Indispensability Arguments—What Mine Was, and Are They Necessary

If one consults the Stanford Encyclopedia of Philosophy on the topic “Indispensability Arguments in the Philosophy of Mathematics,”[1] one finds (as part of a moderately lengthy entry written by Mark Colyvan) the following statements:

From the rather remarkable but seemingly uncontroversial fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine...[2] and Putnam...[3] have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities.. ..This argument is known as the Quine-Putnam indispensability argument for mathematical realism.

From my point of view, Colyvan’s description of my argument(s) is far from right. In “What is Mathematical Truth” what I argued was that the internal success and coherence of mathematics is evidence that it is true under some interpretation, and that its indispensability for physics is evidence that it is true under a realist interpretation— the antirealist interpretation I considered there was Intuitionism. This is a distinction that Quine nowhere draws. It is true that in Philosophy of Logic I argued that at least some set theory is indispensable in physics as well as logic (Quine had a very different view on the relations of set theory and logic, by the way), but both “What Is Mathematical Truth?” and “Mathematics without Foundations” were published in Mathematics, Matter and Method together with “Philosophy of Logic,” and in both of those essays I said that set theory did not have to be interpreted Platonistically. In fact, in “What Is Mathematical Truth?”[4]1 said, “the main burden of this essay is that one does not have to ‘buy’ Platonist epistemology to be a realist in the philosophy of mathematics. The modal logical picture shows that one doesn’t have to ‘buy’ Platonist ontology either.” Obviously, a careful reader of Mathematics, Matter and Method would have had to know that I was in no way giving an argument for realism about sets as opposed to realism about truth values on a modal interpretation.

Unlike my argument in “What is Mathematical Truth”, Davis’s argument against Godel’s version of “Platonism” does not mention “indispensability for physics”, and this raised for me the question I mentioned at the beginning of this essay, the question as to whether my “indispensability argument is needed, or whether the considerations Davis offers in favor of regarding mathematical truth as objective are actually sufficient.” To discuss this question, we have to return to the notion of objectivity.

Assuming that Davis and I are on the same wavelength with respect to that notion, and recalling that antirealist philosophies of mathematics all identify truth with provability, in one sense or another of “provability”, this reduces to the question as to whether his arguments really rule out the possibility that mathematical truth is the same thing as provability. Let us begin with two clarifications.

First, the question isn’t whether mathematical truth = provability in some one fixed formal system that “we can see to be correct”. Even before the Godel Incompleteness Theorems were proved, Brouwer’s Intuitionism did not depend on assuming—in fact, Brouwer didn’t believe—that constructive provability could be captured by any one formal system. And after the Godel theorems were proved, Turing thought that we can see from Godel’s argument that reflection[5] on any formal system that is strong enough for arithmetic and that we can intuitively see to be correct will enable us to find a more powerful system, in fact a constructive transfinite sequence of stronger and stronger systems, such that a proof in any one of them would still intuitively count as an acceptable “mathematical proof”.[6] (However, it had better not be possible to “see from below” just how far up the hierarchy of constructive ordinals such metamathematical reflection can take us, if we are not to run into contradiction.) So there may be a sense of “proof” in which what is “provable” outruns what is formally provable in one system that we can see to be correct.

And the Godel results are not enough to exclude identification of mathematical truth with provability is such a sense. Do Davis’s arguments exclude such an identification?

I assume they are meant to. In philosopher’s jargon, views according to which mathematical truth is just provability (from axioms human mathematicians can see to be correct) count as antirealist and I have been assuming that, like me, Davis is a realist. It would be a disappointment to find out he isn’t, and I have been laboring under a serious misconception!

One reason for supposing that Davis is not an antirealist is that he clearly thinks that our means of mathematical discovery are often “inductive”, that is, they are not just deductions from self-evident axioms. Indeed, that is the main point of his essay. So I am not seriously worried that I have misunderstood ’’Pragmatic Platonism”. But the question now arises in another form: if “Pragmatic Realism” is an argument against identifying truth with provability is the argument good enough? In “What is Mathematical Truth” I had written that “the consistency and fertility of classical mathematics is evidence that it—or most of it—is true under some interpretation. But the interpretation might not be a realist interpretation.”[7] And I went on to rule out this possibility with the aid of two arguments I have already mentioned: the indispensability argument and the argument that an antirealist interpretation of pure mathematics does not fit together with a scientific realist interpretation of physics. Was this last step actually unnecessary?

What Davis’s arguments show is that mathematicians do not proceed by proof alone. They also use “induction”, that is, quasi-empirical methods. But does the fact that mathematical discovery is first made—in many cases—without formal proof show that correctness of the result isn’t simply provability? Arguably, Torricelli’s results were correct in the sense that, and only in the sense that, they were provable from acceptable axioms, even if Torricelli himself didn’t have either the proof or the axioms.

Well, both Davis and I (in “What is Mathematical Truth”) mention that new axioms sometimes get accepted in mathematics. But (1) this does not happen very often; and (2) when it does happen, it happens because the new axioms are appealing for mathematical reasons. The indispensability argument considers the need for an interpretation of the mathematical concepts when they function in empirical science, and argues that antirealism has no satisfactory account of this. Davis’ argument considers only what goes on in pure mathematics. It certainly confirms a claim I made in “What is Mathematical Truth”, the claim that mathematics does not only use deduction, but is full of quasi-inductive elements. But is that something an antirealist need be impressed by? Couldn’t an antirealist say that Davis’s essay has to do with the context of discovery, and, perhaps, has also to do with an often-unrecognized context of “quasi-empirical justification”, but not with the question of realism. If mathematics yields, as Davis says, “objective information”, is that “objective information” about more than what we humans count as proof? We both think it is about more than that, but I still think that the success of applied mathematics needs to be brought into the picture in order to make the best case. But I look forward happily and affectionately to Davis’s response.

  • [1] Mark Colyvan, “Indispensability Arguments in the Philosophy of Mathematics,” in E.N. Zalta, ed.,The Stanford Encyclopedia of Philosophy (Fail 2004 Edition), Colyvan is also the author of The Indispensability of Mathematics(Oxford: Oxford University Press, 2001).
  • [2] The author of this entry, Mark Colyvan, is referring to W.V. Quine, “Carnap and Logical Truth,”reprinted in The Ways of Paradox and Other Essays, revised edition (Cambridge, Mass.: HarvardUniversity Press, 1976), 107-132 and in Paul Benacerraf and Hilary Putnam, eds., Philosophy ofMathematics, Selected Readings (Cambridge: Cambridge University Press,1983), 355-376; W.V.Quine, “On What There Is,” Review of Metaphysics, 2 (1948): 21-38; reprinted in From a LogicalPoint of View (Cambridge, Mass.: Harvard University Press, 19802), 1-19; W.V. Quine, “TwoDogmas of Empiricism,” Philosophical Review, 60, 1 (January 1951): 20-43; reprinted in hisFrom a Logical Point of View (Cambridge, Mass.: Harvard University Press, 1961), 20-46; W. V.Quine, “Things and Their Place in Theories,” in Theories and Things (Cambridge, Mass.: HarvardUniversity Press, 1981), 1-23; W.V. Quine, “Success and Limits of Mathematization,” in Theoriesand Things (Cambridge, Mass.: Harvard University Press, 1981), 148-155.
  • [3] Colyvan is referring to “What is Mathematical Truth” and Hilary Putnam, Philosophy ofLogic(New York: Harper and Row, 1971), reprinted in Mathematics, Matter and Method: PhilosophicalPapers Vol. 1, 2nd edition, (Cambridge: Cambridge University Press, 1979), 323-357.
  • [4] “What is Mathematical Truth?”, 72.
  • [5] “Reflection” here denotes producing a stronger system by adding a consistency statement for agiven system. If the systems are indexed by notations for constructive ordinals—that is, elementsof a recursive well-ordering—and the ordering is already proved to be a well-ordering, one cancontinue “reflection” into the transfinite, when one comes to a “limit notation” by adding a suitablyformalized statement to the effect that the union of the systems with indexes below the limit notationis a consistent system.
  • [6] Turing, A.M. (1939), ‘Systems of Logic Based on Ordinals’, Proceedings of the London Mathematical Society, Ser. 2 45, pp. 161-228.
  • [7] “What is Mathematical Truth”, 73.
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