FOM: Re: How clear was Hilbert?

[dubucs]

The present posting refers to my current exchange with Neil Tennant about
Hilbert attitude w.r.t. the principle of solvability of any mathematical
question.

Tennant asked: "Does Dubucs or any other fom-er have any interesting
textual evidence from Hilbert dating from *before 1917* which would show
that Hilbert had distinguished clearly between the following claims of
Mathematical
>Monism and Mathematical Optimism?

Let me begin by a general remark. It's a pity that no complete edition of
Hilbert's work is now available. The third volume of the so-called
"Gesammelte Abhandlungen" (subtitle: "Analysis, Grundlagen der Mathematik,
Physik, Verschiedenes"; first publication in Berlin, 1935; reprint by
Chelsea, 1965) is very significant for anyone interested in f.o.m..
Moreover, this volume contains an interesting survey by Paul Bernays, the
well-known co-author of the Grundlagen der Mathematik. But this edition
falls very short of the academic standard of completness, rigour, and
critical apparatus that are satisfied, e.g., by Feferman and alii edition
of Goedel. For example, the text I excerpted in my last posting from
Hilbert's conference in Bologna is not reprinted in this edition. Viewing
the paramount importance of Hilbert's ideas in the development of f.o.m.
research in many sectors (model theory, proof theory, calculability
theory), professionals of f.o.m. should reflect on the great utility of
starting a similar enterprise concerning Hilbert.

Viewing the lack of sources I just alluded to, I confess that I'm unable to
directly answer to Tennant's question. But the following point may be an
indirect way of answering.
Look at the consistency proof sketched by Hilbert before the Heidelberg
congress (1904; edited in the proceedings of the congress (Leipzig, 1905);
not reprinted in the "Gesammelte Abhandlungen").

This consistency proof runs roughly as follows. Number theory is formalized
in such a way that

a) Every axiom has a certain morphological (decidable) property P
b) The property P is preserved by applying the inference rules of the system
c) The property P is not preserved by negation

(Whence consistency immediately follows, for the negation of an axiom can't
be proved). Keeping apart the objection of circularity formulated by
Poincare (one uses the induction principle to show that any theorem enjoys
the property P, but induction is just one of the principles one wants to
show the consistency of), this consistency proof, if completely achieved,
would provide us with an effectively recognizable condition of provability
in number theory. Thus, we were in position of deciding, for any
number-theoretical conjecture, between irrefutability (if the formula has
the property P) and unprovability (if the formula has not P). Of course,
this algorithm is not exactly a "decision algorithm", but it is very near
to that (imagine that we have the hilbertian algorithm at our disposal in
number theory!).  That is the sort of developments I termed the "flirt" of
Hilbert with the general idea of the mechanization of mathematics. To sum
up, Tennant is perfectly right in conjecturing that Hilbert was not clear,
in his early f.o.m. writings, about the distinction between "Optimism" and
"Mechanism".

Anyway, Tennant's distinction between "Optimism", "Mechanism" and "Monism"
raises some of the most interesting and difficult issues in the
philosophical approach of f.o.m., and I'm sure that other listmembers will
contribute also to the discussion. But I can't resist to start it by two
related remarks and questions addressed to Tennant.

1) You wrote:

>Here, Mathematical Optimism would be the AE-claim "for every
>conjecture P, there is some [suitably constrained] axiomatic system S
>that either proves or refutes P", where the constraints on S would be
>designed to rule out the trivial case of taking P as an axiom, and
>would ensure consistency.
>Mathematical Monism would be the EA-claim "there is some [suitably
>constrained] axiomatic system S such that every conjecture P is either
>provable or refutable in S".

But it's not enough to exclude the case where P is an axiom. We have
certainly to exclude also the case where P & X is an axiom, as well as the
case where X and 'X implies P' are axioms, a.s.o.. In short, we have to
exclude the case where the proof of P from the axioms is trivial. But it's
not trivial to say what 'trivial' is. Or we have to exclude that the axioms
have been expressely concocted to proof P ? But that would be psychology,
not logic ! How to properly define "suitably constrained" ?

2) "(Rationalistic) optimism" is of course a very convenient label for
Hilbert's train of thoughts ("Non ignorabimus"!). But the label has been
also applied by Wang to Goedel's own conception, which seems quite distinct
of Hilbert's one. Goedel is confident in our ability to find an "absolute"
proof or disproof of any mathematical conjecture. How to clarify this
notion of "absoluteness", which seems absent from Hilbert's formulations ?

JD

Jacques Dubucs
IHPST CNRS Paris I
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75006 Paris
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