Although the concept of proof is the heart of mathematics, science, logic and generally all rational human activity, there is no generally accepted definition of “what proof is” and what has it been in history and across different cultures. In different fields, there are distinct definitions or requirements as to what constitutes proof.

In mathematics, proof establishes the truth of a proposition on the grounds of already established true propositions or axioms. However, what constitutes proof for a classical mathematician, maybe not acceptable by another (e.g. a constructive) mathematician. What constitutes proof for a mathematician of antiquity (e.g. Euclid) maybe not rigorous enough for a mathematician of modern times (e.g. Hilbert). Truth in Greek geometry is established by geometric axiom-based reasoning over abstract entities, whereas in the East (China, Japan) truth is never founded on axiomatic assumptions. On the other hand, the appearance of computer-assisted proofs in the 1970s highlighted the role of human understanding of proof as a means of its validation and recognition by the mathematical community.

In the physical sciences, scientific proof can be grounded on experimental data and observations. The philosophical proof is an inference concluded from a series of fundamental, plausible arguments that can typically be considered persuasive. Legal proofs are reached by a jury on the grounds of allowable evidence presented at a trial. Proofs in computing can be programs that prove the properties of systems.

In the history of mathematics, mathematical proofs involve many informal components, a kind of rigour that is independent of complete formalization and some kind of “meaning” or semantic content that is transmitted through a “text” and calls its reader for understanding and verification. Moreover, proofs are often conducted under different (local) logics and formulated in distinct styles of reasoning by using diverse mediums and codes of communication in different cultures in history.

If proof is part of logic, then the problem is ultimately reducible to the question “what logic is?” But there is no consensus either on the question of what logic is. For instance, the model-theoretic understanding of logic (“a logic is something that has syntax and semantics”) is different from the proof-theoretic understanding (“logic is a deductive system that has the cut-elimination property”).

Furthermore, proofs can be carried out within different logics, and thereby establishing different kinds of truth, for instance, classical truths, constructive truths, probabilistic (statistical) truths, modal truths, paraconsistent truths, etc., which might be understood and accepted only by the community, who reason within the corresponding logic. On the other hand, proofs can be codified and communicated in different styles: Hilbert-style proofs, natural-deduction proofs, sequent-calculus proofs, tableau proofs, etc., also informal and meta-mathematical proofs, philosophical argumentation written up in a blend of natural and sign languages. The same proof can be exposed in different formal or informal ways, but even in a single formalism, the same proof can take different forms.