FOM: June 1 - June 24, 1999

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FOM: How clear was Hilbert?

In an earlier message, I claimed that Hilbert's phrase 

	"problem of the decidability of any mathematical question
	by means of a finite number of operations"

(from "Axiomatisches Denken", a lecture delivered to the Swiss
Mathematical Society in September 1917) posed "not quite the question
of whether there is a single effectively decidable system of
mathematics", and I claimed that "it could instead be construed as the
question whether every mathematical question can be decided by means
of a finite proof in some system or other." (Call this latter view
Mathematical Optimism.) 

Jacques Dubucs wrote ( Fri Jun 11 16:31:01 1999):

"The question about Hilbert's attitude w.r.t. the Entscheidungsproblem
is by no way [so] easy to adjudicate."

First, let me point out that I was not claiming to "adjudicate"
Hilbert's attitude. I was simply pointing out a difficulty in doing
so, within the limited context provided by that 1917 formulation of his.
Dubucs goes on to say

"The expression "in some system or other" in your last sentence is
obviously in need of some further qualification, even to be just candidate
for an explanation of Hilbert's genuine train of thoughts. For each
mathematical sentence can be decided by means of a finite (actually, of
lenght 1) proof in a system to which it belongs as an axiom. Such an
uninformative claim can't be seriously attributed to Hilbert."

I agree; and the preceding discussion within the 1917 lecture would
make it perfectly clear that the system consisting of the axiom P to
"decide" a difficult conjecture P would not be one that Hilbert would
contemplate as a verifier of the P-instantiation of Mathematical
Optimism. Hilbert had set up some strong demands on what could play
the role of axioms in such systems.

The fact that "Axiomatisches Denken" was delivered in 1917 shows that
Hilbert's *subsequent* "long-run flirt with the idea of a general method of
decision for mathematics" (as Dubucs put it) in the 1920s could well
have been the result of a later attempt on his part to get clearer
about the ambiguity to which I was drawing attention.

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? 

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".

Once the idea of an effective procedure was formulated, Mathematical
Monism would be seen to imply the following claim of Mathematical
Mechanism: "there is some effective method M such that for every
conjecture P, M(P) will be either a proof or a refutation of P from
suitably constrained axioms."

It would not surprise me if the relationships among Monism, Optimism
and Mechanism were not properly clarified until the (late) 1920s.

Neil Tennant

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