[FOM] Concerning proof, truth, and certainty in mathematics

[ARF (Richard L. Epstein)]

Concerning proof, truth, and certainty in mathematics

On Jul 21, 2010, at 8:59 PM, Vaughan Pratt wrote:

"The critical difference as I see it is that even a very strong argument
may nevertheless not be sufficient to establish the truth of a proposition."

It seems that Dr. Pratt believes that proofs in mathematics are meant to lead to knowledge of
truths, and hence that strong proofs are not acceptable in mathematics.  This can only be if proofs
in mathematics are meant to yield certain knowledge, as opposed to proofs in other intellectual
endeavors.

If mathematics is a body of truths, with certain knowledge of them, then such knowledge is the same
kind as theological knowledge.  That places Dr. Pratt in the company of the born-again Christian,
fanatical muslim, or devout Catholic, knowing what cannot be experienced but can only be intuited,
at least for the basic truths of mathematics.  This is the view of Godel.  It also is the view of
G.H. Hardy and L.E.J. Brouwer both of whom thought that proofs were only stimulants to our
imagination.

The other kind of certain knowledge that might arise in mathematics is that afforded by proof.  A
proof in mathematics is intended to show that the inference from the premises to the conclusion is
valid.  If it is indeed valid, it rules out all other possibilities.  However, it is extraordinarily
rare for a proof that such an inference is valid is itself valid.  It is not just that
mathematicians leave lots to the reader.  It is far too hard to fill in all the steps, as anyone who
has tried to formalize a proof in mathematics knows.  It is even harder to read such a proof, and
hence less likely to lead to certain knowledge.  Hence, a proof in mathematics purports or attempts
to show that an inference is valid, but, except in rare instances, is itself at best a strong
argument for that.  Such proofs cannot yield certain knowledge.

It cannot be held that mathematics is aloof from all other human endeavors in yielding certain
knowledge unless one is a feist.  I have set out this view along with an analysis of the nature of
mathematical reasoning and mathematics in my essay "On mathematics" which can be found in the 3rd
edition of the book Computability by Walter Carnielli and me or directly from me.

Richard L. "Arf" Epstein