Method
Axiom
Overview
An axiom, postulate, or assumption, is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
The precise definition varies across fields of study. In classical philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning.
In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example a + 0 = a in integer arithmetic.
Non-logical axioms may also be called "postulates", "assumptions" or "proper axioms". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain.
Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Axiom is discussed.
So far as this axiom applies to space, it is superfluous. For the property follows from our previous axioms. It is however necessary for time as a supplement to the axiom of kinetic symmetry. The meaning of the axiom is that if the time-unit of system {alpha} is congruent to the time-unit of system {beta}, and the time-unit of system {beta} is congruent to the time-unit of system {gamma}, then the time-units of…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
names the idea; states what it is; author linked in the graph; opens on it
The above axiom is the axiom of Free Mobility[151]. I propose to prove (1) that the denial of this axiom would involve logical and philosophical absurdities, so that it must be classed as wholly _à priori_; (2) that metrical Geometry, if it refused this axiom, would be unable, without a logical absurdity, to establish the notion of spatial magnitude at all. The conclusion will be, that the axiom cannot be proved or…
Bertrand Russell, An essay on the foundations of geometry
names the idea; a chapter is titled for it; commentary markers; states what it is; author linked in the graph; opens on it
Thus accidents are not produced by accidents promiscuously; quality cannot be the origin of quantity, nor quantity that of quality; a form cannot emanate from matter, nor matter from form. A third axiom is this: A single agent that acts with design and will, and not merely by the force of the laws of Nature, can produce different objects. A fourth axiom is as follows: An object, whose several elements are only…
Moses Maimonides, The guide for the perplexed
names the idea; states what it is; opens on it
For it is a defect, in an argument, to employ more premisses than the conclusion demands: what mathematicians call elegance results from employing only the essential principles in virtue of which the thesis is true. It is a merit in Euclid that he advances as far as he is able to go without employing the axiom of parallels--not, as is often said, because this axiom is inherently objectionable, but because, in…
Bertrand Russell, Mysticism and Logic and Other Essays
states what it is; names the idea; author linked in the graph
CIV. Nor can we suffer the understanding to jump and fly from particulars to remote and most general axioms (such as are termed the principles of arts and things), and thus prove and make out their intermediate axioms according to the supposed unshaken truth of the former. This, however, has always been done to the present time from the natural bent of the understanding, educated too, and accustomed to this very…
Francis Bacon, Novum organum
names the idea; states what it is
Philosophers who discussed Axiom
- Avicenna
- Carl Gustav Hempel
- Giambattista Vico
- Murray Rothbard
- Henry Sidgwick
- Jan Łukasiewicz
- Ayn Rand
- Baruch Spinoza
- Ruth Barcan Marcus
- Imre Lakatos
- Blaise Pascal
- Alfred North Whitehead
- Jeremy Bentham
- Carl Stumpf
- Rudolf Carnap
- Arthur Schopenhauer
- Alain Badiou
- Penelope Maddy
- Ernst Mally
- Jean Nicod
- F. C. S. Schiller
Related ideas
- Formal language
- First-order logic
- Falsifiability
- Self-evidence
- Ancient philosophy
- Greek language
- General relativity
- Georg Cantor
- Gödel's completeness theorem
- Giuseppe Peano
- Philosophy
- Infinite set
- Integer
- Free variables and bound variables
- Ancient Greek
- Ancient Greek mathematics
- Zermelo–Fraenkel set theory
- Propositional variable
- Negation
- Kurt Gödel
- Posterior Analytics
- Logical connective
- Propositional logic
- David Bohm
Place in the tradition
Part of British philosophy.