FOM: more on Hilbert, Monism and Optimism

[Neil Tennant]

Jacques Dubucs suggests (Jun 13 12:30:52 1999) that the following
considerations lend some credence to the claim that the early Hilbert
had global decidability in mind ("Mathematical Monism"), rather than
piecemeal solubility of mathematical conjectures ("Mathematical
Optimism"): 

_____________________

"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."

_____________________

With respect, I disagree. If we take an arbitrary sentence S and
decide (effectively) whether it has the property P in question, and
happen to return a negative verdict, then this simply tells us that P
is not provable in the system in question. It does not suffice to show
that P is refutable---unless we know, on independent grounds, that the
system is theoretically complete. Then, but only then, would the
unprovability of S entail the refutability of S.

Is there any clue in Hilbert's early writings that he countenanced the
following possibility, with respect to a sentence S and a given theory?:

(1) the effective test (for the presence of property P) shows that S
    is not provable; and
(2) the effective test shows that not-S is not provable; while yet
(3) the logic underlying the theory is known to be complete
    (i.e. every logical consequence admits of finitary proof) ?

In any event, I welcome Dubucs's gracious concession that "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"." 

I agree with Dubucs that it is a highly non-trivial venture to specify
the epistemic constraints on a well-motivated axiomatic system, so as
to rule out the trivial addition of P as an axiom in order to "settle"
the conjecture P ('P' here is now a sentence, not a property), and in
order also to rule out other unmotivated additional axioms from which
P would trivially follow.  But I do think that the early foundational
writings of a figure like Hilbert (at least, circa 1917) show that he
has some very exigent conception (epistemically) on what would count
as a good set of axioms, and would not for a moment brook the trivial
verifiers of the claim of Mathematical Optimism.

One possible way to constrain axiomatic extensions would be by
insisting that they take the form of reflection principles, starting
with some synthetic base theory and iterating on its reflective
extensions. But that would require arithmetization (coding of syntax)
or an equivalent device, which had not yet been thought of in 1917; so
it would a trifle anachronistic to suggest that Hilbert might have
been contemplating such a constraint.

As to the matter of an edition of Hilbert's writings: I believe
Wilfried Sieg and others are busy on this. Wilfried, if you are out
there and listening, can you report on any progress in this
connection?

Neil Tennant