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From a formalist’s point of view Abraham ROBINSON

0. The great fascination that contemporary mathematical logic has for its devotees is due, in large measure, to the ever increasing sophisti- cation of its techniques rather than to any definitive contribution to vur understanding of the foundations of mathematics. Nevertheless, the achievements of logic in recent years are relevant to foundational ques- tions and it behooves the logician, at least once in a while, to reflect on the basic nature of his subject and perhaps even to report his con- clusions. In an address given some years ago (ref. 2), the present writer stated a point of view on the foundations of mathematics which may be summed up as follows. (1) Infinite totalities do not exist and any purported reference to them is, literally, meaningless ; (2) this should not prevent us from developing mathematics in the classical vein, involving the free use of infinitary concepts ; and (3) although an infinitary framework such as set theory, or even only Peano number theory cannot be regarded as the ultimate foundation for mathematics, it appears that we have to accept at least a rudimentary form of logic and arithmetic as common to all mathematical reasoning.

Taken together, these three points may be regarded as a good summary of the formalist point of view. More particularly, (1) distin- guishes formalism sharply from platonic realism, (2) rejects the solu- tions proposed by various groups of constructivists, and (3) recoils from the bottomless pragmatism of the logical positivist. In the present paper, I propose to continue the discussion of ref. 2 in the light of more recent developments in set theory and by way of reply to debating points made in connection with one or the other of the three tenets stated above.

1. The most frequently heard criticism of the point of view of ref. 2 concerns the assertion that phrases involving the intended nega- tion of infinite totalities are strictly meaningless. I made it clear at the time that in this connection I took the intended mention of an infinite set to imply, as it would for a finite set, the existence of a

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collection of distinct objects which are the elements of the set. It is n matter of experience that such is indeed the picture of an infinite set in the minds of many working mathematicians of our time and it is this kind of picture that seems to me illusory. I certainly did not wish to denigrate the value of theories which, in a sense, involve meaningless phrases. However it is perhaps natural that a mathematician should resent suggestions which deprive him of the comforting feeling that he, like the physicist or biologist, spends his life in the exploration of some form of reality.

As long as it appeared that the accepted axiomatic systems of set theory such as the system of Zermelo-Fraenkel were able to cope with all set theoretical problems which are of interest to the working mathe- matician, belief in the existence of a unique << universe of sets )> was almost unanimous. However, this simple view of the situation has been severely shaken in recent years by two distinct developments. One of these was sparked by Paul Cohen’s proof of the independence of the continuum hypothesis and has, by now, led to the realization that the relation between the scale of ordinals and the scale of power sets is so flexible that it seems to be quite beyond control, at least for the time being. The second development is due to the emergence of new and varied axioms of infinity. At present one cannot see any compelling arguments for or against the adoption of one or the other of these axioms. Nevertheless, the orthodox Platonist believes that in the real world such an axiom must be either true or false; thus, for example, to the Platonist the real world either does or does not comprise mea- surable cardinals. He may or may not feel confident that in this and similar issues we shall finally arrive at a decision even if no decision is forced upon us by a compelling logical argument. And he may, with some justification, point to the historical development of Mathematics and of Logic, which shows that, time and again, new basic principles have been put forward and have found almost universal acceptance for no compelling logical reason.

According to a point of view emphasized particularly by G. Kreisel we should, without necessarily committing ourselves in the matter of the reality of the entities of set theory, rely at least on the objectivity of set theorical truths. And we should, by mature reflection, try to arrive at new basic principles that may stop the gaping holes in our present understanding of the subject. However, it would seem to me that while such reflection may indeed be profitable in leading to a clarification of present difficulties, this does not exclude a solutioii

From a formalist’s point of view 47

which involves the equal acceptance of several kinds of set theory. At this point, it is natural to recall the historical development of Euclidean geometry whose analogy with the recent development of axiomatic set theory is perhaps closer than is generally appreciated. The realization that the axiom of parallels might be, and indeed is, independent of the remaining axioms, following a long series of attempts to derive the former from the latter, was not then regarded as a matter of purely technical interest. For, at that time, geometry was still regarded, by long tradition, as the foundation of all mathematics, and the “arith- metization” of analysis was only in its infancy. Thus, the removal of geometry from its pre-eminent place in the foundations of mathematics, was concurrent with and perhaps influenced by, the emergence of other kinds of metric geometry as co-equal disciplines. Is it not possible that we are just now witnessing a corresponding development in the ca5e of abstract set theory?

2. The question what kind (or kinds) of mathematics is to be reconi- mended as a worthwhile occupation. is of an entirely different character. It is part of the deliberate policy of the formalist to deal with all sorts of infinite totalities as i f they were real, and so his everyday mathe- matics does not differ from that of the platonic realist. Nevertheless it is inconsistent to adhere at the same time both to the Platonist and to the formalist outlook on mathematics. On the other hand, the for- malist shares with the various types of constructivists a basic distrust of infinity and, indeed, is more finitistic than most of them. Although he has opted for the classical way of doing mathematics he need not object to certain constructive versions of arithmetic and analysis and may well admit that they are more concrete and possess a greater immediate empirical content. In practice, he may be reluctant to engagc in, say, intuitionistic mathematics since he believes that he has found a simpler way and also because he may have the uneasy feeling that the intuitionist, in his desire to recapture analysis, has after all gone beyond the bounds of strict finitism into the realm of the infinite. At the same time, the intuitionist may believe that the classical mathe- matican, whatever his underlying philosophy, is wasting his time in developing uninterpretable games with symbols, the sin of the formalist being the greater because he does so deliberately and consciously.

It must be admitted that the absence of a satisfactory intellectual motivation for his mathematical activities does in fact place the for- malist, as conceived in the present paper, at a disadvantage compared with the proponents of other philosophical views on the foudations of

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mathematics, And it has been suggested that the founder of formalism, D. HILBERT, adopted a more moderate point of view. Thus, it has been said that HILBERT insisted on the finitistic point of view only in order to establish the consistency of mathematics beyond a shadow of doubt and that he did not wish to imply that only finitistic concepts are meaningful. The enthusiasm with which HILBERT defended abstract set theory is in fact well known and is epitomised in the famous remark that “nobody shall drive us out of Cantor’s paradise”. However, in the same address in which this remark occurs HILBERT also said that “the Infinite is not realized anywhere; it does not exist in the physical world, nor is it admissible as a foundation for our intelligent thinking-which is a remarkable harmony between Being and Thought.” (ref. 1, trans- 1 a ted) .

In our time, infinitary concepts and, more particularly, refecences to infinite totalities (which are, in our sense, meaningless) have been introduced deliberately also into metamathematics. This is noticeable not only in model theory where these concepts rule supreme but also in proof theory and, in recent years, even in the various generalizations of recursion theory. This is a perfectly natural extension of the formalist approach to mathematics, which has already proved its fruitfulness. However, the tendency has been sharply attacked by SKOLEM (not a formalist) who objected to the use of such dubious methods in the very field whose purpose it is to make the foundations of mathematics secure.

3. I t is not a trivia1 question where to draw the line between infini- tistic and essentially uninterpreted mathematics on one hand and fini- tistic and intuitive mathematics on the other hand. For example, the present writer would agree that any universal sentence of elementary arithmetic - i.e. a sentence in prenex normal form formulated in the first order predicate calculus in terms of addition, multiplication and equality and containing universal quantifiers only-has a direct intui- tive meaning although it refers to an unlimited domain of individuals. But already a single change of quantifiers may take us to the realm of make believe. In addition we have the question what rules of logic belong to the framework that controls our finite thinking and is prior to the development of our formal theories, in other words, what kind of arguments are admissible in our basic-and meaningful-metama- thematics. Logical positivism, in its most consistent form, maintains that all logic, as part of language, is ultimately arbitrary and is regu- lated only by its usefulness in coping with the empirical world. Thus, even the simplest rules of the propositional calculus or the most basic

From a formalist’s point of view 49

laws of arithmetic, such as the commutative law of addition are not regarded as valid a priori but are thought to be replaceable by other arrays of rules with entirely different features. The issue is not merely whether such alternative systems of logic are possible, since the answer to this question is clearly affirmative, but whether they can serve as the ultimate foundation of logical thinking just as our familiar two- valued logic and basic arithmetic. Can we imagine an intelligent (human) being who has a different logic from ours a t all levels of the hierarchy of languages, yet is capable of comprehending our kind of logic, just as we are capable of comprehending his, and who is able to cope with empirical phenomena more or less satisfactorily, like ourselves? The most obvious rival to classical logic which comes to mind here-intui- tionistic logic- is inappropriate since it agrees with the formalist’s classical logic at the level of concrete metamathematics. There are, however, other forms of logic, e.g. some that have been suggested for quantum mechanics, that deserve further consideration in this connec- tion.

T o end on a personal note, while I do not at present adhere to the positivistic view just outlined I cannot exclude the possibility that I will be persuaded by it some day. Equally the development of “mean- ingless” infinitistic theories may at some future date become so unsat- isfactory to me that I shall be willing to acknowledge the greater intellectual seriousness of some form of constructivism. But I cannot imagine that I shall ever return to the creed of the true Platonist, who sees the world of the actual infinite spread out before him and believes that he can comprehend the incomprehensible.

A. Robinson University of California, Los Angeles and Yale University

REFERENCES 1 . D. HILBERT, Uber das Unendliche, Jahresberichte der Deutschen Mathematiket-

vereinigung, vol. 36, 1927, pp. 209-215. 2. A. ROBINSON, Formalism 64, Proceedings of the 1964 International Congress for

Logic, Methodology and Phi’losophy of Science, (Jerusalem), Amsterdam, 1966.