Tuesday, December 9, 2014

Prequel to one or more posts on my modal-logical interpretation of mathematics
In 1976, when I delivered the John Locke Lectures at Oxford, I often spent time with Peter Strawson, and one day at lunch he made a remark I have never been able to forget. He said, "Surely half the pleasure of life is sardonic comment on the passing show".  This blog is devoted to comments, not all of them sardonic, on the passing philosophical show.
Hilary Putnam

For readers of the post (or posts) that will follow this one I repeat a section from chapter 11 of my Philosophy in an Age of Science (pp. 223-226) that briefly explains what the modal logical interpretation of mathematics was. (The interested reader is strongly advised to also read the section of that chapter that follows the one below (ibid, pp. 227-233), The concept of a “Set”.)

The “modal-logical interpretation”

In “Mathematics Without Foundations,” a paper I published in The Journal of Philosophy in 1967, I proposed an interpretation of mathematics (and particularly of set theory) that I called “mathematics as modal logic.” [In recent decades the modal interpretation has been developed in detail by Geoffrey Hellman.[i] In “Mathematics Without Foundations” I explained the idea thus:[ii]
 My purpose is not to start a new school in the foundations of mathematics (say, “modalism”). Even if in some contexts the modal-logical picture is more helpful than the mathematical-objects picture, in other contexts the reverse is the case. Looking at things from the standpoint of many different “equivalent descriptions,” considering what is suggested by all the pictures, is both a healthy antidote to foundationalism and of real heuristic value in the study of scientific questions.
 Now, the natural way to interpret set-theoretic statements in the modal-logical language is to interpret them as statements of what would be the case if there were standard models for the set theories in question. Since the models for Zermelo-Fraenkel set theory and its strengthenings are also models for Zermelo set theory, let me concentrate on Zermelo set theory. In order to “concretize” the notion of a model, let us think of the model as a graph. The “sets” of the model will then be pencil points (or some higher-dimensional analogue of pencil points, in the case of models of large cardinality), and the relation of membership will be indicated by “arrows.” (I assume there is nothing inconceivable in the idea of a space of arbitrarily high cardinality[iii]; so models of this kind need not be denumerable, and may even be standard.) The model will be called a “standard concrete model” if (1) there are no infinite-descending “arrow” paths; and (2) it is not possible to extend the model by adding more “sets” without adding some “ranks” in the model[iv]. If we regard the “points” and “arrows” of the graph as possible individuals, then the statement that a graph G is a standard concrete model for Zermelo set theory can be expressed using no “non-nominalistic” notions except the “□” [the modal-logical symbol for “it is necessarily the case that”].
 If S is a statement of bounded rank, and if we can characterize the “given rank” in question in some invariant way (invariant with respect to standard models of Zermelo set theory), then the statement S can easily be translated into modal-logical language. The translation is just the statement that if G is any standard model for Zermelo set theory and G contains the invariantly characterized rank in question, then necessarily S holds in G. (It is trivial to express “S holds in G” for any particular S without employing the set-theoretic notion of “holding.”) Our problem, then, is how to translate statements of unbounded rank into modal-logical language.
 The method is best indicated by means of an example. If the statement has the form (x)(Ey)(z)Mxyz, where M is quantifier-free, then the translation is this:

Necessarily: If G is any standard concrete model for Zermelo set theory and if x is any point in G, then it is possible[v] that there is a graph G' that extends G and is a standard concrete model for Zermelo set theory and a point y in G' such that (if G'' is any standard concrete model for Zermelo set theory that extends G' and z is any point in G'', then Mxyz holds in G'').
I do not propose the modal-logical interpretation as a step to arguing that numbers don’t really exist, as the title of Hellman’s book “Mathematics Without Numbers” unfortunately suggests. The idea that we are saying something false when we say things like “There is always a prime between n and 2n” seems obviously wrong to me. And I have argued elsewhere that the idea that using quantifiers always “commits us” to the existence of objects is at best unclear and at worst hopelessly misleading. Nor, by the way, do I think that the modal-logical interpretation yields an “epistemology” for mathematics. On that (the epistemology of mathematics), I agree with a view that John Burgess has expressed, that the best way to find out how mathematical knowledge is obtained is to look at what mathematicians do. I suspect that Burgess is also right in thinking that what we will find won’t fit any epistemological picture so far proposed. So why do I like the modal-logical interpretation?
Well, one thing I like about it is that it yields a natural resolution of Benacerraf’s famous problem about “what numbers could not be” and its generalizations. Benacerraf’s Problem is that while the natural numbers can, as is well known, be identified with sets—e.g., with the von Neumann ordinals, Ø,{Ø},{Ø,{Ø}},{Ø,{Ø},{Ø{Ø}}},...—they can be identified with sets in infinitely many ways. [For example they could also be identified with the Hao Wang ordinals, Ø,{Ø},{{Ø}},{{{Ø}}}...] And to stamp one’s feet and insist that “the natural numbers are not identical with sets at all” seems a bit arbitraryin set theory we do often identify them with progressions of sets after all, from the first days of modern mathematical logic.
Shall we then say that quantification over natural numbers is a sort of deliberately ambiguous quantification over the successive elements of any infinite series of sets (any “ω-sequence”) you like? The problem, as I am sure Benacerraf well knows, is that the same problem arises with quantification over sets. Sets can, after all, be identified with “characteristic functions” (functions whose range is {0,1}), as is standardly done in a good deal of recursion theory and hierarchy theory, for example. On the other hand, functions can be identified with ordered n-tuples. And ordered n-tuples can be identified with sets in infinitely many different ways. Can there really be a “fact of the matter” as to whether sets are a kind of function or functions are a kind of set? Can there really be a “fact of the matter” as to what the “correct” definition of “ordered pair” is? The mind boggles.
If, however, quantification over sets, functions, etc., is simply quantification over possibilia and not over actually existing entities, then the problem disappears in the sense that all of the different “translations” of number theory into set theory, and all of the different translations of set theory into function theory, and all of the different translations of function theory into set theory, are just different ways of showing what structures have to possibly exist in order for our mathematical assertions to be true. In my view, then, what the modal-logical “translation” of a mathematical statement gives us is a statement with the same mathematical content which does not have even the appearance of being about the actual existence of “immaterial objects.”





[i] Geoffrey Hellman, Mathematics Without Numbers; Towards a Modal-Structural Interpretation (Oxford: Clarendon Press, 1989)
[ii] Hilary Putnam, “Mathematics without Foundations,” Journal of Philosophy, 64, 1 (1967): 5-22; reprinted in Mathematics, Matter and Method (Cambridge: Cambridge University Press, 1979), pp 43-59; quotation from p. 57. I have abbreviated the passage and also slightly revised the wording).
[iii] In the original I had “nothing inconceivable in the idea of a physical space of arbitrarily high cardinality.” “Physical” was a blunder; nothing hangs on whether a such spaces could or could not be “physical,” what matter is only their mathematical possibility.
[iv] In Zermelo-Frankel set theory and its successors, the ranks are defined thus: the collection of all hereditarily finite sets (the finite sets all of whose members, and members of members. and members of members of members and so on are also finite) are rank 0, and the set of all subsets of a given rank k (its “power set”) is rank k+1. When λ is a limit ordinal, rank λ is defined to be the union of all the ranks below λ. Thus ranks, unlike Russellian “types,” are cumulative the sets of a given rank also belong to all higher ranks.
[v] The modal-logical notion “it is possible that p” (symbolized “◊p”) is defined as “~□~p.”

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