[FOM] Are proofs in mathematics based on sufficient evidence?
jimhardy at isu.edu
Tue Jul 13 12:51:00 EDT 2010
On 7/11/2010 6:53 AM, hendrik at topoi.pooq.com wrote:
> Because the other notions of proof do not "meet that high standard", you
> even find that if you present a long, careful, mathematical, deductively
> valid proof to someone whoo follows one of the other notions of proof,
> you'll find that he jusn't believe it, because in his experience, long,
> detailed proofs often come to wrong conclusions.
> Perhaps this is eveidence that they are really other notions of proof.
I don't think this is evidence of other notions of proof in the sense of
the original post.
Certainly, different disciplines count different things as proof. One
discipline may count something as proof while another dismisses it
entirely. But this doesn't show that the underlying concepts are
different, that "proof" is ambiguous between them and the dispute is
merely apparent rather than substantive. When the mathematician and the
jurist, for example, disagree about whether something is a proof they
aren't simply confused in the way that a mathematician puzzled by the
baker's use of "proof" in "proof box", or a doctor's use of it in
"proof against heart attacks". Rather there is a core concept they
largely agree on, and the dispute centers on whether this or that thing
falls under the concept. To be sure, there may be some conceptual
disagreements at the borders of the concept also, e.g need a proof be
understood in order to count as a proof, but these are indicative of
different uses to which the concept may be put, not of different
concepts altogether. As an analogy, consider whether the archaeologist
and the roofer have different concepts of hammer. If a roofer asks for
a hammer and the archaeologist presents a roughly worked oval stone, the
roofer may insist that it's not a hammer while the archaeologist insists
that it is. I submit that in such a case the roofer and archaeologist
share a common concept but differ in the use to which they put the
concept. The roofer is interested in hammers as a way of pounding nails
to attach shingles, the archaeologist has other interests, so they
differ in what they are willing to call a hammer. The difference is
primarily in the pragmatics of "hammer", not its semantics. (This isn't
quite right, but I hope it's close enough for explanatory purposes.)
Similarly, the mathematician and the jurist may use "proof" under
somewhat different circumstances because they have different goals even
while sharing a core core concept.
I think Vaughan's original post was addressing this core notion of
"proof". The mathematician and the jurist are both broadly interested
in whether certain propositions may be accepted or affirmed
non-provisionally. A proof, in both cases, is something that licenses
the affirmation of the proposition. Put more closely to Vaughan's
formulation, it is something that provides sufficient basis for
accepting the proposition. Evidence and argument provide a basis, but
you only have a proof when that basis is sufficient. Standards of
sufficiency vary across uses, but that is not to say that the concept of
As a slight tangent, there's an interesting question about whether
"argument" has a common meaning in logical and rhetorical disciplines.
My take on "proof" above is somewhat similar to Joe Wenzel's position
regarding "argument" in "Perspectives on Argument", roughly that the
difference is one of perspective rather than denotation.
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