[FOM] What is a proof?
JoeShipman@aol.com
JoeShipman at aol.com
Fri Oct 31 02:49:16 EST 2003
Baldwin suggests a proof is either a means of establishing the truth of a
proposition or an explanation of why a proposition is true.
The former obviously includes the latter -- an explanatory proof is just a
particularly satisfactory kind of proof. Of course, these definitions have
problems -- there is an undefined term ("truth") and it is not at all clear how to
formalize the notion of "explanation" (though we have some useful ways of
formalizing "proof"). Furthermore, the subjective aspect of proofs is ignored.
Phenomenologically, a proof is a proof OF a proposition, and it is generated
by a "prover" and presented to a "reader" (or "listener" in seminar and
classroom settings), and it has the practical aim of soliciting ASSENT to the
proposition.
The best definition of "proof" that I know of (which works in nonmathematical
contexts as well) is "a completely convincing argument". To understand a
proof is to be convinced by an argument and to assent to the proposition proved
(with the necessary bracketing when the prover uses an assumption A not
accepted by the reader, who then regards the prover has having proved "A implies P"
rather than P). No concept of "truth" is necessary.
The "gold standard" for proofs of propositions which are mathematical in
content is "formalizable in ZFC". But there are proofs which have been generally
accepted by the mathematical community, which are far from any sort of
formalizability in ZFC. It is really an act of faith (so far, well-founded) to
suppose that all the proofs in mathematical journals can be turned into formal
ZFC-proofs. The situation is comparable to, though more extreme than, the
situation in computer science, where any effective procedure is taken to be
performable by a Turing machine. There is much more evidence for Church's thesis ("any
mathematical function humans can compute can be computed by a Turing
Machine") then for its mathematical analogue, which I will call "Zermelo's thesis"
("any mathematical proof humans can produce and agree about can be formalized in
ZFC").
-- JS
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