Lewis on Set Theory David Lewis in the short monograph


§2. Objections to Singletons



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§2. Objections to Singletons

Mereology was founded in Poland by the nominalist Stanisław Leśniewski and popularized in the West by the nominalist Nelson Goodman precisely as a partial substitute for the set theory that they as nominalists rejected. Though Lewis is, by contrast, anything but a nominalist, still he does very much sympathize with traditional nominalistically-inclined mereologists' complaints about set theory.

The background is as follows. Cantor's work on trigonometric series led him to move from thinking of the points (plural) where a function misbehaves to thinking the set (singular) of points of misbehavior as a single object, to which operations can be applied, notably the operation of throwing away isolated points. Repeated application of this operation may have the result that only one point remains or none at all, and so it is a simple and natural step to admit singleton or unit sets and a null or empty set as "ideal elements" or limiting or degenerate cases, though in fact singletons and the empty set only really came to play an important role in set theory with Zermelo's axiomatization of the subject.

There is a certain kind of philosopher addicted to quibbling and querulous objections to mathematicians' habitual practice of rounding out systems of entites by positing "ideal objects," and more generally of counting in limiting or degenerate cases. Lewis gives a good impersonation of such a philosopher in some of his remarks on set theory, when he observes that Cantor's definition of set, which in free translation runs "any collection into a whole of definite, well-distinguished sensible or intelligible objects," is difficult to reconcile with reckoning in singletons, and impossible to reconcile with reckoning in an empty set.

Lewis considers a counterargument along the following lines. Suppose we have a stamp collection, complete with a catalogue, and then, finding ourselves in reduced circumstances, have to begin selling off our stamps, deleting their listings from the catalogue. This may go on until we have only one stamp left or none at all, leaving us with a singleton collection or an empty one. We need not even have a physical collection, but just a catalogue, if we switch the example from stamp collections to Facebook selections of "favorites" in one or another category, if as fans we are fickle. Lewis rejects the argument on the grounds that all this talk of "collecting" is merely metaphorical. Well, of course it is: Cantor's notion of set was new — he did not suppose that what appears on the surface to be plural talk of points was deep down secretly singular talk of point-sets all along, and that he was merely making explicit what was already implicit —and a new notion can only be introduced by heuristic metaphors.

One would hardly expect such quibbling over minor ontological assumptions from, of all persons, David Lewis, he of the incredulous-stare-inducing ontology of real, concrete possible worlds. And the querulous objection is in any case pointless, since set theory with the limiting or degenerate cases can easily be interpreted in set theory without them. (As Lewis surely knew, at least by the time of MM, from Hazen 1991.) Indeed, assuming the existence of at least two individuals a and b, the "pure" sets of ordinary set theory can be mapped one-to-one onto those sets x such that x itself, all sets that are elements of x, all sets that are elements of elements of x, and so on, have among their elements a, b, and no other individuals.

The real objection lies elsewhere. Nominalists have traditionally objected less to the part of Cantor's definition quoted so far, than to the additional clause "which are called the elements of the set," with its implication that set-formation is less a process of merger, like that by which Italy was formed from various minor states, than a process of federation, by which thirteen colonies became the United States. The implication, to be more explicit, is that even after the many have been collected together into a one, it is still discernible which many they were: that just as the set is determined by its elements, so also the elements are determined by the set. Mereological fusion, by contrast, obliterates the separate identities of the fused: A single whole can be taken apart in many ways, and there is no one way of taking it apart of which it can be said that the genuine parts of which the set is composed are just those pieces into which it is disassembled when taken apart in that way and no other.

Given Lewis's theses, to be an element of a set or member of class is just to have a singleton that is a part thereof. Grant the notion of singleton, and you have granted the notion of element or member, and the traditionally objectionable part of Cantor's definition. That is the real source of Lewis's objection to singletons, or rather, that together with the observation that many of the categories of metaphysics in the Australian style do not apply in any obvious way to singletons. (Is the relation of a singleton to its single member and internal or an external relation?) But though Lewis grouses and kvetches about such matters almost as much as a Leśniewskian or a Goodmanian might, to the point that one is expecting his discussion to issue in a proposal that if not literally nominalist would at least be in spirit nominalistical, on the contrary he ends by affirming that we must accept set theory like it or not.

For in the most memorable passage in all this material (PC 59, MM 15), Lewis writes that he laughs to think how presumptuous it would be to reject mathematics for philosophical reasons, and goes on the review the "great discoveries" of philosophy in the past, beginning with the proof of the impossibility of motion. This is perhaps a bit unfair, in that natural science after all emerged from natural philosophy; but the point stands that it is comically immodest for the part of philosophy that is still struggling, and therefore still called "philosophy," to seek to "correct" the part of philosophy that has succeeded, and is now called "science," and especially for anything as soft as philosophy to seek to "correct" mathematics, the hardest of the hard sciences. At any rate, with a forceful profession of faith in mathematics — he heads the relevant section "Credo" and might almost have followed the Tertullianists in adding "quia absurdum" — Lewis renounces renunciation of mathematics.

He remains tempted not by renunciation but by reinterpretation of certain kind. The reinterpretation in question is generally known in the contemporary literature as "structuralism," though it goes back (strictly speaking only in the case of arithmetic, though that case is easily adaptable others) to Benacerraf (1965), rather than to any Parisian theoretician fashionable during in the sixties, apart perhaps from a very tenuous link to Bourbaki. The "structuralist" idea, which for Lewis is inspired by his reading of Ramsey, would be this, that instead of accepting a specific singleton-forming function, of philosophically inscrutible nature, simply to posit that there exists at least one function having the properties orthordox set theory ascribes to the singleton function. Lewis fears that even this degree of departure from strict and literal acceptance of set theory might constitute an unacceptable philosophical revisionism, but he perhaps need not have worried so much, for there are historical precedents.

In the seventeenth and eighteenth centuries, leading mathematicians (among them Descartes and Newton) had a more or less definite idea what (positive) real numbers were: ratios of magnitudes, such as lengths. In the nineteenth century, however, mathematicians came to feel that this geometric conception of the continuum needed to be replaced by something more purely arithmetic, and the constructions of Dedekind (his "cuts") and Cantor (equivalence classes of Cauchy sequences) eventually emerged. By the early twentieth century such constructions were beginning to appear in undergraduate textbooks. G. H. Hardy, in his Cambridge freshman calculus textbook Pure Mathematics (in the second edition of 1914 and all subsequent ones), expounds Dedekind's construction, and then remarks that alternatives are possible, and that no great importance should be attached to the particular form of definition he as just finished presenting. He formulates — and quotes Russell as endorsing — the general principle that in mathematics it matters that our symbols should be susceptible to some interpretation, but that if several are possible, it does not matter which we choose. Hardy's principle would seem to be just as applicable to set theory as to the calculus, and if so one has it on very high mathematical authority that there is nothing objectionable in the course that so tempts Lewis. Let us, in any case, see what that course involves.




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