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

Arnold Neumaier Arnold.Neumaier at univie.ac.at
Thu Jul 8 07:18:51 EDT 2010

Vaughan Pratt wrote:

> 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.
> 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."

I wonder why you put mathematical proof and logical proof into the same
category, as opposed to legal or other kinds of proofs.

There are worlds between these two notions of proof, in spite of the
common ground these notions have.

Let me mention three points that I find important, and give a
reference to a well-known 1979 paper that makes the same basic point.

A logical proof is a detailed justification that reduces the proved
claim to a sequence of atomic steps verified directly by the axioms
or already proved things.

But a mathematical proof is a sequence of arguments intended to give
another mathematician insight into the validity of the proved claim,
independently of whether it achieves this goal (whch often depends
on who the recipient is). In this respect it resembles more a legal 
proof (trying to convince a judge) than a formal, logical proof.

Few mathematicians want to read the logical proofs produced by
automatic proof systems like COQ.

Conversely, logical framework cannot understand most proofs that
mathematicians find fully adequate.

Proofs given in mathematical textbooks can be correct or wrong,
be detailed or sketchy.

But I cannot imaginge a logical proof being wrong or having gaps.

The frequently cited paper
     R.A. de Millo, R.J. Lipton and A.J. Perlis,
     Social processes and proofs of theorems and programs
     Communications of the ACM 22 (1979), 271-280.
contains an interesting discussion related to the topic,
from which I excerpted some important paragraphs in

Arnold Neumaier

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