basic constructions:
strong axioms
further
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
In constructive mathematics, proof relevance refers to the concept that mathematical proofs are mathematical objects themselves and that besides just knowing that a proposition has some proof it is relevant to remember the way a given proof is constructed (and to remember possibly different proofs of the same proposition).
This idea is formalized notably by the idea of propositions as types in type theory where a proof is an actual term (the proof term) of some type.
For instance logical disjunction “A or B” in type theory (in particular in homotopy type theory) may be exhibited by the sum type $A + B$ of the types representing the individual propositions, and a proof/term of $A + B$ carries in it the information of whether it proved $A$ or $B$ to (hence) prove their disjunction. Similarly, a proof of $C \to (A + B)$ will include the information of which proofs of $C$ lead to proofs of $A$ and which proofs of $C$ lead to proofs of $B$.
In
the following complaint about mathematical proof (in section 12 Historical and mathematical proof of the Preface) might be read as being a complaint about the traditional non-constructive concept of proof and about the traditional lack of proof relevance:
All the same, while proof is essential in the case of mathematical knowledge, it still does not have the significance and nature of being a moment in the result itself; the proof is over when we get the result, and has disappeared. The process of mathematical proof does not belong to the object; it is a function that takes place outside the matter in hand.
footnote 42: Mathematical truths are not thought to be known unless proved true. Their demonstrations are not, however, kept as parts of what they prove, but are only our subjective means towards knowing the latter. In philosophy, however, consequences always form part of the essence made manifest in them, which returns to itself in such expressions.
See also earlier conceptions of proofs expressing the ‘cause’ of a theorem, where proofs by contradiction in particular were taken generally to fail. Such an idea goes back to Aristotle for whom a proper answer to the question “Why is the angle in a semicircle a right-angle?” gives its cause.