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

[Vaughan Pratt]

There's an interesting dispute just started on Wikipedia concerning 
whether it is reasonable to see some commonality of meaning between the 
concept of proof in mathematics and in other areas such as rhetoric, 
law, philosophy, religion, science, etc.  The dispute is at one or both of

http://en.wikipedia.org/wiki/Talk:Proof_(informal)#Disambig_page

(Editors keep changing the name of the article, which was Proof (truth) 
when I wrote it and others have replaced "truth" first by "logic" and 
then by "informal", neither of which are an improvement.)

The origin of the article in dispute, which Wikipedia editor Gandalf61 
has now changed by deleting the mathematical content, is as follows. 
Some months ago I went to Wikipedia to look up what it considered to be 
a proof and found only a dab (disambiguation) page listing ten articles 
that seemed to about proof as applied to propositions and about as many 
more to do with testing and quality control as in galley proof, proof 
spirit, etc.

It seemed to me that the former kind were not so much different meanings 
of the notion of proof as the same meaning arising in different areas 
all depending on that meaning.  So, still some months ago, I wrote an 
article on that common notion which began

   "A proof is sufficient evidence for the truth of a proposition,"

which as it happens is essentially the first entry in the definition at 
dictionary.com.

The article enumerated the various notions of proof arising in different 
disciplines (all of which have their own Wikipedia articles with much 
more detail), and made a start on characterizing the scope of "evidence" 
(need not be verbal, and need not contain the asserted proposition) and 
"sufficient" (strict for formal proofs, less so elsewhere, to different 
degrees).

The main dispute at the moment is Gandalf61's insistence that "Proof in 
mathematics is not based on 'sufficient evidence' - it is based on 
logical deductions from axioms. It is an entirely different concept from 
proof in rhetoric, law and philospohy."  He backs this up with quotes 
from Krantz---"The unique feature that sets mathematics apart from other 
sciences, from philosophy, and indeed from all other forms of 
intellectual discourse, is the use of rigorous proof" and 
Bornat---"Mathematical truths, if they exist, aren't a matter of 
experience. Our only access to them is through reasoned argument."

My position is that logical and mathematical proofs differ from proofs 
in other disciplines in the provenance of their evidence and the rigor 
of their arguments as parametrized by "sufficient."  Whereas evidence in 
mathematics is drawn from the mathematical world, evidence in science is 
drawn from our experience of nature.  And whereas formal logic sets the 
sufficiency bar very high, mathematics sets it lower and other 
disciplines lower still, at least according to the conventional wisdom.

Whereas I find my position in complete accord with the quotes of both 
Krantz and Bornat when interpreted as in the preceding paragraph, 
Gandalf61 does not.

My questions are

1.  Is mathematical proof so different from say legal proof that the two 
notions should be listed on a disambiguation page as being unrelated 
meanings of the same word, or should they be treated as essentially the 
same notion modulo provenance of evidence and strictness of sufficiency, 
both falling under the definition "sufficient evidence of the truth of a 
proposition."

2.  Gandalf61 evidently feels his sources, Krantz and Bornat, prove the 
notions are incomparable.  Are there suitable sources for the opposite 
assertion, that they are comparable?

3.  Someone with a very heavy hand has tagged practically every sentence 
with a "citation needed" tag.  For those that genuinely do need a 
source, what would you recommend?

Vaughan Pratt

PS.  I hope this sort of argument doesn't put anyone off volunteering to 
help out on Wikipedia.