Subject: Re: [HM] Mistakes in mathematics
From: Jeremy Gray (j.j.gray@open.ac.uk)
Date: Tue Dec 21 1999 - 12:04:16 EST
Dear All,
More about mistakes, and thanks to Julio for reminding me what John Conway's
very useful remarks actually were.
I don't mean to deprecate Hilbert when I say his position was extreme, as
perhaps Martin Davis thinks I do. Not at all. But when we look at what
Hilbert called for in Axiomatisches Denken in 1917, we find that he raised
the problem of the solvability in principle of every mathematical question,
the problem of the checkability of every result, the need for a criterion of
simplicity of proofs, the relationship between content and formalism in
mathematics and logic, and finally the problem of the decidability of a
mathematical question in a finite number of steps (can a given statement be
shown to be provable/refutable in a given theory, or is it independent?). He
indeed concluded his lecture with an illustration of the nature and
importance of the last of these, the decision problem.
I have no doubt at all that these questions go back in Hilbert's mind to his
famous Paris address of 1900 and his _Grundlagen der Geometrie_ of the
previous year. They are different in 1917, and would be different again as
the years went by. But in Paris in 1900 and Ko"nigsberg in 1931 he urged that
there is no Ignorabimus in mathematics, and I think that demonstrates a fair
degree of continuity.
It seems reasonably clear, doesn't it, that Hilbert's energy and optimism
led him to proclaim standards for at least parts of mathematics in 1900 that
became standards for all of mathematics later on that were eventually shown
to be unobtainable. (I don't want to get drawn into questions, left open by
Goedel, about what might be done in some unexpected way.) It also seems
clear that Hilbert had personal, non-mathematical reasons for throwing
himself into this task in the 1920s.
Such beliefs, on Hilbert's part, are surely more than enough to get him
started and keep him going. But two things strike me. One is that Hilbert's
position was very different from remarks about rigour, or writing style, or
clarity. I can't name any because I don't know such people well enough, but
I can easily imagine there are and were analysts and algebraists every bit
as keen on precision in mathematics as Hilbert ever was, who nonetheless did
not go after solvability, checkability, criteria of simplicity, and
decidability at such a programmatic (or, if you prefer, explicit, or
profound, or trenchant) way as Hilbert did. That's why I called it extreme.
Apologies if my choice of word caused problems.
The other thing that strikes me is that there just might be some celebrated
areas of mathematics where these aspects are so deficient that the
enterprise is subject to confusion, if not outright errors. I thought I
would ask you all if you knew of any.
There are some areas where I think one might look, but I think the
chronology is against it. Among the Hilbert problems, the tenth changes its
'feel' after the great growth of mathematical logic and theories of
algorithmic processes in the 1930s. The fifteenth, calling for rigorous
foundations of the Schubert calculus, was already contested earlier, and
later became a struggle between Severi and van der Waerden and others. The
22nd, the uniformisation theorem, was the subject of a proof by Poincar/e
with a crucial gap, as Hilbert pointed out in Paris. The whole subject of
algebraic topology and Poincar/e's part in it might be a case in point.
It would be nice - were it to be true - to be able to point out to people
that another of Hilbert's reasons for pursuing these questions was his
belief that without such standards ordinary working mathematicians were
making problems for themselves.
None of which prevents me from wishing you all a good Christmas and a very
good New Year.
Jeremy Gray
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