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OUTLINES OF A FORMALIST PHILOSOPHY OF MATHEMATICS HASKELL B. CURRY ProfessorofMathematics State College, Pa. U.S.A. ~c m ~ 195 1 NORTH·HOLLAND PUBLISHING COMPANY AMSTERDAM PRINTED IN THE NETHERLANDS DRUKKERIJ HOLLAND N.V., AMSTERDAM PREFACE In the opening days of September, 1939, just as the Second World War was breaking out, an International Congress for the Unity of Science was held at Cambridge, Massachusetts. The committee in charge of that congress invited me to submit a paper there. This invitation suggested that I write out, in a more or less systematic form, the philosophical opinions which I had formed in a little more than a decade of study in mathematical logic. I found the task a longer and more difficult one than I had anticipated; nevertheless I persisted in it. The resulting manuscript was, of course, too long for presentation at the congress; but it formed the basis of the paper presented there, and of several other papers, cited below, which have been written since. The present monograph represents the publication of the manuscript prepared in 1939. It is published here in essentially the same form in which it was originally written down. The only changes are a revision of Example 9 in ChapterV and some foot- notes. These changes date from 1942. In view of the foregoing origin of this monograph, the reader should bear in mind the following points concerning it. In the first place, this monograph represents the views which I held in 1939; it does not represent accurately the views which I would defend right now. The later papers, already mentioned, overlap with it greatly and supersede some parts of it. However, the more amplified presentation which is possible in this monograph will presumably be an advantage for some readers; further- more, the statement of what lies back of the opinions expressed in the later papers may help to make them better understood. A similar caution should be observed in regard to the views of other persons which are quoted or criticized here. This is particularly true in the case of Carnap and other leaders of the VI PREFACE Unity of Science movement. Later publications of these persons show considerable changes in their views also. The citations of other papers are made in a special manner which is explained at the end of Chapter I. For the most part, however, acquaintance with these references is not presupposed. Except for the fact that criticism of views of other persons requires some acquaintance with those views as expressed by the authors themselves, this monograph is self-contained, and thus is in accordance with the aim of this series. For this reason it does not seem to be necessary to replace these citations with others made in a more orthodox form. The matters of notation request comment. In a review published in the Journal of Symbolic Logic, vol. 6, pp. 100-102, Kleene pointed out that the uses of the prefix "meta-" and the adjective "recursive" differ from those which are made elsewhere. (The review had reference to a later paper, but it applies to this one, which Kleene had seen in May, 1940). To remedy any possible ambiguity in regard to the first point, I shall in the future use the prefix "epi-" to replace the prefix "meta-" in the sense used here (Chapter IX), reserving "meta-" for use in connection with semiotical matters. It would, however, only add to the confusion to make this change in the present text. In regard to the word "recursive", the point seems to be that Kleene (and others), taking the notion of recursive function as fundamental, applies the term "recursive" to a class only when its characteristic function is recursive; whereas here a class is referred to as recursive when it can be exhausted by a recursive enumeration or construction. This is an oversight; it would indeed be better to call this kind of class "recursively generated". Thus, in a definite formal system, as defined in Chapter IV, the notions of term, proposition, proof, and other definite concepts could be described as recursive, while the theorems are recursively generated. Here again it seems undesirable to change the present text, but merely to call attention to the situation and bear itin mindforfuture use. The later papers referred to above are the following, with place of publication or presumed publication: PREFACE VII "Remarks on the Nature and Definition of Mathematics", (Abstract of address before International Congress for the Unity of Science, September 5, 1939), Journal of Unified Science, 9, pp. 164-169. "Some Aspects of the Problem ofMathematical Rigor", Bulletin of the American Mathematical Society, (1941), 47, pp. 221-24I. "The Paradox of Kleene and Rosser", Transactions of the American Mathematical Society, (1941), 50, pp. 454-516, (especially pp. 457-462). "Mathematics, Syntactics, and Logic", (Address before Inter- national Congress for the Unity of Science, Chicago, 1941). To appear (1) in Journal of Unified Science. "Languagesand Formal Systems", (Abstractofaddressdelivered August 16, 1948), Proceedings Tenth International Congress of Philosophy, 1949, pp. 770-772. "Language, Metalanguage, and Formal System", (Full form of the preceding). Philosophical Review (1950) 59, pp. 346-353. "A Theory of Formal Deducibility", Notre Dame Mathematical Lectures No.6, (Delivered April 12-15, 1948). Notre Dame, Indiana. 1950, 126 pp. "L-Semantics as a Formal System", Congres International de Philosophie des Sciences, Paris, 1949 (Actual. Sci. et Ind., no. 1134), Paris, 1951. "Las Systemes formals et les langues", Colloque de Logique Symbolique, Paris, 1950 (forthcoming). "Leeons sur la logique algebrique" Paris (Gauthier Villers) and Louvain (Nauwelaerts) (forthcoming). HASKELL B. CURRY State College, Pennsylvania, U.S.A. May, 1951 I INTRODUCTION This book deals with the philosophy of mathematics, not from the point of view of a philosopher, but from that of a mathematician who has had occasion to reflect on the nature of his science. That there is a danger in writing such a book lam fully aware; philosophers will doubtless find that these views are naive, and that too much space is devoted to formal matters, whereas mathematicians may feel that their time is better spent in creating new mathematics. But, aside from the fact that criticism has from time to time had a stimulating influence on research, there appears at the present time to be a special need for discussions of this kind. Owing to the departmentalization of our educational system, mathematicians and philosophers have little contact with one another; on the other hand the rise of mathematical logic has made especially patent the need of a rapprochement between the two. Such a rapprochement was indeed one of the aims which the founders of the Association for Symbolic Logic had in mind. If, then, this book ventures a bit from safe, familiar shores, it is because an attempt to meet the philosophers half way is made in it. I hope that the result will be interesting to both mathematicians and philosophers. This book does not attempt to cover the whole range of mathematical philosophy, but contains merely a discussion of certain special topics. The choice of these topics has been determined partly by my own interests, partly by contacts - oral or other- wise - with my colleagues; and I have laid special emphasis on questions in which these contacts showed that comment would be appropriate. Subject to these limitations, however, the book is intended to be self-contained. There is no attempt to confine attention to what is novel. Moreover I have not tried to do justice to the history of the subject, nor to trace ideas to their sources; the book is based more on direct reflection than on a survey of 2 INTRODUCTION the literature. References are given here and there in the text; but I may have been influenced by contacts long since forgotten. Finally, although the book carries the word 'philosophy' in its title, yet considerable space is taken up by essentially mathematical considerations for which a philosophical interest is presumed. Since we have in various journals excellent bibliographical information concerning the literature, it is possible to abbreviate citations. I shall use the following scheme: The letter 'J' refers to the Journal oj Symbolic Logic; it is followed by volume and page reference either to the original paper or to a review of it. The letter 'C' refers to Church's bibliography in J 1, 121-218 and J 3, 178-212; this 'C' is followed by symbols referring to the individual authors or papers according to the scheme of Church's indices (e.g. J 3, 193). The letter 'Z' refers to the Zentral- blatt j1lr Mathematik, the reference being by volume and page. II THE PROBLEM OF MATHEMATICAL TRUTH The central problem in the philosophy of mathematics is the definition of mathematical truth. Ifmathematics is to be a science, then it must consist of propositions concerning a subject matter, which propositions are true in so far as they correspond with the facts. We are concerned with the nature of this subject matter and these facts. The ordinary mathematician bases the idea of truth upon that of rigor; he regards a mathematical proposition as true when he has a rigorous proof of it. But when we examine the nature of this "rigor", we find there is something vague and subjective about it. This is true to such an extent that intelligent persons have maintained that mathematics is not a science at all; that it has no subject matter nor any criterion of truth; and that in the last analysis it is based on purely aesthetic considerations. Now although there are doubtless aesthetic elements in mathematics, yet most of us cannot be satisfied with such an account of our subject. There must be an objective criterion of truth, and the first task of the mathematical logician is to find it. The first thesis of this book is that such a definition can be found. Indeed mathematics can be conceived as a science in such a way as to be independent of any except the most rudimentary philosophical hypotheses. So conceived, mathematics forms, like every other science, part of the data of philosophy. There are three main types of opinion as to the nature of this subject matter, viz.: 1) realism, or the view that mathematical propositions are true insofar as they correspond with our physical environment; 2) idealism, which relates mathematics to mental objects of one sort or another; and 3) formalism. The realist point of view is now not taken seriously by most mathematicians. Of course it must have been the original view as to mathematics; among primitive peoples mathematics is 4 THE PROBLEl'.I OF MATHEMATICAL TRUTH essentially empirical. Today, however, the view is untenable; for one reason because there is nothing corresponding to infinity in the external environment. It is unnecessary to go into this here; for the point is generally recognized. Everybody thinks of mathematics as "abstract"; and it is a truism for every serious student of the subject that its theorems hold independently ofany relation to external reality. This leaves us with the idealist and formalist positions. The idealist position will concern us in the next section; after some discussion of the nature of formalism we shall return to the formalist definition of mathematical truth in Chapter X. The remaining sections of this book will deal with the relations of mathematics to its applications. III IDEALISTIC VIEWS OF MATHEMATICS On the idealistic view mathematics deals with mental objects of some sort. There are different varieties of this view according to the nature of these mental objects. On the one hand there is the view, here called Platonism1, which ascribes an independent existence to all the infinitistic conceptions of classical mathematics; on the other hand there is intuitionism, which makes everything depend on an a priori basic intuition of temporal succession 2. All forms of idealism are subject to the same fundamental criticism: viz., that the resulting criterion of truth is vague at best, and depends on metaphysical assumptions from which mathematics, if it is to have the pre-philosophical character above mentioned, must be free. That this criticism applies against Platonism has been shown very effectively by its intuitionist critics. But it also applies against intuitionism itself3, at least in the form espoused by Brouwer and his school. In fact, as to the vagueness we have Heyting's own statement4, "Uber dies ist es an sich widersinnig, die Mogliohkeiten des Denkens in das Mieder bestimmter zuvor angegebener Konstruktionsprinzipien zwangen zu wollen. Man muss sich also darauf besohranken, durch mehr oder weniger vage Umschreibungen in dem Horer die mathematische Geistes- haltung hervorzurufen". (Cf. also Church's criticism, J 2, 89). As to the metaphysical character it is clear from the intuitionist writings that their fundamental intuition must have the following properties: 1) it is essentially a thinking activity ("eine kon- 1 This name was suggested by Bernays (0 287.17). 2 Both Platonism and intuitionism have many sub-varieties. In either case the different writers often disagree with one another. 3 Of. Hahn, 0 419.2. 4 0 385.10, p. 12.