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*To*: fom@math.psu.edu*Subject*: FOM: recent fom-exchange on HILBERT*From*: Bernd Buldt <Bernd.Buldt@uni-konstanz.de>*Date*: Wed, 23 Jun 1999 17:46:37 +0000*Sender*: owner-fom@math.psu.edu

greetings, fom-ers! in order to keep my mailbox uncluttered i receive the fom-postings in digest-from. maybe i should change this, but for today i reply to digests, not to individual mails. but for the convenience of those who are not interested in all of it, i've broken down my posting into several sections, each prefixed with a reference. time doesn't permit to reply to everything that caught my eyes, but only for a few comments that i have 'right at my hands'. - RE: the dubucs/tennant/tait(/sieg)-exchange on decidability. i'm unable to present definite answers to the questions raised, but what follows are pieces of evidence, that, i think, should be taken into account. (i) though i'm unable to provide some early textual evidence, i.e., before 1917 as requested, i nevertheless think the following quote from bernays should shed some light on the issue. its background is the following: heinrich scholz, the founder of the 'muenster school of logic', used to write letters to bernays in order to clarify the questions he or his group had either concerning logical questions in general and concerning hilbert's program(me) in particular. in a letter (dated zuerich, march 13th, 1939) bernays wrote (quoted in german in a forthcoming paper, the translation below is a little 'ad hoc'): "it seems to me, that one cannot claim, that the assumption that every mathematical problem is solvable, as it is mentioned in hilbert's paris' address you cite, is refuted by goedel's incompleteness theorem. this is so, because hilbert's conjecture does not draw on the solvability within the frame of a particular formalized deductive system (which obeys the conditions imposed by goedel). this is especially clear from the fact, that the validity of an undecidable sentence of the goedel-type emerges from a contentual reflection on its undecidability within prinicipia mathematica or the formal system z_mu." the second sentence, i think, makes it clear, that at least bernays thought hilbert would subscribe to mathematical optimism only, not to math. monism, to use notions defined by neil tennant earlier in this list. and also that, according to bernays, hilbert had not conceived of this problem as been one of mechanical solvability or something even close to it, as it was, i think rightly, emphasized by mic detlefsen in the last digest. (in contrast, bernays stresses, as von neuman did, that a general solving algorithm, even for 1st order logic, would trivialize mathematics and hence was unprobabale from the outset.) this point of view is further supported by the list of paradigmatic questions, "decided by a finite number of operations", hilbert himself gives in his "axiomatic thinking", pp. 154ff., whose solutions are neither algorithmic, or mechanical, or even stated within a formalized theory. that means also, that i'm unable to agree with william tait, who poposed in the last digest that hilbert's "verfahren" (solving procedure for diophantine equations), at least in general, has "clearly the sense of a decison procedure in our sense" (for more evidence against this identification, see below). (ii) interestingly, a little later in the same letter, bernays remarks: "also the results of church concerning the unsolvability of certain problems do not refute hilbert's conjecture, because those [unsolvable] problems are such that they can't be answered with "yes" or "no", since they contain a variable numberparameter." (iii) with respect to the general context of decidability questions at that time let me also remark, first, that (in another letter to scholz, dated, zuerich, october 12th, 1936) bernays distinguishes between set-theoretical on the one hand and proof-theoretical procedures on the other hand. and that he was not opposed the former in principle, but that they simply failed to be in accordance with the methodology underlying the finitist attitude (cf. Hilbert/Bernays, Grundlagen d. Math. I, 1st ed., 190-196). second, as bernays explicitly mentions, this whole early work was done along the line of logic initiated in germany by ernst schroeder's "algebra of logic". (and maybe it was even done in the spirit of his program(me), that of an "absolute algebra", which was intended to realize the leibniz project of a characteristica universalis -- i don't know.) (iv) in the text book "grundzuege der theoretischen logik" by hilbert/ackermann, published in 1928 (for whose precursor, see sieg's posting), the decidability problem apparently is understood as an algorithm: "the decision problem is solved when one knows a procedure, that allows for deciding the validity resp. the satisfiability of any given expression with a finite number of operations" (p. 73). but further, it is called -- in italics -- the "main problem of math. logic" (1st. ed. p. 77, cf. also the whole of p.III§11, or 2nd. ed. (1938), p. 90, p.III§12). taken this literally, btw, means, that even though the consistency problem might have been considered as the most pressing one, as sieg stressed, but only from a practical point of view; but considered from a systematical point of view, it, apparently, was not considered as the most fundamental one. but maybe more interestingly is, that in the 2nd. ed. from 1938 ackermann doesn't say that the church-turing results provide a fatal blow for hilbert's optimism. instead he points out that they do not provide instances of 'absolute undecidable' sentences, which, in fact, would lead to a contradiction. while ackermann prepared this 2nd. edition he was only a 'corresponding member' of the hilbert school, i.e., he discussed the revision with bernays in letters. but at least in his view (just like bernays did, see above), recursive unsolvability doesn't undercut hilbertian optimism. thus, some kind of solvability/decidability-request runs through all of hilbert's writings: from hilbert's 'problem address' at paris (1900), over 'axiomatic thinking' (1917/18) and his text book (1928) up to one of his last lectures "naturerkennen und logik" from 1930, where, towards the end, hilbert takes up again the ignorabimus-question. gathering all these evidences, what remains unclear to me, is, then, what was the exact relationship between hilbert's "axiom of solvability", his demand for "decidability with a finite number operations", both stressed in his more popular lectures from 1900 throughout to 1930, and the "main problem of math. logic" from his more technical writings. i hesitate to subscribe to dubucs conclusion that hilbert was simply confused on this matters, rather, i'm inclined to think that hilbert himself considered this as an open problem, since he lists on p. 153 of his "axiomatic thinking" the "solvability in principle of every math. question" and the "decidability of a math. question in a finite number of steps" as two separate problems; two problems btw, that should be solved with the help of his proof theory. - RE: the dubucs on post-completeness. this is only a short note to call back to memory, that to think of completeness in terms of post-completeness was kind of 'natural' for hilbert, since he stated the completeness of his axioms for geometry (from the second ed. onwards, missing in the first ed. of his "foundations of geometry") and for the real number field in 1900 via a kind of post-completeness, i.e., that any addition would render the system inconsistent. but what strikes me as odd is that initially the hilbert circle apparently searched for this kind of completeness also for 1st order predicate logic. but suppose it were, then we couldn't have formalized axiomatic systems, since additions of axioms (unprovable within 1st order logic) could render the system inconsistent because of post-completeness, which, in my view, would have partly threatened hilbert's program(me). (e.g., think of an axiom system for some entities x containing an axiom, expressable within 'pure logic', to the effect that there are, say, n entities of sort x.) maybe someone on the list has a clue on this. (for more on completeness see the next section.) - RE: tennant's request for the bernays quotation first, one finds a comprehensive though incomplete bernays bibliograhy in the book "sets and classes. on the work of paul bernays", ed. by g.h. mueller, amsterdam 1976. second, the requested quotation is from "die philosophie der mathematik und die hilbertsche beweistheorie" (philosophy of mathematics and hilbert's proof-theory), in: blaetter fuer deutsche philosophie 4 (1930), 326-367, reprinted, with a postscript, in p. bernays, "abhandlugen zur philosophie der mathematik", darmstadt 1976, 17-61. on p. 58 bernays denies the possibility that an end to the creation of new concepts in mathematics is within sight. nevertheless, he states on p. 59, formalized theories can enjoy this possibility, viz., that no new concepts furnish new results. he then defines a theory to be deductively closed iff the theory is post-complete (see above), or, as some would prefer to say, iff it is maximal. before he goes on to conjecture that the then current formalization of arithmetic of the hilbert school, i.e., the peano axioms with recursive definitions, is deductively closed, he adds footnote nr. 19: "please note that the demand for deductively closedness is weaker then [geht nicht so weit wie] the demand for decidability of each question of the theory in question, viz., that there should be a procedure to decide of any pair of contradictory assertions belonging to this theory, which one is provable (true)." this is the first mentioning of this modern notion of completeness (i.e., being closed under ...) i'm aware of in either hilbert or bernays. possibly, bernays learned it from herz or zermelo, who both used similar closing conditions in papers that were published in 1929. (whereas our modern notion, i think, comes from tarski's papers on the 'methododology of the deductive sciences', that appeared around the same time.) - RE: the sieg/detlefsen-exchange on logicism in hilbert mic detlefsen states he hasn't found any evidence of hilbert's logicism around 1917-1920, as claimed by wilfried sieg in his recent paper in the bsl. well, even before learning of sieg's papers two years ago or so, i thought it uncontroversial that there was a logicist stage in hilbert's development; especially so, because hilbert himself says so. in his "axiomatic thinking" (again p. 153) hilbert says: "... hence it seems necessary, to axiomatize logic itself und to establish, that number theory and set theory are only parts of logic. this path, prepared since long -- not in the least through the penetrating investigations of frege -- was finally and most successfully followed by the brilliant mathematician and logician russell." thus we see hilbert -- contrary to earlier remarks of his own -- taking an explicit logicist stance (number and set theory being simply logic). he then goes on with acknowledging, that in order to bring this frege-russell-project to a happy ending, there still needs a lot of work to be done, which he plans to pursue in the years to come (a first step of which was to hire bernays right away from zuerich, where he gave that lecture). bernays confirms on p. 202 of his 1935-report on hilbert's foundational work (enclosed to vol. III of hilbert's "gesammelte abhandlungen"), that hilbert started out from the frege-russell-project, trying to supply only the missing consistency proof: "thus hilbert was left with the task of providing a consistency proof for these [i.e., frege's and russell's unproved] assumptions." but in order to do so, he envisaged a proof-theory, which was also designed to meet the constrcutive demands put forward by weyl and brouwer. bernays then turns to the more 'riper' developments of hilbert's program(me), leaving us in the dark concerning the details. looking for more details, one can consult the first report ever given on the then on-going hilbert program(me). it is a lecture given by bernays in september 1921, received in october that year, and published in 1922 (jahresberichte der deutschen mathematiker-vereinigung 31 (1922), 10-19). from this lecture it is clear, that hilbert/bernays found serious problems in the logicist program(me) they started with, but also in trying to meet the constructive demands. hence they settled on a new version, later called hilbert's program(me), which, according to p. 15, was conceived of as saving the best of both sides, but with having a heavy constructive list (with later became the finitist attitude). turning to the influence of russell in particular, i'd like to add that hilbert tried hard to get him for a series of lectures to goettingen. why should a mathematician of hilbert's stature should have tried to do so, if not for exchange and 'influence'? (the mathematical circle at goettingen also studied frege's foundational works, as is evident from the reports of sessions held there, published in the "jahresberichte der deutschen mathematiker-vereinigung".) wilfried sieg has corrected and improved much on what was extractable from the hitherto published sources like those mentioned. and my reason to hint at these things is only, that even without sieg's recent paper, and even without drawing on nachlass-material, one find clues in both, hilbert's as well as bernays papers, that there had been some serious logicist flirtations until hilbert's program(me) emerged into the shape we all know it. - RE: detlefsen/tennant/dubucs on rationalistic optimism i won't deal here in extenso with mic detlefsen's claim that hilbert was a kantian, but focus instead on one point only, that is connected also to rationalistic optimism, brought up earlier by neil tennant and jacques dubucs. mic detlefsen suggests that we should place hilbert's axiom of solvability in a kantian context, because of striking parallels from kant's critique of pure reason. there are surely better quotation from kant's critique then the one he gave, because there (p. A vii) kant states that there are UNanswerable questions ("questions ...[that] ... reason ... is also not able to answer"). one good example is found at B 508, where kant says: "it is not so extraordinary as it first seems the case, that a science be in a position to demand and expect none but assured answers to all questions within its domain ... although up to the present they have perhaps not been found." kant goes then on to say (and prove), that pure reason (i.e., metaphysics), pure mathematics, and pure ethics are examples of such sciences. is it wise to interpret this as a kantianism? jacques dubucs has mentioned in digest #216, that this solving optimism can also be found in goedel. but goedel, as we know, was a student of leibniz. so what we have here, is, i think, some kind of a 'common cause', namely, both, kant and leibniz, were within the rationalistic line of western philosophy. and that the products of the mind are transparent to itself, i.e., that there are no undecidable questions in this area, has been a prejudice common to (more or less) all members of that rationalistic line. so what we find in goedel or in hilbert is no peculiar "-ism", but simply that they belong to this rationalistic line of western thought. in addition, whether they studied philosophy (like goedel did) or not (like i assume hilbert did) doesn't -- imho -- matter; mathematicians are anyway, by default, so to speak, members of this rationalistic tradition and hence there no need to explain it by assigning them any '-ism'. ******************************************************* Bernd Buldt FG Philosophie Universitaet Konstanz D-78457 Konstanz Germany Fon: [+49] (0)7531/88-2794 (Office) [+49] (0)7531/88-2750 (Secretary) Fax: [+49] (0)7531/88-4121 Email: bernd.buldt@uni-konstanz.de Fon: [+49] (0)7731/25486 (Private) Fon/Fax:[+49] (0)7731/24116 (Study) *******************************************************

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