Subject: Re: [HM] Mistakes in mathematics
From: Martin Davis (martin@eipye.com)
Date: Tue Dec 21 1999 - 20:15:31 EST
At 05:04 PM 12/21/99 +0000, Jeremy Gray wrote:
> 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.
These were all good problems, and all though to some (e.g., Hardy) it was
evident that the Entscheidungsproblem could not have an algorithmic
solution, to others it was not so obvious. Indeed the cases for which an
algorithm has been shown to exist and those to which the general case is
reducible differ only by a single quantifier.
It's worth recalling that in his 1900 address Hilbert said:
... the conviction (which every mathematician shares, but which no one has
yet supported by a proof) that every definite mathematical problem must
necessarily be susceptible of an exact settlement, either in the form of an
actual answer to the question asked, or
***by the proof of the impossibility of its solution ...
> 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.
In 1900, Hilbert's main call was for the necessity of rigor in mathematical
proofs. It's not clear to me that this standard in "unobtainable"; but it
all depends on how you understand rigor. Proofs using the axiom of choice
were once held to be dubious; today we don't bat an eyelash. Are proofs
using AC not rigorous? If Goedel's finding that some arithmetic truths will
require for their proof, principles independent of the usual axioms for set
theory, and if such principles become as accepted as AC, will those proofs
not be rigorous?
Hilbert's work in the 1920s in effect proposed to explicate "rigorous
proof" as proof formalizable in what has come to be called first-order
logic, and that standard has stuck. No knowledgeable expert doubts that in
principle Hilbert was right about that.
I know of nothing that suggests that Hilbert thought of his program as
seriously effecting the day-to-day practice of working mathematicians. What
he principally sought was a way to guarantee that new antinomies would not
arise.
> None of which prevents me from wishing you all a good Christmas and a very
> good New Year.
> Jeremy Gray
Sentiments that I heartily echo while watching a beautiful sunset over San
Francisco Bay!
Martin Davis
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin@eipye.com
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http://www.eipye.com
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