in the case of Peano).
Cantor ::: 1. (person, mathematics) A mathematician.Cantor devised the diagonal proof of the uncountability of the real numbers:Given a function, f, from the natural numbers to the real numbers, consider the real number r whose binary expansion is given as follows: for each natural number i, r's i-th digit is the complement of the i-th digit of f(i).Thus, since r and f(i) differ in their i-th digits, r differs from any value taken by f. Therefore, f is not surjective (there are values of its result type which it cannot return).Consequently, no function from the natural numbers to the reals is surjective. A further theorem dependent on the axiom of choice turns this result into the statement that the reals are uncountable.This is just a special case of a diagonal proof that a function from a set to its power set cannot be surjective:Let f be a function from a set S to its power set, P(S) and let U = x in S: x not in f(x) . Now, observe that any x in U is not in f(x), so U != f(x); and any x not in U is in f(x), so U != f(x): whence U is not in f(x) : x in S . But U is in P(S). Therefore, no function from a set to its power-set can be surjective.2. (language) An object-oriented language with fine-grained concurrency.[Athas, Caltech 1987. Multicomputers: Message Passing Concurrent Computers, W. Athas et al, Computer 21(8):9-24 (Aug 1988)]. (1997-03-14)
Cantor 1. "person, mathematics" A mathematician. Cantor devised the diagonal proof of the uncountability of the {real numbers}: Given a function, f, from the {natural numbers} to the {real numbers}, consider the real number r whose binary expansion is given as follows: for each natural number i, r's i-th digit is the complement of the i-th digit of f(i). Thus, since r and f(i) differ in their i-th digits, r differs from any value taken by f. Therefore, f is not {surjective} (there are values of its result type which it cannot return). Consequently, no function from the natural numbers to the reals is surjective. A further theorem dependent on the {axiom of choice} turns this result into the statement that the reals are uncountable. This is just a special case of a diagonal proof that a function from a set to its {power set} cannot be surjective: Let f be a function from a set S to its power set, P(S) and let U = { x in S: x not in f(x) }. Now, observe that any x in U is not in f(x), so U != f(x); and any x not in U is in f(x), so U != f(x): whence U is not in { f(x) : x in S }. But U is in P(S). Therefore, no function from a set to its power-set can be surjective. 2. "language" An {object-oriented language} with {fine-grained concurrency}. [Athas, Caltech 1987. "Multicomputers: Message Passing Concurrent Computers", W. Athas et al, Computer 21(8):9-24 (Aug 1988)]. (1997-03-14)
Chanyuan zhuquanji duxu. (J. Zengen shosenshu tojo; K. Sonwon chejonjip toso 禪源諸詮集都序). In Chinese, lit., "Prolegomenon to the 'Collected Writings on the Source of Chan'"; composed by the CHAN and HUAYAN exegete GUIFENG ZONGMI sometime between 828 and 835; typically known by its abbreviated title of "Chan Prolegomenon" (C. Duxu; J. Tojo; K. Toso) and often referred to in English as the "Chan Preface." The text is a comprehensive overview of the Chan collection (Chanyuan zhuquanji), which is said to have been one hundred rolls (juan) in length, but is now entirely lost. Pei Xiu's (787?-860) own preface to Zongmi's "Prolegomenon" describes this collection as a massive anthology of essential prose and verse selections drawn from all the various Chan schools, which was so extensive that Pei says it deserves to be designated as a separate "Chan basket" (Chanzang; see PItAKA), complementing the other "three baskets" (TRIPItAKA) of the traditional Buddhist canon. In order to provide a comprehensive overview of this massive collection of Chan material, Zongmi seeks to assess in his "Prolegomenon" the teachings of eight representative schools of Tang-dynasty Chan: JINGZHONG ZONG, Northern school (BEI ZONG), BAOTANG ZONG, Nanshan Nianfo men Chan zong, the Shitou school of SHITOU XIQIAN (which would eventually evolve into the CAODONG and YUNMEN schools), NIUTOU ZONG, the Heze school of HEZEI SHENHUI, and the HONGZHOU ZONG (or "Jiangxi" as it is called in the text) of MAZU DAOYI. In an effort to bridge both the ever-growing gap between the contending Chan lineages and also their estranged relations with the doctrinal schools (C. jiao, see K. KYO) that derive from the written scriptures of Buddhism, Zongmi provides in his "Prolegomenon" an overarching hermeneutical framework (see JIAOXIANG PANSHI) through which to evaluate the teachings of both the Chan and doctrinal schools. This framework is built around a series of polarities, such as the three core teachings of the scriptures and the three axiomatic perspectives of Chan, the words of the Chan masters and the mind of the Buddha, sudden awakening and gradual practice, and original enlightenment (BENJUE) and nonenlightenment. In order to demonstrate the continuities between Chan and jiao, Zongmi proceeds to demonstrate how various doctrinal traditions align with the three core teachings of the scriptures and how the eight representative Chan schools correlate with the three axiomatic perspectives of Chan. He then correlates the three doctrinal teachings with the three Chan perspectives, thus demonstrating the fundamental correspondence between the Chan and the scriptures. The last polarity he examines, that between original enlightenment and nonenlightenment, also enables Zongmi to outline an etiology of both delusion and awakening, which provides the justification for a soteriological schema that requires an initial sudden awakening followed by continued gradual cultivation (DUNWU JIANXIU). Zongmi's luster faded in China during the Song dynasty, but his vision of the Chan tradition as outlined in his "Prolegomenon" was extremely influential in YONGMING YANSHOU's ZONGJING LU; indeed, it is now believed that the Zongjing lu subsumes a substantial part of Zongmi's lost "Chan Canon" (viz., his Chanyuan zhuquanji). Zongmi and his "Prolegomenon" found a particularly enthusiastic proponent in Korean Son in the person of POJO CHINUL, who placed Zongmi's preferred soteriological schema of sudden awakening followed by gradual cultivation at the core of Korean Son practice. Zongmi's works continued to be widely read in Korea after Chinul's time and, since the seventeenth century, Korean Buddhist seminaries (kangwon) included the "Prolegomenon" (K. Toso) in the SAJIP ("Fourfold Collection"), the four key texts of the Korean monastic curriculum.
Choice, axiom of, or Zermelo's axiom, is the name given to an assumption of logical or logico-mathematical character which may be stated as follows: Given a class K whose members are non-empty classes, there exists a (one-valued) monadic function f whose range is K, such that f(x) &isin: x for all members x of K. This had often been employed unconsciously or tacitly by mathematicians -- and is apparently necessary for the proofs of certain important mathematical theorems -- but was first made explicit by Zermelo in 1904, who used it in a proof that every class can be well-ordered. Once explicitly stated the assumption was attacked by many mathematicians as lacking in validity or as not of legitimately mathematical character, but was defended by others, including Zermelo.
complete metric space "theory" A {metric space} in which every sequence that converges in itself has a limit. For example, the space of {real numbers} is complete by {Dedekind's axiom}, whereas the space of {rational numbers} is not - e.g. the sequence a[0]=1; a[n_+1]:=a[n]/2+1/a[n]. (1998-07-05)
complete metric space ::: (theory) A metric space in which every sequence that converges in itself has a limit. For example, the space of real numbers is complete by Dedekind's axiom, whereas the space of rational numbers is not - e.g. the sequence a[0]=1; a[n_+1]:=a[n]/2+1/a[n]. (1998-07-05)
Consul "language" A {constraint}-based {declarative language} based on {axiomatic set theory} and designed for {parallel} execution on {MIMD} architectures. Consul's fundamental {data type} is the {set} and its fundamental {operators} are the {logical connectives} ("and", "or", "not") and {quantifiers} ("forall", "exists"). It is written in {Lisp}-like {syntax}, e.g., (plus x y z) which means the relation x = y+z (not an {assignment statement}). {["Design of the CONSUL Programming Language", D. Baldwin, C. A. Quiroz Gonzalez, University of Rochester. Computer Science Department, TR208, 1987 Feb (http://hdl.handle.net/1802/6372)]} {["Consul: A Parallel Constraint Language", D. Baldwin, IEEE Software 6(4):62-71, 1989 July (http://dx.doi.org/10.1109/52.31653)]} (2014-10-04)
Conventionalism: Any doctrine according to which a priori truth, or the truth of propositions of logic, or the truth of propositions (or of sentences) demonstrable by purely logical means, is a matter of linguistic or postulational convention (and thus not absolute in character). H. Poincare (q.v.) regarded the choice of axioms as conventional (cf. Science et hypothese, p. 67). -- A.C.
Cudworth, Ralph: (1617-1688) Was the leading Cambridge Platonist (q.v.). His writings were devoted to a refutation of Hobbesean materialism which he characterized as atheistic. He accepted a rationalism of the kind advanced by Descartes. He found clear and distinct fundamental notions or categories reflecting universal reason, God's mind, the nature and essence of things and the moral laws, which he held to be as binding on God as the axioms of mathematics. His two most important works are The True Intellectual System of the Universe, and A Treatise concerning Eternal and Immutable Morality. -- L.E.D.
denotational semantics "theory" A technique for describing the meaning of programs in terms of mathematical {functions} on programs and program components. Programs are translated into functions about which properties can be proved using the standard mathematical theory of functions, and especially {domain theory}. Compare {axiomatic semantics}, {operational semantics}, {standard semantics}. (1996-08-21)
denotational semantics ::: (theory) A technique for describing the meaning of programs in terms of mathematical functions on programs and program components. Programs are translated into functions about which properties can be proved using the standard mathematical theory of functions, and especially domain theory.Compare axiomatic semantics, operational semantics, standard semantics. (1996-08-21)
Deus Emnim et Circulus Est (Latin) “For God is indeed a circle”; a Hermetic axiom ascribed to Pherecydes, a Greek philosopher of the 6th century b.c., said to be the teacher of Pythagoras. The circle is a symbol of the Boundless and also of repetitive cycles; and circular motions and attitudes were prescribed in rituals by Pythagoras, Numa, and many others as being symbolic of divine and celestial concerns.
D. Interpretations of Probability. The methods and results of mathematical probability (and of probability in general) are the subject of much controversy as regards their interpretation and value. Among the various theories proposed, we shall consider the following Probability as a measure of belief, probability as the relative frequency of events, probability as the truth-frequency of types of argument, probability as a primitive notion, probability as an operational concept, probability as a limit of frequencies, and probability as a physical magnitude determined by axioms. I. Probability as a Measure of Belief: According to this theory, probability is the measure or relative degree of rational credence to be attached to facts or statements on the strength of valid motives. This type of probability is sometimes difficult to estimate, as it may be qualitative as well as quantitative. When considered in its mathematical aspects, the measure of probable inference depends on the preponderance or failure of operative causes or observed occurrences of the case under investigation. This conception involves axioms leading to the classic rule of Laplace, namely: The measure of probability of any one of mutually exclusive and apriori equiprobable possibilities, is the ratio of the number of favorable possibilities to the total number of possibilities. In probability operations, this rule is taken as the definition of direct probability for those cases where it is applicable. The main objections against this interpretation are: that probability is largely subjective, or at least independent of direct experience; that equiprobability is taken as an apriori notion, although the ways of asserting it are empirical; that the conditions of valid equiprobability are not stated definitely; that equiprobability is difficult to determine actually in all cases; that it is difficult to attach an adequate probability to a complex event from the mere knowledge of the probabilities of its component parts, and that the notion of probability is not general, as it does not cover such cases as the inductive derivation of probabilities from statistical data. II. Probability as a Relative Frequency. This interpretation is based on the nature of events, and not on any subjective considerations. It deals with the rate with which an event will occur in a class of events. Hence, it considers probability as the ratio of frequency of true results to true conditions, and it gives as its measure the relative frequency leading from true conditions to true results. What is meant when a set of calculations predict that an experiment will yield a result A with probability P, is that the relative frequency of A is expected to approximate the number P in a long series of such experiments. This conception seems to be more concerned with empirical probabilities, because the calculations assumed are mostly based on statistical data or material assumptions suggested by past experiments. It is valuable in so far as it satisfies the practical necessity of considering probability aggregates in such problems. The main objections against this interpretation are: that it does not seem capable of expressing satisfactorily what is meant by the probability of an event being true; that its conclusions are more or less probable, owing to the difficulty of defining a proper standard for comparing ratios; that neither its rational nor its statistical evidence is made clear; that the degree of relevance of that evidence is not properly determined, on account of the theoretical indefinite ness of both the true numerical value of the probability and of the evidence assumed, and that it is operational in form only, but not in fact, because it involves the infinite without proper limitations. III. Probability as Truth-Frequency of Types of Arguments: In this interpretation, which is due mainly to Peirce and Venn, probability is shifted from the events to the propositions about them; instead of considering types and classes of events, it considers types and classes of propositions. Probability is thus the ability to give an objective reading to the relative tiuth of propositions dealing with singular events. This ability can be used successfully in interpreting definite and indefinite numerical probabilities, by taking statistical evaluations and making appropriate verbal changes in their formulation. Once assessed, the relative truth of the propositions considered can be communicated to facts expressed by these propositions. But neither the propositions nor the facts as such have a probability in themselves. With these assumptions, a proposition has a degree of probability, only if it is considered as a member of a class of propositions; and that degree is expressed by the proportion of true propositions to the total number of propositions in the class. Hence, probability is the ratio of true propositions to all the propositions of the class examined, if the class is finite, or to all the propositions of the same type in the long run, if the class is infinite. In the first case, fair sampling may cover the restrictions of a finite class; in the second case, the use of infinite series offers a practical limitation for the evidence considered. But in both cases, probability varies with the class or type chosen, and probability-inferences are limited by convention to those cases where numerical values can be assigned to the ratios considered. It will be observed that this interpretation of probability is similar to the relative frequency theory. The difference between these two theories is more formal than material in both cases the probability refers ultimately to kinds of evidence based on objective matter of fact. Hence the Truth-Frequency theory is open to the sime objections as the Relative-Frequency theory, with proper adjustments. An additional difficulty of this theory is that the pragmatic interpretation of truth it involves, has yet to be proved, and the situation is anything but improved by assimilating truth with probability.
empty set: The set with no elements. (Any two sets with no elements must be the same set due to the Axiom of Extensionality.)
ESOTERICS, BASIC AXIOM OF: There are laws in everything and everything is expressive of law. K 4.11.7
(Et)[xεt ≡x (Ey)[xεy][yεz]] (Axiom of summation)
(Et)[xεt ≡x [x=y ∨ x=z]]. (Axiom of pairing)
(Et)[xεt ≡x x ⊂z]. (Axiom of the set of subsets)
(Et)[xεt ≡x [xεz]A]. (Axiom of subset formation)
(Et)[zεt][xεt ⊃x' (Ey)[yεt][xεy ≡x x=x']]. (Axiom of infinity)
Euclidean geometry: The geometry system defined by the following axioms and "notions".
Extended Self-containing Prolog "language" (ESP) An {object-oriented} extension of {KL0} by Chikayama. ESP has {backtracking}-based control, {unification}-based parameter passing and {object-oriented} calling. An {object} in ESP is an {axiom} set. A {class} definition consists of nature definitions ({inheritance}), slot definitions ({class variables}) and {clause} definitions. ESP has {multiple inheritance} similar to {Flavors}. It has been implemented for {ICOT}'s {PSI} Sequential Inference machine. See also {CESP}. E-mail: "k-hata@air.co.jp". ["Unique Features of ESP", T. Chikayama, Proc Intl Conf 5th Gen Comp Sys, ICOT 1984]. (1994-12-08)
Extension: See Intension and Extension. Extensionality, axiom of: See Logic, formal, § 9. Extensity: A rudimentary spatiality alleged to characterize all sensation. See J. Ward, article "Psychology" in Encyclopaedia Bntannica, 9th Ed. pp. 46, 53 -- L.W.
fallibilism ::: The doctrine that absolute certainty about knowledge is impossible, or at least that all claims to knowledge could, in principle, be mistaken. As a formal doctrine, it is most strongly associated with Charles Sanders Peirce, who used it in his attack on foundationalism. Unlike skepticism, fallibilism does not imply the need for humans to abandon their knowledge: humans need not have logically conclusive justifications for what they know. Rather, it is an admission that because empirical knowledge can be revised by further observation, all knowledge, excepting that which is axiomatically true (such as mathematical and logical knowledge) exists in a constant state of flux.
F. B. Fitch, The consistency of the ramified Principia, The Journal of Symbolic Logic, vol. 3 (1938), pp. 140-149. Ramsey, Frank Plumpton: (1903-1930) In the light of Wittgenstein's work, he proposed several modifications in the Principia Mathematica treatment of functions. These, he urged, made possible the omission of the Axiom of Reducibility, a simplification of the Theory of Types and an improved definition of identity. In stimulating philosophical papers he denied any ultimate distinction between particulars and universals, defended a Wittgensteinian interpretation of general propositions, proposed a subjective theory of probability and a pragmatic view of induction, and offered a theory of theories and a theory of the nature of causal propositions. Most of his work is included in The Foundations of Mathematics, London, Kegan Paul, 1931. -- C.A.B.
formalism ::: 1. A certain school in the philosophy of mathematics, stressing axiomatic proofs through theorems specifically associated with David Hilbert. ::: 2. A school of thought in law and jurisprudence that emphasises the fairness of process over substantive outcomes. See Legal formalism. ::: 3. In economic anthropology, the theoretical perspective that the principles of neoclassical economics can be applied to humans' understanding of all human societies. ::: 4. A certain rigorous mathematical method: see formal system. ::: 5. A set of notations and rules for manipulating them that yield results in agreement with experiment or other techniques of calculation. These rules and notations may or may not have a corresponding mathematical semantics. In the case no mathematical semantics exists, the calculations are often said to be purely formal. See for example scientific formalism. ::: 6. A style of literary and artistic criticism that focuses on artistic or literary techniques in themselves, in separation from the work's social and historical context. See formalism (art), formalism (literature). ::: 7. A style of film criticism that focuses on the technical aspects of filmmaking (e.g., lighting, sets, costumes, etc.). The term may also refer to an avant-garde experimental film movement, often seen as odd or extremist, that was concerned with the beauty of the actual physical form of film (i.e., the celluloid itself).
Formalism (mathematical) is a name which has been given to any one of various accounts of the foundations of mathematics which emphasize the formal aspects of mathematics as against content or meaning, or which, in whole or in part, deny content to mathematical formulas. The name is often applied, in particular, to the doctrines of Hilbert (see Mathematics), although Hilbert himself calls his method axiomatic, and gives to his syntactical or metamathematical investigations the name Beweistheorie (proof theory, (q. v.). -- A.C.
For many purposes, however, it is necessary to add to the functional calculus of order omega the axiom of infinity, requiring the domain of individuals to be infinite. -- This is most conveniently done by adding a single additional primitive formula, which may be described by referring to § 3 above, taking the formula, which is there given as an example of a formula satisfiable in an infinite domain of individuals but not in any finite domain, and prefixing the quantifier (EF) with scope extending to the end of the formula. This form of the axiom of infinity, however, is considerably stronger (in the absence of the axiom of choice) than the "Infin ax" of Whitehead and Russell.
foundation ::: The axiom of foundation states that the membership relation is well founded, i.e. that any non-empty collection Y of sets has a member y which is disjoint from Y. This rules out sets which contain themselves (directly or indirectly).
foundation The axiom of foundation states that the membership relation is well founded, i.e. that any non-empty collection Y of sets has a member y which is disjoint from Y. This rules out sets which contain themselves (directly or indirectly).
frame problem ::: The problem of finding adequate collections of axioms for a viable description of a robot environment.[176]
fundamental ::: a. --> Pertaining to the foundation or basis; serving for the foundation. Hence: Essential, as an element, principle, or law; important; original; elementary; as, a fundamental truth; a fundamental axiom. ::: n. --> A leading or primary principle, rule, law, or article,
Giuseppe Peano "person, mathematics, logic" (1858-08-27 - 1932-04-20) An Italian mathematician who wrote over 200 books and papers, was a founder of {mathematical logic} and {set theory} and taught at the University of Turin. He contributed to mathematical {analysis}, {logic}, the teaching of {calculus}, {differential equations}, {vector analysis} and the axiomatization of mathematics. The standard {axiomatization} of the {natural numbers} is named {Peano arithmetic} or the {Peano axioms} after him. He also invented the {Peano curve}, an early example of a {fractal}. (2013-03-23)
Gödel, Kurt, 1906-, Austrian mathematician and logician -- educated at Vienna, and now located (1941) at the Institute for Advanced Study in Princeton, N. J. -- is best known for his important incompleteness theorem, the closely related theorem on the impossibility (under certain circumstances) of formalizing a consistency proof for a logistic system within that system, and the essentially simple but far-reaching device of arithmetization of syntax which is emploved in the proof of these theorems (see Logic, formal, § 6). Also of importance are his proof of the completeness of the functional calculus of first order (see Logic, formal, § 3), and his recent work on the consistency of the axiom of choice (q. v.) and of Cantor's continuum hypothesis. -- A.C.
Hermetic Axiom “As it is above, so it is below; as it is below, so it is above.” See also SMARAGDINE TABLET
homaloidal ::: a. --> Flat; even; -- a term applied to surfaces and to spaces, whether real or imagined, in which the definitions, axioms, and postulates of Euclid respecting parallel straight lines are assumed to hold true.
infinite "mathematics" 1. Bigger than any {natural number}. There are various formal set definitions in {set theory}: a set X is infinite if (i) There is a {bijection} between X and a {proper subset} of X. (ii) There is an {injection} from the set N of natural numbers to X. (iii) There is an injection from each natural number n to X. These definitions are not necessarily equivalent unless we accept the {Axiom of Choice}. 2. The length of a line extended indefinitely. See also {infinite loop}, {infinite set}. [{Jargon File}] (1995-03-29)
infinite ::: (mathematics) 1. Bigger than any natural number. There are various formal set definitions in set theory: a set X is infinite if(i) There is a bijection between X and a proper subset of X.(ii) There is an injection from the set N of natural numbers to X.(iii) There is an injection from each natural number n to X.These definitions are not necessarily equivalent unless we accept the Axiom of Choice.2. The length of a line extended indefinitely.See also infinite loop, infinite set.[Jargon File] (1995-03-29)
infinite set "mathematics" A set with an infinite number of elements. There are several possible definitions, e.g. (i) ("Dedekind infinite") A set X is infinite if there exists a {bijection} (one-to-one mapping) between X and some proper subset of X. (ii) A set X is infinite if there exists an {injection} from N (the set of {natural numbers}) to X. In the presence of the {Axiom of Choice} all such definitions are equivalent. (1995-03-27)
infinite set ::: (mathematics) A set with an infinite number of elements. There are several possible definitions, e.g.(i) (Dedekind infinite) A set X is infinite if there exists a bijection (one-to-one mapping) between X and some proper subset of X.(ii) A set X is infinite if there exists an injection from N (the set of natural numbers) to X.In the presence of the Axiom of Choice all such definitions are equivalent. (1995-03-27)
Infinity, axiom of: See Logic, formal, §§ 6, 9. Ingression: According to A. N. Whitehead, participation of potentialities in the creation of complex actualities; "a concretion -- that is, a growing together -- of diverse elements." -- R.B.W.
In Germany, the movement was initiated by G. W. Leibniz whose writings reveal another motive for the cult of pure reason, i.e. the deep disappointment with the Reformation and the bloody religious wars among Christians who were accused of having forfeited the confidence of man in revealed religion. Hence the outstanding part played by the philosophers of ''natural law", Grotius, S. Pufendorf, and Chr. Thomasius, their theme being advanced by the contributions to a "natural religion" and tolerance by Chr. Wolff, G. E. Lessing, G. Herder, and the Prussian king Frederik II. Fr. v. Schiller's lyric and dramas served as a powerful commendation of ideal freedom, liberty, justice, and humanity. A group of educators (philanthropists) designed new methods and curricula for the advancement of public education, many of them, eg. Pestalozzi, Basedow, Cooper, A. H. Francke, and Fr. A. Wolf, the father of classic humanism, having achieved international recognition. Although in general agreement with th philosophical axioms of foreign enlighteners, the German philosophy decidedly opposed the English sensism (Hume) and French scepticism, and reached its height in Kant's Critiques. The radical rationalism, however, combined with its animosity against religion, brought about strong philosophical, theological, and literal opposition (Hamann, Jacobi, Lavater) which eventually led to its defeat. The ideals of the enlightenment period, the impassioned zeal for the materialization of the ideal man in an ideal society show clearly that it was basically related to the Renaissance and its continuation. See Aufklärung. Cf. J. G. Hibben, The Philosophy of the Enlightenment, 1910. -- S.v.F.
In his chief work, the Ethica, Spinoza's teaching is expressed in a manner for which geometry supplies the model. This expository device served various purposes. It may be interpreted as a clue to Spinoza's ideal of knowledge. So understood, it represents the condensed and ordered expression, not of 'philosophy' alone, but rather of all knowledge, 'philosophy' and 'science', as an integrated system. In such an ideal ordering of ideas, (rational) theology and metaphysics provide the anchorage for the system. On the one hand, the theology-metaphysics displays the fundamental principles (definitions, postulates, axioms) upon which the anchorage depends, and further displays in deductive fashion the primary fund of ideas upon which the inquiries of science, both 'descriptive' and 'normative' must proceed. On the other hand, the results of scientific inquiry are anchored at the other end, by a complementary metaphysico-theological development of their significance. Ideally, there obtains, for Spinoza, both an initial theology and metaphysics -- a necessary preparation for science -- and a culminating theology and metaphysics, an interpretative absorption of the conclusions of science.
In many (interpreted) logistic systems -- including such as contain, with their usual interpretations, the Zermelo set theory, or the simple theory of types with axiom of infinity, or the functional calculus of second order with addition of Peano's postulates for arithmetic -- it is impossible without contradiction to introduce the numerical name relation with its natural properties, because Grelling's paradox or similar paradoxes would result (see paradoxes, logical). The same can be said of the semantical name relation in cases where symbols for formulas are present.
In particular, it is normally possible -- at least it does not obviously lead to contradiction in the case of such systems as the Zermelo set theory or the simple theory of types (functional calculus of order omega) with axiom of infinity -- to extend a system L1 into a system L2 (the semantics of L1 in the sense of Tarski), so that L2 shall contain symbols for the formulas of L1, and for the essential syntactical relations between formulas of L1, and for a relation which functions as a name relation as regards all the formulas of L1 (or, in the case of the theory of types, one such relation for each type), together with appropriate new primitive formulas. Then L2 may be similarly extended into L3, and so on through a hierarchy of systems each including the preceding one as a part.
interface analysis "testing" A software test which checks the interfaces between program elements for consistency and adherence to predefined rules or {axioms}. (1996-07-09)
interface analysis ::: (testing) A software test which checks the interfaces between program elements for consistency and adherence to predefined rules or axioms. (1996-07-09)
In the axiom of subset formation, A is any formula not containing t as a free varibale (in general, A will contain x as a free variable). In the axiom of replacement, A is any formula which contains neither t nor x' as a free variable (in general, A will contain x and y as free variables). These two axioms are thus represented each by an infinite list of primitive formulas -- the remaining axioms each by one primitive formula.
In the Frege-Russell derivation of arithmetic from logic (see the article Mathematics) necessity for the postulates of Peano is avoided. If based on the theory of types, however, this derivation requires some form of the axiom of infinity -- which may be regarded as a residuum of the Peano postulates.
I. Period of Preparation (9-12 cent.). Though he does not belong in time to this period, the most dominant figure in Christian thought was St. Augustine (+430), who constructed the general framework within which all subsequent Scholastic speculation operated. Another influential figure was Boethius (+525) whose opuscula sacra established the Scholastic method and who furnished many of the classical definitions and axioms. The first great figure of this period was John Scottus Erigena (+c. 877) who introduced to Latin thought the works of Denis the Pseudo-Areopagite, broadened the Scholastic method by his glossary on Boethius' opuscule sacra and made an unfruitful attempt to interest his contemporaries in natural philosophy by his semi-pantheistic De Divisione Naturae. Other figures of note: Gerbert (+1003) important in the realm of mathematics and natural philosophy; Fulbert of Chartres (+1028) influential in the movement to apply dialectics to theology; Berengar of Tours (+1088) Fulbert's disciple, who, together with Anselm the Peripatetic, was a leader in the movement to rationalize theology. Peter Damiani (+1072), preached strongly against this rationalistic spirit. More moderate and more efficacious in his reaction to the dialectical spirit of his age was Lawfranc (+1089), who strove to define the true boundaries of faith and reason.
K. Gödel, Die Vollständigkeit der Axiome des logiscben Funktionen-kalküls, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349-360.
K. Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton, N.J., 1940. Chou Tun-i: (Chou Lien-hsi, Chou Mao-shu, 1017-1073) Was active in government and was a renowned judge. He was the pioneer of Neo-Confucianism (li hsueh), anticipating the Ch'eng brothers. He wrote the T'ung-shu (explanation of the Book of Changes) and the T'aichi T'u-shu (explanation of the diagram of the Great Ultimate), fundamental texts of Neo-Confucian philosophy. -- W.T.C.
laissez faire ::: --> Noninterference; -- an axiom of some political economists, deprecating interference of government by attempts to foster or regulate commerce, manufactures, etc., by bounty or by restriction; as, the doctrine of laissez faire; the laissez faire system government.
Law of Thelema ::: A principal Thelemic axiom that states: "Do What Thou Wilt Shall Be The Whole Of The Law. Love Is The Law. Love Under Will." To translate tersely: Doubt not. Regret not. Move where Spirit guides and where Love thrives.
Logic An attempt to formulate the processes of the ratiocinative mind, connecting idea with idea in a causal sequence, leading from predicate to conclusion. When the predicate consists of axioms, the species of logic is called deductive, or reasoning from the general to the particular; when the predicate is facts of experience, the logic is called inductive, or proceeding from particulars to generals. As a means of arriving at truth it alone is quite unreliable, as it is but a body of rules based on human experiences, and hence it is often rather a means of justifying conclusions after they have already been formed. This unreliability arises both from the difficulty of applying the process with rigid precision, and also from the uncertainty of the predicates in both systems. A study of what is written on logic will show that there is no agreement as to what constitutes an axiom — whether it is an intuitive perception of truth, or whether it is merely an inference from experience. The same uncertainty exists as to the validity of the assumptions from which inductive chains of reasoning are drawn.
Macrocosm ::: The Cosmos in the Large. The Universe that the Mind is a part of. The connection between macrocosm and microcosm is best depicted in the Hermetic Axiom. See also Microcosm.
maxim ::: n. --> An established principle or proposition; a condensed proposition of important practical truth; an axiom of practical wisdom; an adage; a proverb; an aphorism.
The longest note formerly used, equal to two longs, or four breves; a large.
Medieval Period. Medieval Christian thought, axiomatically idealistic, united the personalism of Israel and the speculative idealism of neo-Platonism and Aristotle. Similarly, Islamic thought, centering at Bagdad and Cordova, attached Mohammedan religious idealism to neo-Platonism and Aristotelianism.
member: Also known as an element. The component of a set considered as a collection. Membership of elements is the only property by which a set is distinguished from another (the Axiom of Extensionality), denoed by the symbol ∈, where x ∈ X denotes that x is a member of the set X. This symbol is almost never used the other way around so that the set is placed before the member.
Methodology: The systematic analysis and organization of the rational and experimental principles and processes which must guide a scientific inquiry, or which constitute the structure of the special sciences more particularly. Methodology, which is also called scientific method, and more seldom methodeutic, refers not only to the whole of a constituted science, but also to individual problems or groups of problems within a science. As such it is usually considered as a branch of logic; in fact, it is the application of the principles and processes of logic to the special objects of the various sciences; while science in general is accounted for by the combination of deduction and induction as such. Thus, methodology is a generic term exemplified in the specific method of each science. Hence its full significance can be understood only by analyzing the structure of the special sciences. In determining that structure, one must consider the proper object of the special science, the manner in which it develops, the type of statements or generalizations it involves, its philosophical foundations or assumptions, and its relation with the other sciences, and eventually its applications. The last two points mentioned are particularly important: methods of education, for example, will vary considerably according to their inspiration and aim. Because of the differences between the objects of the various sciences, they reveal the following principal methodological patterns, which are not necessarily exclusive of one another, and which are used sometimes in partial combination. It may be added that their choice and combination depend also in a large degree on psychological motives. In the last resort, methodology results from the adjustment of our mental powers to the love and pursuit of truth. There are various rational methods used by the speculative sciences, including theology which adds certain qualifications to their use. More especially, philosophy has inspired the following procedures: The Soctattc method of analysis by questioning and dividing until the essences are reached; the synthetic method developed by Plato, Aristotle and the Medieval thinkers, which involves a demonstrative exposition of the causal relation between thought and being; the ascetic method of intellectual and moral purification leading to an illumination of the mind, as proposed by Plotinus, Augustine and the mystics; the psychological method of inquiry into the origin of ideas, which was used by Descartes and his followers, and also by the British empiricists; the critical or transcendental method, as used by Kant, and involving an analysis of the conditions and limits of knowledge; the dialectical method proceeding by thesis, antithesis and synthesis, which is promoted by Hegelianlsm and Dialectical Materialism; the intuitive method, as used by Bergson, which involves the immediate perception of reality, by a blending of consciousness with the process of change; the reflexive method of metaphysical introspection aiming at the development of the immanent realities and values leading man to God; the eclectic method (historical-critical) of purposive and effective selection as proposed by Cicero, Suarez and Cousin; and the positivistic method of Comte, Spencer and the logical empiricists, which attempts to apply to philosophy the strict procedures of the positive sciences. The axiomatic or hypothetico-deductive method as used by the theoretical and especially the mathematical sciences. It involves such problems as the selection, independence and simplification of primitive terms and axioms, the formalization of definitions and proofs, the consistency and completeness of the constructed theory, and the final interpretation. The nomological or inductive method as used by the experimental sciences, aims at the discovery of regularities between phenomena and their relevant laws. It involves the critical and careful application of the various steps of induction: observation and analytical classification; selection of similarities; hypothesis of cause or law; verification by the experimental canons; deduction, demonstration and explanation; systematic organization of results; statement of laws and construction of the relevant theory. The descriptive method as used by the natural and social sciences, involves observational, classificatory and statistical procedures (see art. on statistics) and their interpretation. The historical method as used by the sciences dealing with the past, involves the collation, selection, classification and interpretation of archeological facts and exhibits, records, documents, archives, reports and testimonies. The psychological method, as used by all the sciences dealing with human behaviour and development. It involves not only introspective analysis, but also experimental procedures, such as those referring to the relations between stimuli and sensations, to the accuracy of perceptions (specific measurements of intensity), to gradation (least noticeable differences), to error methods (average error in right and wrong cases), and to physiological and educational processes.
Microcosm ::: The Cosmos in the Small. The Universe that is contained within the Mind. The connection between microcosm and macrocosm is best depicted in the Hermetic Axiom. See also Macrocosm.
Multiplicative axiom: See choice, axiom of. Multiplicity: The doctrine of the plurality of beings, or the manifoldness of the real, denied by the Eleatics, who contended that the multiplicity of things was but an illusion of the senses, was defended by Aristotle who maintained that the term, being, is only a common predicate of many things which become out of that which is relatively not-being by making the transition from the potential to the actual. -- J.J.R.
non-Euclidean geometry: Any system of geometry not based on (all 5 of the) Euclidean axioms/postulates. e.g. hyperbolic geometry, spherical geometry.
No proof of consistency of the functional calculus of order omega (or even of lower order) wiih the axiom of infinity added is known, except by methods involving assumptions so strong as to destroy any major significance.
Objecting to Fichte, his master's method of deducing everything from a single, all-embracing principle, he obstinately adhered to the axiom that everything is what it is, the principle of identity. He also departed from him in the principle of idealism and freedom. As nnn is not free in the sense of possessing a principle independent of the environment, he reverted to the Kantian doctrine that behind and underlying the world of appearance there is a plurality of real things in themselves that are independent of the operations of mind upon them. Deserving credit for having developed the realism that was latent in Kant's philosophy, he conceived the ''reals" so as to do away with the contradictions in the concepts of experience. The necessity for assuming a plurality of "reals" arises as a result of removing the contradictions in our experiences of change and of things possessing several qualities. Herbart calls the method he applies to the resolution of the contradictions existing between the empirically derived concepts, the method of relations, that is the accidental relation between the different "reals" is a question of thought only, and inessential for the "reals" themselves. It is the changes in these relations that form the process of change in the world of experience. Nothing can be ultimately real of which two contradictory predicates can be asserted. To predicate unity and multiplicity of an object is to predicate contradictions. Hence ultimate reality must be absolutely unitary and also without change. The metaphysically interpreted abstract law of contradiction was therefore central in his system. Incapability of knowing the proper nature of these "reals" equals the inability of knowing whether they are spiritual or material. Although he conceived in his system that the "reals" are analogous with our own inner states, yet his view of the "reals" accords better with materialistic atomism. The "reals" are simple and unchangeable in nature.
One By itself the One represents not pure unalloyed spirit, which is signified by the zero — the all-containing womb of space and being — but is the First Logos or Pythagorean Monas monadum (monad of monads). From this monad of monads flows forth through emanation the duad, then the triad, and then the entire manifested universe of interlocking hierarchies, emanated from the cosmic womb of being or the zero through the First Logos or the One of primordial manifested spirit. “The sacredness of numbers begins with the great First — the one, and ends only with the nought or zero — symbol of the infinite and boundless circle which represents the universe. All the intervening figures, in whatever combination, or however multiplied, represent philosophical ideas, from vague outlines down to a definitely-established scientific axiom, relating either to a moral or a physical fact in nature. They are a key to the ancient views on cosmogony, in its broad sense, including man and beings, and the evolution of the human race, spiritually as well as physically” (IU 2:407).
ontology 1. "philosophy" A systematic account of Existence. 2. "artificial intelligence" (From philosophy) An explicit formal specification of how to represent the objects, concepts and other entities that are assumed to exist in some area of interest and the relationships that hold among them. For {AI} systems, what "exists" is that which can be represented. When the {knowledge} about a {domain} is represented in a {declarative language}, the set of objects that can be represented is called the {universe of discourse}. We can describe the ontology of a program by defining a set of representational terms. Definitions associate the names of entities in the {universe of discourse} (e.g. classes, relations, functions or other objects) with human-readable text describing what the names mean, and formal {axioms} that constrain the interpretation and well-formed use of these terms. Formally, an ontology is the statement of a {logical theory}. A set of {agents} that share the same ontology will be able to communicate about a domain of discourse without necessarily operating on a globally shared theory. We say that an agent commits to an ontology if its observable actions are consistent with the definitions in the ontology. The idea of ontological commitment is based on the {Knowledge-Level} perspective. 3. "information science" The hierarchical structuring of knowledge about things by subcategorising them according to their essential (or at least relevant and/or cognitive) qualities. See {subject index}. This is an extension of the previous senses of "ontology" (above) which has become common in discussions about the difficulty of maintaining {subject indices}. (1997-04-09)
operational semantics ::: (theory) A set of rules specifying how the state of an actual or hypothetical computer changes while executing a program. The overall state is Each rule specifies certain preconditions on the contents of some components and their new contents after the application of the rule.It is similar in spirit to the notion of a Turing machine, in which actions are precisely described in a mathematical way.Compuare axiomatic semantics, denotational semantics. (1996-08-21)
operational semantics "theory" A set of rules specifying how the state of an actual or hypothetical computer changes while executing a program. The overall state is typically divided into a number of components, e.g. {stack}, {heap}, {registers} etc. Each rule specifies certain preconditions on the contents of some components and their new contents after the application of the rule. It is similar in spirit to the notion of a {Turing machine}, in which actions are precisely described in a mathematical way. Compuare {axiomatic semantics}, {denotational semantics}. (1996-08-21)
Other primitive formulas (possibly involving new primitive notations) which may be added correspond to the axiom of choice (q. v.) or are designed to introduce classes (q. v.) or descriptions (q. v.). Functional abstraction (q. v.) may also be Introduced by means of additional primitive formulas or primitive rules of inference, or it may be defined with the aid of descriptions. Whitehead and Russell employ the axiom of infinity and the axiom of choice but avoid the necessity of special primitive formulas in connection with classes and descriptions by introducing classes and descriptions as incomplete symbols.
Paul Bernays, A system of axiomatic set theory, The Journal of Symbolic Logic, vol. 2 (1937), pp. 65-77, and vol. 6 (1941), pp. 1-17.
postulate: Can be taken to mean an axiom, although some maintain a distinction between the two. Although there may not be any consenses in such distinctions.
postulate ::: n. --> Something demanded or asserted; especially, a position or supposition assumed without proof, or one which is considered as self-evident; a truth to which assent may be demanded or challenged, without argument or evidence.
The enunciation of a self-evident problem, in distinction from an axiom, which is the enunciation of a self-evident theorem.
principle ::: n. --> Beginning; commencement.
A source, or origin; that from which anything proceeds; fundamental substance or energy; primordial substance; ultimate element, or cause.
An original faculty or endowment.
A fundamental truth; a comprehensive law or doctrine, from which others are derived, or on which others are founded; a general truth; an elementary proposition; a maxim; an axiom; a
proof 1. "logic" A {finite} sequence of {well-formed formulas}, F1, F2, ... Fn, where each Fi either is an {axiom}, or follows by some rule of inference from some of the previous F's, and Fn is the statement being proved. See also {proof theory}. 2. A left-associative {natural language} {parser} by Craig R. Latta "latta@xcf.berkeley.edu". Ported to {Decstation 3100}, {Sun-4}. {(ftp://scam.berkeley.edu/pub/src/local/proof/)}. E-mail: "proof@xcf.berkeley.edu". Mailing list: proof-requestf@xcf.berkeley.edu (Subject: add me). (1994-11-29)
proof ::: 1. (logic) A finite sequence of well-formed formulas, F1, F2, ... Fn, where each Fi either is an axiom, or follows by some rule of inference from some of the previous F's, and Fn is the statement being proved.See also proof theory.2. A left-associative natural language parser by Craig R. Latta . Ported to Decstation 3100, Sun-4. .E-mail: . Mailing list: (1994-11-29)
proof: A sequence of finite number of statements, each of which is either an axiom or the result from rules of inference on statements that appeared before.
Radioactivity Scientific discovery has done much to verify the occult axiom that there are no permanent bodies, but that everything is in a state of flux and interchange. Theosophy views the physical universe as an ocean of life, partly imbodied and partly noncorporeal, and regards such terms as matter, energy, wave, and particle as descriptive of various manifestations of this life. The chemical elements are now considered by science to be centers or vortices in a fluid ocean, continually giving and receiving emanations from each other. Thus all forms of physical matter emit radiation and radioactive phenomena are instances of a general law. The emanations studied by science are described partly as actinic rays and partly as emitted particles; and the disintegration series results in a continual emission of both these forms of emanation, accompanied by an elevation of the temperature of the radioactive body above that of its surroundings, a loss of its own mass, the formation of temporary unstable elements of lower atomic weight, until an end-product is reached. Calculations as to the age of the solid crust of the earth, based on disintegration rates, are extremely unreliable, as they involve unverified assumptions as to the rate of this process in past ages. Theosophy states that during the descending arc of cosmic evolution, the process of concretion is predominant, and during the ascending arc the process of disintegration or etherealization is predominant. This indicates that the rate of radioactive disintegration has been on the increase in comparatively recent times, and will continue at an enlarging rate into the geologic future.
Recently, the Polish logician St. Lesniewski has developed a formal theory of the part-whole relationship within the framework of a so-called calculus of individuals, one of the theorems of this theory states that every object is identical with the sum of its parts. This is, of course, a consequence of the way in which the axioms of that calculus were chosen, but that particular construction of the theory was carried out with an eye to applications in logical and epistemological analysis, and the calculus of individuals has already begun to show its value in these fields. See Leonard and Goodman, The Calculus of Individuals and Its Uses, The Journal of Symbolic Logic, 5 (1940, pp. 45-55. -- C.G.H.
Reducibility, axiom of: An axiom which (or some substitute) is necessary in connection with the ramified theory of types (q.v.) if that theory is to be adequate for classical mathematics, but the admissibility of which has been much disputed (see Paradoxes, logical). An exact statement of the axiom can be made only in the context of a detailed formulation of the ramified theory of types -- which will not here be undertaken. As an indication or rough description of the axiom of reducibility, it may be said that it cancels a large part of ihe restrictive consequences of the prohibition against impredicative definition (q.v.) and, in approximate effect, reduces the ramified theory of types to the simple theory of types (for the latter see Logic, formal, § 6). -- A.C.
Rejected in particular by intuitionism are the use of impredicative definition (q. v.); the assumption that all things satisfying a given condition can be united into a set and this set then treated as an individual thing --or even the weakened form of this assumption which is found in Zermelo's Aussonderungsaxiom or axiom of subset formation (see logic, formal, § 9); the law of excluded middle as applied to propositions whose expression lequires a quantifier for which the variable involved has an infinite range. As an example of the rejection of the law of excluded middle, consider the proposition, "Either every even number greater than 2 can be expressed as the sum of two prime numbers or else not every even number greater than 2 can be expressed as the sum of two prime numbers." This proposition is intuitionistically unacceptable, because there are infinitely many even numbers greater than 2 and it is impossible to try them all one by one and decide of each whether or not it is the sum of two prime numbers. An intuitionist would accept the disjunction only after a proof had been given of one or other of the two disjoined propositions -- and in the present state of mathematical knowledge it is not certain that this can be done (it is not certain that the mathematical problem involved is solvable). If, however, we replace "greater than 2" by "greater than 2 and less than 1,000,000,000," the resulting disjunction becomes intuitionistically acceptable, since the number of numbers involved is then finite. The intuitionistic rejection of the law of excluded middle is not to be understood as an assertion of the negation of the law of excluded middle; on the contrary, Brouwer asserts the negation of the negation of the law of excluded middle, i.e., ∼∼[p ∨ ∼p]. Still less is the intuitionistic rejection of the law of excluded middle to be understood as the assertion of the existence of a third truth-value intermediate between truth and falsehood.
Religious A Priori: A separate, innate category of the human consciousness, religious in that it issues certain insights and indisputable certainties concerning God or a Superhuman Presence. Man's religious nature rests upon the peculiar character of his mind. He possesses a native apprehension of the Divine. God's existence is guaranteed as an axiomatic truth. For Ernst Troeltsch (1865-1923) this a priori quality of the mind is both a rational intuition and an immediate experience. God is present as a real fact both rationally and empirically. For Rudolf Otto this a priori quality of the mind is a non-rational awareness of the holy, mysterious and awe-inspiring divine Reality. Man posesses a kind of eerie sense of a Presence which is the basis of the genuinely religious feeling. See Numinous. -- V.F.
Russell's Attic "mathematics" An imaginary room containing {countably many} pairs of shoes (i.e. a pair for each {natural number}), and countably many pairs of socks. How many shoes are there? Answer: countably many (map the left shoes to even numbers and the right shoes to odd numbers, say). How many socks are there? Also countably many, we want to say, but we can't prove it without the {Axiom of Choice}, because in each pair, the socks are indistinguishable (there's no such thing as a left sock). Although for any single pair it is easy to select one, we cannot specify a general method for doing this. (1995-03-29)
Russell's Paradox ::: (mathematics) A logical contradiction in set theory discovered by Bertrand Russell. If R is the set of all sets which don't contain themselves, does R contain itself? If it does then it doesn't and vice versa.The paradox stems from the acceptance of the following axiom: If P(x) is a property then {x : P} i.e. something clearly false. Thus any theory built on this axiom must be inconsistent.In lambda-calculus Russell's Paradox can be formulated by representing each set by its characteristic function - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself: r = \ x . not (x x) If we now apply r to itself, r r = (\ x . not (x x)) (\ x . not (x x))= not ((\ x . not (x x))(\ x . not (x x))) So if (r r) is true then it is false and vice versa.An alternative formulation is: if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the whereas seemingly obvious axioms of set theory suggest the existence of the paradoxical set R.Zermelo Fr�nkel set theory is one solution to this paradox. Another, type theory, restricts sets to contain only elements of a single type, (e.g. integers or sets of integers) and no type is allowed to refer to itself so no set can contain itself.A message from Russell induced Frege to put a note in his life's work, just before it went to press, to the effect that he now knew it was inconsistent but he hoped it would be useful anyway.(2000-11-01)
Russell's Paradox "mathematics" A {paradox} (logical contradiction) in {set theory} discovered by {Bertrand Russell}. If R is the set of all sets which don't contain themselves, does R contain itself? If it does then it doesn't and vice versa. The paradox stems from the acceptance of the following {axiom}: If P(x) is a property then {x : P} is a set. This is the {Axiom of Comprehension} (actually an {axiom schema}). By applying it in the case where P is the property "x is not an element of x", we generate the paradox, i.e. something clearly false. Thus any theory built on this axiom must be inconsistent. In {lambda-calculus} Russell's Paradox can be formulated by representing each set by its {characteristic function} - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself: r = \ x . not (x x) If we now apply r to itself, r r = (\ x . not (x x)) (\ x . not (x x)) = not ((\ x . not (x x))(\ x . not (x x))) = not (r r) So if (r r) is true then it is false and vice versa. An alternative formulation is: "if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the barber?" This can be taken simply as a proof that no such barber can exist whereas seemingly obvious axioms of {set theory} suggest the existence of the paradoxical set R. {Zermelo Fränkel set theory} is one "solution" to this paradox. Another, {type theory}, restricts sets to contain only elements of a single type, (e.g. {integers} or sets of integers) and no type is allowed to refer to itself so no set can contain itself. A message from Russell induced {Frege} to put a note in his life's work, just before it went to press, to the effect that he now knew it was inconsistent but he hoped it would be useful anyway. (2000-11-01)
Russell's solution of the paradoxes is embodied in what is now known as the ramified theory of types, published by him in 1908, and afterwards made the basis of Principia Mathematica. Because of its complication, and because of the necessity for the much-disputed axiom of reducibility, this has now been largely abandoned in favor of other solutions.
schema 1. "database" {database schema}. 2. "logic" {axiom schema}. 3. "data" {XML schema}.
Science [from Latin scientia from scire to know] In its widest sense formulated knowledge, a knowledge of structure, laws, and operations. The unity of human knowledge may be artificially divided into religion, philosophy, and science. Science and philosophy, as presently understood, have in common the quality of being speculative, as opposed to religion, which in the West is supposed to be founded merely on faith and moral sentiments. The present distinction between science and philosophy lies largely in their respective fields of speculation. What is known as modern science investigates the phenomena of physical nature and by inferential reasoning formulates general laws therefrom. Its method is called inductive and its data are so-called facts — i.e., sensory observations; whereas deductive philosophy starts from axioms. Yet a scientist, in order to reason from his data at all, must necessarily use both induction and deduction.
Scratchpad II ::: See Scratchpad I, AXIOM.[Scratchpad II Programming Language Manual, R.D. Jenks et al, IBM, 1985].[Scratchpad II Newsletter: Computer Algebra Group, TJWRC, Box 218, Yorktown Hts, NY 10598].
Scratchpad II See {Scratchpad I}, {AXIOM}. ["Scratchpad II Programming Language Manual", R.D. Jenks et al, IBM, 1985]. [Scratchpad II Newsletter: Computer Algebra Group, TJWRC, Box 218, Yorktown Hts, NY 10598].
sententious ::: a. --> Abounding with sentences, axioms, and maxims; full of meaning; terse and energetic in expression; pithy; as, a sententious style or discourse; sententious truth.
Comprising or representing sentences; sentential.
Serpent One of the most fundamental and prolific symbols of the mystery-language. Its most basic meaning is of the eternal, alternating, cyclic motion during cosmic manifestation. For motion, which to the physicist and the philosopher alike seems an abstraction, is for the ancient wisdom a primordial principle or axiom, of the same order as space and time, existing per se. Never does motion cease utterly even during kosmic pralaya. And motion is essentially circular: where physics would derive circular motion from a composition of rectilinear motions, the opposite procedure would be that of the ancient wisdom. This circular motion, compounding itself into spirals, helixes, and vortices, is the builder of worlds, bringing together the scattered elements of chaos; motion per se is essential cosmic intelligence. This circular motion, returning upon itself like a serpent swallowing its tail, represents the cycles of time. This conscious energy in spirals whirls through all the planes of cosmos as fohat and his innumerable sons — the cosmic energies and forces, fundamentally intelligent, operating in every scale or grade of matter. The caduceus of Hermes, twin serpents wound about a staff, represents cosmically the mighty drama of evolution, in its twin aspects, the staff or tree standing for the structural aspect, the serpent for the fohatic forces that animate the structure.
set theory "mathematics" A mathematical formalisation of the theory of "sets" (aggregates or collections) of objects ("elements" or "members"). Many mathematicians use set theory as the basis for all other mathematics. Mathematicians began to realise toward the end of the 19th century that just doing "the obvious thing" with sets led to embarrassing {paradox}es, the most famous being {Russell's Paradox}. As a result, they acknowledged the need for a suitable {axiomatisation} for talking about sets. Numerous such axiomatisations exist; the most popular among ordinary mathematicians is {Zermelo Fränkel set theory}. {The beginnings of set theory (http://www-groups.dcs.st-and.ac.uk/~history/HistoryTopics.html)}. (1995-05-10)
symbolic mathematics ::: (mathematics, application) (Or symbolic math) The use of computers to manipulate mathematical equations and expressions in symbolic form, as opposed of one expression into another, simplification of an expression, change of subject etc.One of the best known symbolic mathematics software packages is Mathematica. Others include ALAM, ALGY, AMP, Ashmedai, AXIOM*, CAMAL, CAYLEY, CCalc, CLAM, Pari, REDUCE, SAC-1, SAC2, SAINT, Schoonschip, Scratchpad I, SHEEP, STENSOR, SYMBAL, SymbMath, Symbolic Mathematical Laboratory, TRIGMAN, UBASIC.Usenet newsgropup: sci.math.symbolic. (1995-04-12)
symbolic mathematics "mathematics, application" (Or "symbolic math") The use of computers to manipulate mathematical equations and expressions in symbolic form, as opposed to manipulating the numerical quantities represented by those symbols. Such a system might be used for symbolic integration or differentiation, substitution of one expression into another, simplification of an expression, change of subject etc. One of the best known symbolic mathematics software packages is {Mathematica}. Others include {ALAM}, {ALGY}, {AMP}, {Ashmedai}, {AXIOM*}, {CAMAL}, {CAYLEY}, {CCalc}, {CLAM}, {CoCoA}(?), {ESP}, {FLAP}, {FORM}, {FORMAL}, {Formula ALGOL}, {GAP}, {JACAL}, {LiE}, {Macaulay}, {MACSYMA}, {Magic Paper}, {MAO}, {Maple}, {Mathcad}, {MATHLAB}, {MuMath}, {Nother}, {ORTHOCARTAN}, {Pari}, {REDUCE}, {SAC-1}, {SAC2}, {SAINT}, {Schoonschip}, {Scratchpad I}, {SHEEP}, {STENSOR}, {SYMBAL}, {SymbMath}, {Symbolic Mathematical Laboratory}, {TRIGMAN}, {UBASIC}. {Usenet} newsgropup: {news:sci.math.symbolic}. (1995-04-12)
The axiom of extensionality as above stated has (incidentally to its principal purpose) the effect of excluding non-classes entirely and assuming that everything is a class. This assumption can be avoided if desired, at the cost of complicating the axioms somewhat -- one method would be to introduce an additional functional constant, expressing the property to be a class (or set), and to modify the axioms accordingly, the domain of individuals being thought of as possibly containing other things besides sets.
The formalization as a logistic system of the functional calculus of order omega with axiom of infinity leads, by a method which cannot be given here, to a (definite but quite complicated) proposition of arithmetic which is equivalent to -- in a certain sense, expresses -- the consistency of the system. This proposition of arithmetic can be represented within the system by a formula A containing no free variables, and the following second form of gödel's incompleteness theorem can then be proved: If the system is consistent, then the formula A, although its meaning is a true proposition of arithmetic, is not a theorem of the system. We might, of course, add A to the system as a new primitive formula -- we would then have a new system, whose consistency would correspond to a new proposition of arithmetic, represented by a new formula B (containing no free variables), and we would still have in the new system, if consistent, that B was not a theorem.
The Greek Skeptics and Pyrrhonists demonstrate that rigid logic leads to contradictory conclusions (antinomies), a fact which led them to doubt the efficacy of the mentality as a means of ascertaining truth. A strictly logical system may be found in pure mathematics, where we lay down axioms and postulates, which are to be treated as not open to question; and then proceed by rigid rules to the inevitable conclusion. But what is possible in an ideal science is not possible in an actual world of infinite variety and fluidity. Theosophy places the subject in a different light, because it recognizes the existence in man of powers of direct cognition by the awakened faculties of buddhi. Thus man has the means of a true deductive system; but even so, deduction must be considered together with induction, analogy, and other methods, as merely one of the various means by which we arrive at a knowledge of truth.
The Hermetic Axiom ::: As Above So Below, As Without So Within. There are several Hermetic principles and axioms that can be discussed but this is the most famous and illustrates the fact that macrocosm and microcosm are intimately linked and that not only does matter generate mind, so too does mind generate matter.
theorem: A statement which has been proven true by sentences already established to be true: i.e. either axioms (assumed to be true) or other theorems
Theory: (Gr. theoria, viewing) The hypothetical universal aspect of anything. For Plato, a contemplated truth. For Aristotle, pure knowledge as opposed to the practical. An abstraction from practice. The principle from which practice proceeds. Opposite of practice. -- J.K.F. Hypothesis. More loosely: supposition, whatever is problematic, verifiable but not verified. (As opposed to practice) systematically organized knowledge of relatively high generality. (See "the theory of light"). (As opposed to laws and observations): explanation. The deduction of the axioms and theorems of one system from assertions (not necessarily verified) from another system and of a relatively less problematic and more intelligible nature. (Note: Since criteria of what is 'intelligible' and 'problematic' are subjective and liable to fluctuation, any definition of the term is bound to be provisional. It might be advisable to distinguish between laws (general statements in a system), principles (axioms), and theories (methods for deriving the axioms by means of appropriate definitions employing terms from other systems). -- M.B.
The precipitates of the propaedeutical effort are to be found, for Spinoza, in the definitions, axioms, postulates, and within the structural plan expressed in the geometrical ordering. It is highly probable that Spinoza would have admitted the tentative character of at least some of the definitions, axioms, and postulates formulated by him. He doubtless saw the possibility that the process of inquiry, revising, augmenting, and re-coordinating the fund of knowledge, might demand alteration in the structural bases of systematic expression as well as in the knowledge to be ordered. Such changes, however, would occur within limits set by the propaedeutical disclosures and the general framework. Advance might require the abandonment of an older metaphysical element, and the substitution of a new one. But with equal likelihood, the advance of knowledge would make possible a richer and deeper apprehension of the content of fixed principles. To illustrate: The first definition of the Ethica, that of Causa sui, might well be for Spinoza a principle that awakened reason must accept, a truth whose priority and validity could not be undermined. He might regard it as a minimal definition of reality, of the nature of the ultimate object of inquiry. On the other hand, Spinoza, it may be conjectured, would not claim for every element of his system a similar finality. Just as he recognizes the role of hypothesis in science, in a similar way, he would recognize the tentative character of some metaphysical and theological elements.
The prohibition against impredicative definition was incorporated by Russell into his ramified theory of types (1908) and is now usually identified with the restriction to the ramified theorv of types without the axiom of reducibility. (Poincare, however, never made his principle exact and may have intended, vaguely, a less severe restriction than this -- as indeed some passages in later writings would indicate.) -- A. C.
The restriction which is imposed in order to avoid paradox can be seen in connection with the axiom of subset formation. Instead of this axiom, an uncritical formulation of axioms for set theory might well have included (Et)[xεt ≡x A], asserting the existence of a set t whose members are the sets x satisfying an arbitrary condition A expressible in the notation of the system. This, however, would lead at once to the Russell paradox by taking A to be ∼ xεx and then going through a process of inference which can be described briefly by saying that x is put equal to t. As actually proposed, however, the axiom of subset formation allows the use of the condition A only to obtain a set t whose members are the sets x which are members of a previously given set z and satisfy A. This is not known to lead to paradox.
Thesis: (Gr. thesis) In Aristotle's logic (1) an undemonstrated proposition used as a premiss in a syllogism, sometimes distinguished from axiom in that it need not be self-evident or intrinsically necessary; (2) any proposition contrary to general opinion but capable of being supported by reasoning. See Antithesis, Dialectic, Synthesis. -- G.R.M.
This "postulate" has resisted proof for many centuries before consideration is given to the possibility that it is simply not necessarily true. This leads to the development of non-Euclidean geometry, while the familiar geometry in which the fifth postulate is true is known as Euclidean Geometry. It should be noted that we now know of the parallel postulate's independence from the other postulates, that is, the parallel postulate cannot be proven from the four other postulates. In that sense, the parallel postulate of Euclid is more of an axiom for a particular geometric system.
Trividya (Sanskrit) Trividyā [from tri three + vidyā knowledge, science] The three knowledges or sciences; the three fundamental axioms in mysticism: “(a) the impermanency of all existence, or Anityata; (b) suffering and misery of all that lives and is, or Dukha [duhkhata]; and (c) all physical, objective existence as evanescent and unreal as a water-bubble in a dream, or Anatmata” (TG 344).
universe of discourse "artificial intelligence" In {ontology}, the set of all {entities} that can be represented in some {declarative language} or other {formal system}. Each entity is represented by a name and may have some human-readable description of its meaning. Formal {axioms} constrain the interpretation and well-formed use of these names. (2005-07-29)
universe of discourse ::: (ontology) In ontology, the set of all entities that can be represented in some declarative language or other formal system.Each entity is represented by a name and may have some human-readable description of its meaning. Formal axioms constrain the interpretation and well-formed use of these names.(2005-07-29)
VII. Probability as a Physical Magnitude determined by Axioms.. This theory, which is favoured mainly by the Intuitionist school of mathematics, considers probability as a physical constant of which frequencies are measures. Thus, any frequency is an approximate measure of one physical constant attached to an event and to a set of trials: this constant is the probability of that event over the set of trials. As the observed frequencies differ little for large numbers of trials from their corresponding probabilities, some obvious properties of frequencies may be extended to probabilities. This is done without proceeding to the limit, but through general approximation as in the case of physical magnitudes. These properties are not constructed (as in the axiomatization of Mises), but simply described as such, they form a set of axioms defining probability. The classical postulates involved in the treatises of Laplace, Bertrand or Poincare have been modified in this case, under the joint influence of the discovery of measure by Borei, and of the use of abstract sets. Their new form has been fully stated by Kolmogoroff and interpreted by Frechet who proposes to call this latest theory the 'modernized axiomatic definition' of probability. Its interpretation requires that it should be preceded by an inductive synthesis, and followed by numerical verifications.
VI. Probability as a Limit of Frequencies. According to this view, developed especially by Mises and by Wald, the probability of an event is equal to its total frequency, that is to the limit, if it exists, of the frequency of that event in n trials, when n tends to infinity. The difficulty of working out this conception led Mises to propose the notion of a collective in an attempt to evolve conditions for a true random sequence. A collective is a random sequence of supposed results of trials when (1) the total frequency of the event in the sequence exists, and (2) the same property holds with the same limiting value when the sequence is replaced by any sequence derived from it. Various methods were devised by Copeland, Reichenbach and others to avoid objections to the second condition: they were generalized by Wald who restricted the choice of the "laws of selection" defining the ranks of the trials forming one of the derived sequences, by his postulate that these laws must form a denumerable set. This modification gives logical consistency to this theory at the expense of its original simplicity, but without disposing of some fundamental shortcomings. Thus, the probability of an event in a collective remains a relative notion, since it must be known to which denumerable set of laws of selection it has been defined relatively, in order to determine its meaning, even though its value is not relative to the set. Controversial points about the axiomatization of this theory show the possibility of other alternatives.
With the aid of the axiom of infinity and a method of dealing with classes and descriptions, the non-negative integers may be introduced in any one of various ways (e.g., following Frege and Russell, as finite cardinal numbers), and arithmetic (elementary number theory) derived formally within the system. With the further addition of the axiom of choice, analysis (real number theory) may be likewise derived.
W. V. Quine, On the axiom of reducibthty, Mind, n. s. vol. 45 (1936), pp. 498-500.
[xεz ≡x xεy] ⊃ z = y. (Axiom of extensionality)
[yεz ⊃x (Ex')[A ≡x x=x']] ⊃ (Et)[xεt ≡x (Ey)[yεz]A]. (Axiom of replacement)
[yεz ⊃y [y'εz ⊃y' [[xεy][xεy'] ⊃x y=y']]] ⊃ (Et)[yεz ⊃y [x''εy ⊃x'' (Ex')[[xεy][xεt] ≡x x=x']]]. (Axiom of choice)
Zermelo, Ernst (Friedrich Ferdinand), 1871-, German mathematician. Professor of mathematics at Zurich, 1910-1916, and at Freiburg, 1926-. His important contributions to the foundations of mathematics are the Zermelo axiomatic set theory (see logic, formal, § 9), and the explicit enunciation of the axiom of choice (q.v.) and proof of its equivalence to the proposition that every class cm be well-ordered. -- A.C.
Zermelo Fränkel set theory "mathematics" A {set theory} with the {axioms} of {Zermelo set theory} (Extensionality, Union, Pair-set, Foundation, Restriction, Infinity, Power-set) plus the Replacement {axiom schema}: If F(x,y) is a {formula} such that for any x, there is a unique y making F true, and X is a set, then {F x : x in X} is a set. In other words, if you do something to each element of a set, the result is a set. An important but controversial {axiom} which is NOT part of ZF theory is the {Axiom of Choice}. (1995-04-10)
Zermelo Fr�nkel set theory ::: (mathematics) A set theory with the axioms of Zermelo set theory (Extensionality, Union, Pair-set, Foundation, Restriction, Infinity, Power-set) plus the Replacement axiom schema:If F(x,y) is a formula such that for any x, there is a unique y making F true, and X is a set, then {F x : x in X} is a set. In other words, if you do something to each element of a set, the result is a set.An important but controversial axiom which is NOT part of ZF theory is the Axiom of Choice. (1995-04-10)
Zermelo set theory "mathematics" A {set theory} with the following set of {axioms}: Extensionality: two sets are equal if and only if they have the same elements. Union: If U is a set, so is the union of all its elements. Pair-set: If a and b are sets, so is {a, b}. Foundation: Every set contains a set disjoint from itself. Comprehension (or Restriction): If P is a {formula} with one {free variable} and X a set then {x: x is in X and P(x)}. is a set. Infinity: There exists an {infinite set}. Power-set: If X is a set, so is its {power set}. Zermelo set theory avoids {Russell's paradox} by excluding sets of elements with arbitrary properties - the Comprehension axiom only allows a property to be used to select elements of an existing set. {Zermelo Fränkel set theory} adds the Replacement axiom. [Other axioms?] (1995-03-30)
Zermelo set theory ::: (mathematics) A set theory with the following set of axioms:Extensionality: two sets are equal if and only if they have the same elements.Union: If U is a set, so is the union of all its elements.Pair-set: If a and b are sets, so is {a, b}. Foundation: Every set contains a set disjoint from itself.Comprehension (or Restriction): If P is a formula with one free variable and X a set then {x: x is in X and P(x)}. is a set.Infinity: There exists an infinite set.Power-set: If X is a set, so is its power set.Zermelo set theory avoids Russell's paradox by excluding sets of elements with arbitrary properties - the Comprehension axiom only allows a property to be used to select elements of an existing set.Zermelo Fr�nkel set theory adds the Replacement axiom.[Other axioms?] (1995-03-30)
ZFC "mathematics" {Zermelo Fränkel set theory} plus the {Axiom of Choice}. A favourite {axiomatisation} of {set theory}. (1995-03-29)
ZFC ::: (mathematics) Zermelo Fr�nkel set theory plus the Axiom of Choice. A favourite axiomatisation of set theory. (1995-03-29)
zong. (J. shu; K. chong 宗). A polysemous term in Chinese, which can refer to the "core teaching," "cardinal doctrine," or central "axiom" of a text or scholastic philosophy, or to a "school," "tradition," or even "lineage." Because of the denotation of zong as a cardinal doctrine, the term is used to translate the Sanskrit SIDDHĀNTA, an "axiom" or principal "tenet" of the various schools of Indian philosophy (both Buddhist and non-Buddhist). Although commonly translated as "school" ("sect" is incorrect in a Chinese Buddhist context), this rendering is only appropriate for those scholastic and practice traditions that trace themselves back through an unbroken lineage of ancestors to a specific founding "patriarch" (ZUSHI), such as the TIANTAI ZONG, HUAYAN ZONG, and CHAN ZONG. Commentarial traditions focused upon the exegesis of a specific text, such as the DI LUN (DAsABHuMIKASuTRA), SHE LUN (MAHĀYĀNASAMGRAHA), or CHENGSHI LUN (*TATTVASIDDHI), do not qualify as true schools of Chinese Buddhism, since they neither claim to derive from a single founder nor posit an exclusive lineage of successors; they might more correctly be termed "scholastic traditions."
Z /zed/ "language, specification" 1. (After {Zermelo-Fränkel set theory}) A {specification language} developed by the {Programming Research Group} at Oxford University around 1980. Z is used for describing and modelling computing systems. It is based on {axiomatic set theory} and {first order predicate logic}. Z is written using many non-{ASCII} symbols. It was used in the {IBM} {CICS} project. See also {Z++}. ["Understanding Z", J.M. Spivey, Cambridge U Press 1988]. 2. "language, simulation" A {stack}-based, complex arithmetic {simulation} language from {ZOLA Technologies}. (1995-08-11)
Z ::: /zed/ language, specification> 1. (After Zermelo-Fr�nkel set theory) A specification language developed by the Programming Research Group at Oxford systems. It is based on axiomatic set theory and first order predicate logic. Z is written using many non-ASCII symbols. It was used in the IBM CICS project.See also Z++.[Understanding Z, J.M. Spivey, Cambridge U Press 1988].2. (language, simulation) A stack-based, complex arithmetic simulation language from ZOLA Technologies. (1995-08-11)
KEYS (10k)
1 Peter J Carroll
1 Aleister Crowley
NEW FULL DB (2.4M)
8 Fyodor Dostoyevsky
5 Eric Metaxas
5 Anonymous
4 Bertrand Russell
3 Thomas Jefferson
3 Stephen King
3 Gustave Flaubert
3 Barbara Demick
3 Audre Lorde
3 Arthur Conan Doyle
3 Albert Einstein
2 Will Durant
2 Wayne W Dyer
2 Victor Hugo
2 Thomas Carlyle
2 Samuel Johnson
2 Mary Karr
2 Julius Evola
2 Idries Shah
2 Harlan Coben
1:We have all a ruling defect, which is for our soul as the umbilical cord of its birth in sin, and it is by this that the enemy can always lay hold upon us: for some it is vanity, for others idleness, for the majority egotism. Let a wicked and crafty mind avail itself of this means and we are lost; we may not go mad or turn idiots, but we become positively alienated, in all the force of the expression - that is, we are subjected to a foreign suggestion. In such a state one dreads instinctively everything that might bring us back to reason, and will not even listen to representations that are opposed to our obsession. Here is one of the most dangerous disorders which can affect the moral nature. The sole remedy for such a bewitchment is to make use of folly itself in order to cure folly, to provide the sufferer with imaginary satisfactions in the opposite order to that wherein he is now lost. Endeavour, for example, to cure an ambitious person by making him desire the glories of heaven - mystic remedy; cure one who is dissolute by true love - natural remedy; obtain honourable successes for a vain person; exhibit unselfishness to the avaricious and procure for them legitimate profit by honourable participation in generous enterprises, etc. Acting in this way upon the moral nature, we may succeed in curing a number of physical maladies, for the moral affects the physical in virtue of the magical axiom: "That which is above is like unto that which is below." This is why the Master said, when speaking of the paralyzed woman: "Satan has bound her." A disease invariably originates in a deficiency or an excess, and ever at the root of a physical evil we shall find a moral disorder. This is an unchanging law of Nature. ~ Eliphas Levi, Transcendental Magic, #KEYS
2:Chapter LXXXII: Epistola Penultima: The Two Ways to Reality
Cara Soror,
Do what thou wilt shall be the whole of the Law.
How very sensible of you, though I admit somewhat exacting!
You write-Will you tell me exactly why I should devote so much of my valuable time to subjects like Magick and Yoga.
That is all very well. But you ask me to put it in syllogistic form. I have no doubt this can be done, though the task seems somewhat complicated. I think I will leave it to you to construct your series of syllogisms yourself from the arguments of this letter.
In your main question the operative word is "valuable. Why, I ask, in my turn, should you consider your time valuable? It certainly is not valuable unless the universe has a meaning, and what is more, unless you know what that meaning is-at least roughly-it is millions to one that you will find yourself barking up the wrong tree.
First of all let us consider this question of the meaning of the universe. It is its own evidence to design, and that design intelligent design. There is no question of any moral significance-"one man's meat is another man's poison" and so on. But there can be no possible doubt about the existence of some kind of intelligence, and that kind is far superior to anything of which we know as human.
How then are we to explore, and finally to interpret this intelligence?
It seems to me that there are two ways and only two. Imagine for a moment that you are an orphan in charge of a guardian, inconceivably learned from your point of view.
Suppose therefore that you are puzzled by some problem suitable to your childish nature, your obvious and most simple way is to approach your guardian and ask him to enlighten you. It is clearly part of his function as guardian to do his best to help you. Very good, that is the first method, and close parallel with what we understand by the word Magick.
We are bothered by some difficulty about one of the elements-say Fire-it is therefore natural to evoke a Salamander to instruct you on the difficult point. But you must remember that your Holy Guardian Angel is not only far more fully instructed than yourself on every point that you can conceive, but you may go so far as to say that it is definitely his work, or part of his work; remembering always that he inhabits a sphere or plane which is entirely different from anything of which you are normally aware.
To attain to the Knowledge and Conversation of the Holy Guardian Angel is consequently without doubt by far the simplest way by which you can yourself approach that higher order of being.
That, then, is a clearly intelligible method of procedure. We call it Magick.
It is of course possible to strengthen the link between him and yourself so that in course of time you became capable of moving and, generally speaking, operating on that plane which is his natural habitat.
There is however one other way, and one only, as far as I can see, of reaching this state.
It is at least theoretically possible to exalt the whole of your own consciousness until it becomes as free to move on that exalted plane as it is for him. You should note, by the way, that in this case the postulation of another being is not necessary. There is no way of refuting the solipsism if you feel like that. Personally I cannot accede to its axiom. The evidence for an external universe appears to me perfectly adequate.
Still there is no extra charge for thinking on those lines if you so wish.
I have paid a great deal of attention in the course of my life to the method of exalting the human consciousness in this way; and it is really quite legitimate to identify my teaching with that of the Yogis.
I must however point out that in the course of my instruction I have given continual warnings as to the dangers of this line of research. For one thing there is no means of checking your results in the ordinary scientific sense. It is always perfectly easy to find a subjective explanation of any phenomenon; and when one considers that the greatest of all the dangers in any line of research arise from egocentric vanity, I do not think I have exceeded my duty in anything that I have said to deter students from undertaking so dangerous a course as Yoga.
It is, of course, much safer if you are in a position to pursue in the Indian Jungles, provided that your health will stand the climate and also, I must say, unless you have a really sound teacher on whom you can safely rely. But then, if we once introduce a teacher, why not go to the Fountain-head and press towards the Knowledge and conversation of the Holy Guardian Angel?
In any case your Indian teacher will ultimately direct you to seek guidance from that source, so it seems to me that you have gone to a great deal of extra trouble and incurred a great deal of unnecessary danger by not leaving yourself in the first place in the hands of the Holy Guardian Angel.
In any case there are the two methods which stand as alternatives. I do not know of any third one which can be of any use whatever. Logically, since you have asked me to be logical, there is certainly no third way; there is the external way of Magick, and the internal way of Yoga: there you have your alternatives, and there they cease.
Love is the law, love under will.
Fraternally,
666 ~ Aleister Crowley, Magick Without Tears,#KEYS
*** WISDOM TROVE ***
1:Axiom : Novel must have either one living character or a perfect pattern: fails otherwise. ~ e-m-forster, @wisdomtrove 2:Be guided by the axiom: There are no limits to the ability to contribute on the part of a properly selected, well-trained, appropriately supported, and, above all, committed person. ~ tom-peters, @wisdomtrove 3:The idea that to make a man work you've got to hold gold in front of his eyes is a growth, not an axiom. We've done that for so long that we've forgotten there's any other way. ~ f-scott-fitzgerald, @wisdomtrove 4:The writer who neglects punctuation, or mispunctuates, is liable to be misunderstood for the want of merely a comma, it often occurs that an axiom appears a paradox, or that a sarcasm is converted into a sermonoid. ~ edgar-allan-poe, @wisdomtrove 5:It may safely be received as an axiom in our political system, that the state governments will in all possible contingencies afford complete security against invasions of the public liberty by the national authority. ~ alexander-hamilton, @wisdomtrove 6:In a conquered country benevolence is not humanitarianism. It is a general political axiom that a conqueror must not inspire a good opinion of his benevolence until he has demonstrated that he can be severe with malefactors. ~ napoleon-bonaparte, @wisdomtrove 7:The maxim is, that whatever can be affirmed (or denied) of a class, may be affirmed (or denied) of everything included in the class. This axiom, supposed to be the basis of the syllogistic theory, is termed by logicians the dictum de omni et nullo. ~ john-stuart-mill, @wisdomtrove 8:We no longer even understand the question whether change is by itself good or bad, ... We start out with the axiom that it is the norm. We do not see change as altering the order... We see change as being order itself - indeed the only order we can comprehend today is a dynamic, a moving, a changing one. ~ peter-drucker, @wisdomtrove *** NEWFULLDB 2.4M ***
1:My axiom is, to succeed in business: avoid my example. ~ Mark Twain, #NFDB
2:The fundamental axiom of economics is the human mercenary instinct. Without ~ Liu Cixin, #NFDB
3:Whatever is referred to must exist. Let us call this the axiom of existence. ~ John Searle, #NFDB
4:Perhaps the truest axiom in baseball is that the toughest thing to do is repeat. ~ Walter Alston, #NFDB
5:keep this ancient axiom in mind: I get what I think about, whether I want it or not. ~ Wayne W Dyer, #NFDB
6:There is a just Latin axiom, that he who seeks a reason for everything subverts reason. ~ Epes Sargent, #NFDB
7:It is a fundamental axiom of married life that you must lie to a woman. She likes it! ~ Agatha Christie, #NFDB
8:Let this serve as an axiom to every lover: A woman who refuses lunch refuses everything. ~ Enid Bagnold, #NFDB
9:Axiom : Novel must have either one living character or a perfect pattern: fails otherwise. ~ E M Forster, #NFDB
10:Aristotle's axiom: The worst form of inequality is to try to make unequal things equal. ~ Laurence J Peter, #NFDB
11:Hm … yes, all is in a man's hands and he lets it all slip from cowardice, that's an axiom ~ Fyodor Dostoyevsky, #NFDB
12:Hm… yes, all is in a man’s hands and he lets it all slip from cowardice, that’s an axiom. ~ Fyodor Dostoyevsky, #NFDB
13:They say that loving eyes can never see, but that's a fool's axiom. Sometimes, they see too much ~ Stephen King, #NFDB
14:They say that loving eyes can never see, but that’s a fool’s axiom. Sometimes they see too much. ~ Stephen King, #NFDB
15:His work seems to confirm my old axiom: it is useless to try to keep the whole body alive. ~ Adolfo Bioy Casares, #NFDB
16:Hm... yes, all is in a man's hands and he lets it all slip from cowardice, that's an axiom. ~ Fyodor Dostoyevsky, #NFDB
17:It has long been an axiom of mine that the little things are infinitely the most important. ~ Arthur Conan Doyle, #NFDB
18:The fundamental axiom, then, for the study of man is the existence of individual consciousness ~ Murray Rothbard, #NFDB
19:Experiment escorts us last-
His pungent company
will not allow an axiom
An opportunity ~ Emily Dickinson,#NFDB
20:The institution of taxation rests foursquare on the axiom that somebody must rule somebody else. ~ Frank Chodorov, #NFDB
21:An old Qabalistic axiom states that "every blade of grass has over it an Angel bidding it 'Grow. ~ Stephen Skinner, #NFDB
22:The axiom of conditioned repetition, like the binomial theorem, is nothing but a piece of insolence. ~ Edward Abbey, #NFDB
23:It is an old psychological axiom that constant exposure to the object of fear immunizes against the fear. ~ Maxwell Maltz, #NFDB
24:I imagine as an axiom you could say that the better the play, the less "creativity" the director need exert. ~ Edward Albee, #NFDB
25:There's no right or wrong way to seek truth, and as the old axiom states, all paths lead to the same place. ~ Skye Alexander, #NFDB
26:My morality, the morality of reason, is contained in a single axiom: existence exists - and in a single choice: to live. ~ Ayn Rand, #NFDB
27:Perhaps I had inadvertently brushed up against the Buddhist axiom, that enlightenment is the ultimate disappointment. ~ Maggie Nelson, #NFDB
28:The simple Francis Bacon axiom, one we constantly stressed to our students, applied here: Knowledge is power. Benedict ~ Harlan Coben, #NFDB
29:This truth must be recognized as a dogma and assume the validity of an axiom in the general understanding of painting ~ Fernand Leger, #NFDB
30:The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes. ~ Bertrand Russell, #NFDB
31:you may take it as an axiom that you cannot profit in Wall Street by continuously doing the obvious or the popular thing ~ Benjamin Graham, #NFDB
32:a straight line is the shortest possible line between any two points - an axiom equally true in morals as in mathematics. ~ Maria Edgeworth, #NFDB
33:Axiome: la haine du bourgeois est le commencement de la vertu. Axiom: Hatred of the bourgeois is the beginning of wisdom. ~ Gustave Flaubert, #NFDB
34:CLASSIC American axiom warns: “Don’t try to think yourself into a new way of acting: Act yourself into a new way of thinking. ~ Ernest Kurtz, #NFDB
35:It is a spiritual axiom that every time we are disturbed, no matter what the cause, there is something wrong with us. ~ Alcoholics Anonymous, #NFDB
36:AXIOM. — Property is the Right of Increase claimed by the Proprietor over any thing which he has stamped as his own. ~ Pierre Joseph Proudhon, #NFDB
37:It is an axiom of our profession that this work is experiential; one is not born to it, one becomes more skillful with time. ~ Jason Matthews, #NFDB
38:(…)man holds the remedy in his own hands, and lets everything go its own way, simply through cowardice- that is an axiom. ~ Fyodor Dostoyevsky, #NFDB
39:The Americans are the living refutation of the Cartesian axiom, "I think, therefore I am": Americans do not think, yet they are. ~ Julius Evola, #NFDB
40:Axiom: the best place to conserve your water is in your body. It keeps your energy up. You’re stronger. Trust your stillsuit.” She ~ Frank Herbert, #NFDB
41:Honesty is the best policy, says the familiar axiom; but people who are honest on that principle defraud no one but themselves. ~ James A Garfield, #NFDB
42:The world is maintained by change—in the elements and in the things they compose. That should be enough for you; treat it as an axiom. ~ Anonymous, #NFDB
43:For somebody who's never run for office before, Donald Trump understands that old axiom, "Define yourself before you're defined." ~ Kellyanne Conway, #NFDB
44:To accept the dignity of another person is an axiom. It has nothing to do with subduing, supporting, or giving charity to other people. ~ Leo Tolstoy, #NFDB
45:To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed. ~ Bertrand Russell, #NFDB
46:That’s when I learned my first hard lesson: Men Never Change. I was still too naïve to realize that the axiom didn’t apply just to men. ~ Laurelin Paige, #NFDB
47:You see, Mersualt, all the misery and cruelty of our civilisation can be measured by this one stupid axiom: happy nations have no history. ~ Albert Camus, #NFDB
48:Government is the perfect portrayer of the accuracy of the axiom that if you lie big enough, long enough, the lie becomes the “truth. ~ Neale Donald Walsch, #NFDB
49:It should be remembered, as an axiom of eternal truth in politics, that whatever power in any government is independent, is absolute also. ~ Thomas Jefferson, #NFDB
50:I have never believed in the axiom that a writer should first and foremost write about what he knows. I think it’s a piece of misinformation. ~ William Trevor, #NFDB
51:Keep in mind this basic axiom—if all that now exists was once imagined, then what you want to exist for you in the future must now be imagined. ~ Wayne W Dyer, #NFDB
52:It was a BuSab axiom that all power blocs tended toward aristocratic forms, that the descendants of decision makers dominated the power niches. ~ Frank Herbert, #NFDB
53:It has long been an axiom of mine that the little things are infinitely the most important." - Sherlock Holmes in A Case of Identity - 1891 ~ Arthur Conan Doyle, #NFDB
54:Remark 1.4.20. The empty set axiom is needed in order to rule out the degenerate situation in which every set (including the empty set) has innite measure ~ Anonymous, #NFDB
55:I found bossing other people about such a delightful novelty that I had to remind myself of Lord Acton's famous axiom about its tendency to corrupt. ~ Victoria Clayton, #NFDB
56:The demand that we love our neighbor as ourselves contains as an axiom the demand that we shall love ourselves, shall accept ourselves as we were created. ~ Max Frisch, #NFDB
57:The great spy novelist John Le Carré suggested this axiom: The cat sat on the mat is not the beginning of a story. The cat sat on the dog's mat, is. ~ James Scott Bell, #NFDB
58:It is an axiom of political science in the United States that the sole means of neutralizing the effects of newspapers is to multiply their number. ~ Alexis de Tocqueville, #NFDB
59:It may be assumed as an axiom that Providence has never gifted any political party with all of political wisdom or blinded it with all of political folly. ~ John George Nicolay, #NFDB
60:Everything is darkest," Xaphen mused, "before the dawn."
"That, my brother, is an axiom that sounds immensely profound until you realize it's a lie. ~ Aaron Dembski Bowden,#NFDB
61:Axiom: you are a sea. Your eye- lids curve over chaos My hands where they touch you, create small inhabited islands soon you will be all earth: a known land, a country. ~ Margaret Atwood, #NFDB
62:At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it. ~ Bertrand Russell, #NFDB
63:Be guided by the axiom: There are no limits to the ability to contribute on the part of a properly selected, well-trained, appropriately supported, and, above all, committed person. ~ Tom Peters, #NFDB
64:There’s an axiom evolutionary biologists used to like: “ontogeny recapitulates phylogeny,” or the development of the embryonic individual repeats that of its species’ evolution. ~ Rebecca Solnit, #NFDB
65:The first axiom of Marx Scientist is that everything they tell you is a lie. The second axiom is that it doesn’t matter, since you are lying too. The third axiom is ‘Kill all liars! ~ Roger Scruton, #NFDB
66:The idea that to make a man work you've got to hold gold in front of his eyes is a growth, not an axiom. We've done that for so long that we've forgotten there's any other way. ~ F Scott Fitzgerald, #NFDB
67:There is no more fundamental axiom of American freedom than the familiar statement: In a free country we punish men for the crimes they commit but never for the opinions they have. ~ Harry S Truman, #NFDB
68:The Convention promulgated this great axiom: "The liberty of one citizen ends where the liberty of another citizen begins," which comprises in two lines the entire law of human society. ~ Victor Hugo, #NFDB
69:The only thing that might have annoyed some mathematicians was the presumption of assuming that maybe the axiom of choice could fail, and that we should look into contrary assumptions. ~ Alonzo Church, #NFDB
70:hold it, then, as an axiom — or else I’d stop writing right now — that our calling is to follow Jesus in our context rather than to retrieve and re-create his context in our world. What ~ Scot McKnight, #NFDB
71:School is an institution built on the axiom that learning is the result of teaching. And institutional wisdom continues to accept this axiom, despite overwhelming evidence to the contrary. ~ Ivan Illich, #NFDB
72:It is an axiom of political life that you never raise expectations, whether in a political or military campaign, because your defeats are then magnified and your victories discounted. ~ Charles Krauthammer, #NFDB
73:A new generation gladly abandons its critical and skeptical faculties. Old slogans and hatreds are dusted off. What was only recently muttered guiltily is now offered as political axiom and agenda. ~ Carl Sagan, #NFDB
74:His axiom is that the soul must empty itself of self in order to be filled with God, that it must be purified of the last traces of earthly dross before it is fit to become united with God. In ~ Juan de la Cruz, #NFDB
75:You want at least 5% of the population being serious. That five, 6% of the population carries the rest of the people. You've heard that old axiom: 5% of the people pull the wagon; 95% are in it. ~ Rush Limbaugh, #NFDB
76:the old axiom that 'all power corrupts' has doubtful validity, because it derives from our neglect of Plato's advice to find men carefully and train them by methods which make them fit for heroes. ~ Oswald Mosley, #NFDB
77:...the mathematician uses an indirect definition of congruence, making use of the fact that the axiom of parallels together with an additional condition can replace the definition of congruence. ~ Hans Reichenbach, #NFDB
78:Many successful people are no more talented than unsuccessful people. The difference between them lies in the old axiom that successful people do those things that unsuccessful people don't like to do. ~ Harvey Mackay, #NFDB
79:The controversy as to whether socialism is possible has been settled by the fact that it exists, and it is a fundamental axiom of my philosophy, at any rate, that anything that exists, is possible. ~ Kenneth E Boulding, #NFDB
80:My fundamental axiom of speculative philosophy is that materialism and spiritualism are opposite poles of the same absurdity-the absurdity of imagining that we know anything about either spirit or matter. ~ Thomas Huxley, #NFDB
81:There are neither good nor bad subjects. From the point of view of pure Art, you could almost establish it as an axiom that the subject is irrelevant, style itself being an absolute manner of seeing things. ~ Gustave Flaubert, #NFDB
82:It may . . . be pronounced as an universal axiom in politics, That an hereditary prince, a nobility without vassals, and a people voting by their representatives, form the best monarchy, aristocracy, and democracy. ~ David Hume, #NFDB
83:The principal axiom in their theory was: Everything can be proved, and everything can be disproved; and in the process, one must profit as much from the folly of others, and from his own superiority, as he can. ~ Moses Mendelssohn, #NFDB
84:The writer who neglects punctuation, or mispunctuates, is liable to be misunderstood for the want of merely a comma, it often occurs that an axiom appears a paradox, or that a sarcasm is converted into a sermonoid. ~ Edgar Allan Poe, #NFDB
85:All is in a man's hands and he lets it all slip from cowardice, that's an axiom. It would be interesting to know what it is men are most afraid of. Taking a new step, uttering a new word is what they fear most… . ~ Fyodor Dostoyevsky, #NFDB
86:There is an ancient Celtic axiom that says 'Good people drink good beer.' Which is true, then as now. Just look around you in any public barroom and you will quickly see: Bad people drink bad beer. Think about it. ~ Hunter S Thompson, #NFDB
87:It may safely be received as an axiom in our political system, that the state governments will in all possible contingencies afford complete security against invasions of the public liberty by the national authority. ~ Alexander Hamilton, #NFDB
88:The opinions that the price of commodities depends solely on the proportion of supply and demand, or demand to supply, has become almost an axiom in political economy, and has been the source of much error in that science. ~ David Ricardo, #NFDB
89:It is practically an axiom in psychiatry that precocious intellect combined with physical weakness can give rise to many unpleasant character traits - avarice, delusions of grandeur , and obsessive masturbation, to name just a few. ~ Sam Savage, #NFDB
90:In a conquered country benevolence is not humanitarianism. It is a general political axiom that a conqueror must not inspire a good opinion of his benevolence until he has demonstrated that he can be severe with malefactors. ~ Napoleon Bonaparte, #NFDB
91:Be willing to dream, and imagine yourself becoming all that you wish to be. Keep in mind the basic axiom -- all that now exists was once imagined. It follows then that what you want to exist for you in the future must now be imagined. ~ Wayne Dyer, #NFDB
92:An Italian proverb says, In men every mortal sin is venial; in woman every venial sin is mortal. And a German axiom, that There are only two good women in the world: one of them is dead, and the other is not to be found. ~ George Augustus Henry Sala, #NFDB
93:Words are really powerful. I don't believe that axiom at all - words can absolutely hurt you. Words can wound. They can do a lot of damage. I think they can do way more damage than sticks and stones. I'll take sticks and stones. ~ Mary Louise Parker, #NFDB
94:One Washington axiom, proved more times than the Pythagorean theorem, states that in the presence of oxygen, one loud fart with an obvious culprit will cover many small emissions in the same room, provided they are nearly simultaneous. ~ Thomas Harris, #NFDB
95:I am not convinced absence makes the heart grow fonder. Perhaps we should test the veracity of this axiom more thoroughly, you and I.” —The Dowager Marchioness of Wallingham to her nephew upon his fourth request for an increase in funds. ~ Elisa Braden, #NFDB
96:At root, vulgar just means popular on a mass scale. It is the semantic opposite of pretentious or snobby. It is humility with a comb-over. It is Nielsen ratings and Barnum's axiom and the real bottom line. It is big, big business. ~ David Foster Wallace, #NFDB
97:Do you think the axiom is true, that ‘all is fair in love and war’?” she asked. “Of course not,” Alex said. “That’s the antithesis of the Golden Rule. The precipice of a very slippery slope. The universal justification for lowering the bar. ~ Tim Tigner, #NFDB
98:For an act may be wrong judged purely by itself, but when the motive that prompted the act is understood, it is construed differently. I lay it down as an axiom, that only that is criminal in the sight of God where crime is meditated. ~ Elizabeth Keckley, #NFDB
99:It is an axiom in my mind that our liberty can never be safe but in the hands of the people themselves, and that too of the people with a certain degree of instruction. This it is the business of the state to effect, and on a general plan. ~ Thomas Jefferson, #NFDB
100:It is an axiom in my mind, that our liberty can never be safe but in the hands of the people themselves, and that too of the people with a certain degree of instruction. This it is the business of the State to effect, and on a general plan. ~ Thomas Jefferson, #NFDB
101:For Stirner, the social axiom of conservative, liberal, and socialist schools of political thought alike is in itself repressive: it disguises as potentially redemptive an order whose central function is inhibitory of the individual's interests. ~ John Carroll, #NFDB
102:– No SF novel ever won the Booker, growls a prowling clansman on his way into the SF Café.
The librarian swings a shotgun from inside her longcoat, blasts the bullshit axiom from the air. Screw the Booker, she thinks. She’d rather have a hookah. ~ Hal Duncan,#NFDB
103:Whence we may draw the general axiom, which never or rarely errs, that he who is the cause of another’s greatness is himself undone, since he must work either by address or force, each of which excites distrust in the person raised to power. ~ Niccol Machiavelli, #NFDB
104:The basis of almost every argument or conclusion I can make is the axiom that the short story can be anything the author decides it shall be;...In that infinite flexibility, indeed lies the reason why the short story has never been adequately defined. ~ H E Bates, #NFDB
105:John Locke’s guiding axiom was that all men have a natural right to the fruits of their labor. A corollary to this logic was that landlords have a right only to what they themselves produce, not to exploit and appropriate the labor of their tenants: ~ Michael Hudson, #NFDB
106:Maybe there is a law after all. Of nature. Like gravity. An unwritten axiom that governs our emotional dealings. What you do comes back to you with twice the force, fuck it, three times the force. We are not punished for our sins we are punished by them. ~ Anonymous, #NFDB
107:The maxim is, that whatever can be affirmed (or denied) of a class, may be affirmed (or denied) of everything included in the class. This axiom, supposed to be the basis of the syllogistic theory, is termed by logicians the dictum de omni et nullo. ~ John Stuart Mill, #NFDB
108:Hm...yes, all is in a man's hands and he lets it all slip from cowardice, that's an axiom. It would be interesting to know what it is men are most afraid of. Taking a new step, uttering a new word is what they fear most...But I am talking too much. ~ Fyodor Dostoyevsky, #NFDB
109:The Americans are the living refutation of the Cartesian axiom, "I think, therefore I am": Americans do not think, yet they are. The American 'mind,' puerile and primitive, lacks characteristic form and is therefore open to every kind of standardization. ~ Julius Evola, #NFDB
110:All this very plausible reasoning does not convince me, as it has not convinced the wisest of our Statesmen, that our ancestors erred in laying it down as an axiom of policy that the toleration of Irregularity is incompatible with the safety of the State. ~ Edwin A Abbott, #NFDB
111:... what I'm doing in here isn't all that different from what I was doing outside. I'll hand you a pretty cynical axiom: the amount of financial help an individual or company needs rises in direct proportion to how many people that person or business is screwing. ~ Stephen King, #NFDB
112:Alyosha was certain that no one in the whole world ever would want to hurt him, and what is more, he knew that no one could hurt him. This was for him an axiom, assumed once and for all without question. And he went his way without hesitation, relying on it. ~ Fyodor Dostoyevsky, #NFDB
113:I want to attempt a thing like that and am frightened by these trifles," he thought, with an odd smile. "Hm … yes, all is in a man's hands and he lets it all slip from cowardice, that's an axiom. It would be interesting to know what it is men are most afraid of ~ Fyodor Dostoyevsky, #NFDB
114:His mother had often said, When you choose an action, you choose the consequences of that action. She had emphasized the corollary of this axiom even more vehemently: when you desired a consequence you had damned well better take the action that would create it. ~ Lois McMaster Bujold, #NFDB
115:To every reversal of people's soveregnity, to every disappearance of the Republic corresponds a frank or disguised restitution in force of the regal justice. "Tell me, according to what you judge and I'll tell you who you are." No axiom in politics is more certain than this. ~ Francois Mitterrand, #NFDB
116:It is considered a rather cheerful axiom that all Americans distrust politicians. (No one takes the further and less cheerful step of considering just what effect this mutual contempt has on either the public or the politicians, who have, indeed, very little to do with one another.) ~ James Baldwin, #NFDB
117:Axiom : hatred of the bourgeois is the beginning of wisdom. But I include in the word bourgeois , the bourgeois in blouses as well the bourgeois in coats. It is we and we alone , that is to say the literary men , who are the people, or to say it better : the tradition of humanity. ~ Gustave Flaubert, #NFDB
118:We cannot wonder enough at the facility with which men resign themselves to continue ignorant of what it is most important that they should know; and we may be certain that such ignorance is incorrigible in those who venture to proclaim this axiom: There are no absolute principles. ~ Fr d ric Bastiat, #NFDB
119:Good point. But I still go with the Sherlock Holmes axiom.” “What’s that?” “I’m paraphrasing, but basically Sherlock warned that you should never theorize before you have the facts because then you twist the facts to suit the theory instead of twisting the theory to suit the facts.” Brandon ~ Harlan Coben, #NFDB
120:In his introduction to Charles M. Doughty’s Travels in Arabia Deserta, T. E. Lawrence attempted to describe the character of the desert Arabs that both he and Doughty had admired. “They are the least morbid of peoples,” Lawrence wrote, “who take the gift of life unquestioningly, as an axiom. ~ David Berlinski, #NFDB
121:There are times like this when I wish I’d gone to the military academy and joined the army.” “Why?” “Because there’s a soldier axiom about not sharing a foxhole with anyone crazier than yourself. If I’d actually joined, I’d be able to quote it precisely. That would be apt right now.” “Ha ha. ~ Lindsay Buroker, #NFDB
122:We're taught Lord Acton's axiom: all power corrupts, absolute power corrupts absolutely. I believed that when I started these books, but I don't believe it's always true any more. Power doesn't always corrupt. Power can cleanse. What I believe is always true about power is that power always reveals. ~ Robert Caro, #NFDB
123:Für mich ist Emersons Axiom, daß gute Bücher die beste Universität ersetzen, unentwegt gültig geblieben, und ich bin noch heute überzeugt, daß man ein ausgezeichneter Philosoph, Historiker, Philologe, Jurist und was immer werden kann, ohne je eine Universität oder sogar ein Gymnasium besucht zu haben. ~ Stefan Zweig, #NFDB
124:But for Aristotle the meaning was hidden in the particulars of experience. The scope of his work was itself witness to his belief in the unity of experience and his confidence that it could somehow be encompassed by the human mind. And so he confirms his axiom that "the actuality of thought is life. ~ Daniel J Boorstin, #NFDB
125:The argument of the invisible hand was so well made that it became an axiom of economics: just get out of the way and let the market work. But, Hardin asked, did the same reasoning still hold true in the economics not of the 1770s, when the world seemed unlimited, but of the 1970s, when it didn’t? ~ Shawn Lawrence Otto, #NFDB
126:We no longer even understand the question whether change is by itself good or bad, ...We start out with the axiom that it is the norm. We do not see change as altering the order... We see change as being order itself - indeed the only order we can comprehend today is a dynamic, a moving, a changing one. ~ Peter Drucker, #NFDB
127:Carlyle's axiom that the true university of these days is a good collection of books has remained valid as far as I'm concerned, and even today I am convinced that one can become an excellent philosopher, historian, philologist, lawyer, or what will you, without having attended a university or even a Gymnasium. ~ Stefan Zweig, #NFDB
128:Frederick the Great once said that the greatest crime in war is not to make the wrong decision, but to make no decision.’ Again the Prince gestured at Sharpe with the brandy glass. ‘You should remember that axiom, Sharpe!’ Sharpe did not even know what an axiom was, but he nodded respectfully. ‘I will, sir. ~ Bernard Cornwell, #NFDB
129:Excellence, in a poem especially, may be considered in the light of an axiom, which need only be properly put, to become self-evident. It is not excellence if it require to be demonstrated its such:—and thus to point out too particularly the merits of a work of Art, is to admit that they are not merits altogether. ~ Edgar Allan Poe, #NFDB
130:The Ruling Principle of the Universe is one of Harmony and Love—God is Love. Therefore, if we wish to embody the most ancient axiom, "as above, so below," we should become unified precisely upon this principle of love and should be subordinate to it, regarding it as our only boundless Ruler. ~ Helena Roerich Letters II, (15 April 1936), #NFDB
131:Your religious beliefs are your business. They are not and should not be the basis for law. If you use them as justification to discriminate against others, don’t be upset when others decide you’re an asshole."
[Blog post of July 26, 2011] ~ Jim C Hines,#NFDB
132:I've made movies that were adaptations and I've been kind of frustrated by the process because, you know that old axiom, 'It's never as good as the book'? It's often true because nothing competes with your own imagination. When you're reading a book and you imagine something in your head, nothing's going to compete with that. ~ Amber Heard, #NFDB
133:a gifted human player could always triumph over the game’s AI, because software couldn’t improvise. It could either react randomly, or in a limited number of predetermined ways, based on a finite number of preprogrammed conditions. This was an axiom in videogames, and would be until humans invented true artificial intelligence. ~ Ernest Cline, #NFDB
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The fact that I suspect I'm an asshole means I probably am not, because a real asshole doesn't think he's an asshole, does he? Therefore, by realizing that I'm an asshole, I am in fact negating that very realization, am I not? Descartes's Asshole Axiom: I think I am; therefor I'm not one. ~ Jonathan Tropper,#NFDB
135:could always triumph over the game’s AI, because software couldn’t improvise. It could either react randomly, or in a limited number of predetermined ways, based on a finite number of preprogrammed conditions. This was an axiom in videogames, and would be until humans invented true artificial intelligence. Our second game came right down ~ Ernest Cline, #NFDB
136:The most sublime labour of poetry is to give sense and passion to insensate things; and it is characteristic of children to take inanimate things in their hands and talk to them in play as if they were living persons... This philological-philosophical axiom proves to us that in the world's childhood men were by nature sublime poets... ~ Giambattista Vico, #NFDB
137:A revered Hollywood axiom warns: “Movies are about their last twenty minutes.” In other words, for a film to have a chance in the world, the last act and its climax must be the most satisfying experience of all. For no matter what the first ninety minutes have achieved, if the final movement fails, the film will die over its opening weekend. ~ Robert McKee, #NFDB
138:This axiom of Aristotelian logic has so deeply imbued our habits of thought that it is felt to be “natural” and self-evident, while on the other hand the statement that X is A and not A seems to be nonsensical. (Of course, the statement refers to the subject X at a given time, not to X now and X later, or one aspect of X as against another aspect.) ~ Erich Fromm, #NFDB
139:At a game like this, a gifted human player could always triumph over the game’s AI, because software couldn’t improvise. It could either react randomly, or in a limited number of predetermined ways, based on a finite number of preprogrammed conditions. This was an axiom in videogames, and would be until humans invented true artificial intelligence. ~ Ernest Cline, #NFDB
140:I put a lot of stock in the written word, and the power of it. That's what I love about acting and reading scripts. Words are really powerful. I don't believe that axiom at all - words can absolutely hurt you. Words can wound. They can do a lot of damage. I think they can do way more damage than sticks and stones. I'll take sticks and stones. ~ Mary Louise Parker, #NFDB
141:It was an axiom of "containment" that no part of the known world could be considered neutral. "Neutralism" was among the Cold Warriors' gravest curse words, applied with caustic hostility to India and even France. Those who were not with were against, subjected to intense economic and ideological and sometimes military pressure to fall into line. ~ Christopher Hitchens, #NFDB
142:Fundamentalists know they are right because they have read the truth in a holy book and they know, in advance, that nothing will budge them from their belief. The truth of the holy book is an axiom, not the end product of a process of reasoning. The book is true, and if the evidence seems to contradict it, it is the evidence that must be thrown out, not the book. ~ Richard Dawkins, #NFDB
143:I am asked if I would not be gratified if my friends would procure me promotion to a brigadier-generalship. My feeling is that I would rather be one of the good colonels than one of the poor generals. The colonel of a regiment has one of the most agreeable positions in the service, and one of the most useful. "A good colonel makes a good regiment," is an axiom. ~ Rutherford B Hayes, #NFDB
144:Yeah, I know what the shrinks say: "Conflict and conflict resolution are the mainstays of human intimacy." That fatuous little axiom may be true, but it presupposes that human intimacy is a desirable thing. I have never been nearly as happy with somebody else in the room as I am when I’m by myself. It seems to me that loneliness is a small price to pay for peace and quiet. ~ Bart Yates, #NFDB
145:In the dark jaws, where all things tumble, where societies crumble and old men stumble, love is the air we breathe, the earth we walk on, the economy we function in. It's not a passion or a fixation or a desire or a guilt. Love should have been a conduct, a process of life, an axiom, a grandeur we evolve...Its connective nature, its transferal powers, make utter sense. ~ Arthur Nersesian, #NFDB
146:In one day I had altered my life; my life, therefore, was alterable. This simple axiom did not call out for exegesis; no, it entered my bloodstream directly, as powerful as heroin. I could feel it pump and surge, the way it brightened my veins to a kind of glass. I had wakened that morning to narrowness and predestination and now I was falling asleep in the storm of my own free will. ~ Carol Shields, #NFDB
147:Its basic axiom is to be followed by individuals as well as great nations, by Losers and Winners alike. We have demonstrated the workability of the axiom in Vietnam, in Bangladesh, in Biafra, in Palestinian refugee camps, in our own ghettos, in our migrant labor camps, on our Indian reservations, in our institutions for the defective and the deformed and the aged. This is it: Ignore agony. ~ Kurt Vonnegut, #NFDB
148:How the mind gears itself for its environment, she thought. And she recalled a Bene Gesserit axiom: “The mind can go either direction under stress—toward positive or toward negative: on or off. Think of it as a spectrum whose extremes are unconsciousness at the negative end and hyperconsciousness at the positive end. The way the mind will lean under stress is strongly influenced by training. ~ Frank Herbert, #NFDB
149:The classical theorists resemble Euclidean geometers in a non-Euclidean world who, discovering that in experience straight lines apparently parallel often meet, rebuke the lines for not keeping straight as the only remedy for the unfortunate collisions which are occurring. Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry. ~ John Maynard Keynes, #NFDB
150:The two basic maxims of the so-called historical criticism are the postulate of the common and the axiom of the ordinary. Postulate of the common: everything really great, good, and beautiful, is improbable, since it is extraordinary and therefore at least suspect. Axiom of the ordinary: our conditions and environment must have existed everywhere, for they are really so natural. ~ Karl Wilhelm Friedrich Schlegel, #NFDB
151:Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty (of the parallel axiom). He went so far as to write a paper, which he took with him to the Institute, and began to read it. But in the first paragraph something struck him that he had not observed: he muttered: 'Il faut que j'y songe encore', and put the paper in his pocket.' [I must think about it again] ~ Augustus De Morgan, #NFDB
152:Any given system of power is built on an assumption (which of course is trying to portray itself as an axiom) that to receive joy you need to pay or obey. The ultimate act of subversion is thus finding joy in a refusal to pay and obey, in an act of living by radically different values. It’s not an act of deprivation or austerity, it’s not a vow, it’s an act that reveals joy that transcends given boundaries. ~ Nadya Tolokonnikova, #NFDB
153:There was always a trick to beating a computer-controlled opponent. At a game like this, a gifted human player could always triumph over the game’s AI, because software couldn’t improvise. It could either react randomly, or in a limited number of predetermined ways, based on a finite number of preprogrammed conditions. This was an axiom in videogames, and would be until humans invented true artificial intelligence. ~ Ernest Cline, #NFDB
154:A crucial difference between lite libertarians and the right kind is that to the former, the idea of liberty is propositional - a deracinated principle, unmoored from the realities of history, hierarchy, biology, tradition, culture, values. Conversely, the paleolibertarian grasps that ordered liberty has a civilizational dimension, stripped of which the libertarian non-aggression axiom, by which we all must live, cannot endure. ~ Ilana Mercer, #NFDB
155:But if men would give heed to the nature of substance they would doubt less concerning the Proposition that Existence appertains to the nature of substance: rather they would reckon it an axiom above all others, and hold it among common opinions. For then by substance they would understand that which is in itself, and through itself is conceived, or rather that whose knowledge does not depend on the knowledge of any other thing. ~ Baruch Spinoza, #NFDB
156:Man is naturally self-centered and he is inclined to regard expediency as the supreme standard for what is right and wrong. However, we must not convert an inclination into an axiom that just as man's perceptions cannot operate outside time and space, so his motivations cannot operate outside expediency; that man can never transcend his own self. The most fatal trap into which thinking may fall is the equation of existence and expediency. ~ Abraham Joshua Heschel, #NFDB
157:When we say the Bible is infallible in its origin, we are merely ascribing its origin to a God who is infallible. This is not to say that the biblical writers were intrinsically or in themselves infallible. They were human beings who, like other humans, proved the axiom Errare humanum est, “To err is human.” It is precisely because humans are given to error that, for the Bible to be the Word of God, its human authors required assistance in their task. ~ R C Sproul, #NFDB
158:...and although she'd glibly remarked that you couldn't stand still, was this actually true or was it a hollow axiom as false and misleading as any other trite saying? Why should one not stand still? If the position in which one found oneself standing was a satisfactory and comfortable one? She felt no need, no need at all to move on from being Mma Ramotswe of The No. 1 Ladies' Detective Agency, wife that great mechanic, Mr. J.L.B. Matekoni. ~ Alexander McCall Smith, #NFDB
159:As a leader... I have always endeavored to listen to what each and every person in a discussion had to say before venturing my own opinion. Oftentimes, my own opinion will simply represent a consensus of what I heard in the discussion. I always remember the axiom; a leader is like a shepherd. He stays behind the flock, letting the most nimble go out ahead, whereupon the others follow, not realizing that all along they are being directed from behind. ~ Nelson Mandela, #NFDB
160:There is a very profound axiom in law, which is consistent with what I told you a short time ago, and it is this: unless an evil thought is born in a twisted mind, human nature is repelled by crime. However, civilization has given us needs, vices and artificial appetites which sometimes cause us to repress our good instincts and lead us to wrongdoing.1 Hence the maxim: if you wish to find the guilty party, first discover whose interests the crime serves! ~ Alexandre Dumas, #NFDB
161:Women don’t use knives,’ Griffoni answered, reciting it as though she were Euclid listing another axiom. Although he agreed with her, Brunetti was curious about the basis for her belief. ‘You offering proof of that?’ ‘Kitchens,’ she said laconically. ‘Kitchens?’ ‘The knives are kept in the kitchen, and their husbands pass through there every day, countless times, yet very few of them get stabbed. That’s because women don’t use knives, and they don’t stab people. ~ Donna Leon, #NFDB
162:Having a moment of clarity was one thing; I'd had moments like that before. It had to be followed with a dedicated push of daily exercise. It's a trite axiom, but practice DOES make perfect. If you want to be a strong swimmer or an accomplished musician, you have to practice. It's the same with sobriety, though the stakes are higher. If you don't practice your program every day, you're putting yourself in a position where you could fly out of the orbit one more time. ~ Anthony Kiedis, #NFDB
163:To embarrass Madison, Elias Boudinot read aloud in Congress some passages about the “necessary and proper” clause from Federalist number 44, notably the following: “No axiom is more clearly established in law or in reason than wherever the end is required, the means are authorized; wherever a general power to do a thing is given, every particular power for doing it is included.”26 Hamilton probably tipped off his old friend that Madison had written these incriminating words. ~ Ron Chernow, #NFDB
164:So great is the force of laws, and of particular forms of government, and so little dependence have they on the humors and tempers of men, that consequences almost as general and certain may sometimes be deduced from them, as any which the mathematical sciences afford us. . . . It may . . . be pronounced as an universal axiom in politics, That an hereditary prince, a nobility without vassals, and a people voting by their representatives, form the best monarchy, aristocracy, and democracy. ~ David Hume, #NFDB
165:The Iranian issue I don't think has much to do with nuclear weapons frankly. Nobody is saying Iran should have nuclear weapons nor should anybody else. But the point in the Middle East, as distinct from North Korea, is that this is center of the world's energy resources. Originally the British and secondarily the French had dominated it, but after the Second World War, it's been a U.S. preserve. That's been an axiom of U.S. foreign policy, that it must control Middle East energy resources. ~ Noam Chomsky, #NFDB
166:In the modern world all terrors could be gutted by simple use of the transitive axiom of quality. Some fears were justified, of course (you don't drive when you're too plowed to see, don't extend the hand of friendship to snarling dogs, don't go parking with boys you don't know - how did the old joke go? Screw or walk?), but until now she had not believed that some fears were larger than comprehension, apocalyptic and nearly paralyzing. This equation was insoluble. The act of moving forward at all became heroism. ~ Stephen King, #NFDB
167:It was uphill work for a foreigner, lame or sound, to make his way with the Bleeding Hearts. In the first place, they were vaguely persuaded that every foreigner had a knife about him; in the second, they held it to be a sound constitutional national axiom that he ought to go home to his own country. They never thought of inquiring how many of their own countrymen would be returned upon their hands from divers parts of the world, if the principle were generally recognised; they considered it particularly and peculiarly British. ~ Charles Dickens, #NFDB
168:Speaking of which, a Dominion brushed past me carrying the final digits to a half-dozen transcendental numbers. It passed them along to a whirling Throne who appeared to be acting as an impromptu sub-foreman, who passed them up the chain to where they could do some good. A cloud of Powers surveyed the damage and orchestrated the repair effort with a thousand-dimensional bird’s-eye view. Somebody had built scaffolding out of a mathematics both consistent and complete (chew on that, Gödel) and now the spackle went on one axiom at a time. ~ Ian Tregillis, #NFDB
169:The old and oft-repeated proposition "Totum est majus sua parte" [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts "totum" and "pars". Unfortunately, however, this "axiom" is used innumerably often without any basis and in neglect of the necessary distinction between "reality" and "quantity", on the one hand, and "number" and "set", on the other, precisely in the sense in which it is generally false. ~ Georg Cantor, #NFDB
170:We may lay it down as an incontestible axiom, that, in all the operations of art and nature, nothing is created; an equal quantity of matter exists both before and after the experiment; the quality and quantity of the elements remain precisely the same; and nothing takes place beyond changes and modifications in the combination of these elements. Upon this principle the whole art of performing chemical experiments depends: We must always suppose an exact equality between the elements of the body examined and those of the products of its analysis. ~ Antoine Lavoisier, #NFDB
171:I want to attempt a thing like that and am frightened by these trifles," he thought, with an odd smile. "Hm … yes, all is in a man's hands and he lets it all slip from cowardice, that's an axiom. It would be interesting to know what it is men are most afraid of. Taking a new step, uttering a new word is what they fear most… . But I am talking too much. It's because I chatter that I do nothing. Or perhaps it is that I chatter because I do nothing. I've learned to chatter this last month, lying for days together in my den thinking … of Jack the Giant ~ Fyodor Dostoyevsky, #NFDB
172:ONE OF THE STURDIEST PRECEPTS of the study of human delusion is that every golden age is either past or in the offing. The months preceding the Japanese attack on Pearl Harbor offer a rare exception to this axiom. During 1941, in the wake of that outburst of gaudy hopefulness, the World’s Fair, a sizable portion of the citizens of New York City had the odd experience of feeling for the time in which they were living, at the very moment they were living in it, that strange blend of optimism and nostalgia which is the usual hallmark of the aetataureate delusion. ~ Michael Chabon, #NFDB
173:The famous axiom "Show, don't tell" is the key. Never force words into a character's mouth to tell the audience about world, history, or person. Rather, show us honest, natural scenes in which human beings talk and behave in honest, natural ways...yet at the same time indirectly pass along the necessary facts. In other words, dramatize exposition. Dramatized exposition serves two ends: Its primary purpose is to further the immediate conflict. Its secondary purpose is to convey information. The anxious novice reverses that order, putting expositional duty ahead of dramatic necessity. ~ Robert McKee, #NFDB
174:The idea that only things established by science and reason are true is expressed in strong form by philosopher W.K. Clifford’s axiom: “It is wrong always, everywhere, for anyone to believe anything upon insufficient evidence.” Of course, Clifford doesn’t answer the pressing question of what counts as sufficient evidence. Taken in a straightforward way, then, his axiom creates a large problem for the scientific community, because unless someone has access to the necessary equipment or data, most of us have to believe the scientific authorities, whom Nietzsche scathingly derided as priests of the modern world. ~ Anonymous, #NFDB
175:Hume concluded that many of the questions that philosophers asked were pseudo questions—that is, questions that cannot be answered with the likes of logic, mathematics, and pure reason because their answers will always be founded at some level on an unprovable belief, on an axiom. He thought that philosophers should stop wasting everyone’s time writing copiously about pseudo questions, dump their a priori assumptions, and rein in their speculations, as the scientists had. They had to reject everything that was not founded on fact or observation, and that included eliminating any appeal to the supernatural. ~ Michael S Gazzaniga, #NFDB
176:However, Nick acted as much as possible under the circumstances, and that was rectifying — it brought with it enjoyment and a working faith. He had not gone counter to the axiom that in a case of doubt one was to hold off; for that applied to choice, and he had not at present the slightest pretension to choosing. He knew he was lifted along, that what he was doing was not first-rate, that nothing was settled by it and that if there was essentially a problem in his life it would only grow tougher with keeping. But if doing one's sum to-morrow instead of to-day does not make the sum easier it at least makes to-day so. ~ Henry James, #NFDB
177:The most viable method of elaborating the natural-rights statement of the libertarian position is to divide it into parts, and to begin with the basic axiom of the “right to self-ownership.” The right to self-ownership asserts the absolute right of each man, by virtue of his (or her) being a human being, to “own” his or her own body; that is, to control that body free of coercive interference. Since each individual must think, learn, value, and choose his or her ends and means in order to survive and flourish, the right to self-ownership gives man the right to perform these vital activities without being hampered and restricted by coercive molestation. ~ Murray N Rothbard, #NFDB
178:In the theory of well-ordered series and compact series, we have followed Cantor closely, except in dealing with Zermelo's theorem (*257—8), and in cases where Cantor's work tacitly assumes the multiplicative axiom. Thus what novelty there is, is in the main negative. In particular, the multiplicative axiom is required in all known proofs of the fundamental proposition that the limit of a progression of ordinals of the second class {i.e. applicable to series whose fields have ^{o terms) is an ordinal of the second class (cf *265). In consequence of this fact, a very large part of the recognized theory of transfinite ordinals must be considered doubtful. ~ Alfred North Whitehead, #NFDB
179:1003
The Skies Can'T Keep Their Secret!
191
The Skies can't keep their secret!
They tell it to the Hills—
The Hills just tell the Orchards—
And they—the Daffodils!
A Bird—by chance—that goes that way—
Soft overhears the whole—
If I should bribe the little Bird—
Who knows but she would tell?
I think I won't—however—
It's finer—not to know—
If Summer were an Axiom—
What sorcery had Snow?
So keep your secret—Father!
I would not—if I could,
Know what the Sapphire Fellows, do,
In your new-fashioned world!
~ Emily Dickinson,#NFDB
180:On the basis of wide clinical experiences, I contend that it is a matter of love in only a few cases when man and woman in our civilization engage in the sexual act. The rage which usurps the initial love impulses, hate, and sadistic emotion are all part and parcel of modern man's contempt for sex. I am not speaking of the clear cases in which the sexual act is performed for profit or subsistence. I am speaking of the majority of people of all social strata. It is on the basis of these clinical findings that the Latin saying, "Omne animal post coitum triste?' has become a scientific axiom. There is only one error in this statement: man ascribes his own disappointment to the animal. ~ Wilhelm Reich, #NFDB
181:Call it a good marriage -
For no one ever questioned
Her warmth, his masculinity,
Their interlocking views;
Except one stray graphologist
Who frowned in speculation
At her h's and her s's,
His p's and w's.
Though few would still subscribe
To the monogamic axiom
That strife below the hip-bones
Need not estrange the heart,
Call it a good marriage:
More drew those two together,
Despite a lack of children,
Than pulled them apart.
Call it a good marriage:
They never fought in public,
They acted circumspectly
And faced the world with pride;
Thus the hazards of their love-bed
Were none of our damned business -
Till as jurymen we sat on
Two deaths by suicide. ~ Robert Graves,#NFDB
182:In Steven Spielberg’s film Lincoln, the screenwriter Tony Kushner has the great emancipator explain Euclid’s axiom in the context of a discussion on the equality of the races: “Euclid’s first common notion is this: Things which are equal to the same thing are equal to each other. That’s a rule of mathematical reasoning. It’s true because it works. Has done and always will do. In his book Euclid says this is self-evident. You see, there it is, even in that 2,000-year-old book of mechanical law it is a self-evident truth.” Although Lincoln never actually uttered those words, there is every reason to think that he would have made just such an argument because it’s precisely what is implied in his 1854 argument that A is interchangeable with B. ~ Michael Shermer, #NFDB
183:She had always consciously or unconsciously formed fear into a simple equation: fear = unknown. And to solve the equation, one simply reduced the problem to simple algebraic terms, thus: unknown = creaky board (or whatever), creaky board = nothing to be afraid of. In the modern world all terrors could be gutted by simple use of the transitive axiom of equality. Some fears were justified, of course (you don’t drive when you’re too plowed to see, don’t extend the hand of friendship to snarling dogs, don’t go parking with boys you don’t know – how did the old joke go? Screw or walk?), but until now she had not believed that some fears were larger than comprehension, apocalyptic and nearly paralyzing. This equation was insoluble. The act of moving forward at all became heroism. ~ Stephen King, #NFDB
184:The fundamental axiom of economics is the human mercenary instinct. Without that assumption, the entire field would collapse. There isn’t any fundamental axiom for sociology yet, but it might be even darker than economics. The truth always picks up dust. A small number of people could fly off into space, but if we knew it would come to that, why would we have bothered in the first place?” “Bothered with what?” “Why would we have had the Renaissance? Why the Magna Carta? Why the French Revolution? If humanity had stayed divided into classes, kept in place by the law’s iron rule, then when the time came, the ones who needed to leave would leave, and the ones who had to stay behind would stay. If this took place in the Ming or Qing Dynasties, then I’d leave, of course, and you’d stay behind. But that’s not possible now. ~ Liu Cixin, #NFDB
185:The overall structure of the calculus is simple. The subject is defined by a fantastic leading idea, one basic axiom, a calm and profound intellectual invention, a deep property, two crucial definitions, one ancillary definition, one major theorem, and the fundamental theorem of the calculus.
The fantastic leading idea: the real world may be understood in terms of the real numbers.
The basic axiom: brings the real numbers into existence.
The calm and profound invention: the mathematical function.
The deep property: continuity.
The crucial definitions: instantaneous speed and the area underneath a curve.
The ancillary definition: a limit
The major theorem: the mean value theorem.
The fundamental theorem of the calculus is the fundamental theorem of the calculus.
These are the massive load-bearing walls and buttresses of the subject. ~ David Berlinski,#NFDB
186:What is this thing called government, which can grant and take away rights? There are all sorts of answers to that question, but all the answers will agree on one point, that government is a social instrument enjoying a monopoly of coercion. The socialist says that the monopoly of coercion is vested in the government in order that it may bring about an ideal social and economic order; others say that the government must have a monopoly of coercion in order to prevent individuals from using coercion on one another. In short, the essential characteristic of government is power. If, then, we say that our rights stem from government, on a loan basis, we admit that whoever gets control of the power vested in government is the author of rights. And simply because he has the power to enforce his will. Thus, the basic axiom of socialism, in all its forms, is that might is right. ~ Anonymous, #NFDB
187:Plato in his Protagoras well saith, a good philosopher as much excels other men, as a great king doth the commons of his country; and again, [2062] quoniam illis nihil deest, et minimè egere solent, et disciplinas quas profitentur, soli à contemptu vindicare possunt, they needed not to beg so basely, as they compel [2063] scholars in our times to complain of poverty, or crouch to a rich chuff for a meal's meat, but could vindicate themselves, and those arts which they professed. Now they would and cannot: for it is held by some of them, as an axiom, that to keep them poor, will make them study; they must be dieted, as horses to a race, not pampered, [2064] Alendos volunt, non saginandos, ne melioris mentis flammula extinguatur; a fat bird will not sing, a fat dog cannot hunt, and so by this depression of theirs [2065] some want means, others will, all want [2066] encouragement, as being forsaken almost; and generally contemned. ~ Robert Burton, #NFDB
188:As for myself, I can only exhort you to look on Friendship as the most valuable of all human possessions, no other being equally suited to the moral nature of man, or so applicable to every state and circumstance, whether of prosperity or adversity, in which he can possibly be placed. But at the same time I lay it down as a fundamental axiom that "true Friendship can only subsist between those who are animated by the strictest principles of honour and virtue." When I say this, I would not be thought to adopt the sentiments of those speculative moralists who pretend that no man can justly be deemed virtuous who is not arrived at that state of absolute perfection which constitutes, according to their ideas, the character of genuine wisdom. This opinion may appear true, perhaps, in theory, but is altogether inapplicable to any useful purpose of society, as it supposes a degree of virtue to which no mortal was ever capable of rising. ~ Marcus Tullius Cicero, #NFDB
189:Hamilton and Madison came to symbolize opposite ends of the political spectrum. At the time of the Federalist essays, however, they were so close in style and outlook that scholars find it hard to sort out their separate contributions. In general, Madison’s style was dense and professorial, Hamilton’s more graceful and flowing, yet they had a similar flair for startling epigrams and piercing insights. At this stage, Madison often sounded “Hamiltonian” and vice versa. Later identified as a “strict constructionist” of the Constitution, Madison set forth the doctrine of implied powers that Hamilton later used to expand the powers of the federal government. It was Madison who wrote in Federalist number 44, “No axiom is more clearly established in law or in reason than that wherever the end is required, the means are authorized.” 31 At this juncture, they could make common cause on the need to fortify the federal government and curb rampant state abuses. Both ~ Ron Chernow, #NFDB
190:A naively formulated goal transmutes, with time, into the sinister form of the life-lie. One forty-something client told me his vision, formulated by his younger self: “I see myself retired, sitting on a tropical beach, drinking margaritas in the sunshine.” That’s not a plan. That’s a travel poster. After eight margaritas, you’re fit only to await the hangover. After three weeks of margarita-filled days, if you have any sense, you’re bored stiff and self-disgusted. In a year, or less, you’re pathetic. It’s just not a sustainable approach to later life. This kind of oversimplification and falsification is particularly typical of ideologues. They adopt a single axiom: government is bad, immigration is bad, capitalism is bad, patriarchy is bad. Then they filter and screen their experiences and insist ever more narrowly that everything can be explained by that axiom. They believe, narcissistically, underneath all that bad theory, that the world could be put right, if only they held the controls. ~ Jordan Peterson, #NFDB
191:Experience day by day protested and showed by infinite examples, that good and evil fortunes fall to the lot of pious and impious alike; still they would not abandon their inveterate prejudice, for it was more easy for them to class such contradictions among other unknown things of whose use they were ignorant, and thus to retain their actual and innate condition of ignorance, than to destroy the whole fabric of their reasoning and start afresh. They therefore laid down as an axiom, that God's judgments far transcend human understanding. Such a doctrine might well have sufficed to conceal the truth from the human race for all eternity, if mathematics had not furnished another standard of verity in considering solely the essence and properties of figures without regard to their final causes. There are other reasons (which I need not mention here) besides mathematics, which might have caused men's minds to be directed to these general prejudices, and have led them to the knowledge of the truth. ~ Baruch Spinoza, #NFDB
192:A naively formulated goal transmutes, with time, into the sinister form of the life-lie. One forty-something client told me his vision, formulated by his younger self: “I see myself retired, sitting on a tropical beach, drinking margaritas in the sunshine.” That’s not a plan. That’s a travel poster. After eight margaritas, you’re fit only to await the hangover. After three weeks of margarita-filled days, if you have any sense, you’re bored stiff and self-disgusted. In a year, or less, you’re pathetic. It’s just not a sustainable approach to later life. This kind of oversimplification and falsification is particularly typical of ideologues. They adopt a single axiom: government is bad, immigration is bad, capitalism is bad, patriarchy is bad. Then they filter and screen their experiences and insist ever more narrowly that everything can be explained by that axiom. They believe, narcissistically, underneath all that bad theory, that the world could be put right, if only they held the controls. ~ Jordan B Peterson, #NFDB
193:Material force is the ultima ratio of political society everywhere. Arms alone can keep the peace." This was and still remains the axiom with men everywhere. The sword is not only the source of security; it is also the symbol of honor and glory; it is bliss and song.
When the prophets appeared, they proclaimed that might is not supreme, that the sword is an abomination, that violence is obscene. The sword, they said, shall be destroyed.
They shall beat their swords into plowshares,
And their spears into pruning hooks;
Nation shall not lift up sword against nation,
Neither shall they learn war any more.
Isaiah 2:4
The prophets, questioning man's infatuation with might, insisted not only on the immorality but also on the futility and absurdity of war.[...] What is the ultimate profit of all the arms, alliances, and victories? Destruction, agony, death.
Peoples labor only for fire,
Nations weary themselves for naught.
Habakkuk 2:13 ~ Abraham Joshua Heschel,#NFDB
194:There is," said he, at the end of his meditations, "a clever maxim, which bears upon what I was saying to you some little while ago, and that is, that unless wicked ideas take root in a naturally depraved mind, human nature, in a right and wholesome state, revolts at crime. Still, from an artificial civilization have originated wants, vices, and false tastes, which occasionally become so powerful as to stifle within us all good feelings, and ultimately to lead us into guilt and wickedness. From this view of things, then, comes the axiom that if you visit to discover the author of any bad action, seek first to discover the person to whom the perpetration of that bad action could be in any way advantageous.
"That alters the case. This man might, after all, be a greater scoundrel than you have thought possible,"
"Upon my word," said Dantes, "you make me shudder. Is the world filled with tigers and crocodiles?"
"Yes; and remember that two-legged tigers and crocodiles are more dangerous than the others. ~ Alexandre Dumas,#NFDB
195:As was mentioned before, this does feel like it's “just philosophy.” You can set things up, though, so that real decisions depend on it. Maybe you've heard of the surefire way of winning the lottery: buy a lottery ticket and if it doesn't win, then you kill yourself. Then, you clearly have to condition on being alive to ask the question of whether you are alive or not, and so because you're asking the question, you must be alive, and thus must have won the lottery. What can you say about this? You can say that in actual practice, most of us don't accept as a decision-theoretic axiom that you're allowed to condition on remaining alive. You could jump off a building and condition on there happening to be a trampoline or something that will rescue you. You have to take into account the possibility that your choices are going to kill you. On the other hand, tragically, some people do kill themselves. Was this in fact what they were doing? Were they eliminating the worlds where things didn't turn out how they wanted? ~ Scott Aaronson, #NFDB
196:...the Kabbalist was interested not in the perfected text whose author is dead and can no longer respond but in contact with the living Author for whom the text is an intermediary. Even when the pneuma was needed in order to better understand the Bible, the content of this deeper apprehension was, in many cases, a better insight into divine matters. According to the French philosopher, the death of the author is a condition for finalizing the text and rendering it into a static perfection, allowing for a "complete" relation. This request is based upon a rigid attitude toward the contents, which are to be approached when they can no longer change. It is an axiom of the Kabbalists that the sacred text is in an ongoing process of change, evidently a symptom of its inherent infinity and divinity. For them, Scripture is a way of overcoming the post-prophetic eclipse of revelation, an endeavor to recapture the presence of the Author and its nature; the biblical text produces a silent dialogue and eventually even union between Author and reader,.. ~ Moshe Idel, #NFDB
197:This development had dramatic philosophical consequences. As in the case of the non-Euclidean geometries in the nineteenth century, there wasn't just one definitive set theory, but rather at least four! One could make different assumptions about infinite sets and end up with mutually exclusive set theories. For instance, once could assume that both the axiom of choice and the continuum hypothesis hold true and obtain one version, or that both do not hold, and obtain an entirely different theory. Similarly, assuming the validity of one of the two axioms and the negation of the other would have led to yet two other set theories.
This was the non-Euclidean crisis revisited, only worse. The fundamental role of set theory as the potential basis for the whole of mathematics made the problem for the Platonists much more acute. If indeed one could formulate many set theories simply by choosing a different collection of axioms, didn't this argue for mathematics being nothing but a human invention? The formalists' victory looked virtually assured. ~ Mario Livio,#NFDB
198:The Hermit
AN ATTACK ON BARBERCRAFT
[Dedicated to George Cecil Jones]
At last an end of all I hoped and feared!
Muttered the hermit through his elfin beard.
Then what art thou? the evil whisper whirred.
I doubt me soerly if the hermit heard.
To all God's questions never a word he said,
But simply shook his venerable head.
God sent all plagues; he laughed and heeded not,
Till people certified him insane.
But somehow all his fellow-luntaics
Began to imitate his silly ticks.
And stranger still, their prospects so enlarged
That one by one the patients were discharged.
God asked him by what right he interfered;
He only laughed and into his elfin beard.
When God revealed Himself to mortal prayer
He gave a fatal opening to Voltaire.
Our Hermi had dispensed with Sinai's thunder,
But on the other hand he made no blunder;
He knew ( no doubt) that any axiom
Would furnish bricks to build some Donkeydom.
But!-all who urged that hermit to confess
Caught the infection of his happiness.
I would it were my fate to dree his weird;
76
I think that I will grow an elfin beard.
~ Aleister Crowley,#NFDB
199:It was uphill work for a foreigner, lame or sound, to make his way with the Bleeding Hearts. In the first place, they were vaguely persuaded that every foreigner had a knife about him; in the second, they held it to be a sound constitutional national axiom that he ought to go home to his own country. They never thought of inquiring how many of their own countrymen would be returned upon their hands from divers parts of the world, if the principle were generally recognised; they considered it particularly and peculiarly British. In the third place, they had a notion that it was a sort of Divine visitation upon a foreigner that he was not an Englishman, and that all kinds of calamities happened to his country because it did things that England did not, and did not do things that England did. In this belief, to be sure, they had long been carefully trained by the Barnacles and Stiltstalkings, who were always proclaiming to them, officially, that no country which failed to submit itself to those two large families could possibly hope to be under the protection of Providence; and who, when they believed it, disparaged them in private as the most prejudiced people under the sun. ~ Charles Dickens, #NFDB
200:What I'm sure of is that you can't be happy without money. That's all. I don't like superficiality and I don't like romanticism. I like to be conscious. And what I've noticed is that there's a kind of spiritual snobbism in certain 'superior beings' who think that money isn't necessary for happiness. Which is stupid, which is false, and to a certain degree cowardly.... For a man who is well born, being happy is never complicated. It's enough to take up the general fate, only not with the will for renunciation like so many fake great men, but with the will for happiness. Only it takes time to be happy. A lot of time. Happiness, too, is a long patience. And in almost every case, we use up our lives making money, when we should be using our money to gain time. That's the only problem that's ever interested me.... To have money is to have time. That's my main point. Time can be bought. Everything can be bought. To be or to become rich is to have time to be happy, if you deserve it.... Everything for happiness, against the world which surrounds us with its violence and its stupidity.... All the cruelty of our civilization can be measured by this one axiom: happy nations have no history. ~ Albert Camus, #NFDB
201:AN ATTACK ON BARBERCRAFT
[Dedicated to George Cecil Jones]
At last an end of all I hoped and feared!
Muttered the hermit through his elfin beard.
Then what art thou? the evil whisper whirred.
I doubt me soerly if the hermit heard.
To all God's questions never a word he said,
But simply shook his venerable head.
God sent all plagues; he laughed and heeded not,
Till people certified him insane.
But somehow all his fellow-luntaics
Began to imitate his silly ticks.
And stranger still, their prospects so enlarged
That one by one the patients were discharged.
God asked him by what right he interfered;
He only laughed and into his elfin beard.
When God revealed Himself to mortal prayer
He gave a fatal opening to Voltaire.
Our Hermi had dispensed with Sinai's thunder,
But on the other hand he made no blunder;
He knew ( no doubt) that any axiom
Would furnish bricks to build some Donkeydom.
But!-all who urged that hermit to confess
Caught the infection of his happiness.
I would it were my fate to dree his weird;
I think that I will grow an elfin beard.
~ Aleister Crowley, The Hermit
,#NFDB
202:The Rules of Misquotation: Axiom 1. Any quotation that can be altered will be. Corollary 1A: Vivid words hook misquotes in the mind. ~ Corollary 1B: Numbers are hard to keep straight. ~ Corollary 1C: Small changes can have a big impact (or: what a difference an a makes). ~ Corollary 1D: If noted figures don't say what needs to be said, we'll say it for them. ~ Corollary 1E: Journalists are a less than dependable source of accurate quotes. Corollary 1F: Famous dead people make excellent commentators on current events. Axiom 2. Famous quotes need famous mouths. ~ Corollary 2A: Well-known messengers get credit for clever comments they report from less celebrated mouths. ~ Corollary 2B: Particularly quotable figures receive more than their share of quotable quotes. ~ Corollary 2C: Comments made about someone might as well have been said by that person. ~ Corollary 2D: Who you think said something may depend on where you live. ~ Corollary 2E: Vintage quotes are considered to be in the public domain. Corollary 2F: In a pinch, any orphan quote can be called a Chinese proverb. ~ Ralph Keyes, "Nice Guys Finish Seventh": False Phrases, Spurious Sayings, and Familiar Misquotations (1992) ISBN 0062700200., #NFDB
203:There's a good general reason to expect that physical theories consistent with special relativity will have to be field theories. Here it comes:
A major result of the special theory of relativity is that there is a limiting velocity: the speed of light, usually denoted c. The influence of one particle on another cannot be transmitted faster than that. Newton's law for the gravitational force, according to which the force due to a distant body is proportional to the inverse square of its distance right now, does not obey that rule, so it is not consistent with special relativity. Indeed the concept "right now" itself is problematic. Events that appear simultaneous to a stationary observer will not appear simultaneous to an observer moving at constant velocity. Overthrowing the concept of a universal "now" was, according to Einstein himself, by far the most difficult step in arriving at special relativity:
[A]ll attempts to clarify this paradox satisfactorily were condemned to failure as long as the axiom of the absolute character of times, viz., of simultaneity, unrecognizedly was anchored in the unconscious. Clearly to recognize this axiom and its arbitrary character really implies already the solution of the problem. ~ Frank Wilczek,#NFDB
204:Recall Marx’s fundamental insight about the “bourgeois” limitation of the logic of equality: capitalist inequalities (“exploitation”) are not the “unprincipled violations of the principle of equality,” but are absolutely inherent to the logic of equality, they are the paradoxical result of its consistent realization. What we have in mind here is not only the wearisome old motif of how market exchange presupposes formally/legally equal subjects who meet and interact in the market; the crucial moment of Marx’s critique of “bourgeois” socialists is that capitalist exploitation does not involve any kind of “unequal” exchange between the worker and the capitalist—this exchange is fully equal and “just,” ideally (in principle), the worker gets paid the full value of the commodity he is selling (his labor-power). Of course, radical bourgeois revolutionaries are aware of this limitation; however, the way they try to counteract it is through a direct “terroristic imposition of more and more de facto equality (equal salaries, equal access to health services…), which can only be imposed through new forms of formal inequality (different sorts of preferential treatments for the underprivileged). In short, the axiom of equality” means either not enough (it remains the abstract form of actual inequality) or too much (enforce “terroristic” equality)— it is a formalistic notion in a strict dialectical sense, that is, its limitation is precisely that its form is not concrete enough, but a mere neutral container of some content that eludes this form. ~ Slavoj i ek, #NFDB
205:It is beyond my power to induce in you a belief in God. There are certain things which are self proved and certain which are not proved at all. The existence of God is like a geometrical axiom. It may be beyond our heart grasp. I shall not talk of an intellectual grasp. Intellectual attempts are more or less failures, as a rational explanation cannot give you the faith in a living God. For it is a thing beyond the grasp of reason. It transcends reason. There are numerous phenomena from which you can reason out the existence of God, but I shall not insult your intelligence by offering you a rational explanation of that type. I would have you brush aside all rational explanations and begin with a simple childlike faith in God. If I exist, God exists. With me it is a necessity of my being as it is with millions. They may not be able to talk about it, but from their life you can see that it is a part of their life. I am only asking you to restore the belief that has been undermined. In order to do so, you have to unlearn a lot of literature that dazzles your intelligence and throws you off your feet. Start with the faith which is also a token of humility and an admission that we know nothing, that we are less than atoms in this universe. We are less than atoms, I say, because the atom obeys the law of its being, whereas we in the insolence of our ignorance deny the law of nature. But I have no argument to address to those who have no faith. ~ Mohandas Karamchand Gandhi in Young India (24 September 1931); also in Teachings Of Mahatma Gandhi (1945), edited by Jag Parvesh Chander, p. 458, #NFDB
206:Agnosticism, in fact, is not a creed, but a method, the essence of which lies in the rigorous application of a single principle. That principle is of great antiquity; it is as old as Socrates; as old as the writer who said, 'Try all things, hold fast by that which is good'; it is the foundation of the Reformation, which simply illustrated the axiom that every man should be able to give a reason for the faith that is in him, it is the great principle of Descartes; it is the fundamental axiom of modern science. Positively the principle may be expressed: In matters of the intellect, follow your reason as far as it will take you, without regard to any other consideration. And negatively: In matters of the intellect, do not pretend that conclusions are certain which are not demonstrated or demonstrable. That I take to be the agnostic position, which if a man keep whole and undefiled, he shall not be ashamed to look the universe in the face, whatever the future may have in store for him.
The results of the working out of the agnostic principle will vary according to individual knowledge and capacity, and according to the general condition of science. That which is unproved today may be proved, by the help of new discoveries, tomorrow. The only negative fixed points will be those negations which flow from the demonstrable limitation of our faculties. And the only obligation accepted is to have the mind always open to conviction.
That it is wrong for a man to say he is certain of the objective truth of a proposition unless he can provide evidence which logically justifies that certainty. This is what agnosticism asserts and in my opinion, is all that is essential to agnosticism. ~ Thomas Henry Huxley,#NFDB
207:We have all a ruling defect, which is for our soul as the umbilical cord of its birth in sin, and it is by this that the enemy can always lay hold upon us: for some it is vanity, for others idleness, for the majority egotism. Let a wicked and crafty mind avail itself of this means and we are lost; we may not go mad or turn idiots, but we become positively alienated, in all the force of the expression - that is, we are subjected to a foreign suggestion. In such a state one dreads instinctively everything that might bring us back to reason, and will not even listen to representations that are opposed to our obsession. Here is one of the most dangerous disorders which can affect the moral nature. The sole remedy for such a bewitchment is to make use of folly itself in order to cure folly, to provide the sufferer with imaginary satisfactions in the opposite order to that wherein he is now lost. Endeavour, for example, to cure an ambitious person by making him desire the glories of heaven - mystic remedy; cure one who is dissolute by true love - natural remedy; obtain honourable successes for a vain person; exhibit unselfishness to the avaricious and procure for them legitimate profit by honourable participation in generous enterprises, etc. Acting in this way upon the moral nature, we may succeed in curing a number of physical maladies, for the moral affects the physical in virtue of the magical axiom: "That which is above is like unto that which is below." This is why the Master said, when speaking of the paralyzed woman: "Satan has bound her." A disease invariably originates in a deficiency or an excess, and ever at the root of a physical evil we shall find a moral disorder. This is an unchanging law of Nature. ~ Eliphas Levi, Transcendental Magic, #NFDB
208:what happens if you're in a relationship with someone and you trust them, then you make certain assumptions about the past, and you make certain assumptions about the present, and you make certain assumptions about the future. And everything's stable, so you're standing on solid ground. And the chaos, it's like you're standing on thin ice. The chaos is hidden. The shark beneath the waves isn't there. You're safe, you're in the lifeboat. But then if the person betrays you — like if you're in an intimate relationship and the person has an affair and you find out about it — then you think, one moment you're one in one place, right? You're where everything is secure because you've predicated your perception of the world on the axiom of trust, and the next second — really, the next second — you're in a completely different place. And not only is that place different right now, the place you were years ago is different, and the place you're going to be in the future years hence is different. And so, all of that certainty that strange certainty that you inhabit can collapse into incredible complexity. And you say, well if someone betrays you, you think: "Okay, who were you? Because you weren't who I thought you were. And I thought I knew you. But I didn't know you at all. And I never knew you, and so all the things we did together, those weren't the things that I thought were happening. Something else was happening! And you're someone else. That means I'm someone else because I thought I knew what was going on, and clearly I don't. I'm some sort of blind sucker, or the victim of a psychopath or someone who's so naive that they can barely live. And I don't understand anything about human beings, and I don't understand anything about myself, and I have no idea where I am now. I thought I was at home, but I'm not. I'm in a house and it's full of strangers. I don't know what I'm going to do tomorrow, or next week, or next year. ~ Jordan Peterson, #NFDB
209:what happens if you're in a relationship with someone and you trust them, then you make certain assumptions about the past, and you make certain assumptions about the present, and you make certain assumptions about the future. And everything's stable, so you're standing on solid ground. And the chaos, it's like you're standing on thin ice. The chaos is hidden. The shark beneath the waves isn't there. You're safe, you're in the lifeboat. But then if the person betrays you — like if you're in an intimate relationship and the person has an affair and you find out about it — then you think, one moment you're one in one place, right? You're where everything is secure because you've predicated your perception of the world on the axiom of trust, and the next second — really, the next second — you're in a completely different place. And not only is that place different right now, the place you were years ago is different, and the place you're going to be in the future years hence is different. And so, all of that certainty that strange certainty that you inhabit can collapse into incredible complexity. And you say, well if someone betrays you, you think: "Okay, who were you? Because you weren't who I thought you were. And I thought I knew you. But I didn't know you at all. And I never knew you, and so all the things we did together, those weren't the things that I thought were happening. Something else was happening! And you're someone else. That means I'm someone else because I thought I knew what was going on, and clearly I don't. I'm some sort of blind sucker, or the victim of a psychopath or someone who's so naive that they can barely live. And I don't understand anything about human beings, and I don't understand anything about myself, and I have no idea where I am now. I thought I was at home, but I'm not. I'm in a house and it's full of strangers. I don't know what I'm going to do tomorrow, or next week, or next year. ~ Jordan B Peterson, #NFDB
210:This and Rothbard’s own life-long cultural conservatism notwithstanding, however, from its beginnings in the late 1960s and the founding of a libertarian party in 1971, the libertarian movement had great appeal to many of the counter-cultural left that had then grown up in the U.S. in opposition to the war in Vietnam. Did not the illegitimacy of the state and the non-aggression axiom imply that everyone was at liberty to choose his very own non-aggressive lifestyle, no matter what it was? Much of Rothbard’s later writings, with their increased emphasis on cultural matters, were designed to correct this development and to explain the error in the idea of a leftist multi-counter-cultural libertarianism, of libertarianism as a variant of libertinism. It was false—empirically as well as normatively—that libertarianism could or should be combined with egalitarian multiculturalism. Both were in fact sociologically incompatible, and libertarianism could and should be combined exclusively with traditional Western bourgeois culture; that is, the old-fashioned ideal of a family-based and hierarchically structured society of voluntarily acknowledged rank orders of social authority. Empirically, Rothbard did not tire to explain, the left-libertarians failed to recognize that the restoration of private-property rights and laissez-faire economics implied a sharp and drastic increase in social “discrimination.” Private property means the right to exclude. The modern social-democratic welfare state has increasingly stripped private-property owners of their right to exclude. In distinct contrast, a libertarian society where the right to exclude was fully restored to owners of private property would be profoundly unegalitarian. To be sure, private property also implies the owner’s right to include and to open and facilitate access to one’s property, and every private-property owner also faces an economic incentive of including (rather than excluding) so long as he expects this to increase the value of his property. ~ Anonymous, #NFDB
211:One assumption that is already being shattered is the idea that only routine, semi-skilled jobs like taxi driving, food delivery, or household chores are susceptible. Even traditional professions like medicine and law are proving to be susceptible to platform models. We’ve already mentioned Medicast, which applies an Uber-like model to finding a doctor. Several platform companies are providing online venues where legal services are available with comparable ease, speed, and convenience. Axiom Law has built a $200 million platform business by using a combination of data-mining software and freelance law talent to provide legal guidance and services to business clients; InCloudCounsel claims it can process basic legal documents such as licensing forms and nondisclosure agreements at a savings of up to 80 percent compared with a traditional law firm.11 In the decades to come, it seems likely that the platform model will be applied—or at least tested—in virtually every market for labor and professional services. How will this trend impact the service industries—not to mention the working lives of hundreds of millions of people? One likely result will be an even greater stratification of wealth, power, and prestige among service providers. Routine and standardized tasks will move to online platforms, where an army of relatively low-paid, self-employed professionals will be available to handle them. Meanwhile, the world’s great law firms, medical centers, consulting partnerships, and accounting practices will not vanish, but their relative size and importance will shrink as much of the work they used to do migrates to platforms that can provide comparable services at a fraction of the cost and with far greater convenience. A surviving handful of world-class experts will increasingly focus on a tiny subset of the most highly specialized and challenging assignments, which they can tackle from anywhere in the world using online tools. Thus, at the very highest level of professional expertise, winner-take-all markets are likely to emerge, with (say) two dozen internationally renowned attorneys competing for the splashiest and most lucrative cases anywhere on the globe. ~ Geoffrey G Parker, #NFDB
212:When Philosophies Sleep
'Everything is fate'
That was father's faith;
He had nothing to do but wait.
'History alone is real
Its developments, all'
The son had his credo;
The hope of the house, the daughter
Remained single, withering,
A plantain one ceased to water,
Daddy had her horoscope read
That's it!
She must wait to wed,
What has been ordained one cannot amend
Even by a dot, try till the days do end.
To substantiate his stand
He could quote Ramayana
From A to Z.
To this axiom of belief
The son put an axe
He can recite Marx
Like nursery rhymes.
The decadent bourgeois order,
Entitled joint-family
To hell, let it go!
A girl is no commodity
To be peddled in market place.
If domestic felicity
Be historic necessity
She can come to agreement
Regarding such arrangements.
She heard them all
But understood none.
When her clothes were torn
21
The daughter darned the lot.
She got up one day,
That is, before
The third quarter of night
And lit the little oil lamp.
She spread the mat, and placed a bowl of water
Her father needs them every morn for his prayer,
A cup of tea she kept
Near her brother's bed
He must have it to be himself.
To the hall she came
And touched the door
A flash of lightning reached her core
Through the doors that gently came apart
The wide world saluted her resolute heart;
Stretching its cool soft hand;
It placed a wreath of thrill
Upon her head.
Once, she turned to big a silent farewell
To her home, its presiding deity
To her brother and sire,
To the loose end of her dhoti
A coil she tied A token offering to the
Lord of Guruvayur.
With a fluttering heart, with steps faltering
She paced down to the yard,
She paused a while.
Years back, her mother, then a bride,
Walked in through the same
Sand-strewn yard
Facing an auspicious lamp.
In darkness the daughter
Crossed the very yard
Her eyes in floods, toes striking stones.
[Translated from the original Malayalam
'Thathwasastrangal Urangumbol'
22
by Madhavan Ayyappath.]
~ Edasseri Govindan Nair,#NFDB
213:If we live in a world of states, and if out-of-state existence is impossible, then we all must live as national citizens. We are the nation, and the nation is us. This is as fundamental as it is an inescapable reality. Nationalism engulfs both the individual and the collective; it produces the 'I' and 'We' dialectically and separately. Not only does nationalism produce the community and its individual members: it is itself the community and its realized individual subjects, for without these there is no nationalism.
"Leading sociologists and philosophers have emphasized the pervasive presence of the community in individual consciousnesses, where the social bond is an essential part of the self. It is not only that the 'I' is a member of the 'We,' but, more importantly, that the 'We' is a necessary member of the 'I.' It is an axiom of sociological theory, writes Scheler, that all human knowledge 'precedes levels of self-contagiousness of one's self-value. There is no "I" without "We." The "We" is filled with contents prior to the "I." ' Likewise, Mannheim emphasizes ideas and thought structures as functions of social relations that exist within the group, excluding the possibility of any ideas arising independently of socially shared meanings. The social reality of nationalism not only generates meanings but is itself a 'context of meaning'; hence our insistence that nationalism constitutes and is constituted by the community as a social order. 'It is senseless to pose questions such as whether the mind is socially determined, as though the mind and society each posses a substance of their own' [citing Pressler and Dasilva's Sociology]. The profound implications of the individual's embeddedness in the national community is that the community's ethos is prior and therefore historically determinative of all socioepistemic phenomena. And if thought structures are predetermined by intellectual history, by society's inheritance of historical forms of knowledge, then those structures are also a priori predetermined by the linguistic structures in which this history is enveloped, cast, and framed.
Like law, nationalism is everywhere: it creates the community and shapes world history even before nationalism comes into it. ~ Wael B Hallaq,#NFDB
214:Stop,” Jesse said.
I stared up at him, almost panting with fear.
“Stop, beloved,” he said more gently, and took up my clenched fist with both hands. “I’ve upset you, and I shouldn’t have. I don’t want you to dread yourself. I don’t want you to dread what is to come. Like I said, you’re exceptional, so there may be nothing to worry about at all. But whatever happens, whatever you face, I’ll face it with you. Do you hear?”
“How can you say it? It nearly happened on the roof today. You can’t know-“
“I will be with you. We’re together now, and the universe knows I won’t let you make your sacrifice alone. Dragon protects star. Star adores dragon. An age-old axiom. Simple as that.”
I looked down at our hands, both of his curled over mine. I unclenched my fist. Blood from the thorn smeared my skin.
“The universe,” I muttered. “The same universe that has produced the Kaiser and bedbugs and Chloe Pemington. How reassuring.”
With the same absolute concentration he might have shown for turning flowers into gold, Jesse Holms smoothed out my fingers between his, wiping away the blood. He turned my hand over and lifted it to his lips. His next words fell soft as velvet into the heart of my palm.
“Those nights, in the sweetest dark, we shared our dreams. That’s you answer. I was stitched into yours, and you were stitched into mind, and that was real, I promise you.” I felt his lips curve into a smile. The unbelievably sensual, ticklish scuff of his whiskers. “Very good dreams they were, too,” he added.
It was no use trying to cling to mortification or fear. He was holding my hand. He was smiling at me past the cup of my fingers, and although I couldn’t see it, the shape of it against my skin was beyond tantalizing, rough and masculine. I was a creature gone hot and cold and light-headed with pleasure. I wanted to snatch back my hand and I wanted him to go on touching me like this forever. I wanted to walk with him back to his cottage, to his bed, and to hell with the Germans and school and all the rest of the world.
But he looked up suddenly.
“They’re searching for you,” he said, releasing me at once, moving away.
They were. I heard my name being called by a variety of voices in a variety of tones, all of them still inside the castle, none of them sounding happy.
“Go on.” With a few quick steps, Jesse was less than a shadow, retreating into the black wall of the woods. “Don’t get into trouble. And, Lora?”
“Yes?”
There was hushed laughter in his voice. “Until we can see each other again, do us both a favor. Keep away from rooftops. ~ Shana Abe,#NFDB
215:And the son bursting into his father's house, killing him, and at the same time not killing him, this is not even a novel, not a poem, it is a sphinx posing riddles, which it, of course, will not solve itself. If he killed him, he killed him; how can it be that he killed him and yet did not kill him--who can understand that? Then it is announced to us that our tribune is the tribune of truth and sensible ideas, and so from this tribune of 'sensible ideas' an axiom resounds, accompanied by an oath, that to call the murder of a father parricide is simply a prejudice! But if parricide is a prejudice, and if every child ought to ask his father, 'Father, why should I love you?'--what will become of us, what will become of the foundations of society, where will the family end up? Parricide--don't you see, it's just the 'brimstone' of some Moscow merchant's wife? The most precious, the most sacred precepts concerning the purpose and future of the Russian courts are presented perversely and frivolously, only to achieve a certain end, to achieve the acquittal of that which cannot be acquitted. 'Oh, overwhelm him with mercy,' the defense attorney exclaims, and that is just what the criminal wants, and tomorrow everyone will see how overwhelmed he is! And is the defense attorney not being too modest in asking only for the defendant's acquittal? Why does he not ask that a fund be established in the parricide's name, in order to immortalize his deed for posterity and the younger generation? The Gospel and religion are corrected: it's all mysticism, he says, and ours is the only true Christianity, tested by the analysis of reason and sensible ideas. And so a false image of Christ is held up to us! With what measure ye mete, it shall be measured to you,' the defense attorney exclaims, and concludes then and there that Christ commanded us to measure with the same measure as it is measured to us--and that from the tribune of truth and sensible ideas! We glance into the Gospel only on the eve of our speeches, in order to make a brilliant display of our familiarity with what is, after all, a rather original work, which may prove useful and serve for a certain effect, in good measure, all in good measure! Yet Christ tells us precisely not to do so, to beware of doing so, because that is what the wicked world does, whereas we must forgive and turn our cheek, and not measure with the same measure as our offenders measure to us. This is what our God taught us, and not that it is a prejudice to forbid children to kill their own fathers. And let us not, from the rostrum of truth and sensible ideas, correct the Gospel of our God, whom the defense attorney deems worthy of being called merely 'the crucified lover of mankind,' in opposition to the whole of Orthodox Russia, which calls out to him: 'For thou art our God...! ~ Fyodor Dostoyevsky, #NFDB
216:The normative principle I am suggesting for the law is simply this: No action should be considered illicit or illegal unless it invades, or aggresses against, the person or just property of another. Only invasive actions should be declared illegal, and combated with the full power of the law. The invasion must be concrete and physical. There are degrees of seriousness of such invasion, and hence, different proper degrees of restitution or punishment. "Burglary," simple invasion of property for purposes of theft, is less serious than "robbery," where armed force is likely to be used against the victim. Here, however, we are not concerned with the questions of degrees of invasion or punishment, but simply with invasion per se.
If no man may invade another person's "just" property, what is our criterion of justice to be? There is no space here to elaborate on a theory of justice in property titles. Suffice it to say that the basic axiom of libertarian political theory holds that every man is a selfowner, having absolute jurisdiction over his own body. In effect, this means that no one else may justly invade, or aggress against, another's person. It follows then that each person justly owns whatever previously unowned resources he appropriates or "mixes his labor with." From these twin axioms — self-ownership and "homesteading" — stem the justification for the entire system of property rights titles in a free-market society. This system establishes the right of every man to his own person, the right of donation, of bequest (and, concomitantly, the right to receive the bequest or inheritance), and the right of contractual exchange of property titles.
Legal and political theory have committed much mischief by failing to pinpoint physical invasion as the only human action that should be illegal and that justifies the use of physical violence to combat it. The vague concept of "harm" is substituted for the precise one of physical violence. Consider the following two examples. Jim is courting Susan and is just about to win her hand in marriage, when suddenly Bob appears on the scene and wins her away. Surely Bob has done great "harm" to Jim. Once a nonphysical-invasion sense of harm is adopted, almost any outlaw act might be justified. Should Jim be able to "enjoin" Bob's very existence?
Similarly, A is a successful seller of razor blades. But then B comes along and sells a better blade, teflon-coated to prevent shaving cuts. The value of A's property is greatly affected. Should he be able to collect damages from B, or, better yet, to enjoin B's sale of a better blade? The correct answer is not that consumers would be hurt if they were forced to buy the inferior blade, although that is surely the case. Rather, no one has the right to legally prevent or retaliate against "harms" to his property unless it is an act of physical invasion. Everyone has the right to have the physical integrity of his property inviolate; no one has the right to protect the value of his property, for that value is purely the reflection of what people are willing to pay for it. That willingness solely depends on how they decide to use their money. No one can have a right to someone else's money, unless that other person had previously contracted to transfer it to him.
Legal and political theory have committed much mischief by failing to pinpoint physical invasion as the only human action that should be illegal and that justifies the use of physical violence to combat it. (1/2) ~ Murray N Rothbard,#NFDB
217:Physical Invasion
The normative principle I am suggesting for the law is simply this: No action should be considered illicit or illegal unless it invades, or aggresses against, the person or just property of another. Only invasive actions should be declared illegal, and combated with the full power of the law. The invasion must be concrete and physical. There are degrees of seriousness of such invasion, and hence, different proper degrees of restitution or punishment. "Burglary," simple invasion of property for purposes of theft, is less serious than "robbery," where armed force is likely to be used against the victim. Here, however, we are not concerned with the questions of degrees of invasion or punishment, but simply with invasion per se.
If no man may invade another person's "just" property, what is our criterion of justice to be? There is no space here to elaborate on a theory of justice in property titles. Suffice it to say that the basic axiom of libertarian political theory holds that every man is a selfowner, having absolute jurisdiction over his own body. In effect, this means that no one else may justly invade, or aggress against, another's person. It follows then that each person justly owns whatever previously unowned resources he appropriates or "mixes his labor with." From these twin axioms — self-ownership and "homesteading" — stem the justification for the entire system of property rights titles in a free-market society. This system establishes the right of every man to his own person, the right of donation, of bequest (and, concomitantly, the right to receive the bequest or inheritance), and the right of contractual exchange of property titles.
Legal and political theory have committed much mischief by failing to pinpoint physical invasion as the only human action that should be illegal and that justifies the use of physical violence to combat it. The vague concept of "harm" is substituted for the precise one of physical violence. Consider the following two examples. Jim is courting Susan and is just about to win her hand in marriage, when suddenly Bob appears on the scene and wins her away. Surely Bob has done great "harm" to Jim. Once a nonphysical-invasion sense of harm is adopted, almost any outlaw act might be justified. Should Jim be able to "enjoin" Bob's very existence?
Similarly, A is a successful seller of razor blades. But then B comes along and sells a better blade, teflon-coated to prevent shaving cuts. The value of A's property is greatly affected. Should he be able to collect damages from B, or, better yet, to enjoin B's sale of a better blade? The correct answer is not that consumers would be hurt if they were forced to buy the inferior blade, although that is surely the case. Rather, no one has the right to legally prevent or retaliate against "harms" to his property unless it is an act of physical invasion. Everyone has the right to have the physical integrity of his property inviolate; no one has the right to protect the value of his property, for that value is purely the reflection of what people are willing to pay for it. That willingness solely depends on how they decide to use their money. No one can have a right to someone else's money, unless that other person had previously contracted to transfer it to him.
"Legal and political theory have committed much mischief by failing to pinpoint physical invasion as the only human action that should be illegal and that justifies the use of physical violence to combat it. ~ Murray N Rothbard,#NFDB
218:If my opinion that substance requires a true unity were founded only on a definition I had formulated in opposition to common usage, *then the dispute would be only one of words*. But besides the fact that most philosophers have taken the term in almost the same fashion, distinguishing between a unity in itself and an accidental unity, between substantial and accidental form, and between perfect and imperfect, natural and artificial mixtures, I take things to a much higher level, and setting aside the question of terminology, *I believe that where there are only beings by aggregation, there aren't any real beings*. For every being by aggregation presupposes beings endowed with real unity, because every being derives its reality only from the reality of those beings of which it is composed, so that it will not have any reality at all if each being of which it is composed is itself a being by aggregation, a being for which we must still seek further grounds for its reality, grounds which can never be found in this way, if we must always continue to seek for them. I agree, Sir, that there are only machines (that are often animated) in all of corporeal nature, but I do not agree that *there are only aggregates of substances, there must also be true substances from which all the aggregates result.
We must, then, necessarily come down to the atoms of Epicurus and Cordemoy (which are things you reject along with me), or else we must admit that we do not find any reality in bodies; or finally, we must recognize some substances that have a true unity. I have already said in another letter that the composite made up of the diamonds of the Grand Duke and of the Great Mogul can be called a pair of diamonds, but this is only a being of reason. And when they are brought closer to one another, it would be a being of the imagination or perception, that is to say, a phenomenon. For contact, common motion, and participation in a common plan have no effect on substantial unity. It is true that there are sometimes more, sometimes fewer, grounds for supposing that several things constitute a single thing, in proportion to the extent to which these things are connected. But this serves only to abbreviate our thoughts and to represent the phenomena.
It also seems that what constitutes the essence of a being by aggregation is only a mode (*maniére d'être*) of the things of which it is composed. For example, what constitutes the essence of an army is only a mode of the men who compose it. This mode therefore presupposes a substance whose essence is not a mode of substance. Every machine also presupposes some substance in the pieces of which it is made, and there is no plurality without true unities. To put it briefly, I hold this identical proposition, differentiated only by the emphasis, to be an axiom, namely, *that what is not truly* one *being is not truly one* being *either*. It has always been thought that one and being are reciprocal things. Being is one thing and beings are another; but the plural presupposes the singular, and where there is no being still less will there be several beings. What could be clearer? [[I therefore believed that I would be allowed to distinguish beings by aggregation from substances, since these beings have their unity in our mind only, a unity founded on the relations or modes [*modes*] of true substances. If a machine is one substance, a circle of men holding hands will also be one substance, and so will an army, and finally, so will every multitude of substances.]]."
—from Letters to Arnauld ~ Gottfried Wilhelm Leibniz,#NFDB
219:If my opinion that substance requires a true unity were founded only on a definition I had formulated in opposition to common usage, *then the dispute would be only one of words*. But besides the fact that most philosophers have taken the term in almost the same fashion, distinguishing between a unity in itself and an accidental unity, between substantial and accidental form, and between perfect and imperfect, natural and artificial mixtures, I take things to a much higher level, and setting aside the question of terminology, *I believe that where there are only beings by aggregation, there aren't any real beings*. For every being by aggregation presupposes beings endowed with real unity, because every being derives its reality only from the reality of those beings of which it is composed, so that it will not have any reality at all if each being of which it is composed is itself a being by aggregation, a being for which we must still seek further grounds for its reality, grounds which can never be found in this way, if we must always continue to seek for them. I agree, Sir, that there are only machines (that are often animated) in all of corporeal nature, but I do not agree that *there are only aggregates of substances, there must also be true substances from which all the aggregates result.
We must, then, necessarily come down to the atoms of Epicurus and Cordemoy (which are things you reject along with me), or else we must admit that we do not find any reality in bodies; or finally, we must recognize some substances that have a true unity. I have already said in another letter that the composite made up of the diamonds of the Grand Duke and of the Great Mogul can be called a pair of diamonds, but this is only a being of reason. And when they are brought closer to one another, it would be a being of the imagination or perception, that is to say, a phenomenon. For contact, common motion, and participation in a common plan have no effect on substantial unity. It is true that there are sometimes more, sometimes fewer, grounds for supposing that several things constitute a single thing, in proportion to the extent to which these things are connected. But this serves only to abbreviate our thoughts and to represent the phenomena.
It also seems that what constitutes the essence of a being by aggregation is only a mode (*maniére d'être*) of the things of which it is composed. For example, what constitutes the essence of an army is only a mode of the men who compose it. This mode therefore presupposes a substance whose essence is not a mode of substance. Every machine also presupposes some substance in the pieces of which it is made, and there is no plurality without true unities. To put it briefly, I hold this identical proposition, differentiated only by the emphasis, to be an axiom, namely, *that what is not truly* one *being is not truly one* being *either*. It has always been thought that one and being are reciprocal things. Being is one thing and beings are another; but the plural presupposes the singular, and where there is no being still less will there be several beings. What could be clearer? [[I therefore believed that I would be allowed to distinguish beings by aggregation from substances, since these beings have their unity in our mind only, a unity founded on the relations or modes [*modes*] of true substances. If a machine is one substance, a circle of men holding hands will also be one substance, and so will an army, and finally, so will every multitude of substances.]]."
—from Letters to Arnauld ~ Gottfried Wilhelm Leibniz,#NFDB
220:Chapter LXXXII: Epistola Penultima: The Two Ways to Reality
Cara Soror,
Do what thou wilt shall be the whole of the Law.
How very sensible of you, though I admit somewhat exacting!
You write-Will you tell me exactly why I should devote so much of my valuable time to subjects like Magick and Yoga.
That is all very well. But you ask me to put it in syllogistic form. I have no doubt this can be done, though the task seems somewhat complicated. I think I will leave it to you to construct your series of syllogisms yourself from the arguments of this letter.
In your main question the operative word is "valuable. Why, I ask, in my turn, should you consider your time valuable? It certainly is not valuable unless the universe has a meaning, and what is more, unless you know what that meaning is-at least roughly-it is millions to one that you will find yourself barking up the wrong tree.
First of all let us consider this question of the meaning of the universe. It is its own evidence to design, and that design intelligent design. There is no question of any moral significance-"one man's meat is another man's poison" and so on. But there can be no possible doubt about the existence of some kind of intelligence, and that kind is far superior to anything of which we know as human.
How then are we to explore, and finally to interpret this intelligence?
It seems to me that there are two ways and only two. Imagine for a moment that you are an orphan in charge of a guardian, inconceivably learned from your point of view.
Suppose therefore that you are puzzled by some problem suitable to your childish nature, your obvious and most simple way is to approach your guardian and ask him to enlighten you. It is clearly part of his function as guardian to do his best to help you. Very good, that is the first method, and close parallel with what we understand by the word Magick.
We are bothered by some difficulty about one of the elements-say Fire-it is therefore natural to evoke a Salamander to instruct you on the difficult point. But you must remember that your Holy Guardian Angel is not only far more fully instructed than yourself on every point that you can conceive, but you may go so far as to say that it is definitely his work, or part of his work; remembering always that he inhabits a sphere or plane which is entirely different from anything of which you are normally aware.
To attain to the Knowledge and Conversation of the Holy Guardian Angel is consequently without doubt by far the simplest way by which you can yourself approach that higher order of being.
That, then, is a clearly intelligible method of procedure. We call it Magick.
It is of course possible to strengthen the link between him and yourself so that in course of time you became capable of moving and, generally speaking, operating on that plane which is his natural habitat.
There is however one other way, and one only, as far as I can see, of reaching this state.
It is at least theoretically possible to exalt the whole of your own consciousness until it becomes as free to move on that exalted plane as it is for him. You should note, by the way, that in this case the postulation of another being is not necessary. There is no way of refuting the solipsism if you feel like that. Personally I cannot accede to its axiom. The evidence for an external universe appears to me perfectly adequate.
Still there is no extra charge for thinking on those lines if you so wish.
I have paid a great deal of attention in the course of my life to the method of exalting the human consciousness in this way; and it is really quite legitimate to identify my teaching with that of the Yogis.
I must however point out that in the course of my instruction I have given continual warnings as to the dangers of this line of research. For one thing there is no means of checking your results in the ordinary scientific sense. It is always perfectly easy to find a subjective explanation of any phenomenon; and when one considers that the greatest of all the dangers in any line of research arise from egocentric vanity, I do not think I have exceeded my duty in anything that I have said to deter students from undertaking so dangerous a course as Yoga.
It is, of course, much safer if you are in a position to pursue in the Indian Jungles, provided that your health will stand the climate and also, I must say, unless you have a really sound teacher on whom you can safely rely. But then, if we once introduce a teacher, why not go to the Fountain-head and press towards the Knowledge and conversation of the Holy Guardian Angel?
In any case your Indian teacher will ultimately direct you to seek guidance from that source, so it seems to me that you have gone to a great deal of extra trouble and incurred a great deal of unnecessary danger by not leaving yourself in the first place in the hands of the Holy Guardian Angel.
In any case there are the two methods which stand as alternatives. I do not know of any third one which can be of any use whatever. Logically, since you have asked me to be logical, there is certainly no third way; there is the external way of Magick, and the internal way of Yoga: there you have your alternatives, and there they cease.
Love is the law, love under will.
Fraternally,
666 ~ Aleister Crowley, Magick Without Tears,#NFDB
22 Occultism
18 Psychology
13 Integral Yoga
10 Philosophy
2 Poetry
1 Christianity
1 Alchemy
19 Carl Jung
11 Nolini Kanta Gupta
11 Aleister Crowley
4 Aldous Huxley
3 Sri Aurobindo
3 Jorge Luis Borges
3 Jordan Peterson
3 Friedrich Nietzsche
9 Magick Without Tears
7 Mysterium Coniunctionis
6 The Secret Doctrine
5 The Practice of Psycho therapy
4 The Perennial Philosophy
4 Aion
3 Twilight of the Idols
3 The Archetypes and the Collective Unconscious
3 Maps of Meaning
3 Collected Works of Nolini Kanta Gupta - Vol 02
3 Collected Works of Nolini Kanta Gupta - Vol 01
2 The Problems of Philosophy
2 The Life Divine
2 Labyrinths
2 Collected Works of Nolini Kanta Gupta - Vol 07
0.00 - The Wellspring of Reality, #Synergetics - Explorations in the Geometry of Thinking, #R Buckminster Fuller, #Science
We must start with scientific fundamentals, and that means with the data of experiments and not with assumed axioms predicated only upon the misleading nature of that which only superficially seems to be obvious. It is the consensus of great scientists that science is the attempt to set in order the facts of experience.
Holding within their definition, we define Universe as the aggregate of allhumanity's consciously apprehended and communicated, nonsimultaneous, and only partially overlapping experiences. An aggregate of finites is finite. Universe is a finite but nonsimultaneously conceptual scenario.
01.07 - Blaise Pascal (1623-1662), #Collected Works of Nolini Kanta Gupta - Vol 02, #Nolini Kanta Gupta, #Integral Yoga
In his inquiry into truth and certitude Pascal takes his stand upon what he calls the geometrical method, the only valid method, according to him, in the sphere of reason. The characteristic of this method is that it takes for granted certain fundamental principles and realitiescalled axioms and postulates or definitionsand proceeds to other truths that are infallibly and inevitably deduced from them, that are inherent and implied in them. There is no use or necessity in trying to demonstrate these fundamentals also; that will only land us into confusion and muddle. They have to be simply accepted, they do not require demonstration, it is they that demonstrate others. Such, for instance, are space, time, number, the reality of which it is foolishness and pedantry to I seek to prove. There is then an order of truths that do not i require to be proved. We are referring only to the order of I physical truths. But there is another order, Pascal says, equally I valid and veritable, the order of the Spirit. Here we have another set of fundamentals that have to be accepted and taken for granted, matrix of other truths and realities. It can also be called the order of the Heart. Reason posits physical fundamentals; it does not know of the fundamentals of the Heart which are beyond its reach; such are God, Soul, Immortality which are evident only to Faith.
But Faith and Reason, according to Pascal, are not contraries nor irreconcilables. Because the things of faith are beyond reason, it is not that they are irrational. Here is what Pascal says about the function and limitation of reason:
0 1962-12-15, #Agenda Vol 03, #The Mother, #Integral Yoga
(Mother shows Satprem some pamphlets printed during Theon's time, "Fundamental axioms of Cosmic Philosophy," which have just been found among some old papers:)
This is pretty funny! (Laughing, Mother reads:)
--
It was in both French and English. He called it Fundamental axioms of Cosmic Philosophy. It was the work of a certain French metaphysician who was well known around the turn of the centuryhis name began with a B. He met Theon in Egypt when Theon was with Blavatski; they started a magazine with an ancient Egyptian name (I cant recall what it was), and then he told Theon (Theon must have already known French) to publish a Cosmic Review and the Cosmic Books. And this B. is the one who formulated all this gobbledygook.
There used to be the name of the printer and the year it was printed, but its not there any more.
--
Yes, Edouard Schur. He was a contemporary of Edouard Schur, a bit older (I met Schur, by the waya rather hollow individual). His name began with a B and hes the one who formulated these axioms.
You once mentioned someone called Barley.
--
(Mother starts leafing through the axioms again)
They make all kinds of recommendations here: for instance, when you go out of your body you should wear a loose-fitting robe, a robe kept specially for that.
02.03 - National and International, #Collected Works of Nolini Kanta Gupta - Vol 01, #Nolini Kanta Gupta, #Integral Yoga
We have just passed through another, a far greater, a catastrophic Kurukshetra, the last Act (Shanti Parvam) of which we are negotiating at the present moment. The significance of this cataclysm is clear and evident if we only allow ourselves to be led by the facts and not try to squeeze the facts into the groove of our past prejudices and set notions. All the difficulties that are being encountered on the way to peace and reconstruction arise mainly out of the failure to grasp what Nature has forced upon us. It is as simple as the first axiom of Euclid: Humanity is one and all nations are free and yet interdependent members of that one and single organism. No nation can hope henceforth to stand in its isolated grandeurnot even America or Russia. Subject or dependent nations too who are struggling to be free will be allowed to work out their freedom and independence, on condition that the same is worked out in furtherance and in collaboration with the ideal of human unity. That ideal has become dynamic and insistent the more man refuses to accept it, the more he will make confusion worse confounded.
***
03.02 - The Philosopher as an Artist and Philosophy as an Art, #Collected Works of Nolini Kanta Gupta - Vol 02, #Nolini Kanta Gupta, #Integral Yoga
I wonder why Philosophy has never been considered as a variety of Art. Philosophy is admired for the depth and height of its substance, for its endeavour to discover the ultimate Truth, for its one-pointed adherence to the supremely Real; but precisely because it does so it is set in opposition to Art which is reputed as the domain of the ideal, the imaginative or the fictitious. Indeed it is the antagonism between the two that has always been emphasised and upheld as an axiomatic truth and an indisputable fact. Of course, old Milton (he was young, however, when he wrote these lines) says that philosophy is divine and charming:
Not harsh, and crabbed as dull fools suppose,
03.05 - The World is One, #Collected Works of Nolini Kanta Gupta - Vol 02, #Nolini Kanta Gupta, #Integral Yoga
We say not only that India is one and indivisible (and for that matter, Bengal too is one and indivisible, since we have to repeat axiomatic truths that have fallen on evil days and on evil tongues) but that also the whole world is one and indivisible. They who seek to drive in a wedge anywhere, who are busy laying some kind of cordon sanitaire across countries and nations or cultures and civilisations, in the name of a bigoted ideology, are, to say the least, doing a disservice to humanity, indeed they are inviting a disaster and catastrophe to the world and equally to themselves. For that is an attempt to stem the high tide of Nature's swell towards a global unity that shall brook no resistance.
The distinctions and differences that held good in other times and climes can have no sense or value in the world of today. Race or religion can divide man no longer; even nationhood has lost much of its original force and meaning. It is strangeperhaps it is inevitable in the secret process of Nature's working that when everything in conditions and circumstances obviously demands and points to an obliteration of all frontiers of division and separationeconomically and politically tooand all drives towards a closer co-operation and intermingling, it is precisely then that the contrary spirit and impulse raises its head and seems even to gather added strength and violence. The fact may have two explanations. First of all, it may mean a defence gesture in Nature, that is to say, certain forces or formations have a permanent place in Nature's economy and when they apprehend that they are being ousted and neglected, when there is a one-pointed drive for their exclusion, naturally they surge up and demand recognition with a vengeance: for things forgotten or left aside that form indissolubly part of Nature's fabric and pattern, one has to retrace one's steps in order to pick them up again. But also the phenomenon may mean a simple case of atavism: for we must know that there are certain old-world aboriginal habits and movements that have to go and have no place in the higher scheme of Nature and these too come up off and on, especially when the demand is there for their final liquidation. They have to be recognised as such and treated as such. Radical and religious (including ideological) egoisms seem to us to belong to this category.
04.03 - The Eternal East and West, #Collected Works of Nolini Kanta Gupta - Vol 01, #Nolini Kanta Gupta, #Integral Yoga
This view finds its justification because of a particular outlook on spirituality and non-spirituality. If the Spirit and things spiritual are taken to mean something transcending and rejecting the world and the things of the world, something exclusive of life and its fulfilment here on earth, if on the other hand, the world and its life are given only their face value emptying them of their deeper and transcendent contentsin the manner of the great Laplace who could find no place for God in his map of the world which seemed to be quite complete in itself, if this trenchant division is made in the very definition of the terms, in our primary axioms and postulates, then, of course, we cannot avoid a scission and an eternal struggle. If you consider the Spirit as only pure spirit, an absolute without any relation, as, an ever-fixed and static entity and if we view Matter as purely material and the law of mechanics as supreme and inviolable, then there cannot be a reconciliation or even a meeting between the two. There are some who have a great goodwill, who wish to avoid clash and quarrel and are for concord and harmony. They have tried the reconciliation, but failed. The two positions being fundamentally exclusive of each other can, at best, be juxtaposed, but not unified or fused together.
And yet mankind has always sought for an integral, an all comprehending fulfilment, a truth and a realisation that would go round his entire existence. Man has always aspired, in the midst of the transience and imperfection that the world is, for something stable and perfect, in the heart of disharmony for some core of perfect harmony. He termed it God, Atman, Summum Bonum and he sought it sometimes, as he thought necessary, even at the cost of the world and the life, if it is to be found elsewhere. Man aspired also always to find this habitation of his made somewhat better. Dissatisfied with his present state, he sought to mould it, remake it, put into it something which his aspiration and inspiration called the True, the Beautiful, the Good. There was always this double aspiration in man, one of ascent and the other of descent, one vertical and the other horizontal, one leading up and beyondtotally beyond, in its extreme urge the other probing into the mystery locked up there below, releasing the power to reform or recreate the world, although he was not always sure whether it was a power of mind or of matter.
05.05 - In Quest of Reality, #Collected Works of Nolini Kanta Gupta - Vol 01, #Nolini Kanta Gupta, #Integral Yoga
A hypothesis, however revolutionary or unorthodox it may seem for the moment, has to be tested by its effective application, in its successful working out. All scientific discoveries in the beginning appear as inconveniences that upset the known and accepted order. Copernicus, Newton, Galileo, Kepler, Maxwell or Einstein in our day enunciated principles that were not obvious sense-given axioms. These are at the outset more or less postulates that have to be judged by their applicability.
Creation as a movement or expression of consciousness need not be dubbed a metaphysical jargon; it can be assumed as a scientific working hypothesis and seen how it affects our view, meets our problems and difficulties, whether it can give a satisfactory clue to some of the riddles of physical and psychical phenomena. A scientific supposition (or intuition) is held to be true if it can be applied invariably to facts of life and experience and if it can open up to our vision and perception new facts. The trend of scientific discoveries today is towards the positing of a background reality in Nature of which energy (radiant and electrical) is the first and overt form. We discarded ether, only to replace it by field and disposition. We have arrived at a point where the question is whether we cannot take courage" in both hands and declare, as some have already done, that the substratum in Nature is consciousness-energy and on that hypothesis better explain certain movements of Matter and Life and Mind in a global unity. Orthodox and die-hard views will always protest and cry that it is a misalliance, a misjoinder to couple together Matter and Consciousness or even Life and Consciousness. But since the light has touched the higher mind even among a few of the positivist type, the few may very well be the precursor of the order of the day.
07.11 - The Problem of Evil, #Collected Works of Nolini Kanta Gupta - Vol 03, #Nolini Kanta Gupta, #Integral Yoga
As I say, the question is wrongly put. The very form of the question already assumes a certain notion about God and creation. Your postulates or axioms themselves are vitiated.
The universe and its creator are not separate things, they are one and identical in their origin. The universe is God himself projected into Space (and Time). So the universe is the Divine in one aspect or another. You cannot divide the two, making one the creator and the other, his work, the watch-maker and his watch. You put your idea of the Divine upon him and ask, why he has created such a nasty world. If the Divine were to answer, It is not I, it is yourself. Become myself again, you will no longer feel and see as you do now you are not yourself, therefore your question and your problem! Indeed, when you unite your consciousness with the divine consciousness there is no longer any problem. Everything appears then natural and simple, and correct and as it should be. It is when you cut yourself from your origin and stand outside, in front of him and against him that all the trouble begins. Of course you may ask, how is it that the Divine has tolerated a part of himself going out and separating itself and creating all this disorder? I would reply on behalf of the Divine, If you want to know, you had better unite yourself with the Divine, for that is the only way of knowing why he has done so. It is not by questioning him by your mind that you will get the answer. The mind cannot know. And repeat, when you come to this identification, all problems are solved. The feeling, one can explain, that things are not all right, that they should be otherwise comes precisely from the fact that there is a divine will unfolding itself in a continuous progression, that things that were and are have to give place to things that shall be and shall be better and better than they have been. The world that was good yesterday will no longer be so tomorrow. The universe might have appeared quite harmonious in some other age but now appears quite discordant: it is because we see the possibility of a better universe. If we found it as it should be, we would not do what we have to do, we would not try to make it better. Even so, we would conceive the Divine in a very human way; for we remain imprisoned within ourselves, confined to this consciousness of ours which is like a grain of sand in the infinite immensity. You want to understand the immensity? That is not possible. It is possible only under one condition; be one with the immensity. The drop of water cannot very well ask how the ocean is: it has to lose itself into the ocean.
1.02 - MAPS OF MEANING - THREE LEVELS OF ANALYSIS, #Maps of Meaning, #Jordan Peterson, #Psychology
unexpected pops us out of unconscious, axiomatic complacency, and forces us (painfully) to think.
The implications of novel or unpredictable occurrences are unknown, by definition. This observation
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regarded as axiomatic, for the purposes of the current operation), has been violated. The story go
downstairs to eat only retained its function in an environment characterized by valid means of betweenfloor transportation. The existence of these means constituted a given I had used the elevator or the stairs
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Their implicit contextual constraints or axioms of these procedures, however, lead researchers to draw odd
conclusions about the nature of the acquisition of fear.
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capacity in part because it actually seems to underly (to serve as a necessary or axiomatic precondition
for) our ability to understand, explicitly.
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most fundamental axioms of the body of law that we imitate (that governs our behavior) are
inextricably embedded in the conception of natural rights (which is to say, is embedded in a statement of
1.02 - The Pit, #A Garden of Pomegranates - An Outline of the Qabalah, #Israel Regardie, #Occultism
Athena. This necessity was emphasized in the most surprising way by the result of the Michelson-Morley experiments, when Physics itself calmly and frankly offered a contradiction in terms. It was not the metaphysicians this time who were picking holes in a vacuum. It was the mathematicians and the physicists who found the ground completely cut away from under their feet. It was not enough to replace the geometry of Euclid by those of Riemann and Lobatchevsky and the mechanics of Newton by those of Einstein, so long as any of the axioms of the old thought and the definitions of its terms survived. They deliberately abandoned positivism and materialism for an indeterminate mysticism, creating a new mathematical philosophy and a new logic, wherein infinite-or rather transfinite-ideas might be made commensurable with those of ordinary thought in the forlorn hope that all might live happily ever after. In short, to use a Qabalistic nomenclature, they found it incumbent upon themselves to adopt for inclusion of terms of Ruach (intellect) concepts which are proper only to Neschamah (the organ and faculty of direct spiritual apperception and intuition). This same process took place in Philosophy years earlier. Had the dialectic of Hegel been only. half understood, the major portion of philosophical speculation from the Schoolmen to
Kant's perception of the Antinomies of Reason would have been thrown overboard.
1.02 - THE PROBLEM OF SOCRATES, #Twilight of the Idols, #Friedrich Nietzsche, #Philosophy
subtle axiom, _that the value of life cannot be estimated._ A living
man cannot do so, because he is a contending party, or rather the very
1.02 - The Three European Worlds, #The Ever-Present Origin, #Jean Gebser, #Integral
Some dozen years later, the three Commentarii of Lorenzo Ghiberti also treat of this same perspective; but despite his attempt to remain within the tradition, his treatises describe in a novel way not only perspective but also anatomy and a theory of drawing (teorica del disegno). It is significant that he corrects his principal model, Vitruvius, by inserting a chapter an "perspective" where Vitruvius would have included a chapter an the "knowledge of rules," and consequently intentionally 'elevates perspectivity to a basic axiom of his time.
There is yet another major artist of that age who continues the discussion of this subject in advance of the definitive statements of Leonardo. Toward the end of his life, Pierodella Francesca furnishes a penetrating theory of perspective compared to which Alberti's seems amateurish and empirical. In his three books De Perspectiva Pingendi based anEuclid, which were written in collaboration with Luca Pacioli, he defines for the first time costruzionepittorica as perspective. He had himself been successful in the practical application of perspective during the time ofFoquet, i.e., the latter half of the fifteenth century, though after the brothers van Eyck (to mention only the outstanding figures). This had facilitated the ultimate achievement of perspectivity, the "aerial perspective" of Leonardo's Last Supper.
1.04 - THE APPEARANCE OF ANOMALY - CHALLENGE TO THE SHARED MAP, #Maps of Meaning, #Jordan Peterson, #Psychology
questions make up the axiomatic statements of the paradigm, which are, according to Kuhn, explicitly
formulated semantically represented, according to the argument set forth here or left implicit
embedded in (episodic) fantasy or embodied behavior. The validity of the axioms must either be accepted
189
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articles of faith are axioms of morality, so to speak some explicit (represented declaratively, in image
and word), most still implicit which evolved in the course of human exploration and social organization,
over the course of hundreds of thousands of years. In their purely implicit states, such axioms are extremely
resistant to alteration. Once made (partially) explicit, however, moral axioms rapidly become subject to
endless careful and thoughtful or casual careless debate. Such debate is useful, for continuance and
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principles, is bound by necessity to accept certain axioms on faith. These axioms follow:
1. A straight line segment can be drawn joining any two points.
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developed. The axioms of primary concerns are the simplest and baldest platitudes it is possible to
formulate: that life is better than death, happiness better than misery; health better than sickness,
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pattern of behaving in the face of the unknown, and the paradigm cannot be shifted (its basic axioms cannot
be modified), without dramatic consequences without dissolution, metaphoric death prior to (potential)
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challenge to the perceived validity of the axioms currently underlying the maintenance of normal sanity
the sociohistorically determined stability of mutually-determined behavioral adaptation and experiential
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historical canon (that established axiomatically-predicated hierarchy of values and assumptions) he takes
inspired action and transcends his culturally-determined limitations. Instead of denying the existence of the
1.05 - CHARITY, #The Perennial Philosophy, #Aldous Huxley, #Philosophy
Our present economic, social and international arrangements are based, in large measure, upon organized lovelessness. We begin by lacking charity towards Nature, so that instead of trying to co-operate with Tao or the Logos on the inanimate and subhuman levels, we try to dominate and exploit, we waste the earths mineral resources, ruin its soil, ravage its forests, pour filth into its rivers and poisonous fumes into its air. From lovelessness in relation to Nature we advance to lovelessness in relation to arta lovelessness so extreme that we have effectively killed all the fundamental or useful arts and set up various kinds of mass production by machines in their place. And of course this lovelessness in regard to art is at the same time a lovelessness in regard to the human beings who have to perform the fool-proof and grace-proof tasks imposed by our mechanical art-surrogates and by the interminable paper work connected with mass production and mass distribution. With mass-production and mass-distribution go mass-financing, and the three have conspired to expropriate ever-increasing numbers of small owners of land and productive equipment, thus reducing the sum of freedom among the majority and increasing the power of a minority to exercise a coercive control over the lives of their fellows. This coercively controlling minority is composed of private capitalists or governmental bureaucrats or of both classes of bosses acting in collaborationand, of course, the coercive and therefore essentially loveless nature of the control remains the same, whether the bosses call themselves company directors or civil servants. The only difference between these two kinds of oligarchical rulers is that the first derive more of their power from wealth than from position within a conventionally respected hierarchy, while the second derive more power from position than from wealth. Upon this fairly uniform groundwork of loveless relationships are imposed others, which vary widely from one society to another, according to local conditions and local habits of thought and feeling. Here are a few examples: contempt and exploitation of coloured minorities living among white majorities, or of coloured majorities governed by minorities of white imperialists; hatred of Jews, Catholics, Free Masons or of any other minority whose language, habits, appearance or religion happens to differ from those of the local majority. And the crowning superstructure of uncharity is the organized lovelessness of the relations between state and sovereign statea lovelessness that expresses itself in the axiomatic assumption that it is right and natural for national organizations to behave like thieves and murderers, armed to the teeth and ready, at the first favourable opportunity, to steal and kill. (Just how axiomatic is this assumption about the nature of nationhood is shown by the history of Central America. So long as the arbitrarily delimited territories of Central America were called provinces of the Spanish colonial empire, there was peace between their inhabitants. But early in the nineteenth century the various administrative districts of the Spanish empire broke from their allegiance to the mother country and decided to become nations on the European model. Result: they immediately went to war with one another. Why? Because, by definition, a sovereign national state is an organization that has the right and duty to coerce its members to steal and kill on the largest possible scale.)
Lead us not into temptation must be the guiding principle of all social organization, and the temptations to be guarded against and, so far as possible, eliminated by means of appropriate economic and political arrangements are temptations against charity, that is to say, against the disinterested love of God, Nature and man. First, the dissemination and general acceptance of any form of the Perennial Philosophy will do something to preserve men and women from the temptation to idolatrous worship of things in timechurch-worship, state-worship, revolutionary future-worship, humanistic self-worship, all of them essentially and necessarily opposed to charity. Next come decentralization, widespread private ownership of land and the means of production on a small scale, discouragement of monopoly by state or corporation, division of economic and political power (the only guarantee, as Lord Acton was never tired of insisting, of civil liberty under law). These social rearrangements would do much to prevent ambitious individuals, organizations and governments from being led into the temptation of behaving tyrannously; while co-operatives, democratically controlled professional organizations and town meetings would deliver the masses of the people from the temptation of making their decentralized individualism too rugged. But of course none of these intrinsically desirable reforms can possibly be carried out, so long as it is thought right and natural that sovereign states should prepare to make war on one another. For modern war cannot be waged except by countries with an over-developed capital goods industry; countries in which economic power is wielded either by the state or by a few monopolistic corporations which it is easy to tax and, if necessary, temporarily to nationalize; countries where the labouring masses, being without property, are rootless, easily transferable from one place to another, highly regimented by factory discipline. Any decentralized society of free, uncoerced small owners, with a properly balanced economy must, in a war-making world such as ours, be at the mercy of one whose production is highly mechanized and centralized, whose people are without property and therefore easily coercible, and whose economy is lop-sided. This is why the one desire of industrially undeveloped countries like Mexico and China is to become like Germany, or England, or the United States. So long as the organized lovelessness of war and preparation for war remains, there can be no mitigation, on any large, nation-wide or world-wide scale, of the organized lovelessness of our economic and political relationships. War and preparation for war are standing temptations to make the present bad, God-eclipsing arrangements of society progressively worse as technology becomes progressively more efficient.
1.05 - Christ, A Symbol of the Self, #Aion, #Carl Jung, #Psychology
81 The earliest authority of all for the later axiom "Omne
bonum a Deo, omne malum ab homine" is Tatian (2nd cen-
--
that we get the axiom "Omne bonum a Deo, omne malum ab
homine." This is a contradiction of the truth that he who
1.05 - Problems of Modern Psycho therapy, #The Practice of Psycho therapy, #Carl Jung, #Psychology
Even if it be a biological axiom that man is a herd animal who only finds
optimum health in living as a social being, the very next case may quite
possibly invert this axiom and show us that he is completely healthy only
when leading an abnormal and unsocial life. It is enough to drive one to
1.05 - THE HOSTILE BROTHERS - ARCHETYPES OF RESPONSE TO THE UNKNOWN, #Maps of Meaning, #Jordan Peterson, #Psychology
therefore, everything defining hope comes to be axiomatically regarded as punishment and threat; makes
life unbearable, as the realm of acceptable action shrinks inexorably. The attendant and unavoidable
--
predeceased us, as paradise has already been spread before us. This is a terrible position, as the axiom of
faith we are redeemed makes human suffering itself (which can never be eradicated, as a consequence of
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revolutionary restructuring of the axioms of Western morality.
Christ has long been considered implicitly contained in the Old Testament. Frye comments:
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described the psychological consequences of axiomatic thinking in great detail, as well. He first posed the
question What happens to the (paradigmatic) representational structure in someones mind (in the
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Science is predicated upon the axiomatic presupposition that it is worthwhile to devote effort towards
analysis of the material or collectively apprehensible sensory world and its transformations. This belief,
--
apply to criticism, when the critic sees in the far distance some such axiom as Criticism can and should make sense
of literature, and refuses to settle for anything less. [Frye, N. (1990). pp. xxii-xxiii].
1.07 - The Three Schools of Magick 2, #Magick Without Tears, #Aleister Crowley, #Philosophy
The basis of the Black philosophy is not impossibly mere climate, with its resulting etiolation of the native, its languid, bilious, anaemic, fever-prostrated, emasculation of the soul of man. We accordingly find few true equivalents of this School in Europe. In Greek philosophy there is no trace of any such doctrine. The poison in its foulest and most virulent form only entered with Christianity.*[AC17] But even so, few men of any real eminence were found to take the axioms of pessimism seriously. Huxley, for all of his harping on the minor key, was an eupeptic Tory. The culmination of the Black philosophy is only found in Schopenhauer, and we may regard him as having been obsessed, on the one hand, by the despair born of that false scepticism which he learnt from the bankruptcy of Hume and Kant; on the other, by the direct obsession of the Buddhist documents to which he was one of the earliest Europeans to obtain access. He was, so to speak, driven to suicide by his own vanity, a curious parallel to Kiriloff in The Possessed of Dostoiewsky.
We have, however, examples plentiful enough of religions deriving almost exclusively from the Black tradition in the different stages. We have already mentioned the Evangelical cults with their ferocious devil-god who creates mankind for the pleasure of damning it and forcing it to crawl before him, while he yells with druken glee over the agony of his only son.[AC18] But in the same class, we must place Christian Science, so grotesquely afraid of pain, suffering and evil of every sort, that its dupes can think of nothing better than to bleat denials of its actuality, in the hope of hypnotizing themselves into anaesthesia.
1.07 - TRUTH, #The Perennial Philosophy, #Aldous Huxley, #Philosophy
In connection with the Mahayanist view that words play an important and even creative part in the evolution of unregenerate human nature, we may mention Humes arguments against the reality of causation. These arguments start from the postulate that all events are loose and separate from one another and proceed with faultless logic to a conclusion that makes complete nonsense of all organized thought or purposive action. The fallacy, as Professor Stout has pointed out, lies in the preliminary postulate. And when we ask ourselves what it was that induced Hume to make this odd and quite unrealistic assumption that events are loose and separate, we see that his only reason for flying in the face of immediate experience was the fact that things and happenings are symbolically represented in our thought by nouns, verbs and adjectives, and that these words are, in effect, loose and separate from one another in a way which the events and things they stand for quite obviously are not. Taking words as the measure of things, instead of using things as the measure of words, Hume imposed the discrete and, so to say, pointilliste pattern of language upon the continuum of actual experiencewith the impossibly paradoxical results with which we are all familiar. Most human beings are not philosophers and care not at all for consistency in thought or action. Thus, in some circumstances they take it for granted that events are not loose and separate, but co-exist or follow one another within the organized and organizing field of a cosmic whole. But on other occasions, where the opposite view is more nearly in accord with their passions or interests, they adopt, all unconsciously, the Humian position and treat events as though they were as independent of one another and the rest of the world as the words by which they are symbolized. This is generally true of all occurrences involving I, me, mine. Reifying the loose and separate names, we regard the things as also loose and separatenot subject to law, not involved in the network of relationships, by which in fact they are so obviously bound up with their physical, social and spiritual environment. We regard as absurd the idea that there is no causal process in nature and no organic connection between events and things in the lives of other people; but at the same time we accept as axiomatic the notion that our own sacred ego is loose and separate from the universe, a law unto itself above the moral dharma and even, in many respects, above the natural law of causality. Both in Buddhism and Catholicism, monks and nuns were encouraged to avoid the personal pronoun and to speak of themselves in terms of circumlocutions that clearly indicated their real relationship with the cosmic reality and their fellow creatures. The precaution was a wise one. Our responses to familiar words are conditioned reflexes. By changing the stimulus, we can do something to change the response. No Pavlov bell, no salivation; no harping on words like me and mine, no purely automatic and unreflecting egotism. When a monk speaks of himself, not as I, but as this sinner or this unprofitable servant, he tends to stop taking his loose and separate selfhood for granted, and makes himself aware of his real, organic relationship with God and his neighbours.
In practice words are used for other purposes than for making statements about facts. Very often they are used rhetorically, in order to arouse the passions and direct the will towards some course of action regarded as desirable. And sometimes, too, they are used poetically that is to say, they are used in such a way that, besides making a statement about real or imaginary things and events, and besides appealing rhetorically to the will and the passions, they cause the reader to be aware that they are beautiful. Beauty in art or nature is a matter of relationships between things not in themselves intrinsically beautiful. There is nothing beautiful, for example, about the vocables, time, or syllable. But when they are used in such a phrase as to the last syllable of recorded time, the relationship between the sound of the component words, between our ideas of the things for which they stand, and between the overtones of association with which each word and the phrase as a whole are charged, is apprehended, by a direct and immediate intuition, as being beautiful.
1.08 - RELIGION AND TEMPERAMENT, #The Perennial Philosophy, #Aldous Huxley, #Philosophy
In traditional Christianity, as in all the great religious formulations of the Perennial Philosophy, it was axiomatic that contemplation is the end and purpose of action. Today the great majority even of professed Christians regard action (directed towards material and social progress) as the end, and analytic thought (there is no question any longer of integral thought, or contemplation) as the means to that end.
In traditional Christianity, as in the other formulations of the Perennial Philosophy, the secret of happiness and the way to salvation were to be sought, not in the external environment, but in the individuals state of mind with regard to the environment. Today the all-important thing is not the state of the mind, but the state of the environment. Happiness and moral progress depend, it is thought, on bigger and better gadgets and a higher standard of living.
1.09 - SKIRMISHES IN A WAY WITH THE AGE, #Twilight of the Idols, #Friedrich Nietzsche, #Philosophy
this piece of ingenuousness, it is the first axiom of this science.
And now let us straightway add the second to it: nothing is ugly save
1.13 - Knowledge, Error, and Probably Opinion, #The Problems of Philosophy, #Bertrand Russell, #Philosophy
In derivative knowledge our ultimate premisses must have some degree of self-evidence, and so must their connexion with the conclusions deduced from them. Take for example a piece of reasoning in geometry. It is not enough that the axioms from which we start should be self-evident: it is necessary also that, at each step in the reasoning, the connexion of premiss and conclusion should be self-evident. In difficult reasoning, this connexion has often only a very small degree of self-evidence; hence errors of reasoning are not improbable where the difficulty is great.
From what has been said it is evident that, both as regards intuitive knowledge and as regards derivative knowledge, if we assume that intuitive knowledge is trustworthy in proportion to the degree of its self-evidence, there will be a gradation in trustworthiness, from the existence of noteworthy sense-data and the simpler truths of logic and arithmetic, which may be taken as quite certain, down to judgements which seem only just more probable than their opposites. What we firmly believe, if it is true, is called _knowledge_, provided it is either intuitive or inferred (logically or psychologically) from intuitive knowledge from which it follows logically. What we firmly believe, if it is not true, is called _error_. What we firmly believe, if it is neither knowledge nor error, and also what we believe hesitatingly, because it is, or is derived from, something which has not the highest degree of self-evidence, may be called _probable opinion_. Thus the greater part of what would commonly pass as knowledge is more or less probable opinion.
1.14 - Bibliography, #Aion, #Carl Jung, #Psychology
x Ripley: Duodecim portarum axiomata philosophica [pp.
123-39]
--
Ripley, George. "Duodecim portarum axiomata philosophica." See
(A) Theatrum chemicum, x.
1.14 - The Limits of Philosophical Knowledge, #The Problems of Philosophy, #Bertrand Russell, #Philosophy
The mathematicians, however, have not been content with showing that space as it is commonly supposed to be is possible; they have shown also that many other forms of space are equally possible, so far as logic can show. Some of Euclid's axioms, which appear to common sense to be necessary, and were formerly supposed to be necessary by philosophers, are now known to derive their appearance of necessity from our mere familiarity with actual space, and not from any _a priori_ logical foundation. By imagining worlds in which these axioms are false, the mathematicians have used logic to loosen the prejudices of common sense, and to show the possibility of spaces differing--some more, some less--from that in which we live. And some of these spaces differ so little from Euclidean space, where distances such as we can measure are concerned, that it is impossible to discover by observation whether our actual space is strictly Euclidean or of one of these other kinds.
Thus the position is completely reversed. Formerly it appeared that experience left only one kind of space to logic, and logic showed this one kind to be impossible. Now, logic presents many kinds of space as possible apart from experience, and experience only partially decides between them. Thus, while our knowledge of what is has become less than it was formerly supposed to be, our knowledge of what may be is enormously increased. Instead of being shut in within narrow walls, of which every nook and cranny could be explored, we find ourselves in an open world of free possibilities, where much remains unknown because there is so much to know.
1.14 - The Structure and Dynamics of the Self, #Aion, #Carl Jung, #Psychology
either by the axiom of Maria, 3 -f- 1, or by the sesquitertian pro-
portion, 3 : 4. This quaternio could therefore replace that of the
1.15 - Index, #Aion, #Carl Jung, #Psychology
"Duodecim portarum axiomata phi-
losophica," 131ft
--
Maria, axiom of, 153, 251
Maria the prophetess, 240
1.27 - CONTEMPLATION, ACTION AND SOCIAL UTILITY, #The Perennial Philosophy, #Aldous Huxley, #Philosophy
IN ALL the historic formulations of the Perennial Philosophy it is axiomatic that the end of human life is contemplation, or the direct and intuitive awareness of God; that action is the means to that end; that a society is good to the extent that it renders contemplation possible for its members; and that the existence of at least a minority of contemplatives is necessary for the well-being of any society. In the popular philosophy of our own time it goes without saying that the end of human life is action; that contemplation (above all in its lower forms of discursive thought) is the means to that end; that a society is good to the extent that the actions of its members make for progress in technology and organization (a progress which is assumed to be causally related to ethical and cultural advance); and that a minority of contemplatives is perfectly useless and perhaps even harmful to the community which tolerates it. To expatiate further on the modern Weltanschauung is unnecessary; explicitly or by implication it is set forth on every page of the advertising sections of every newspaper and magazine. The extracts that follow have been chosen in order to illustrate the older, truer, less familiar theses of the Perennial Philosophy.
Work is for the purification of the mind, not for the perception of Reality. The realization of Truth is brought about by discrimination, and not in the least by ten millions of acts.
1.29 - What is Certainty?, #Magick Without Tears, #Aleister Crowley, #Philosophy
How how curious it must appear at the first glance to note that the truths of this order should prove to be what we call axioms or even Platitudes
. . . . . . What's that noise?
1.35 - The Tao 2, #Magick Without Tears, #Aleister Crowley, #Philosophy
The least abject asset in the intellectual bankruptcy of European thought is the Hebrew Qabalah. Properly understood, it is a system of symbolism indefinitely elastic, assuming no axioms, postulating no principles, asserting no theorems, and therefore adaptable, if managed adroitly, to describe any conceivable doctrine. It has been my continual study since 1898, and I have found it of infinite value in the study of the "Tao Teh King." By its aid I was able to attribute the ideas of Lao Tze to an order with which I was exceedingly familiar, and whose practical worth I had repeatedly proved by using it as the basis of the analysis and classification of all Aryan and Semitic religions and philosophies. Despite the essential difficulty of correlating the ideas of Lao Tze with any others, the persistent application of the Qabalistic keys eventually unlocked his treasure-house. I was able to explain to myself his teachings in terms of familiar systems.
This achievement broke the back of my Sphinx. Having once reduced Lao Tze to Qabalistic form, it was easy to translate the result into the language of philosophy. I had already done much to create a new language based on English with the assistance of a few technical terms borrowed from Asia, and above all by the use of a novel conception of the idea of Number and of algebraic and arithmetical procedure to convey the results of spiritual experience to intelligent students.
1.50 - A.C. and the Masters; Why they Chose him, etc., #Magick Without Tears, #Aleister Crowley, #Philosophy
I know this sounds mad; but it's true. Well, then, I set myself to repair the omission with Part III; this should be a really complete treatise on the Art and Science of magick, and it should be worked out from the beginning, a logical sequence like Euclid. Hence axiom, Postulate and Theorems. I supposed even then that I could cover the field with another volume comparable in size with the former two.
I did indeed "finish" this, even announced publication; it was just going to Press when War (also announced five years before by Bartzabel, the Spirit of Mars) came along in 1914. I toted the rod around the world with me (excuse my American!) and in a fatal hour of weakness, self-mistrust, took to shewing it to some of my students. Of course I might have known they all with one accord began: "Oh, but you haven't said anything about " all the subjects in the world. So I started to fill in the gaps. As I did so, I found any amount more to do on my own. It went on like that for 14 years! Since it came out the voices of detraction have been dumb. I really do believe that I've covered the ground at last. Of course, time shewed that Part I, although it did really give the essentials of Yoga in the simplest possible language, was hardly more than an outline. More, it did not correlate Yoga with general philosophy. Eight Lectures have, I believe, remedied this.
1.65 - Man, #Magick Without Tears, #Aleister Crowley, #Philosophy
In your last letter you thank me for having made clear to you the initiated teaching with regard to the Universe; and you now very rightly enquire "this being so, where do we come in?" You hold up to me one of the oldest axioms of the Qabalah. "That which is above is like that which is below," and you ask me for details. What, you enquire, is the constitution of Man? With what parts of the Great System is the Little System to coincide?
Perhaps I could hardly do better than call your attention to the description given in my essay on Man in my small book Little Essays Toward Truth.
1.70 - Morality 1, #Magick Without Tears, #Aleister Crowley, #Philosophy
Presently, I hope, you will begin to wonder whether, after all, the "morality" of the middle classes of the nineteenth century, in Anglo-Saxon countries, is quite as axiomatic as you were taught to suppose.
Please let me emphasize the fact that I have heard and seen these conditions in Eastern countries with my own ears and eyes. Vivekananda certainly the best of the modern Indian writes on Yoga complained bitterly that the old greymalkin witches of New York who called themselves his disciples had to be dodged with infinite precaution whenever he wanted to spend an evening in the Tenderloin. On the other hand, the Sheikh of Mish and a very holy Sheikh he was introduced his "boy friend" as such to me when I visited him in the Sahara, without the slightest shame or embarrassment.
1.77 - Work Worthwhile - Why?, #Magick Without Tears, #Aleister Crowley, #Philosophy
So much for the argument; it will be agreed readily enough that to put it into practice we shall need an Alphabet, a Grammar and a Dictionary. Follow the axioms, the Postulates, the Theorems; finally, the Experiments.
And that is what all these letters are about.
1.82 - Epistola Penultima - The Two Ways to Reality, #Magick Without Tears, #Aleister Crowley, #Philosophy
There is however one other way, and one only, as far as I can see, of reaching this state. It is at least theoretically possible to exalt the whole of your own consciousness until it becomes as free to move on that exalted plane as it is for him. You should note, by the way, that in this case the postulation of another being is not necessary. There is no way of refuting the solipsism if you feel like that. Personally I cannot accede to its axiom. The evidence for an external universe appears to me perfectly adequate.
Still there is no extra charge for thinking on those lines if you so wish.
1.83 - Epistola Ultima, #Magick Without Tears, #Aleister Crowley, #Philosophy
It may seem to you strange as you read this letter to have watched how the pendulum has swung always a little more and more towards the side of Magick. I do not know why this should have been, but that it is so I have no doubt whatever. I see quite clearly now that Yoga from its very first beginnings is liable to lead the mind away into a condition of muddle, and though for each such state Yoga itself provides the necessary cure, may not one ask oneself if it is really wise to begin one's work with axioms and postulates which are inherently dangerous. The whole controversy might be expressed as a differential equation. Their curves become identical only at infinity, and there is no doubt, at least to my mind, that the curve of Magick follows a more pleasant track than that of Yoga.
To take one point alone: it is evidently more satisfactory to have one's malignant demons external to oneself.
1.ac - The Hermit, #Crowley - Poems, #Aleister Crowley, #Philosophy
He knew ( no doubt) that any axiom
Would furnish bricks to build some Donkeydom.
1.poe - Eureka - A Prose Poem, #Poe - Poems, #unset, #Zen
"Do you know, my dear friend," says the writer, addressing, no doubt, a contemporary -"Do you know that it is scarcely more than eight or nine hundred years ago since the metaphysicians first consented to relieve the people of the singular fancy that there exist but two practicable roads to Truth? Believe it if you can! It appears, however, that long, long ago, in the night of Time, there lived a Turkish philosopher called Aries and surnamed Tottle." [Here, possibly, the letter-writer means Aristotle; the best names are wretchedly corrupted in two or three thousand years.] "The fame of this great man depended mainly upon his demonstration that sneezing is a natural provision, by means of which over-profound thinkers are enabled to expel superfluous ideas through the nose; but he obtained a scarcely less valuable celebrity as the founder, or at all events as the principal propagator, of what was termed the de ductive or a priori philosophy. He started with what he maintained to be axioms, or self-evident truths: -and the now well-understood fact that no truths are self -evident, really does not make in the slightest degree against his speculations: -it was sufficient for his purpose that the truths in question were evident at all. From axioms he proceeded, logically, to results. His most illustrious disciples were one Tuclid, a geometrician," [meaning Euclid] "and one Kant, a Dutchman, the originator of that species of Transcendentalism which, with the change merely of a C for a K, now bears his peculiar name.
"Well, Aries Tottle flourished supreme, until the advent of one Hog, surnamed 'the Ettrick shepherd,' who preached an entirely different system, which he called the a posteriori or in ductive. His plan referred altogether to sensation. He proceeded by observing, analyzing, and classifying facts -instantiae Naturae, as they were somewhat affectedly called -and arranging them into general laws. In a word, while the mode of Aries rested on noumena, that of Hog depended on phenomena; and so great was the admiration excited by this latter system that, at its first introduction, Aries fell into general disrepute. Finally, however, he recovered ground, and was permitted to divide the empire of Philosophy with his more modern rival: -the savans contenting themselves with proscribing all other competitors, past, present, and to come; putting an end to all controversy on the topic by the promulgation of a Median law, to the effect that the Aristotelian and Baconian roads are, and of right ought to be, the sole possible avenues to knowledge: -'Baconian,' you must know, my dear friend," adds the letter-writer at this point, "was an adjective invented as equivalent to Hog-ian, and at the same time more dignified and euphonious.
--
"Nor had our forefathers any better right to talk about certainty, when pursuing, in blind confidence, the a priori path of axioms, or of the Ram. At innumerable points this path was scarcely as straight as a ram's-horn. The simple truth is, that the Aristotelians erected their castles upon a basis far less reliable than air; for no such things as axioms ever existed or can possibly exist at all. This they must have been very blind, indeed, not to see, or at least to suspect; for, even in their own day, many of their long-admitted ' axioms' had been abandoned: -'ex nihilo nihil fit,' for example, and a 'thing cannot act where it is not,' and 'there cannot be antipodes,' and 'darkness cannot proceed from light.' These and numerous similar propositions formerly accepted, without hesitation, as axioms, or undeniable truths, were, even at the period of which I speak, seen to be altogether untenable: -how absurd in these people, then, to persist in relying upon a basis, as immutable, whose mutability had become so repeatedly manifest!
"But, even through evidence afforded by themselves against themselves, it is easy to convict these a priori reasoners of the grossest unreason -it is easy to show the futility -the impalpability of their axioms in general. I have now lying before me" it will be observed that we still proceed with the letter -"I have now lying before me a book printed about a thousand years ago. Pundit assures me that it is decidedly the cleverest ancient work on its topic, which is 'Logic.' The author, who was much esteemed in his day, was one Miller or Mill; and we find it recorded of him, as a point of some importance, that he rode a mill-horse whom he called Jeremy Bentham: -but let us glance at the volume itself!
"Ah! -'Ability or inability to conceive,' says Mr. Mill very properly, 'is in no case to be received as a criterion of axiomatic truth.' Now, that this is a palpable truism no one in his senses will deny. Not to admit the proposition, is to insinuate a charge of variability in Truth itself, whose very title is a synonym of the Steadfast. If ability to conceive be taken as a criterion of Truth, then a truth to David Hume would very seldom be a truth to Joe; and ninety-nine hundredths of what is undeniable in Heaven would be demonstrable falsity upon Earth. The proposition of Mr. Mill, then, is sustained. I will not grant it to be an axiom; and this merely because I am showing that no axioms exist; but, with a distinction which could not have been cavilled at even by Mr. Mill himself, I am ready to grant that, if an axiom there be, then the proposition of which we speak has the fullest right to be considered an axiom -that no more absolute axiom is -and, consequently, that any subsequent proposition which shall conflict with this one primarily advanced, must be either a falsity in itself -that is to say no axiom -or, if admitted axiomatic, must at once neutralize both itself and its predecessor.
"And now, by the logic of their own propounder, let us proceed to test any one of the axioms propounded. Let us give Mr. Mill the fairest of play. We will bring the point to no ordinary issue. We will select for investigation no common-place axiom -no axiom of what, not the less preposterously because only impliedly, he terms his secondary class -as if a positive truth by definition could be either more or less positively a truth: -we will select, I say, no axiom of an unquestionability so questionable as is to be found in Euclid. We will not talk, for example, about such propositions as that two straight lines cannot enclose a space, or that the whole is greater than any one of its parts. We will afford the logician every advantage. We will come at once to a proposition which he regards as the acme of the unquestionable -as the quintessence of axiomatic undeniability. Here it is: -'Contradictions cannot both be true that is, cannot coexist in nature.' Here Mr. Mill means, for instance, -and I give the most forcible instance conceivable -that a tree must be either a tree or not a tree -that it cannot be at the same time a tree and not a tree: -all which is quite reasonable of itself and will answer remarkably well as an axiom, until we bring it into collation with an axiom insisted upon a few pages before -in other words -words which I have previously employed -until we test it by the logic of its own propounder. 'A tree,' Mr. Mill asserts, 'must be either a tree or not a tree.' Very well: -and now let me ask him, why. To this little query there is but one response: -I defy any man living to invent a second. The sole answer is this: 'Because we find it impossible to conceive that a tree can be anything else than a tree or not a tree.' This, I repeat, is Mr. Mill's sole answer: -he will not pretend to suggest another: -and yet, by his own showing, his answer is clearly no answer at all; for has he not already required us to admit, as an axiom, that ability or inability to conceive is in no case to be taken as a criterion of axiomatic truth? Thus all -absolutely his argumentation is at sea without a rudder. Let it not be urged that an exception from the general rule is to be made, in cases where the 'impossibility to conceive' is so peculiarly great as when we are called upon to conceive a tree both a tree and not a tree. Let no attempt, I say, be made at urging this sotticism; for, in the first place, there are no degrees of 'impossibility,' and thus no one impossible conception can be more peculiarly impossible than another impossible conception: -in the second place, Mr. Mill himself, no doubt after thorough deliberation, has most distinctly, and most rationally, excluded all opportunity for exception, by the emphasis of his proposition, that, in no case, is ability or inability to conceive, to be taken as a criterion of axiomatic truth: -in the third place, even were exceptions admissible at all, it remains to be shown how any exception is admissible here. That a tree can be both a tree and not a tree, is an idea which the angels, or the devils, may entertain, and which no doubt many an earthly Bedlamite, or Transcendentalist, does.
"Now I do not quarrel with these ancients," continues the letter-writer, "so much on account of the transparent frivolity of their logic -which, to be plain, was baseless, worthless and fantastic altogether -as on account of their pompous and infatuate proscription of all other roads to Truth than the two narrow and crooked paths -the one of creeping and the other of crawling -to which, in their ignorant perversity, they have dared to confine the Soul -the Soul which loves nothing so well as to soar in those regions of illimitable intuition which are utterly incognizant of 'path.'
--
Referring to the Newtonian Gravity, Dr. Nichol, the eloquent author of "The Architecture of the Heavens," says: -"In truth we have no reason to suppose this great Law, as now revealed, to be the ultimate or simplest, and therefore the universal and all-comprehensive, form of a great Ordinance. The mode in which its intensity diminishes with the element of distance, has not the aspect of an ultimate principle; which always assumes the simplicity and self-evidence of those axioms which constitute the basis of Geometry."
Now, it is quite true that "ultimate principles," in the common understanding of the words, always assume the simplicity of geometrical axioms -(as for "self-evidence," there is no such thing) -but these principles are clearly not "ultimate;" in other terms what we are in the habit of calling principles are no principles, properly speaking -since there can be but one principle, the Volition of God. We have no right to assume, then, from what we observe in rules that we choose foolishly to name "principles," anything at all in respect to the characteristics of a principle proper. The "ultimate principles" of which Dr. Nichol speaks as having geometrical simplicity, may and do have this geometrical turn, as being part and parcel of a vast geometrical system, and thus a system of simplicity itself -in which, nevertheless, the TRuly ultimate principle is, as we know, the consummation of the complex -that is to say, of the unintelligible -for is it not the Spiritual Capacity of God?
I quoted Dr. Nichol's remark, however, not so much to question its philosophy, as by way of calling attention to the fact that, while all men have admitted some principle as existing behind the Law of Gravity, no attempt has been yet made to point out what this principle in particular is: -if we except, perhaps, occasional fantastic efforts at referring it to Magnetism, or Mesmerism, or Swedenborgianism, or Transcendentalism, or some other equally delicious ism of the same species, and invariably patronized by one and the same species of people. The great mind of Newton, while boldly grasping the Law itself, shrank from the principle of the Law. The more fluent and comprehensive at least, if not the more patient and profound, sagacity of Laplace, had not the courage to attack it. But hesitation on the part of these two astronomers it is, perhaps, not so very difficult to understand. They, as well as all the first class of mathematicians, were mathematicians solely: their intellect, at least, had a firmly-pronounced mathematico-physical tone. What lay not distinctly within the domain of Physics, or of Mathematics, seemed to them either Non-Entity or Shadow. Nevertheless, we may well wonder that Leibnitz, who was a marked exception to the general rule in these respects, and whose mental temperament was a singular admixture of the mathematical with the physico-metaphysical, did not at once investigate and establish the point at issue. Either Newton or Laplace, seeking a principle and discovering none physical, would have rested contentedly in the conclusion that there was absolutely none; but it is almost impossible to fancy, of Leibnitz, that, having exhausted in his search the physical dominions, he would not have stepped at once, boldly and hopefully, amid his old familiar haunts in the kingdom of Metaphysics. Here, indeed, it is clear that he must have adventured in search of the treasure: -that he did not find it after all, was, perhaps, because his fairy guide, Imagination, was not sufficiently well-grown, or well-educated, to direct him aright.
--
Now, the laws of irradiation are known. They are part and parcel of the sphere. They belong to the class of indisputable geometrical properties. We say of them, "they are true -they are evident." To demand why they are true, would be to demand why the axioms are true upon which their demonstration is based. Nothing is demonstrable, strictly speaking; but if anything be, then the properties -the laws in question are demonstrated.
But these laws -what do they declare? Irradiation -how -by what steps does it proceed outwardly from a centre?
--
And if here, for the mere sake of cavilling, it be urged, that although my starting-point is, as I assert, the assumption of absolute Simplicity, yet Simplicity, considered merely in itself, is no axiom; and that only deductions from axioms are indisputable -it is thus that I reply:
Every other science than Logic is the science of certain concrete relations. Arithmetic, for example, is the science of the relations of number -Geometry, of the relations of form -Mathematics in general, of the relations of quantity in general -of whatever can be increased or diminished. Logic, however, is the science of Relation in the abstract -of absolute Relation -of Relation considered solely in itself. An axiom in any particular science other than Logic is, thus, merely a proposition announcing certain concrete relations which seem to be too obvious for dispute -as when we say, for instance, that the whole is greater than its part: -and, thus again, the principle of the Logical axiom -in other words, of an axiom in the abstract is, simply, obviousness of relation. Now, it is clear, not only that what is obvious to one mind may not be obvious to another, but that what is obvious to one mind at one epoch, may be anything but obvious, at another epoch, to the same mind. It is clear, moreover, that what, to-day, is obvious even to the majority of mankind, or to the majority of the best intellects of mankind, may to-morrow be, to either majority, more or less obvious, or in no respect obvious at all. It is seen, then, that the axiomatic principle itself is susceptible of variation, and of course that axioms are susceptible of similar change. Being mutable, the "truths" which grow out of them are necessarily mutable too; or, in other words, are never to be positively depended upon as truths at all -since Truth and Immutability are one.
It will now be readily understood that no axiomatic idea -no idea founded in the fluctuating principle, obviousness of relation -can possibly be so secure -so reliable a basis for any structure erected by the Reason, as that idea -(whatever it is, wherever we can find it, or if it be practicable to find it anywhere) -which is ir relative altogether -which not only presents to the understanding no obviousness of relation, either greater or less, to be considered, but subjects the intellect, not in the slightest degree, to the necessity of even looking at any relation at all. If such an idea be not what we too heedlessly term "an axiom," it is at least preferable, as a Logical basis, to any axiom ever propounded, or to all imaginable axioms combined: -and such, precisely, is the idea with which my deductive process, so thoroughly corroborated by induction, commences. My particle proper is but absolute Irrelation. To sum up what has been advanced: -As a starting point I have taken it for granted, simply, that the Beginning had nothing behind it or before it -that it was a Beginning in fact -that it was a beginning and nothing different from a beginning -in short, that this Beginning was -that which it was. If this be a "mere assumption" then a "mere assumption" let it be.
To conclude this branch of the subject: -I am fully warranted in announcing that the Law which we have been in the habit of calling Gravity exists on account of Matter's having been irradiated, at its origin, atomically, into a limited sphere of Space, from one, individual, unconditional, irrelative, and absolute Particle Proper, by the sole process in which it was possible to satisfy, at the same time, the two conditions, irradiation, and generally-equable distribution throughout the sphere -that is to say, by a force varying in direct proportion with the squares of the distances between the irradiated atoms, respectively, and the Particular centre of Irradiation.
2.03 - THE ENIGMA OF BOLOGNA, #Mysterium Coniunctionis, #Carl Jung, #Psychology
[68] Malvasius goes out of his way to be fair to the author of the epitaph. He calls Agatho very skilled in this science and that;167 indeed he compares him, as being a pre-eminent worshipper of the exceedingly auspicious number Three,168 to Hermes Trismegistus, and calls him Thrice-Greatest, an allusion to the concluding sentence of the Tabula smaragdina.169 He does this because the inscription is divided into three parts,170 to which he devotes a long dissertation. Here he gets into difficulties with the four elements and the four qualities, and, like all the alchemists, flounders about in his attempts to interpret the axiom of Maria.171 His idea of a miscarriage likewise comes within the sphere of alchemy (not to mention Gnosticism),172 for we read in the Tractatus Aristotelis: 173This serpent is impetuous, seeking the issue [death] before birth, wishing to lose the foetus and desiring a miscarriage.174 This refers, of course, to the Mercurial serpent or prima materia, which, the treatise maintains,175 strives to pass quickly through the transformation process and to force the light-seeds of the anima mundi hidden within it into flower.
[69] Of the numerous interpretations made by the commentators I would like to mention one which seems to me worth rescuing from oblivion. This is the view expressed by the two friends of Malvasius (see n. 127), namely that Lucius Agatho was a real person, but that Aelia was a fictitious woman, or perhaps an evil genius in female form or an ungodly spirit, who in the opinion of one of them flies about in the air, and according to the other dwells in the earth and was enclosed and affixed in a Junonian oak; a sylvan sprite, nymph, or hamadryad who, when the oak was cut down and burnt, was obliged to seek another dwelling-place and so was found, as if dead, in this sarcophagus. Thus it was that she was praised, described, and commemorated by the loved and loving Agatho.176
2.0 - THE ANTICHRIST, #Twilight of the Idols, #Friedrich Nietzsche, #Philosophy
should "sin." ... Supreme axiom: "God forgiveth him that repenteth"--in
plain English: _him that submitteth himself to the priest._
2.10 - On Vedic Interpretation, #Evening Talks With Sri Aurobindo, #unset, #Zen
Disciple: In a new geometry there are different postulates and axioms not the same that we find in ordinary geometry.
Disciple: Nowadays all can be included in geometry. There exist imaginary numbers; e.g., the square root of minus 1. It is an imaginary number which the mind cannot conceive and represent in some substance, but yet it indicates some reality.
2.15 - Reality and the Integral Knowledge, #The Life Divine, #Sri Aurobindo, #Integral Yoga
This ego-centric attitude has in recent times been elevated into a valid standard of knowledge; it has been implicitly or explicitly held as an axiom that all truth must be referred to the judgment of the personal mind, reason and experience of every man or else it must be verified or at any rate verifiable by a common or universal experience in order to be valid. But obviously this is a false standard of reality and of knowledge, since this means the sovereignty of the normal or average mind and its limited capacity and experience, the exclusion of what is supernormal or beyond the average intelligence. In its extreme, this claim of the individual to be the judge of everything is an egoistic illusion, a superstition of the physical mind, in the mass a gross and vulgar error. The truth behind it is that each man has to think for himself, know for himself according to his capacity, but his judgment can be valid only on condition that he is ready to learn and open always to a larger knowledge. It is reasoned that to depart from the physical standard and the principle of personal or universal verification will lead to gross delusions and the admission of unverified truth and subjective phantasy into the realm of knowledge. But error and delusion and the introduction of personality and one's own subjectivity into the pursuit of knowledge are always present, and the physical or objective standards and methods do not exclude them. The probability of error is no reason for refusing to attempt discovery, and subjective discovery must be pursued by a subjective method of enquiry, observation and verification; research into the supraphysical must evolve, accept and test an appropriate means and methods other than those by which one examines the constituents of physical objects and the processes of Energy in material Nature.
To refuse to enquire upon any general ground preconceived and a priori is an obscurantism as prejudicial to the extension of knowledge as the religious obscurantism which opposed in Europe the extension of scientific discovery. The greatest inner discoveries, the experience of self-being, the cosmic consciousness, the inner calm of the liberated spirit, the direct effect of mind upon mind, the knowledge of things by consciousness in direct contact with other consciousness or with its objects, most spiritual experiences of any value, cannot be brought before the tribunal of the common mentality which has no experience of these things and takes its own absence or incapacity of experience as a proof of their invalidity or their non-existence. Physical truth or formulas, generalisations, discoveries founded upon physical observation can be so referred, but even there a training of capacity is needed before one can truly understand and judge; it is not every untrained mind that can follow the mathematics of relativity or other difficult scientific truths or judge of the validity either of their result or their process. All reality, all experience must indeed, to be held as true, be capable of verification by a same or similar experience; so, in fact, all men can have a spiritual experience and can follow it out and verify it in themselves, but only when they have acquired the capacity or can follow the inner methods by which that experience and verification are made possible. It is necessary to dwell for a moment on these obvious and elementary truths because the opposite ideas have been sovereign in a recent period of human mentality, - they are now only receding, - and have stood in the way of the development of a vast domain of possible knowledge. It is of supreme importance for the human spirit to be free to sound the depths of inner or subliminal reality, of spiritual and of what is still superconscient reality, and not to immure itself in the physical mind and its narrow domain of objective external solidities; for in that way alone can there come liberation from the Ignorance in which our mentality dwells and a release into a complete consciousness, a true and integral self-realisation and self-knowledge.
2.21 - The Order of the Worlds, #The Life Divine, #Sri Aurobindo, #Integral Yoga
Three questions then arise, interrelated or interdependent: - whether there is any evidence or any true intimation of the existence of such other worlds; whether, if they exist, they are of the nature we have indicated, arising or descending in the order and within the rationale of a hierarchical series between Matter and Spirit; if that is their scale of being, are they otherwise quite independent and unconnected, or is there a relation and interaction of the higher worlds on the world of Matter? It is a fact that mankind almost from the beginning of its existence or so far back as history or tradition can go, has believed in the existence of other worlds and in the possibility of communication between their powers and beings and the human race. In the last rationalistic period of human thought from which we are emerging, this belief has been swept aside as an age-long superstition; all evidence or intimations of its truth have been rejected a priori as fundamentally false and undeserving of inquiry because incompatible with the axiomatic truth that only Matter and the material world and its experiences are real; all other experience purporting to be real must be either a hallucination or an imposture or a subjective result of superstitious credulity and imagination or else, if a fact, then other than what it purported to be and explicable by a physical cause: no evidence could be accepted of such a fact unless it is objective and physical in its character; even if the fact be very apparently supraphysical, it cannot be accepted as such unless it is totally unexplainable by any other imaginable hypothesis or conceivable conjecture.
It should be evident that this demand for physically valid proof of a supraphysical fact is irrational and illogical; it is an irrelevant attitude of the physical mind which assumes that only the objective and physical is fundamentally real and puts aside all else as merely subjective. A supraphysical fact may impinge on the physical world and produce physical results; it may even produce an effect on our physical senses and become manifest to them, but that cannot be its invariable action and most normal character or process. Ordinarily, it must produce a direct effect or a tangible impression on our mind and our life-being, which are the parts of us that are of the same order as itself, and can only indirectly and through them, if at all, influence the physical world and physical life. If it objectivises itself, it must be to a subtler sense in us and only derivatively to the outward physical sense. This derivative objectivisation is certainly possible; if there is an association of the action of the subtle body and its sense-organisation with the action of the material body and its physical organs, then the supraphysical can become outwardly sensible to us. This is what happens, for example, with the faculty called second sight; it is the process of all those psychic phenomena which seem to be seen and heard by the outer senses and are not sensed inwardly through representative or interpretative or symbolic images which bear the stamp of an inner experience or have an evident character of formations in a subtle substance. There can, then, be various kinds of evidence of the existence of other planes of being and communication with them; objectivisation to the outer sense, subtle-sense contacts, mind contacts, life contacts, contacts through the subliminal in special states of consciousness exceeding our ordinary range.
3.01 - The Mercurial Fountain, #The Practice of Psycho therapy, #Carl Jung, #Psychology
the number 4 to 3 to 2 to 1 is the axiom of Maria, which runs in various
forms through the whole of alchemy like a leitmotiv. If we set aside the
--
Fundamentally it is the same theme, the same axioma Mariae, telling
how Rosencreutz is transformed out of his former unenlightened condition
3.03 - The Naked Truth, #The Practice of Psycho therapy, #Carl Jung, #Psychology
the inscription. In accordance with the axiom of Maria, the elementary
quaternity has become the active triad, and this will lead to the coniunctio
3.04 - LUNA, #Mysterium Coniunctionis, #Carl Jung, #Psychology
[180] This text abounds in obscurities. In the preceding section Ventura discusses the unity of the lapis and the medicina, mentioning the axioms Introduce nothing alien and Nothing from outside300 with quotations from Geber, the Turba, and the Thesaurus thesaurorum of Arnaldus.301 Then he turns to the superfluities to be removed.302 The lapis, he says, is by nature most pure. It is therefore sufficiently purified when it is led out of its proper house and enclosed in an alien house. The text continues:
In the proper house the flying bird is begotten, and in the alien house303 the tincturing stone. The two flying birds304 hop on to the tables and heads of the kings,305 because both, the feathered bird and the plucked,306 have given [us] this visible art307 and cannot relinquish the society of men.308 The father309 of [the art] urges the indolent to work, its mother310 nourishes the sons who are exhausted by their labours, and quickens and adorns their weary limbs.
3.05 - SAL, #Mysterium Coniunctionis, #Carl Jung, #Psychology
[235] Owing to the theory of correspondentia, regarded as axiomatic in the Middle Ages, the principles of each of the four worlds the intelligible or divine, the heavenly, the earthly, and the infernal379corresponded to each other. Usually, however, there was a division into three worlds to correspond with the Trinity: heaven, earth, hell.380 Triads were also known in alchemy. From the time of Paracelsus the most important triad was Sulphur-Mercurius-Sal, which was held to correspond with the Trinity. Georg von Welling, the plagiarist of Johann Rudolf Glauber, still thought in 1735 that his triad of fire, sun, and salt381 was in its root entirely one thing.382 The use of the Trinity formula in alchemy is so common that further documentation is unnecessary. A subtle feature of the Sulphur-Mercurius-Sal formula is that the central figure, Mercurius, is by nature androgynous and thus partakes both of the masculine red sulphur and of the lunar salt.383 His equivalent in the celestial realm is the planetary pair Sol and Luna, and in the intelligible realm Christ in his mystical androgyny, the man encompassed by the woman,384 i.e., sponsus and sponsa (Ecclesia). Like the Trinity, the alchemical triunity is a quaternity in disguise owing to the duplicity of the central figure: Mercurius is not only split into a masculine and a feminine half, but is the poisonous dragon and at the same time the heavenly lapis. This makes it clear that the dragon is analogous to the devil and the lapis to Christ, in accordance with the ecclesiastical view of the devil as an autonomous counterpart of Christ. Furthermore, not only the dragon but the negative aspect of sulphur, namely sulphur comburens, is identical with the devil, as Glauber says: Verily, sulphur is the true black devil of hell, who can be conquered by no element save by salt alone.385 Salt by contrast is a light substance, similar to the lapis, as we shall see.
[236] From all this we get the following schema:
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[269] Psychologically the vision of Ezekiel is a symbol of the self consisting of four individual creatures and wheels, i.e., of different functions. Three of the faces are theriomorphic and only one anthropomorphic, which presumably means that only one function has reached the human level, whereas the others are still in an unconscious or animal state. The problem of three and four (trinity and quaternity) plays a great role in alchemy as the axiom of Maria497 and, like the vision of Ezekiel, is concerned with the God-image. The symbols of the self are as a rule symbols of totality, but this is only occasionally true of God-images. In the former the circle and the quaternity predominate, in the latter the circle and the trinity and this, moreover, only in the case of abstract representations, which are not the only ones to occur.
[270] These hints may throw a little light on the strange idea of the serpent-chariot. It is a symbol of the arcane substance and the quintessence, of the aether that contains all four elements, and at the same time a God-image or, to be more accurate, an image of the anima mundi. This is indicated by the Mercurial serpent, which in its turn was interpreted by the alchemists as the spirit of life that was in the wheels (DV).498 We should also mention that according to Ezekiel 1 : 18 the inter-revolving wheels were full of eyes round about. The old illustrators therefore produced something like an astrolabe in their attempts to depict the vision. The notion of wheels is naturally connected with movement in all directions, for the eyes of the Lord run to and fro through the whole earth (Zech. 4 : 10). It is said of the horses, too, that they walk to and fro through the earth (Zech. 6 : 7). Eyes are round and in common speech are likened to cart-wheels. They also seem to be a typical symbol for what I have called the multiple luminosities of the unconscious. By this I mean the seeming possibility that complexes possess a kind of consciousness, a luminosity of their own, which, I conjecture, expresses itself in the symbol of the soul-spark, multiple eyes (polyophthalmia), and the starry heaven.499
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[278] We have seen earlier that the Erythraean Sea is a mysterious place, but here we meet with some noteworthy details. To begin with, our author reaches this sea just when he has completed the journey through the three continents and is about to enter the critical fourth region. We know from the axiom of Maria and from Faust the crucial importance of that seemingly innocent question at the beginning of the Timaeus:
SOCRATES: One, two, three but where, my dear Timaeus, is the fourth of those guests of yesterday who were to entertain me today?
3.05 - The Conjunction, #The Practice of Psycho therapy, #Carl Jung, #Psychology
ut duo qui fuerant, unum quasi corpore fiant. Thus the axiom of Maria
is fulfilled. In this union the Holy Ghost disappears as well, but to make up
31.04 - Sri Ramakrishna, #Collected Works of Nolini Kanta Gupta - Vol 07, #Nolini Kanta Gupta, #Integral Yoga
The first word about spirituality is this that it is a clear realisation, an unveiled intuition of truth, God-attainment. It is not a mere concept or a doctrine or an intellectual conclusion, but a living experience, a concrete realisation. Philosophers and scholars are at pains to prove the existence of God and the Self with the help of reason, argumentation and syllogism. But it is ridiculous to try that way. You stand before me face to face. Yet should I try to prove your presence? Or to prove my own? Truth is self-existent. It is rear since it exists. The existence of God, the Self, is the truth of all truths and is axiomatic. It is a matter of experience, insight and intuition.
The spiritual world is as real as this physical world - even more real, Ramakrishna says. It is a different sphere of consciousness. One has to come up to this level to know of it and one has to settle here for good, leaving behind one's earthly dwelling. Not merely the field of action in life and not just one part of the being but the whole life and being have to be consecrated to the attainment of that only Goal. To know the Truth, the Self or God, one has first to realise them, one has to merge in them, one has to become these - as the Upanishad says, "Verily the knower of the Brahman becomes the Brahman Itself." The essential truth of spirituality consists not even in "doing" something but in this 'becoming'. Man manifests his inner soul through his actions. His external conduct reflects his inner becoming. In the words of Sri Ramakrishna, a man breathes out what he eats.
31.08 - The Unity of India, #Collected Works of Nolini Kanta Gupta - Vol 07, #Nolini Kanta Gupta, #Integral Yoga
The collective union of Europe, its living truth, is not the unity that exists among the countries or nations of Europe. Of course an attempt is being made to bring about a sort of oneness through the length and breadth of Europe; but that oneness is still a conception, still an ideal, not even a living ideal yet. The inner being has not as yet become conscious and alert about its own personality. But in the case of India the awakening of her inner being has stood by far the first. India's personality of collective existence is indeed a living truth. The unity of the whole of India is, as it were, axiomatic and God-given. The oneness that is trying to become awake and conscious in India is not the unity and the personality of the whole of India, but rather, the oneness of the different personalities of the states of India. Therefore we notice contrary ways of life-manifestation in India and in Europe. In Europe the collective personality has awakened and the sense of unity has been established first within the limited areas of different countries. Having established these two things Europe is progressing towards the larger unity in its totality. The trend of Europe is the gradual ascension to the total consciousness of the inner being from the consciousness of the external physical parts. But in India what happened is the gradual descent to the external physical parts from the total consciousness of the inner subtle being. India first discovered her own inner being. This being of India is ever awake, so in India the manifestation of that being and the descent of her power are making the different states gradually become conscious and discover their own individualities. Perhaps Europe has had to start from a political union or oneness in order gradually and finally to attain to the unity-of the soul, and achieve her own individuality. India has long acquired the unity of the soul and this self-conscious individuality is the last word of India's oneness.
***
3.10 - The New Birth, #The Practice of Psycho therapy, #Carl Jung, #Psychology
shown that the axiom of Maria consists of 4, 3, 2, 1; the sum of these
numbers is 10, which stands for unity on a higher level. The unarius
--
heads; in the other, a single snake. This is an obvious allusion to the axiom
of Maria and the old dilemma of 3 and 4, and also to the mystery of the
3.2.09 - The Teachings of Some Modern Indian Yogis, #Letters On Yoga II, #Sri Aurobindo, #Integral Yoga
As for my writings, I dont know if there is any that would clear up the difficulty. You would find mostly the statement of the Vedantic experience, for it is that through which I passed and, though now I have passed to something beyond, it seems to me the most thorough-going and radical preparation for whatever is Beyond, though I do not say that it is indispensable to pass through it. But whatever the solution, it seems to me that the Vedantin is right in insisting that one must, to arrive at it, admit the two facts, the prevalence of evil and suffering here and the experience of that which is free from these thingsand it is only by the progressive experience that one can get a solutionwhe ther through reconciliation, a conquering descent or an escape. If we start from the basis taken as an axiom that the prevalence of suffering and evil in the present and in the hard, outward fact of things, disproves of itself all that has been experienced by sages and mystics of the other side, the realisable Divine, then no solution seems possible.
***
3-5 Full Circle, #unset, #Arthur C Clarke, #Fiction
The mutual exclusiveness of traditional scientific categories--such as the kingdoms of man, of animals, of plants, or of atoms--is of course recognized and honored. (Thought itself requires that it be.) But Unified Science affirms the strategic prepotency of inclusiveness, demanded by understanding of the System-hierarchy. The basic unit of Unisci theory is the system, not one or another of its parts. (Those are the main thought units of unorganized one-field specialists.) Unified Science affirms as axiomatic that every natural system except the lowest includes systems lower than itself in the System-hierarchy. Inclusiveness is implicit in Unisci's fifth paradigm: this axiom afhrms the preponderance of positive coaction. And its acceptance as a strategic attitude involves our thought-parts in positive, moral coaction over-all.
IV. The lowest members of the System-hierarchy are certainly
36.07 - An Introduction To The Vedas, #Collected Works of Nolini Kanta Gupta - Vol 08, #unset, #Zen
The angle of vision from which the Europeans look at the Vedas has to be traced to its starting-point in the modern theory of evolution. Europe has been a victim to this theory. It has coloured the entire outlook of Europe. Evolution means gradual progression. Man and human society are undergoing a change for the better. In antiquity man was just a little remote from the animal. His intelligence gradually developed. His conduct has become polished. Thus he has grown into what he is today. The more we cast our glance into the past, the more shall we come across man's original, primitive and immature nature. As the Vedas owe their origin to a hoary past, it is a axiomatic that there can be no solid philosophical truth and spiritual experience in them. It is vain to seek for something in the Vedas that can satisfy the modern scientific mind. Hence any such attempt will end in utter failure.
In modern times those very scientists are confronted with an anomalous phenomenon supported by irrefutable evidence. Many scientific theories are going to be upset by the new discoveries. Archaeological excavation has been furnishing more and more evidence of ancient culture and education. These discoveries go to prove that the ancients were not immature in the least in their mental faculties, education and culture. On the contrary, we find in them signs of superior qualities and endeavours. Strangely enough, these archaeological finds are found in the places which were so long considered by us to be inhabited by barbarians. The wonderful artistic works and remnants of scientific achievements that we meet with among the discoveries made in the dense forests of America, in the archipelago of the Pacific, beneath the desert of Central Asia have hardly any parallel in this much vaunted scientific age. The Egyptians and the Babylonians have created a tradition. But the hoary past of their source is just being revealed. Greece was considered to be the mainspring of European culture and civilisation. But that a still more civilised race had inhabited the neighbouring island of Crete can by no means be denied now. The older civilisations of Atlantis, Sumeria, Akad, Aztec, Maya and Toltec no longer appear to be mere poetical imaginations. We are wonder-struck by such amazing prehistoric achievements. We can hardly assert that we possess a culture and civilisation superior to theirs. According to the Biblical statement the world came into existence only four thousand years ago. This statement had left its stamp unawares on the mind of the European savants. At present, not to speak of the age of the world or of the advent of man, the age of civilised man can itself be put at about a lakh of years.
--
If we consider man to be a sufficiently old creature on earth and that his evolution runs in a spiral movement, then the statement that the Aryans of the Vedic age were not highly advanced cannot be regarded as an axiomatic truth. Of course, there is no hard and fast rule that the education, culture and realisation of the Vedic age should have been similar to those of modern times. But their widely differing outlook and activities need not be inferior to ours. True, Valmiki and Rabindranath are not peers of the same grain. On that account we cannot definitely assign a higher status to Rabindranath. To consider the Vedic seers inferior to the modern scientists simply because they do not resemble there is nothing but a stark superstition.
As a matter of fact, here lies the greatest folly of the moderns. We fail to arrive at the angle of vision of the ancients. We fail to comprehend that there was a time when this ancient culture was as living as that of today. As the Europeans used to take us for rustics because of our bare body and eating with hands and. such other habits, even so we conclude from the words go (cow), asva (horse), somarasa (wine) and devas (gods) etc., that the Vedic seers were no better than primitives. For in our conception the men of knowledge speak of no such material subjects. They would rather deal with metaphysical discourses and scientific researches. We want to measure the ferment in the brain of the ancients by that of our own. We forget the very fact that they had a culture of their own which need not tally with ours. In fact, the truth attained by the ancients was not the outcome of an intellect given to mundane things. Rather the criticism may be applied to our present-day intellect.
4.02 - The Psychology of the Child Archetype, #The Archetypes and the Collective Unconscious, #Carl Jung, #Psychology
is forgotten. Now it is an axiom of psychology that when a part
of the psyche is split off from consciousness it is only apparently
5.01 - ADAM AS THE ARCANE SUBSTANCE, #Mysterium Coniunctionis, #Carl Jung, #Psychology
[552] We must now turn to the question of why it was that Adam should have been selected as a symbol for the prima materia or transformative substance. This was probably due, in the first place, to the fact that he was made out of clay, the ubiquitous materia vilis that was axiomatically regarded as the prima materia and for that very reason was so tantalizingly difficult to find, although it was before all eyes. It was a piece of the original chaos, of the massa confusa, not yet differentiated but capable of differentiation; something, therefore, like shapeless, embryonic tissue. Everything could be made out of it.17 For us the essential feature of the prima materia is that it was defined as the massa confusa and chaos, referring to the original state of hostility between the elements, the disorder which the artifex gradually reduced to order by his operations. Corresponding to the four elements there were four stages of the process (tetrameria), marked by four colours, by means of which the originally chaotic arcane substance finally attained to unity, to the One, the lapis, which at the same time was an homunculus.18 In this way the Philosopher repeated Gods work of creation described in Genesis 1. No wonder, therefore, that he called his prima materia Adam and asserted that it, like him, consisted or was made out of the four elements. For out of the four elements were created our Father Adam and his children, says the Turba.19 And Gabir ibn Hayyan (Jabir)20 says in his Book of Balances:
The Pentateuch says, regarding the creation of the first being, that his body was composed of four things, which thereafter were transmitted by heredity: the warm, the cold, the moist, and the dry. He was in fact composed of earth and water, a body and a soul. Dryness came to him from the earth, moisture from the water, heat from the spirit, and cold from the soul.21
5.03 - ADAM AS THE FIRST ADEPT, #Mysterium Coniunctionis, #Carl Jung, #Psychology
[574] The series of eight incarnations of the true prophet is distinguished by the special position of the eighth, namely Christ. The eighth prophet is not merely the last in the series; he corresponds to the first and is at the same time the fulfilment of the seven, and signifies the entry into a new order. I have shown in Psychology and Alchemy (pars. 200ff.), with the help of a modern dream, that whereas the seven form an uninterrupted series, the step to the eighth involves hesitation or uncertainty and is a repetition of the same phenomenon that occurs with three and four (the axiom of Maria). It is very remarkable that we meet it again in the Taoist series of eight immortals (hsien-yn): the seven are great sages or saints who dwell in heaven or on the earth, but the eighth is a girl who sweeps up the fallen flowers at the southern gate of heaven.114 The parallel to this is Grimms tale of the seven ravens: there the seven brothers have one sister.115 One is reminded in this connection of Sophia, of whom Irenaeus says: This mother they also call the Ogdoad, Sophia, Terra, Jerusalem, Holy Spirit, and, with a masculine reference, Lord.116 She is below and outside the Pleroma. The same thought occurs in connection with the seven planets in Celsuss description of the diagram of the Ophites, attacked by Origen.117 This diagram is what I would call a mandalaan ordering pattern or pattern of order which is either consciously devised or appears spontaneously as a product of unconscious processes.118 The description Origen gives of the diagram is unfortunately not particularly clear, but at least we can make out that it consisted of ten circles, presumably concentric, since he speaks of a circumference and a centre.119 The outermost circle was labelled Leviathan and the innermost Behemoth, the two apparently coinciding, for Leviathan was the name for the centre as well as the circumference.120 At the same time, the impious diagram said that the Leviathan . . . is the soul that has permeated the universe.121
[575] Origen had got hold of a diagram like the one used by Celsus and discovered in it the names of the seven angels Celsus alludes to. The prince of these angels was called the accursed God, and they themselves were called sometimes gods of light and sometimes archons. The accursed God refers to the Judaeo-Christian world-creator, as Origen duly notes. Yahweh appears here obviously as the prince and father of the seven archons.122 The first of them had a lions form and was named Michael; the second was a bull and was named Suriel, the bull-formed; the third, Raphael, had the form of a snake; the fourth, named Gabriel, the form of an eagle; the fifth, Thauthabaoth, the form of a bear; the sixth, Erataoth, the form of a dog; and the seventh had the form of an ass and was called Onol or Taphabaoth or Thar thataoth.123
5.06 - THE TRANSFORMATION, #Mysterium Coniunctionis, #Carl Jung, #Psychology
[619] In the Cabalistic view Adam Kadmon is not merely the universal soul or, psychologically, the self, but is himself the process of transformation, its division into three or four parts (trimeria or tetrameria). The alchemical formula for this is the axiom of Maria: One becomes two, two becomes three, and out of the Third comes the One as the Fourth.218 The treatise of Rabbi Abraham Cohen Irira (Hacohen Herrera) says: Adam Kadmon proceeded from the Simple and the One, and to that extent he is Unity; but he also descended and fell into his own nature, and to that extent he is Two. And again he will return to the One, which he has in him, and to the Highest; and to that extent he is Three and Four.219 This speculation refers to the essential Name, the Tetragrammaton, which is the four letters of Gods name, three different, and the fourth a repetition of the second.220 In the Hebrew word YHVH (written without vowels), he is feminine and is assigned as a wife to yod221 and to vau. As a result yod222 and vau223 are masculine, and the feminine he, though doubled, is identical and therefore a single unit. To that extent the essential Name is a triad. But since he is doubled, the Name is also a tetrad or quaternity224a perplexity which coincides most strangely with the axiom of Maria. On the other hand the Tetragrammaton consists of a double marriage and thus agrees in an equally remarkable manner with our Adam diagrams. The doubling of the feminine he is archetypal,225 since the marriage quaternio presupposes both the difference and the identity of the feminine figures. This is true also of the two masculine figures, as we have seen, though here their difference usually predominatesnot surprisingly, as these things are mostly products of the masculine imagination. Consequently the masculine figure coincides with mans consciousness, where differences are practically absolute. Though the feminine figure is doubled it is so little differentiated that it appears identical. This double yet identical figure corresponds exactly to the anima, who, owing to her usually unconscious state, bears all the marks of non-differentiation.
[620] If we apply these considerations to the alchemical schema, we shall be able to modify it in a way that was not possible with the psychological one. We thus arrive at a formula which reduces both to the same denominator:
5 - The Phenomenology of the Spirit in Fairytales, #The Archetypes and the Collective Unconscious, #Carl Jung, #Psychology
problem is known as the axiom of Maria and runs all through
alchemical philosophy for more than a thousand years, finally
--
other three, it would make a whole. The enigmatic axiom of
Maria runs: ". . . from the third comes the one as the fourth"
--
the theory of psychological functions, but also of the axiom of
Maria Prophetissa, which plays a considerable role in alchemy.
6.01 - THE ALCHEMICAL VIEW OF THE UNION OF OPPOSITES, #Mysterium Coniunctionis, #Carl Jung, #Psychology
[656] The synthesis of the incorruptible One or quintessence follows the axiom of Maria, the earth representing the fourth. The separation of the hostile elements corresponds to the initial state of chaos and darkness. From the successive unions arise an active principle (sulphur) and a passive (salt), as well as a mediating, ambivalent principle, Mercurius. This classical alchemical trinity then produces the relationship of male to female as the supreme and essential opposition. Fire comes at the beginning and is acted on by nothing, and earth at the end acts on nothing. Between fire and earth there is no interaction; hence the four elements do not constitute a circle, i.e., a totality. This is produced only by the synthesis of male and female. Thus the square at the beginning corresponds to the quaternio of elements united in the quinta essentia at the endquadrangle will answer to quadrangle.
[657] The alchemical description of the beginning corresponds psychologically to a primitive consciousness which is constantly liable to break up into individual affective processesto fall apart, as it were, in four directions. As the four elements represent the whole physical world, their falling apart means dissolution into the constituents of the world, that is, into a purely inorganic and hence unconscious state. Conversely, the combination of the elements and the final synthesis of male and female is an achievement of the art and a product of conscious endeavour. The result of the synthesis was consequently conceived by the adept as self-knowledge,18 which, like the knowledge of God, is needed for the preparation of the Philosophers Stone.19 Piety is needed for the work, and this is nothing but knowledge of oneself.20 This thought occurs not only in late alchemy but also in Greek tradition, as in the Alexandrian treatise of Krates (transmitted by the Arabs), where it is said that a perfect knowledge of the soul enables the adept to understand the many different names which the Philosophers have given to the arcane substance.21 The Liber quartorum emphasizes that there must be self-observation in the work as well as of events in due time.22 It is evident from this that the chemical process of the coniunctio was at the same time a psychic synthesis. Sometimes it seems as if self-knowledge brought about the union, sometimes as if the chemical process were the efficient cause. The latter alternative is decidedly the more frequent: the coniunctio takes place in the retort23 or, more indefinitely, in the natural vessel or matrix.24 The vessel is also called the grave, and the union a shared death.25 This state is named the eclipse of the sun.26
6.0 - Conscious, Unconscious, and Individuation, #The Archetypes and the Collective Unconscious, #Carl Jung, #Psychology
legs (The axiom of Maria! Cf. Psychology and Alchemy, pars. 209L)
37 Hist, nat., Lib. XXXIII, cap. vii.
--
we have here stumbled again on the axiom of Maria, that pe-
culiar dilemma of three and four, 66 which I have discussed
--
16 An instance of the axiom of Maria. Other well-known examples are Horus
and his 4 (or 3 -f- 1) sons, the 4 symbolical figures in Ezekiel, the 4 evangelists
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Western alchemical tradition as the axiom of Maria. It also plays
a not inconsiderable role in dream symbolism. 4
--
role in alchemy, as the axiom of Maria. 32
30 The Secret of the Golden Flower (1962), p. 22.
--
Maria, axiom of, 234, 237, 245, 300/2,
310,346/2,360,378
7 - Yoga of Sri Aurobindo, #unset, #Arthur C Clarke, #Fiction
God and creation. Your postulates or axioms themselves
are vitiated.
APPENDIX I - Curriculum of A. A., #Liber ABA, #Aleister Crowley, #Philosophy
The object of this course of reading is to familiarize the student with all that has been said by the Great Masters in every time and country. He should make a critical examination of them; not so much with the idea of discovering where truth lies, for he cannot do this except by virtue of his own spiritual experience, but rather to discover the essential harmony in those varied works. He should be on his guard against partisanship with a favourite author. He should familiarize himself thoroughly with the method of mental equilibrium, endeavouring to contradict any statement soever, although it may be apparently axiomatic.
The general object of this course, besides that already stated, is to assure sound education in occult matters, so that when spiritual illumination comes it may find a well-built temple. Where the mind is strongly biased towards any special theory, the result of an illumination is often to inflame that portion of the mind which is thus overdeveloped, with the result that the aspirant, instead of becoming an Adept, becomes a bigot and fanatic.
Avatars of the Tortoise, #unset, #Arthur C Clarke, #Fiction
more. In the axiom "the part is less than the whole" he does not perceive two
terms and the relation "less than"; he perceives three ("part," "less than,"
BOOK II. -- PART I. ANTHROPOGENESIS., #The Secret Doctrine, #H P Blavatsky, #Theosophy
become almost an axiom with the men of science, that those who wrote in antiquity upon various
sacred Dragons and Serpents either were superstitious and credulous people, or were bent upon
BOOK II. -- PART III. ADDENDA. SCIENCE AND THE SECRET DOCTRINE CONTRASTED, #The Secret Doctrine, #H P Blavatsky, #Theosophy
the rising Sun, if the following esoteric axioms were admitted: (a) the enormous antiquity (and the
existence) of our planetary chain; (b) the actuality of the Seven Rounds; (c) the separation of human
--
spirit," to complete the Kabalistic axiom. The more so, since on page 82 of the same work we read the
following admission: . . . "Produced in the way of spontaneous generation . . . it is by the aid of intense
--
as coming from such high scientific quarters, is an eligible candidate for axiomatic truth and law, a
theory people are in honour bound to accept, if they would be on a right level with modern Science.
--
Finally Virchow's opinion of the giant tombs of Germany is now accepted as an axiom: -- "The tombs
alone are gigantic, and not the bones they contain" -- says that German biologist; and archaeology has
BOOK II. -- PART II. THE ARCHAIC SYMBOLISM OF THE WORLD-RELIGIONS, #The Secret Doctrine, #H P Blavatsky, #Theosophy
the same, only a still more mystic, reassertion of the Kabalistic axiom, "Deus est Demon inversus"; the
word "demon," however, as in the case of Socrates, and in the spirit of the meaning given to it by the
--
Deus enim et circulus est, says Pherecydes, in his hymn to Jupiter. It was a Hermetic axiom, and
Pythagoras prescribed such a circular prostration and posture during the hours of contemplation. "The
--
discern them at first. The incommunicable axiom is kabalistically contained therein, and
this is what is called the magic arcanum by the masters.' " ("Isis Unveiled.")
BOOK I. -- PART I. COSMIC EVOLUTION, #The Secret Doctrine, #H P Blavatsky, #Theosophy
sufficiently graphic to need no further elucidation. Thus, then, the first fundamental axiom of the
Secret Doctrine is this metaphysical ONE ABSOLUTE -- BE-NESS -- symbolised by finite
--
Samvriti -- is a philosophical axiom.*
------STANZA I. -- Continued.
--
of the last few centuries have come to the same ideas and conclusions that were taught as axiomatic
truths in the secrecy of the Adyta dozens of
--
Kabalistic axiom: 'A stone becomes a plant; a plant, a beast; a beast, a man; a man, a spirit; and the
spirit, a god.'" (Vol. I., p. 301, note.)
--
long ago. At the same time it is now a theory that has lately become an axiom, that the phenomenon of
polar lights is accompanied by, and productive of, strong sounds, like whistling, hissing, and cracking.
BOOK I. -- PART III. SCIENCE AND THE SECRET DOCTRINE CONTRASTED, #The Secret Doctrine, #H P Blavatsky, #Theosophy
the speculations of modern science. Archaic axioms must be placed side by side with modern
hypotheses and comparison left to the sagacious reader.
--
consists, in its fullness, of atoms and vacuity. Even leaving aside the axiom -- now absolutely
demonstrated by telescope and microscope -- taught by the ancients, that nature abhors
--
the principle of the immutable law of analogy -- "as it is above, so it is below" -- that other axiom, that
http://www.theosociety.org/pasadena/sd/sd1-3-09.htm (2 von 15) [06.05.2003 03:33:33]
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hitherto esoteric axiom, that nothing -- whether in the spiritual, psychic, or physical realm of being -could come into existence out of nothing. There is no cause in the manifested universe without its
adequate effects, whether in space or time; nor can there be an effect without its primal cause, which
--
On this point Materialistic Science is inexorable. To support its position, it upsets its own axiomatic
law of uniformity in the laws of nature, that of continuity, and all the logical sequence of analogies in
--
he fearlessly proclaims as a fundamental axiom that Science has not made itself acquainted, so far,
with real simple elements. For Prof. Crookes tells his audience:
--
suspected that he was repeating an axiom of the Occultists. It is quite true also, as Burmeister (quoted
in "Force and matter") remarks, that
--
Eastern axiom. Therefore, the sidereal "prophecies" of the zodiac, as they are called by Christian
mystics, never point to any one particular event, however solemn and sacred it may be for some one
BOOK I. -- PART II. THE EVOLUTION OF SYMBOLISM IN ITS APPROXIMATE ORDER, #The Secret Doctrine, #H P Blavatsky, #Theosophy
chemical and physical sciences progress, does this occult axiom find its corroboration in the world of
knowledge: the scientific hypothesis, that even the simplest elements of matter are identical in nature
BS 1 - Introduction to the Idea of God, #unset, #Arthur C Clarke, #Fiction
You may know, you may not, that Im an admirer of Nietzsche. Nietzsche was a devastating critic of dogmatic ChristianityChristianity as it was instantiated in institutions. Although, he is a very paradoxical thinker. One of the things Nietzsche said was that he didnt believe the scientific revolution would have ever got off the ground if it hadnt been for Christianityand, more specifically, for Catholicism. He believed that, over the course of a thousand years, the European mind had to train itself to interpret everything that was known within a single coherent frameworkcoherent if you accept the initial axioms. Nietzsche believed that the Catholicization of the phenomena of life and history produced the kind of mind that was then capable of transcending its dogmatic foundations, and concentrating on something else. In this particular case, it happened to be the natural world.
Nietzsche believed that Christianity died of its own hand, and that it spent a very long time trying to attune people to the necessity of the truth, absent the corruption, and all that thats always part of any human endeavour. The truth the spirit of truth that was developed by Christianity turned on the roots of Christianity. Everyone woke up and said, or thought, something like, how is it that we came to believe any of this? Its like waking up one day and noting that you really dont know why you put a Christmas tree up, but youve been doing it for a long time, and thats what people do. There are reasons Christmas trees came about. The ritual lasts long after the reasons have been forgotten.
--
Its really not a good thing, because it manifests itself not only in individual psychopathologies, but also in social psychopathologies. Thats this proclivity of people to get tangled up in ideologies, and I really do think of them as crippled religions. Thats the right way to think about them. Theyre like religion thats missing an arm and a leg, but can still hobble along. It provides a certain amount of security and group identity, but its warped and twisted and demented and bent, and its a parasite on something underlying thats rich and true. Thats how it looks to me, anyways. I think its very important that we sort out this problem. I think that there isnt anything more important that needs to be done than that. Ive thought that for a long, long timeprobably since the early 80s, when I started looking at the role that belief systems played in regulating psychological and social health. You can tell that they do that because of how upset people get if you challenge their belief systems. Why the hell do they care, exactly? What difference does it make if all of your ideological axioms are 100 percent correct?
People get unbelievable upset when you poke them in the axioms, so to speak, and it is not by any stretch of the imagination obvious why. Theres a fundamental truth that theyre standing on. Its like theyre on a raft in the middle of the ocean. Youre starting to pull out the logs, and theyre afraid theyre going to fall in and drown. Drown in what? What are the logs protecting them from? Why are they so afraid to move beyond the confines of the ideological system? These are not obvious things. Ive been trying to puzzle that out for a very long time. Ive done some lectures about that that are on YouTube. Most of you know that. Some of what Im going to talk about in this series youll have heard, if youve listened to the YouTube videos, but Im trying to hit it from different angles.
Nietzsche's idea was that human beings were going to have to create their own values. He understood that we had bodies, motivations, and emotions. He was a romantic thinker, in some sense, but way ahead of his time. He knew that our capacity to think wasnt some free-floating soul, but was embedded in our physiology, constrained by our emotions, shaped by our motivations, and shaped by our body. He understood that. But he still believed that the only possible way out of the problem would be for human beings themselves to become something akin to God, and to create their own values. He talked about the person who created their own values as the Overman, or the Superman. That was one part of the Nietzschean philosophy that the Nazis took out of context and used to fuel their superior man ideology. We know what happened with that. That didnt seem to turn out very well. Thats for sure.
--
Were biological creatures. When we formulated our strange capacity to abstract and use language, we still had all those underlying systems that were there when we were only animals. We have to use those systems that are there. Part of the emotional and motivational architecture of our thinking, part of the reason why we can demonize our enemies who upset our axioms, is because we perceive them as if theyre carnivorous predators. We do it with the same system. Thats chaos itself, the thing that always threatens us the snakes that came to the trees when we lived in them, like 60 million years ago. Its the same damned systems.
The Marduk story is partly the story of using attention and language to confront those things that most threaten us. Some of those things are real world threats, but some of them are psychological threats, which are just as profound but far more abstract. But we use the same system to represent them. Thats why you freeze, if you're frightened. Youre a prey animal. Youre like a rabbit, and youve seen something that's going to eat you. You freeze, and youre paralyzed. Youre turned to stone, which is what you do when you see a Medusa with a head full of snakes. You turn to stone. Youre paralyzed, and the reason you do that is because youre using the predator detector system to protect yourself. Your heart rate goes way up, and you get ready to move.
ENNEAD 05.05 - That Intelligible Entities Are Not External to the Intelligence of the Good., #Plotinus - Complete Works Vol 02, #Plotinus, #Christianity
On the other hand, the intelligible entities are either deprived of feeling, life and intelligence, or they are577 intelligent. If they be intelligent, they, like truth, fuse with intelligence into the primary Intelligence. In this case we shall have to inquire into the mutual relations of intelligence, intelligible entity, and truth. Do these constitute but one single entity, or two? What in the world could intelligible entities be, if they be without life or intelligence? They are surely neither propositions, axioms, nor words, because in this case they would be enunciating things different from themselves, and would not be things themselves; thus, when you say that the good is beautiful, it would be understood that these two notions are foreign to each other. Nor can we think that the intelligibles for instance, beauty and justiceare entities that are simple, but completely separate from each other; because the intelligible entity would have lost its unity, and would no longer dwell within a unitary subject. It would be dispersed into a crowd of particular entities, and we would be forced to consider into what localities these divers elements of the intelligible were scattered. Besides, how could intelligence embrace these elements and follow them in their vicissitudes? How could intelligence remain permanent? How could it fix itself on identical objects? What will be the forms or figures of the intelligibles? Will they be like statues of gold, or like images and effigies made of some other material? In this case, the intelligence that would contemplate them would not differ from sensation. What would be the differentiating cause that would make of one justice, and of the other something else? Last, and most important, an assertion that the intelligible entities are external to Intelligence would imply that in thus contemplating objects exterior to itself Intelligence will not gain a genuine knowledge of them, having only a false intuition of them. Since, under this hypothesis, true realities will remain exterior to Intelligence, the latter, while contemplating578 them, will not possess them; and in knowing them will grasp only their images. Thus reduced to perceiving only images of truth, instead of possessing truth itself, it will grasp only deceptions, and will not reach realities. In this case (intelligence will be in the dilemma) of either acknowledging that it grasps only deceptions, and thus does not possess truth; or intelligence will be ignorant of this, being persuaded it possesses truth, when it really lacks it. By thus doubly deceiving itself, intelligence will by that very fact be still further from the truth. That is, in my opinion, the reason why sensation cannot attain the truth. Sensation is reduced to opinion248 because it is a receptive249 poweras indeed is expressed by the word "opinion"250;and because sensation receives something foreign, since the object, from which sensation receives what it possesses remains external to sensation. Therefore, to seek truth outside of intelligence is to deprive intelligence of truth or verity of intelligence. It would amount to annihilating Intelligence, and the truth (which was to dwell within it) will no longer subsist anywhere.
THE NOTION OF INTELLIGENCE IMPLIES ITS POSSESSION OF ALL INTELLIGIBLES.
Liber 111 - The Book of Wisdom - LIBER ALEPH VEL CXI, #unset, #Arthur C Clarke, #Fiction
Angel did declare unto Kelly the very axiomata of our Law of Thelema,
in good Measure, and plainly; but Dee, afflicted by the Fixity of his
Liber 46 - The Key of the Mysteries, #unset, #Arthur C Clarke, #Fiction
axiom
The spirit clothes itself to descend, and strips itself to rise.
--
axiom I
Nothing resists the will of man, when he knows the truth, and wills the
--
axiom II
To will evil, is to will death. A perverse will is a beginning of
--
axiom III
To will good with violence, is to will evil, for violence produces
--
axiom IV
One can, and one should, accept evil as the means of good; but one must
--
axiom V
To have the right to possess always, one must will patiently and long.
axiom VI
To pass ones life in willing that it is impossible to possess always,
--
axiom VII
The more obstacles the will surmounts, the stronger it is. It is for
--
axiom VIII
When the will is vowed to the absurd, it is reproved by eternal reason.
axiom IX
The will of the just man is the will of God himself, and the law of
--
axiom X
It is by the will that the intelligence sees. If the will is healthy,
--
axiom XI
When one creates phantoms for oneself, one puts vampires into the
--
axiom XII
To affirm and to will what ought to be is to create; to affirm and will
--
axiom XIII
Light<---
--
axiom XIV
The empire of the world is the empire of the light.<--
axiom XV
Great intellects whose wills are badly balanced are like comets which
--
axiom XVI
To do nothing is as fatal as to do evil, but it is more cowardly. The
--
axiom XVII
To suffer is to work. A great sorrow suffered is a progress
--
axiom XVIII
Voluntary death from devotion is not suicide; it is the apotheosis of
--
axiom XIX
Fear is nothing but idleness of the will, and for that reason public
--
axiom XX
Succeed in not fearing the lion, and the lion will fear you. Say to
--
axiom XXI
A chain of iron is easier to break than a chain of flowers.
axiom XXII
Before saying that a man is happy or unhappy, find out what the
LUX.07 - ENCHANTMENT, #Liber Null, #Peter J Carroll, #Occultism
From a magical point of view, it is axiomatic that we have created the world in which we exist. Looking about himself, the magician can say "thus have I willed," or "thus do I perceive," or more accurately, "thus does my Kia manifest."
It may seen strange to have willed such limiting circumstances, but any form of dualistic manifestation or existence implies limits. If the Kia had willed a different set of limitations, it would have incarnated elsewhere. The tendency of things to continue to exist, even when unobserved, is due to their having their being in Chaos. The magician can only change something if he can "match" the Chaos which is upholding the normal event.
The Act of Creation text, #The Act of Creation, #Arthur Koestler, #Psychology
names. Certainly, Aut Caesar aut nullus is not an axiom to which the
modern historian would subscribe either in these or any other in-
--
are axiomatic beliefs and prejudices which are taken for granted and
implied in the code. Further implied, hidden in the space between the
--
implied, as hidden axioms, and taken for granted the code must be
de-coded. The rest is easy: find the 'link* the focal concept, word, or
--
tacit axioms and habits of thought which were implied in the code and
taken for granted; the un-covering of what has always been there.
--
derived from the axiom of absolute time, which had been built into
the codes of 'rational* meaning post-Newtonian thinking about
--
concepts, of the axioms and prejudices engrained in the very texture of
specialized ways of thought. It allows the rnind to discard the strait-
--
various ways and degrees. Hidden axioms, implied in the old codes,
suddenly stand revealed and are subsequently dropped; the rules of the
--
It is the axiomatic belief that the pointers on his ^ afc do not move
at random, which makes the readings of his instruments meaningful
--
built-in axioms, which determine the rules of the game, and make most
of us run, most of the time, in the grooves of habit reducing us to
--
the immortal axioms of Euclidean geometry.
However, the ideal to which the bloated Venus of Willendorf testi-
--
de-differentiation of thought-matrices, a dismantling of its axioms, a
new innocence of the eye; followed by the liberation from restraint
--
stated or tacidy implied axioms, general oudook and selective em-
phasis. These can be briefly schematized as follows:
--
looks at the axioms of Principia Mathematical and that the physico-
chemical equilibration of these traces should be capable of producing
--
some hidden axiom, idee recue, unwarranted assumption.
(h) Over-explicit, rigid definitions which explain away problems as
--
tions and 'self-evident axioms' which as often as not are specious contra-
band; and the empirical strategies are often weighted by a stubborn
--
supposedly self-evident axioms. In other respects, however, they are
remarkably adaptable, and their dialectical strategies are of great
--
plausible axioms of Aristotelian physics.
'Thinking aside' also occurs on all intermediate levels of difficulty.
Theaetetus, #unset, #Arthur C Clarke, #Fiction
(b) The fixedness of impressions of sense furnishes a link of connexion between ancient and modern philosophy. The modern thinker often repeats the parallel axiom, 'All knowledge is experience.' He means to say that the outward and not the inward is both the original source and the final criterion of truth, because the outward can be observed and analyzed; the inward is only known by external results, and is dimly perceived by each man for himself. In what does this differ from the saying of Theaetetus? Chiefly in thisthat the modern term 'experience,' while implying a point of departure in sense and a return to sense, also includes all the processes of reasoning and imagination which have intervened. The necessary connexion between them by no means affords a measure of the relative degree of importance which is to be ascribed to either element. For the inductive portion of any science may be small, as in mathematics or ethics, compared with that which the mind has attained by reasoning and reflection on a very few facts.
II. The saying that 'All knowledge is sensation' is identified by Plato with the Protagorean thesis that 'Man is the measure of all things.' The interpretation which Protagoras himself is supposed to give of these latter words is: 'Things are to me as they appear to me, and to you as they appear to you.' But there remains still an ambiguity both in the text and in the explanation, which has to be cleared up. Did Protagoras merely mean to assert the relativity of knowledge to the human mind? Or did he mean to deny that there is an objective standard of truth?
--
SOCRATES: These three axioms, if I am not mistaken, are fighting with one another in our minds in the case of the dice, or, again, in such a case as thisif I were to say that I, who am of a certain height and taller than you, may within a year, without gaining or losing in height, be not so tallnot that I should have lost, but that you would have increased. In such a case, I am afterwards what I once was not, and yet I have not become; for I could not have become without becoming, neither could I have become less without losing somewhat of my height; and I could give you ten thousand examples of similar contradictions, if we admit them at all. I believe that you follow me, Theaetetus; for I suspect that you have thought of these questions before now.
THEAETETUS: Yes, Socrates, and I am amazed when I think of them; by the Gods I am! and I want to know what on earth they mean; and there are times when my head quite swims with the contemplation of them.
The Dwellings of the Philosophers, #unset, #Arthur C Clarke, #Fiction
dissociation. It is a common axiom in spagyrics that it is easier to make gold than to destroy
it. But here we must add a brief remark.
--
fundamental axiom which teaches that bodies have no action on bodies would be false and
paradoxical. Take the time and the trouble to experiment, and you will recognize that metals
--
Besides, it is in formal opposition to the philosophical axiom we have often stated: bodies
have no action on bodies; only spirits are active and acting.
--
Thus is accomplished the first part of the axiom: solve el coagula, by the constant
volatilization of the fixed and by its combination with the volatile; the body spiritualized itself
--
In being born, we die every day. A serious thought of Seneca, the philosopher, an axiom
which we would hardly expect to find here. Evidently, this profound albeit ethical truth,
--
this masterly work. We think, in fact, that the Latin axiom borrowed by Louis dEstissac from
Neros stoical governor, was not inappropriately put there. It is the only written word written
--
the mystery, free from its matrix, gives away the hidden reason for the amphibological axiom.
And the superficial formula, reminding man of his mortal origin is erased and disappears, now
--
quotidie morimur, and demonstrated how this classical axiom, skillfully used by Louis
dEstissac throws a new light on the lapidary work of the hermetic scientist.
--
outstanding virtue of higher minds. It was an axiom that masters used to repeat to their
disciples, and through which they signified to them the only means of attaining the highest
--
This curious bas-relief is characterized by the axiom:
153
--
Therefore, during the entire course of the work the hermetic axiom told by Lintaut must be
remembered which teaches that "gold, once dissolved into spirit, if it feels the cold, is lost
--
to it that they apply the old hermetic axiom: una res, una via, una dispositione. One matter,
one vessel, one furnace. Such is our ear then vase, a despised, plain vase of common use,
--
subject. This is what the Spanish axiom quite exactly justified, for the more reiterations, the
more the broken and dissociated body is wronged and the less the quintessence which comes
--
to be killed for the dead to be resuscitated. The practical application of this axiom assures the
sage of the possession of the live sulphur, principal agent of the stone and of the
--
criterion then is insufficient, although it justifies the well known axiom that all dry matter
dissolves and corrupts in the humidity which is natural and homogeneous to it. This is the
--
An image and reaction just like the hermetic axiom: Solve et coagula, dissolve and coagulate.
A similar subject can be found at Bourges on one of the ceiling panels of the Chapel
--
Uncommon axiom of which the truth is questionable when applied to true merit where
wealth quite seldom regards virtue that it would be appropriate to look elsewhere for its
--
hidden under the enigmatic axiom solve el coagula : dissolve the body) and coagulate (the
spirit). This can be done in one operation including two dissolutions, one violent, dangerous,
--
nature. "Every dry thing avidly drinks its own humidity", says an old alchemical axiom. But
this sulfur, during its first extraction, is never stripped of the metallic mercury with which it
The Library Of Babel 2, #Labyrinths, #Jorge Luis Borges, #Poetry
tory), I wish to recall a few axioms.
First: The Library has existed ab ternitate. That truth, whose immedi
The Library of Babel, #Labyrinths, #Jorge Luis Borges, #Poetry
a few axioms.
First: The Library exists ab aeterno. This truth, whose immediate corollary is
The Monadology, #unset, #Arthur C Clarke, #Fiction
34. It is thus that in Mathematics speculative Theorems and practical Canons are reduced by analysis to Definitions, axioms and
Postulates.
35. In short, there are simple ideas, of which no definition can be given; there are also axioms and postulates, in a word, primary principles, which cannot be proved, and indeed have no need of proof; and these are identical propositions, whose opposite involves an express contradiction. (Theod. 36, 37, 44, 45, 49, 52, 121-122,
337, 340-344.)
--- Overview of noun axiom
The noun axiom has 2 senses (first 1 from tagged texts)
1. (1) maxim, axiom ::: (a saying that is widely accepted on its own merits)
2. axiom ::: ((logic) a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident)
--- Synonyms/Hypernyms (Ordered by Estimated Frequency) of noun axiom
2 senses of axiom
Sense 1
maxim, axiom
=> saying, expression, locution
=> speech, speech communication, spoken communication, spoken language, language, voice communication, oral communication
=> auditory communication
=> communication
=> abstraction, abstract entity
=> entity
Sense 2
axiom
=> proposition
=> statement
=> message, content, subject matter, substance
=> communication
=> abstraction, abstract entity
=> entity
--- Hyponyms of noun axiom
2 senses of axiom
Sense 1
maxim, axiom
=> aphorism, apothegm, apophthegm
=> gnome
=> moralism
Sense 2
axiom
=> Euclid's axiom, Euclid's postulate, Euclidean axiom
--- Synonyms/Hypernyms (Ordered by Estimated Frequency) of noun axiom
2 senses of axiom
Sense 1
maxim, axiom
=> saying, expression, locution
Sense 2
axiom
=> proposition
--- Coordinate Terms (sisters) of noun axiom
2 senses of axiom
Sense 1
maxim, axiom
-> saying, expression, locution
=> Beatitude
=> logion
=> calque, calque formation, loan translation
=> advice and consent
=> ambiguity
=> euphemism
=> dysphemism
=> shucks
=> tongue twister
=> anatomical reference, anatomical
=> southernism
=> motto, slogan, catchword, shibboleth
=> maxim, axiom
=> epigram, quip
=> proverb, adage, saw, byword
=> idiom, idiomatic expression, phrasal idiom, set phrase, phrase
=> agrapha
=> sumpsimus
Sense 2
axiom
-> proposition
=> particular, particular proposition
=> universal, universal proposition
=> negation
=> converse
=> lemma
=> theorem
=> conclusion, ratiocination
=> postulate, posit
=> axiom
--- Grep of noun axiom
axiom
euclid's axiom
euclid's fifth axiom
euclid's first axiom
euclid's fourth axiom
euclid's second axiom
euclid's third axiom
euclidean axiom
parallel axiom
Wikipedia - Affine plane (incidence geometry) -- Euclidean space of dimension 2 that is axiomatically defined
Wikipedia - Algebraic structure -- Set with one or more operations that satisfy a given set of axioms
Wikipedia - Armstrong's axioms
Wikipedia - Axiomatic semantics
Wikipedia - Axiomatic set theory
Wikipedia - Axiomatic system -- Mathematical term; any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems
Wikipedia - Axiomatic
Wikipedia - Axiomatization of Boolean algebras
Wikipedia - Axiomatization
Wikipedia - Axiom A
Wikipedia - AXIOM (camera) -- Series of open-hardware cinema cameras
Wikipedia - Axiom (computer algebra system)
Wikipedia - Axiom of adjunction
Wikipedia - Axiom of categoricity
Wikipedia - Axiom of Choice
Wikipedia - Axiom of choice -- axiom of set theory
Wikipedia - Axiom of constructibility
Wikipedia - Axiom of countable choice
Wikipedia - Axiom of dependent choice
Wikipedia - Axiom of determinacy
Wikipedia - Axiom of extensionality
Wikipedia - Axiom of global choice
Wikipedia - Axiom of infinity
Wikipedia - Axiom of limitation of size
Wikipedia - Axiom of non-choice -- Axiom of set theory
Wikipedia - Axiom of pairing
Wikipedia - Axiom of power set
Wikipedia - Axiom of real determinacy -- Axiom of set theory
Wikipedia - Axiom of reducibility
Wikipedia - Axiom of regularity -- Axiom of set theory
Wikipedia - Axiom of union
Wikipedia - Axiom Orbital Segment -- Planned orbital segment designed by Axiom Space
Wikipedia - Axiom S5
Wikipedia - Axiom schema of replacement
Wikipedia - Axiom schema of separation
Wikipedia - Axiom schema of specification
Wikipedia - Axiom schema
Wikipedia - Axiom's End -- 2020 science fiction novel by Lindsay Ellis
Wikipedia - Axioms (journal) -- mathematics journal
Wikipedia - Axiom Space -- Private American aerospace company
Wikipedia - Axioms
Wikipedia - Axiom -- Statement that is taken to be true
Wikipedia - Basic belief -- The axioms under the epistemological view called foundationalism
Wikipedia - Beevor's axiom
Wikipedia - Blum axioms
Wikipedia - Blum complexity axioms
Wikipedia - Construction of the real numbers -- Axiomatic definitions of the real numbers
Wikipedia - Draft:Axiomatik (band) -- Canadian rock band
Wikipedia - Formal proof -- Establishment of a theorem using inference from the axioms
Wikipedia - Frame problem -- The problem of finding adequate collections of axioms for a viable description of a robot environment using first-order logic
Wikipedia - Godel's speed-up theorem -- There are theorems whose proofs can be shortened in more powerful axiomatic systems
Wikipedia - Hilbert's axioms
Wikipedia - Kolmogorov axioms
Wikipedia - Laws of thermodynamics -- Axiomatic basis of thermodynamics
Wikipedia - List of axioms -- Wikipedia list article
Wikipedia - Luce's choice axiom
Wikipedia - Martin's axiom
Wikipedia - Mathematical theory -- Mathematical model that is based on axioms
Wikipedia - Non-Euclidean geometry -- Two geometries based on axioms closely related to those specifying Euclidean geometry
Wikipedia - Parallel axiom
Wikipedia - Parallel postulate -- Geometric axiom
Wikipedia - Peano axioms -- axioms for the natural numbers
Wikipedia - Peripatetic axiom
Wikipedia - Probability axioms
Wikipedia - Robinson arithmetic -- Axiomatic logical system
Wikipedia - Separation axiom
Wikipedia - Social Axioms Survey
Wikipedia - SpaceX Axiom Space-1 -- Mission
Wikipedia - Typographical Number Theory -- Axiomatic system
Wikipedia - Von Neumann-Morgenstern utility theorem -- Any individual whose preferences satisfy four axioms has a utility function
Wikipedia - Wholeness axiom -- Axiom of set theory
Wikipedia - Zermelo-Fraenkel set theory -- Standard system of axiomatic set theory
Wikipedia - Zorn's lemma -- mathematical proposition equivalent to the axiom of choice
https://www.goodreads.com/book/show/10393449-defending-the-axioms
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https://www.goodreads.com/book/show/15913224-an-axiomatic-basis-for-quantum-mechanics
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https://www.goodreads.com/book/show/22658824-an-axiomatic-basis-for-quantum-mechanics
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https://www.goodreads.com/book/show/39892820-axiomatic
https://www.goodreads.com/book/show/4582712-axiomatic-heresy
https://www.goodreads.com/book/show/7781892-an-axiomatic-basis-for-quantum-mechanics---volume-2
https://www.goodreads.com/book/show/992703.Axiomatic_Models_Of_Bargaining
https://religion.wikia.org/wiki/Peripatetic_axiom
Integral World - Integral Theory and Its Discontents, On reification, the limits of map making, reductive coloring, axiomatic argumentation, and the dangers of being cross-eyed, David Lane
Stanford Encyclopedia of Philosophy - axiom-choice
Stanford Encyclopedia of Philosophy - truth-axiomatic
https://tvtropes.org/pmwiki/pmwiki.php/Literature/AxiomsEnd
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https://tvtropes.org/pmwiki/pmwiki.php/VideoGame/SonicAxiom
https://en.wikiquote.org/wiki/Axiom
https://en.wikiquote.org/wiki/Axiomatic_system
https://disney.fandom.com/wiki/Axiom
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https://eq2.fandom.com/wiki/Axiom_of_the_Great_Bear_(Collection)
https://eq2.fandom.com/wiki/Axiom_of_the_Great_Chokidai_(Collection)
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https://yoyo.fandom.com/wiki/YoYoJam_Axiom
Sword Art Online: Alicization - War of Underworld -- -- A-1 Pictures -- 12 eps -- Light novel -- Action Game Adventure Romance Fantasy -- Sword Art Online: Alicization - War of Underworld Sword Art Online: Alicization - War of Underworld -- Despite the defeat of Quinella—the pontifex of the Axiom Church—things have not seemed to calm down yet. Upon contacting the real world, Kazuto "Kirito" Kirigaya finds out that the Ocean Turtle—a mega-float controlled by Rath—was raided. Due to a sudden short-circuit caused by the raiders, Kirito's fluctlight is damaged, leaving him comatose. Feeling insecure about the people at the Axiom Church, Alice brings the unconscious Kirito back to their hometown—Rulid Village, disregarding her banishment due to an unabsolved crime. Now, Alice is living an ordinary and peaceful life close by the village, wishing for Kirito to wake up. -- -- However, tragedy strikes when Alice notices that the Dark Territory has already started to invade the Human Empire. Reassuming her previous alias, Alice Synthesis Thirty, she promises to defeat the Dark Territory in order to defend the world that Kirito and Eugeo worked so hard to protect. -- -- -- Licensor: -- Aniplex of America -- 466,598 7.60
A (Axiom)
Aczel's anti-foundation axiom
Armstrong's axioms
Axiom
Axiom A
Axiom (album)
Axiomatic (book)
Axiomatic geometry
Axiomatic system
Axiom (band)
AXIOM (camera)
Axiom Collection
Axiom (computer algebra system)
Axiom (disambiguation)
Axiom Films
Axiom independence
Axiom of adjunction
Axiom of choice
Axiom of constructibility
Axiom of countability
Axiom of countable choice
Axiom of Cumulative Inertia
Axiom of dependent choice
Axiom of determinacy
Axiom of empty set
Axiom of equity
Axiom of extensionality
Axiom of global choice
Axiom of infinity
Axiom of limitation of size
Axiom of Maria
Axiom of pairing
Axiom of power set
Axiom of projective determinacy
Axiom of regularity
Axiom of union
Axiom Orbital Segment
Axiom (rapper)
Axiom (record label)
Axioms (album)
Axiom schema
Axiom schema of replacement
Axiom schema of specification
Axiom Space
Axiom Verge
Baumgartner's axiom
Birkhoff's axioms
Blum axioms
CantorDedekind axiom
Diracvon Neumann axioms
EilenbergSteenrod axioms
Ensemble axiom
Freiling's axiom of symmetry
Gluing axiom
Group structure and the axiom of choice
Hilbert's axioms
HuzitaHatori axioms
Kuratowski closure axioms
List of axioms
Luce's choice axiom
Martin's axiom
Minimal axioms for Boolean algebra
Open coloring axiom
Pasch's axiom
Peano axioms
Playfair's axiom
Probability axioms
Proper forcing axiom
Scale-space axioms
Separation axiom
Social Axioms Survey
Soul Axiom
SpaceX Axiom Space-1
Tarski's axiomatization of the reals
Tarski's axioms
Tychonoff axiom
User:Streetsoda/sandbox/Axiom's End
Wholeness axiom
Wightman axioms
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