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*To*: dubucs@idf.ext.jussieu.fr, fom@math.psu.edu*Subject*: FOM: more on Hilbert, Monism and Optimism*From*: Neil Tennant <neilt@mercutio.cohums.ohio-state.edu>*Date*: Sun, 13 Jun 1999 18:42:34 -0400 (EDT)*Cc*: neilt@mercutio.cohums.ohio-state.edu*Sender*: owner-fom@math.psu.edu

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

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