What Is an Axiom? The Starting Points Nothing Else Justifies

A proof derives a conclusion from earlier statements, and those statements must themselves be derived from something or be assumed. Requiring everything to be proved leads to infinite regress, circularity or an arbitrary stop, which is the regress problem in general epistemology appearing in mathematics. Mathematics resolves it by choosing the stopping points explicitly and stating them, which is a strength rather than an evasion, since the assumptions are visible and can be examined. Classically, axioms were understood as self-evident truths that no reasonable person would deny, and Euclid presented his as exactly that. The modern understanding is quite different and treats them as assumptions adopted for a system, assessed by whether the resulting structure is consistent, useful and interesting rather than by whether they are obviously true. That shift, from axioms as evident truths to axioms as stipulations, is one of the most consequential changes in the history of mathematics and it happened largely in the nineteenth century.

One assumption drove that change, and its history is the standard illustration of what axioms are:

• •Euclid's fifth postulate, which states in effect that through a point not on a line there is exactly one parallel line, is longer and less obvious than his other four
• •Mathematicians spent two thousand years attempting to derive it from the others, believing it must be a theorem rather than an assumption
• •Every attempt failed, and several supposed proofs turned out to assume something equivalent to what they were proving
• •In the nineteenth century, Bolyai, Lobachevsky and Gauss independently explored what follows if it is denied, expecting a contradiction
• •No contradiction appeared, and the result was consistent geometries in which no parallel exists or infinitely many do, which describe surfaces of positive and negative curvature
• •Those geometries turned out to describe physical reality, since general relativity models gravity as curvature of spacetime, so the non-Euclidean case is not an abstraction but the actual geometry of the universe

Several properties are wanted and they cannot all be guaranteed. Consistency means no contradiction can be derived, and it is essential, since from a contradiction anything follows and the system becomes worthless. Independence means no axiom can be derived from the others, which is a matter of economy rather than necessity. Completeness means every statement expressible in the system can be proved or disproved within it. Godel's incompleteness theorems, published in 1931, established that any consistent formal system rich enough to express basic arithmetic contains statements that are true and unprovable within it, and that such a system cannot prove its own consistency, which ended the programme of establishing mathematics on a complete and self-verifying foundation. That result is frequently overinterpreted in popular writing, and what it does not say is that mathematics is unreliable or that anything can be believed, since the vast majority of mathematical work is unaffected in practice.

Because axioms are stipulations, genuine choices exist and mathematicians argue about them. The axiom of choice, which states that a selection can be made from each of infinitely many sets simultaneously, is independent of the standard set theory axioms, meaning it can be adopted or denied without contradiction. Adopting it enables a great deal of standard mathematics and also implies results many find deeply counterintuitive, most famously a decomposition of a sphere into pieces that reassemble into two spheres of the same size. Most mathematicians adopt it and some work explicitly without it. Similar independence results hold for other statements including the continuum hypothesis, which was shown to be neither provable nor refutable from the standard axioms. The practical attitude that emerged is pluralist: different axiom systems describe different mathematical structures, all may be studied, and the question of which is true is either meaningless or a question about physics rather than about mathematics.

Justification must stop somewhere, so mathematics chooses its starting points explicitly and states them, which makes the assumptions visible. Two thousand years of failure to derive Euclid's parallel postulate ended when denying it produced consistent geometries, which general relativity later showed describe the actual universe. Godel proved that any system rich enough for arithmetic contains true statements it cannot prove.