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

[Nick Nielsen]

This offers a fascinating insight into the Wikipedia editing process,
and I can easily imagine someone being put off by it.

There is an implicit philosophical thesis in the position of the
editor Gandalf61, and one can find it expressed, for example, in
William Barrett, who wrote that anyone who has studied mathematics
knows that there are no proofs in philosophy. (The Illusion of
Technique, Garden City: Anchor Books, 1979, p. 180) In other words,
rigorous proof is unique to mathematics (and logic).

I would say, on the contrary, that anyone who has studied philosophy
knows that there are no proofs in mathematics, and this too is a
philosophical thesis regarding the nature of proof, though it assumes
the non-uniqueness of proof.

In the Dictionary of Philosophy written by Mario Bunge, there is this
definition of proof: "Logically valid derivation of a theorem from
assumptions or definitions with the help of rules of inference." There
is a sense in which this is a generic, non-unique characterization of
proof, but there is also terminology that connects this definition to
formal systems. Given considerations of this nature, Gandalf61 could
take just about any definition of proof and interpret it in terms of
the uniqueness of proof in the formal sciences.

This points to the underlying issue: Gandalf61 is conflating the ideas
of proof and formal proof. While it is possible to construct a formal
proof in non-formal areas of human knowledge (rhetoric, law,
philosophy, religion, etc.), it is highly unusual to do so, whereas it
is much more common to construct a formal proof within the formal
sciences (logic, mathematics, mereology, etc.). Moreover, it is also
possible, and in fact quite common, to sketch informal proofs within
the formal sciences (even "proof by hand-waving"), which further
points to the non-uniqueness of proof.

Best wishes,

Nick Nielsen





On Wed, Jul 7, 2010 at 6:10 PM, Vaughan Pratt <pratt at cs.stanford.edu> 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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