[FOM] Are proofs in mathematics based on sufficient evidence?
Jim Hardy
jimhardy at isu.edu
Thu Jul 8 11:49:34 EDT 2010
Vaughan,
I'm in agreement with your general approach to the subject. That's
certainly how I've always understood and taught it. There is a core
notion of proof running across the disciplines though the disciplines
differ on the level of rigor required and on what count as acceptable
premises.
One possible source of confusion is your use of "sufficient evidence"
rather than something like "sufficient reason". There is a reading of
"evidence" which is restricted to empirical domains. On such a reading,
mathematics and logics have nothing whatsoever to do with evidence,
sufficient or not. Perhaps shifting to a slightly different formulation
of the definition could help dissipate a disagreement more apparent than
real.
Jim Hardy
On 7/7/2010 7:10 PM, Vaughan Pratt wrote:
> 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.
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