FOM: certainty/belief

[Martin Davis]

A few remarks on the indubitability question:

1. Harvey's comments on the science behind the standards of top research
mathematical journals are (as usual) very interesting. I'm not as persuaded
as he that there really is any hard science to be done there. But let's see ...

2. Hilbert of course knew perfectly well that people makes mistakes and that
"proofs" offered to the public subsequently had to be withdrawn. What upset
him was that there seemed to be no clear line between correct modes of proof
and those that lead to contradiction. He didn't know what to make of
Russell's paradox and such, and he wanted to restore basic confidence in
mathematical method.

3. Indubitability: this is a true straw man. Are predictions of forthcoming
eclipses "indubitable"? Well, an omnipotent God could certainly decide to
turn off the solar system, or to reverse the sign of gravitational
attraction or whatever. But such talk is properly received as sophmoric and
silly. What about my favorite example: Lagrange's theorem that every
positive integer is the sum of 4 squares. Is that indubitable? You bet! The
prime number theorem? Yes! Is there anyone (other than Professor Sazonov) on
fom who has the slightest doubt about the correctness of either of these?

4. Computer aided proofs: When I was working on a proof that the Riemann
Hypothesis has a simple Pi-0-1 form(*), I needed a numerical estimate for
the logarithmic derivative of the zeta funtion at -1. I looked it up in a
table. With a little effort, I could have written a computer program to do
the work. Does this really diminish our confidence in the correctness of the
result? Is the fact that the proof of the 4 color theorem uses computers to
do combinatorial calculations instead of the more traditional use for
numerical computation evidence of some conceptual distinction? Perhaps, if
anything, the combinatorial use has a better case: no error analysis needed,
no round-off error.

Martin

(*) Martin Davis,  Yuri Matijasevic & Julia Robinson ``Hilbert's Tenth
Problem: Diophantine Equations: Positive Aspects of a Negative Solution,''
Proceedings of Symposia in Pure Mathematics, vol.28(1976), pp. 323-378.