Re: Are mathematical proofs incomparable with proofs in other disciplines?

[Vaughan Pratt <pratt@cs.stanford.edu>]

On 7/9/2010 7:10 AM, Michael Barr wrote:
> Nonetheless, the biggest difference between mathematical
> proof and, say, legal proof, is that the latter depends on real world
> evidence.

Yes, exactly.  Mathematical proof depends on evidence (or whatever you
feel it should be renamed as in that setting) found in mathematical worlds.

> Legal terms do not have definitions that can be understood
> without reference to the real world.

What about intellectual property?  Granted ideas can't be patented, but
is a method or algorithm (the case of software patents) a thing in the
real world or the conceptual world?

> we believe that, in principle,
> every proof we publish is a surrogate for a formal logical deduction,
> but I once tried that for a simple argument and gave up after I had
> filled a couple pages with indecipherable chicken scratchings (cf.
> Russell & Whitehead).

But R&W didn't give up, they published the full proof that 1 +_c 1 = 2
(the _c denotes cardinal addition), which they were able to complete as
*110.643 by the top of p. 83 of Volume II, with the modest remark, "The
above proposition is occasionally useful.  It is used at least three
times, in *113.66 and *120.123.472."  I always regretted that they
didn't use Roman instead of Arabic numerals and follow the convention of
omitting plus rather than times, so that *110.643 would then read

      II = II.

But I agree it's tedious.  In 1975 I published 23 lines of chicken
scratchings constituting a complete formal proof that 474,397,531 was
prime.  (Obviously this couldn't have been just an enumeration of its
possible factors up to its square root.)  This was not terribly tedious.
   Two years later, with Stephen Litvintchouk I published a complete
formal proof of correctness of the Knuth-Morris-Pratt pattern matching
algorithm, using a broadly construed notion of inference rule that
allowed the use of decision methods in checking some steps.  This was
somewhat more tedious, but it would have been worse without the decision
methods.

> I agree, however, that they [legal and mathematical proofs] are incomparable.

Along the dimension of provenance of evidence, yes.  And there is an
understandable reluctance on the part of some to refer to premises of
mathematical arguments as "evidence," which can be addressed by
broadening the notion to "evidence or argument."

Vaughan


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