An axiom is an initial statement of a theory, accepted as true within that theory without the requirement of its proof and used as the basis for the proof of its other statements according to its rules of logical deduction. Logical deduction makes it possible to transfer the truth of axioms to the consequences deduced from them. The propositions deducted from the axioms are called theorems. The set of initial axioms and propositions deduced from them form an axiomatically constructed theory. In science, an axiom is understood as a position of a scientific theory that is accepted as an initial one, and the question of the truth of an axiomatic position is solved either within the framework of other scientific theories or through the interpretation of this theoretical system: the realization of some formalized axiomatic system in a particular subject area indicates the truth of the axioms accepted in it.
The need to accept axioms without proof follows from an inductive consideration: any proof has to rely on some statements, and if one requires its own proof for each of them, the chain “statement – proof” would become infinite. To avoid infinity, it is necessary to break this chain somewhere, that is, to accept some statements without proofs as initial. Exactly such statements, accepted as initial, are called axioms.
Typical examples of axioms:
Some expression of the symbolic language of the calculus, if by further reasoning is understood to use its conclusions within that calculus. In this case, the reason for accepting axioms is the very definition of the calculus in question. In this case, doubts about the acceptance of the axioms are meaningless.
Some empirical hypothesis, if by further reasoning we mean, for example, a section of physics systematically developed on its basis. In this case the reason for accepting an axiom is the belief in the regularity of nature expressed by this hypothesis. In this case, doubts about the acceptance of the axiom are not only meaningful, but also desirable.
The agreement to understand the terms involved in the formulation of some judgement as one wishes, but still in such a way that with this understanding the formulation in question expresses a true judgement. This is the case in which further reasoning is understood as the derivation of knowingly true consequences from an ambiguously understood initial judgment. In this case, doubts about accepting the axiom are meaningless. When this kind of axiom is used within a scientific theory, it is often called a postulate of meaning.
A statement evaluated as necessarily true (apodictic) if further reasoning is understood as some systematically developed doctrine that claims to be epistemologically perfect. In this case, the reason for accepting an axiom is evidence of a special cognitive capacity (intuition) to directly discern certain (often called self-evident) truths. Within this claim, doubts about the acceptance of an axiom are meaningless, but the question of the validity of the claim itself is one of the most significant problems in philosophy (see Philosophy).
Axioms emerge in the long and complex development of scientific cognition. From antiquity until the nineteenth century, axioms were seen not simply as a starting point of evidence, but as intuitively obvious or a priori true propositions. The importance of axioms was substantiated by Aristotle who believed that axioms do not require proof because they are clear and simple because they “possess the highest degree of generality and represent the beginning of everything”.
Euclid considered axioms accepted by him within his geometrical system as self-evident truths, sufficient to deduce all other truths of geometry. On the basis of the accumulated by that time facts and knowledge he singled out and formulated several axiomatic statements (postulates), accepted without proofs, from which their logical consequences were deduced in the form of theorems. At the same time axioms were often treated as eternal and immutable truths, known before any experience and not depending on it, the attempt to justify which could only undermine their obviousness. Kant’s doctrine of the a priori nature of axioms, that is, that they precede all experience and do not depend on it, was the culmination of such views on axioms.
The rethinking of axioms is connected with the discovery in the 19th century of non-Euclidean geometry (C. F. Gauss, N. I. Lobachevsky, J. Boiai); the appearance in abstract algebra of new number systems, and their whole families at once; the appearance of variable structures like groups; finally, the wide discussion of questions like “which geometry is true?” All this contributed to the realization of two new statuses of axioms: axioms as descriptions (classes of possible universes of reasoning) and axioms as assumptions rather than self-evident assertions.