Ramsey and the Vienna Circle 03

M ICHAEL D UMMETT

THE VICIOUS CIRCLE PRINCIPLE According to Frank Ramsey, and likewise to Gödel, the validity of Russell’s Vicious Circle Principle depends on whether mathematical objects exist independently of us in an abstract realm, or whether they are human creations, brought into being by intellectual constructions we have effected and sustained in being by our ability to repeat those constructions. (A work of fiction or a poem is a human creation, but, once written down or printed, is not sustained in being by human intellectual activity.) The example always given to illustrate the former alternative is that of picking out a particular man in a room as the tallest man in the room. This specification involves quantifying over the set of men in the room, a set of which the individual thus picked out is a member. This apparently violates the Vicious Circle Principle, but is quite evidently legitimate. On the diagnosis of Ramsey and of Gödel, its legitimacy derives from the fact that the men in the room exist independently of any observer or commentator. It is at first sight paradoxical to hold that, in order to determine the validity of a logical principle such as the Vicious Circle Principle, we should first have to settle a grand metaphysical question such as the ontological status of mathematical objects. Surely the validity of the Vicious Circle Principle for any given domain of quantification depends, not on the solution of so large a metaphysical problem, but on how that domain is to be specified. If we try to specify it by appeal to quantification over that very domain, we shall have violated the Vicious Circle Principle: we shall have committed a genuinely vicious circle. Quantification over a domain assumes a prior conception of what belongs to that domain: by trying to specify what belongs to the domain by using quantification over that same domain, we assume as already known what we are attempting to specify. But if we can specify the extent of the domain in some manner which determines the scope of quantification over it without assuming that as already given and available for use in our specification, we shall be free to pick out a particular element of the domain by quantifying over it. Suppose, for example, that we start with some first-order theory, our individual variables ranging over some domain, such as the real numbers, which we take to be well defined. We want to extend our theory by expanding the language to admit a new sort of variables for classes of elements of the original domain. If we explain the notion of such a class predicatively, that is, as associated with a formula of the original unextended theory with one free variable, membership of the associated class being determined by satisfaction of the corresponding formula, there will be no circularity in our manner of specifying the domain of the

29 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 29–33. © 2006 Springer. Printed in the Netherlands.

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new class-variables. But if we were to make the classes correspond in the same sense to formulas of the expanded language, we should by this impredicative specification have violated the Vicious Circle Principle, since formulas of the expanded language will include ones involving quantification over classes; our explanation will indeed be viciously circular. If, on the other hand, we specify that the class-variables are to range over every possible function from elements of the base domain – real numbers, in the example – to the truth-values true and false, we escape this difficulty: we may specify a particular class by quantification over classes. There is much room for doubt whether the intended sense of “possible” is a legitimate or truly comprehensible one, namely, when we speak of all possible functions from real numbers to truth-values, that is, of all possible determinations, for every real number, of whether it belongs to the associated class or not. But there can be no doubt that, if it is legitimate, we shall in this way have evaded the Vicious Circle Principle. Why, then, do Ramsey and Gödel insist that the validity of the Vicious Circle Principle depends upon whether mathematical objects exist independently of us or are created by us in thought? Their idea is surely this. When we wish to specify a domain of quantification consisting of physical objects, all we need to do is to select a suitable concept, in Frege’s sense of “concept”. A Fregean concept is determined by a precise condition for an arbitrary object to fall under the concept. If we wish to quantify over the mammals in the London Zoo, we shall select the concept mammal in the London Zoo; if we wish our domain of quantification to comprise just the elephants in the London Zoo, we shall select the concept elephant in the London Zoo. Making a choice of the relevant concept is all that we need to do in order completely to determine the domain of quantification. We do not need, in addition to selecting the concept that fixes the condition for membership of the domain, to lay down what objects there are which fall under the concept, or how many of them exist: external reality does that for us. Sometimes, indeed, it is like this in the mathematical realm. If, for some reason, we wanted to quantify over prime numbers of the form 2n - 1, we should need simply to specify that our domain was to consist of numbers falling under the concept prime number equal to 2n - 1 for some n. Reality would, as it were, run through the numbers 2n -1 for n = 0, 1, 2, 3, ..., checking each one to see whether it was prime or not, and consigning each such prime number to our domain. We can leave reality to separate out the elements of our domain because we are forming that domain from within a larger, already determinate, domain, that of the natural numbers. The natural numbers form the zoo from among whose denizens we pick out the elements of our new, smaller, domain. If, however, we want our domain to consist of the real numbers, or of the ordinal numbers, the matter appears quite differently to most of us. Associated with the expressions “real number” and “ordinal number” are concepts, just as there are with the words “mammal” and “elephant”. The concept real number is determined by the condition that a mathematical object that is given to us by

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means of some definition or construction rollst satisfy if we are to recognise it as being a real number; we might take this condition to be that of having a determinate position with respect to the rationals, categorising each rational number as being less than or equal to it or greater than it, and some as one and some as the other. (The final phrase is to exclude infinitary end-points of the rational line.) Similarly, the concept ordinal number is determined by the condition for a given mathematical object to be recognised by us as being an ordinal number; we might take this condition to be that of being the order-type of some well-ordered sequence. Such concepts do not, however, appear to most of us to be adequate to determine a domain of quantification. We do not seem to be in a position to require reality to run through all the mathematical objects that there are in order to decide of each one whether it falls under the concept real number or not, or under the concept ordinal number or not. If we are not in that position, then we must do more to fix a definite domain of quantification than to specify a concept under which the elements of our domain must fall. We must also stipulate a criterion for the existence of objects that are to be elements of our domain, or at least for their belonging to the domain over which we intend to quantify. Why this difference? Why do we think that we need to do more than simply to specify the concept under which the elements of our domain are to fall? The reason is that only very few of us are full-fledged realists about the mathematical realm. If we were, we should assume that there is an absolutely determinate totality of mathematical objects, as determinate as the totality of molecules in a particular glass of water or of monkeys in a certain jungle at a given time. In such a case, it would be enough, in order to specify a domain of quantification, to select a concept under which some mathematical objects fall and others do not; for then reality could decide whether any given object fell under the concept as well as we can, once the object is given to us. If we were total or full-fledged realists about the mathematical realm, that is to say, if we interpreted it in just the same way as realists about the physical world interpret that world, we should think that we could specify a domain of quantification simply as consisting of those mathematical objects that fall under the concept real number. It would consist of just those objects that effect a Dedekind cut in the rational line and do nothing else. Whether or not it would contain an element effecting every possible such cut, in the sense intended by platonists who speak of “every possible Dedekind cut”, would depend upon just which mathematical objects reality comprises – which mathematical objects the Creator has chosen to bring into existence. It would be unnecessary, from this uncompromisingly realist standpoint, to specify the domain by speaking of every possible Dedekind cut: rather, it would consist of every mathematical object that there actually was which effected such a cut. Similarly, we could specify a domain as consisting of all the ordinal numbers there are. There need be no fear that this would lead to contradiction via the Burali-Forti paradox. We could modify the concept ordinal number so as to apply only to those order-types of

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well-ordered sequences that had a successor. There would then be just one wellordered sequence that had no ordinal number as its order-type – the sequence of all ordinal numbers; or we might hold that this sequence has no order-type. But this would involve no contradiction. We – or at any rate, all but a small minority of us – are not normally disposed to think of the existence of mathematical objects in this resolutely full-fledged realist manner. Think, for example, of the odd impression made on us by the question whether there exist large cardinals of some particular type. The question may be intended as asking whether it is consistent to assume the existence of such large cardinals. But, if more is being asked than this, what can the question mean? If it is allowed that there may be no contradiction in postulating the existence of large cardinals of that type, but held that there still remains a question whether there really are such cardinals, what could determine the answer to such a question? It is not, surely, a matter of whether God chose to create such cardinals. Kronecker told us that the existence of any mathematical objects other than the natural numbers is the work of man. The existence of monkeys of one or another kind is a contingent matter, a matter of what God has chosen to create. But the existence of mathematical objects should surely be a matter of necessity, of what we could not have found to be otherwise. How should we determine what mathematical objects exist of necessity save by fastening upon same criterion of our own for their existence? Hilbert held that those mathematical objects exist whose existence may be consistently assumed. The mathematical realm, on this view, is maximally full: there are all the mathematical objects that there can be. One who believes that all mathematical theories can be captured in a first-order formalisation may appeal to the completeness theorem in support of this. But this consistency criterion of existence is itself inconsistent. The existence of objects of each of two types may be consistently postulated, and yet a contradiction may result from postulating the existence of both together. It is presumably consistent to assume the existence of the sets which constitute a model of Quine’s NF, but that cannot be combined with the existence of the sets required by ZF, that is, if both are to be sets in the same sense. If we do not think that a first-order axiomatisation can embody every one of our mathematical conceptions, we must allow that there are consistent theories which hold good only if there are only finitely many objects altogether, and others which have only infinite models. Theories which are individually consistent may not be collectively so. I think that full-fledged realism about mathematical objects is a view that is barred to us. We cannot explain what, on such a view, determines which mathematical objects exist; we cannot treat their existence as a matter of what reality holds. I cannot claim that no one is a full-fledged mathematical realist. Professor Timothy Williamson has explained to me in personal correspondence that he hesitates between structuralism and full-fledged realism. In his view, quantification over the elements of a domain has always to he understood as quantification over every object there is, restricted in the standard way by some suitable

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predicate (“is an elephant” or “is a real number”, etc.). Thus if mathematical realism is correct, it must be full-fledged realism. This is certainly a minority view, and I think it untenable. If I am right, we are in no position to declare the Vicious Circle Principle generally irrelevant, as it is when we are quantifying over physical objects The first diagnosis was the right one The Vicious Circle Principle may prohibit us from specifying a domain of quantification in certain ways; it may not apply when we specify the domain in other ways. It is not a matter of whether the elements of the domain exist independently of us or are created by us in thought. It is a matter of the means we adopt to specify that domain and lay down what its elements are to be.

Sir Michael Dummett 54 Park Town Oxford OX2 6XJ UK