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*To*: fom@math.psu.edu, historia-matematica@chasque.apc.org*Subject*: FOM: Hilbert and solvability, etc.*From*: Michael Detlefsen <Detlefsen.1@nd.edu>*Date*: Mon, 21 Jun 1999 08:38:38 -0500*Sender*: owner-fom@math.psu.edu

Neil Tennant, Jacques Dubucs and others have recently been discussing the matter of Hilbert's early understanding of notions of solvability and decidability. I have a few comments to make on this exchange. (1): A careful reading of the 1900 'Mathematical Problems' sheds at least a little light. In speaking of the solvability of every mathematical problem, H says (my translation): "Occasionally it is the case that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense entertained. Such proofs of impossibility were produced by the ancients ... In more recent mathematicsm the question as to the impossibility of certain solutions plays a dominant role, and we ascertain in this way that old and difficult problems ... have at last received fully satisfactory and rigorous solutions, although not in the sense originally intended. Along with other philosophical reasons, it is in all likelihood this important fact that induces the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of a precise resolution, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts." Here, I believe, we find fairly convincing indication that H did NOT think of the solvability of every mathematical problem (what he sometimes referred to as "the axiom of solvability") in terms of a universal decision method for all mathematical problems. We also, I believe, see the influence of Kant. For, after the passage just noted, H goes on to write ... in striking echo of Kant ... "Is this axiom of the solvability of every problem a characteristic that is peculiar to mathematical thought alone, or is there possibly a general law inherent in the nature of the mind that all questions it asks must be answerable?" H goes on to answer the question in the affirmative. But the thing I want to call attention to is the parallel with the opening paragraph(s) of the preface to the first edition of the Critique of Pure Reason. No one who has read that can fail to think of it when reading the above passages (and other accompanying remarks) in the 'Problems' address. For your convenience, I quote the paragraph here: "Human reason has this peculiar fate that in one species of its knowledge it is burdened by questions which, as prescribed by the very nature of reason itself, it is not able to ignore, but which, as transcending all its powers, it is also not able to answer." H's project was, I believe, to show a way out of the dilemma posed by Kant ... and to do so in a way which was itself faithful to the general structure of critical epistemology. Since I have argued this at length elsewhere, I will not repeat the argument again here. H is reacting to Kant (and some other thinkers of certain neo-Kantian tendencies) but also accepting Kant. He urges that there is in mathematics "no ignorabimus', but he does so on the strength of his advocacy (in Kantian fashion) of 'transcendental' solutions (i.e. solutions establishing the unsolvability of the problem under the methods according to which a solution had been sought). (2) There is another ground for H's advocacy of "no ignorabimus" as well, and that is the influence (clear in the 1904 Heidelberg adress and the 1905 lectures 'Logische Prinzipien des mathematischen Denkens') that Dedekind had on his conception of the axiomatic method. (This was also present in at least embryonic form in the 'Problems' address in the several places where H emphasized the 'creative freedom' of the mathematician.) In the 1904 address to the Heidelberg congress, H sets out to develop the 'fundamental idea' of what he refers to as 'a method I would call axiomatic' (Hilbert 1904, 131). He sets out three basic laws that he takes to govern this method. The first of them is something he refers to as the creative principle. He states it as follows: "Once arrived at a certain stage in the development of the theory, I may say that a further proposition is true as soon as we recognize that no contradiction results if it is added to the propositions previously found true ..." Hilbert 1904, 135 H then elaborates this a bit further by saying that "the creative principle ..., in its freest use, justifies us in forming ever new notions, with the sole restriction that we avoid contradiction" (l.c., 136). In this conception of axioms as conditions which themselves implicitly defining the properties of the 'objects' being axiomatically described, H was following Dedekind, who related his axiomatic description of the whole numbers to the ideal of free creation in the following way: "If in the consideration of a simply infinite system N set in order by a transformation f we entirely neglect the special character of the elements; simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformation f, then are these elements called natural numbers or ordinal numbers or simply numbers, and the base element 1 is called the base-number of the number-series N. With reference to this freeing the elements from every other content (abstraction) we are justified in calling the numbers a free creation of the human mind. The relations or laws which are derived entirely from the conditions a, b, g, d ... form the first object of the science of numbers or arithmetic." WsuwsdZ, 68 It was thus by a free exercise of a certain power of abstraction, that Dedekind took himself to have arrived at the first axiomatization of arithmetic. His is no mere 'genetic' logicist attempt to define the numbers in terms of more basic entities. Indeed, he expressly repudiates this suggestion and says that the numbers are to be freed from all content save that which is laid down for them in their defining 'axioms' a, b, g, d. For that reason, I believe it proper to regard him as having linked the free creation of concepts to the axiomatic method in such a way as to suggest the former as the (or at least a) typifying characteristic of the latter. Further support for this view comes from his well-known letter to Weber of 1888. There he warns against confusing his proposal with one which merely seeks to define the numbers as classes. He wrote: "There is much to say about such a class (e.g. that it is a system of infinitely many elements, namely, all similar systems), a weight one would not gladly hang about the neck of the number itself. But doesn't everyone gladly soon forget that the number four is a system of infinitely many things? (That the number 4 is the child of the number 3 and the mother of the number 5, however, will remain in everyone's consciousness.)" Dedekind 1932, 490 To relate this to the proper understanding of H's conception of solvability/decidability, I would note this: There is (and would have been for H) no reason to think of a set of axioms as having the power to decide every question frameable in its language. Nor would this even have been desirable since not all questions frameable are automatically interesting. Indeed, that's the general type of thing from which the 'freedom' of our power to abstract is supposed to free us. In his early (pre-1917) writings, then, H was not, I believe, at all of the view that even a 'local' axiom system ought ideall to furnish answers to all questions frameable in its language ... except in the sense described in (1) above, according to which a proof of undecidability counts as an 'answer'. By the way, I would like to register my reservations regarding Wilfried's suggestion that H was pursuing a logicist project around the time of the 1917 essay and that he was, in that, heavily influenced by Russell. I think there's little in the way of evidence to support such a view. He expressly rejected logicism in either its Fregean, its Dedekindean or its Cantorian forms in the 1904 essay. This rejection appeared again, in expanded form, in his work in the 20s. The chief influence of Russell was, I believe, in his (Russell's) having provided the vehicle of formalization which was needed to give the 'third' option of H's 'kein ignorabimus' view (i.e. the option of 'unsolvable under such-and-such conditions') a precise form. I've looked at the lecture notes from this period and read parts of the material of H's student Behmann that are supposed to 'evidence' the connection with Russell but see nothing in any of this that gives much credence to the idea that H was influenced by the substantive logicist views of Russell during this period. ************************** Michael Detlefsen Department of Philosophy University of Notre Dame Notre Dame, Indiana 46556 U.S.A. e-mail: Detlefsen.1@nd.edu FAX: 219-631-8609 Office phones: 219-631-7534 219-631-3024 Home phone: 219-273-2744 **************************

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