[FOM] Are proofs in mathematics based on sufficient evidence?
Vaughan Pratt
pratt at cs.stanford.edu
Wed Jul 7 21:10:05 EDT 2010
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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