Relations Between Davis’s and My Forms of Mathematical Realism

Strange as it may seem, the largest difference between Davis’s “Pragmatic Platonism” and my “Mathematics without Foundations” and its “modal logical interpretation” is not metaphysical but mathematical. It is not metaphysical, because both essays, my 1967 essay and Davis’s recent on-line essay, reject the idea that mathematics must be interpreted as referring to immaterial objects, a la either Godel or Quine.[1] Both essays argue that mathematics can and does discover objective truths about what would be the case if certain abstract structures were real, and that the success of mathematics and the consilience of results obtained by different mathematical methods, including ones whose justification is “quasi-empirical” (my term) or “inductive” (Davis’s term), justifies the belief that this is so. This is common ground between us, and it is substantial.

However, my “Mathematics without Foundations” sketched a program for “translating” assertions that quantify over set into explicitly modal statements, a program carried out and then extended in new directions by Hellman; a program that suggests new ways of motivating key axioms of set theory and some of its large cardinal extensions, while Davis’s brief essay basically leaves set theory as it is.

This difference exists because already in “Mathematics without Foundations” I was concerned to be consistent with the idea, that I believe to be correct, that there is no such thing as “the totality of all sets”—not even in a “hypothetical world”. Any hypothetical world (to use Davis’s language) of sets is only an initial segment of another possible world of sets. Possible models of set theory are inherently extendable. (This is the idea that led to Hellman’s current efforts to deploy extendability principles to motivate possible existence of large cardinals in standard models of set theory.) The key idea of my “Mathematics without Foundations” was to reformulate statements of set theory that are “unbounded”, in the sense of quantifying over sets of all ranks, without assuming the existence of even a possible totality of such sets. As Hellman describes[2]:

Putnam took initial steps in illustrating how modal translation would proceed without falling back on set-theoretic language normally associated with “models.” To this end, he introduced models of Zermelo set theory as “concrete graphs” consisting of “points” and “arrows” indicating the membership relation, so that, except for the modal operators, the modal logical translation required only nominalistically acceptable language. Finally (in this brief summary), Putnam proposed an intriguing translation pattern[3] for set-theoretic sentences of unbounded rank (standardly understood as quantifying over arbitrary sets of the whole cumulative hierarchy or set-theoretic universe) in which all quantifiers are restricted to items of a (concrete, standard[4]) model but the effect of “unbounded rank” is got by modally quantifying over arbitrary possible extensions of models.

But it is time to return to philosophy.

Both the differences and the similarities between Davis’s views and mine, in the essays I have been discussing stem from the fact that Davis considers whole hypothetical worlds without discussing relations (such as one world’s being an extension of another) between these “worlds”.

On the side of “similarities”. First, there is the already-emphasized fact that both of us believe that it is right to recognize the objectivity, the “there-being-a-fact-of- the-matter”, of mathematical statements, and that this is realism enough. Coupled with rejection of the claim that all mathematical knowledge is apriori,[5] and our (independent) emphasis on the use of quasi-empirical argument in mathematics, this is a large measure of agreement indeed.

It may seem to be a difference that I worry about Benacerraf’s problem (or paradox) and Davis does not, but in fact once one gives up the idea that talk of numbers and sets is talk of real objects in favor of a conception of them as elements of a possible (or “hypothetical”) model, there is no call to worry about which otherwise-specified object the number two is (or the square root of minus one is, although that one did worry British algebraists for a hundred years[6]). Such problems simply disappear.

What may be a difference is that Davis only worries about pure mathematics, and from the beginning I am concerned with finding an interpretation of mathematical truth that is consistent with a scientific realist interpretation of empirical statements, of what I called “mixed statements” above. How to interpret mixed statements in a modal-logical framework is something that Hellman and I discussed over the years, and some of the most ingenious work in Mathematics without Numbers is devoted to it. For example, in applied mathematics one needs to talk (to put it heuristically[7]) about possible worlds in which the physical objects are as they actually are, not about possible worlds simpliciter, and formalizing this is non-trivial. And I repeat, extension relations among possible worlds (or “hypothetical worlds”, or “idealized worlds”, or whatever you want to call them), and between possible worlds and the actual world, need to be considered if potentialist approaches are to be spelled out in a rigorous way.

  • [1] Quine is often described as a “reluctant” Platonist because of statements like this one: “I have feltthat if I must come to terms with Platonism, the least I can do is keep it extensional”, Theories andThings (Cambridge, MA: Harvard University Press, 1990), 100.
  • [2] Hellman, ibid, second page (the page proofs I have seen do not indicate the forthcoming pagenumbers).
  • [3] An example of my translation method (from “Mathematics without Foundations”) is this: If thestatement has the form (x)(Ey)(z)Mxyz, where M is quantifier-free, then the translation is:Necessarily: If G is any graph that is a standard model for Zermelo set theory and if x is any pointin G, then it is possible that there is a graph G' that extends G and is a standard concrete modelfor Zermelo set theory and a point y in G' such that ? (if G" is any standard concrete model forZermelo set theory that extends G' and z is any point in G", then Mxyz holds in G").
  • [4] A model of Zermelo (or Zermelo-Fraenkel) set theory is standardjustincase (1) it is well-founded(no infinite descending membership chains), and (2) power sets are maximal.
  • [5] Actually, I believe that all so-called “a priori” truths presuppose a background conceptual system,and that no conceptual system is guaranteed to never need revision. For this reason, I prefer tospeak of truths being conceptually necessary relative to a conceptual background. I would not besurprised if Martin Davis agreed with this.
  • [6] Menahem Fisch, “The Emergency Which has Arrived: The Problematic History of 19th CenturyBritish Algebra—A Programmatic Outline”, The British Journal for the History of Science, 27:247-276, 1994.
  • [7] Officially, Hellmann and I avoid literal quantification over possible worlds or possibilia, relyingentirely on modal operators that officially we avoid literal quantification over possible worlds orpossibilia, relying entirely on modal operators.
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