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*To*: Neil Tennant <neilt@mercutio.cohums.ohio-state.edu>*Subject*: FOM: Re: more on Hilbert, Monism and Optimism*From*: dubucs <dubucs@ext.jussieu.fr>*Date*: Mon, 14 Jun 1999 10:35:33 +0200*Cc*: fom@math.psu.edu*In-Reply-To*: <199906132242.SAA00854@mercutio.cohums.ohio-state.edu>*Sender*: owner-fom@math.psu.edu

The present posting pursues the current exchange with Neil Tennant about Hilbert's attitude towards decidability of mathematics. I can't manage to make it intelligible without reproducing large parts of Tennant's last posting. Jacques Dubucs suggested (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: Look at the consistency proof sketched by Hilbert before the Heidelberg congress (1904). 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). This consistency proof, if completely achieved, would provide us with an effectively recognizable condition of provability in number theory. Neil Tennant answered (Jun 13 18:42 1999): 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. Here my answer (JD): I never said that Hilbert's sketched consistency proof, if carried through, would have provided an effective algorithm of *decision* for number theory. Of course no, for the property P is only a *necessary* condition of provability, not a sufficient one. Nevertheless, the possibility of effectively deciding between unprovability and irrefutability, while obviously falling short of a full method of decision, is a non-trivial step in this direction. Viewing that Hilbert found this possibility very plausible in 1904, we are entitled to attribute him, even in his early foundational writings, the kind of favouring attitude towards the idea of mechanization of mathematics that I described in a former posting (13 Jun 1999 18:42:34) Incidentally, Johannes von Neumann, who was one of the most lucid members of the Hilbertian School (and, anyway, the first to grasp the significance of Goedel's incompletness results), found the kind of *morphological* consistency proof that Hilbert attempted in 1904 very dubious, just because its non-trivial implications concerning mechanizability (cf *Zur hilbertschen Beweistheorie* (1927), in Collected Works, Pergamon Press, 1961, vol. 1, p. 276). ----------------------- Tennant pursued (Jun 13 18:42 1999): 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) ? Here my answer (JD): I suspect that there is no such possible clue, for a simple and well-known reason. As lenthly and convincingly explained by Goedel in several places, the very notions in terms of which your (3) is expressed were not available to Hilbert, who was always very reluctant (and, by the way, unable) to formulate the completness problems in such semantical terms. Moreover, one can wonder if Hilbert's views favouring decidability were not a simple by-product of his propensity of thinking the foundational problems in exclusively syntactical terms (cf my posting of Jun 11 22:34:58 1999). --------------------------- Tennant pursued (Jun 13 18:42 1999): 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 (...) 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. Here my answer (JD): I agree with Tennant that, if one accepts the idea that the ways of not-trivially solving mathematical conjectures cannot be framed in one single formal system, one is driven towards the idea of (a sequence of) extensions of a "synthetic base theory". Of course, the rigourous idea of a reflection principle was not at Hilbert's disposal in the tweenties to do that. But note that he proposes, in his latest *pre-goedelian* paper (*Die Grundlegung der elementaren Zahlenlehre* (1931), Ges. Abh. III, 194), another coarser way of dealing with the same kind of limits, namely by introducing his omega-rule, that he warmly recommands as a "new FINITE inference rule" ... JD Jacques Dubucs IHPST CNRS Paris I 13, rue du Four 75006 Paris Tel (33) 01 43 54 60 36 (33) 01 43 54 94 60 Fax (33) 01 44 07 16 49

**References**:**Neil Tennant**- FOM: more on Hilbert, Monism and Optimism

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