[FOM] Are proofs in mathematics based on sufficient evidence?

[Jim Hardy]

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 
proof differs.

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.