On Davis’s “Pragmatic Platonism”
Abstract (added by editor). A comparison is made between Martin Davis’s realism in mathematics and the forms of mathematical realism defended by Hilary Putnam. Both reject the idea that mathematics should be interpreted as referring to immaterial objects belonging to a “second plane of reality” and put emphasis on the use of quasi-empirical arguments in mathematics. The author defends Hellman’s use of the formalism of modal logic to explicate his own modal realism.
Keywords Mathematical truth • Objectivity • Modal realism • Indispensability argument • Consilience
When Martin Davis learned that I was going to write about his fascinating essay,1 he emailed me as follows, “Reading your old ‘What is Mathematical Truth?’2 (which was new to me) it seems to me that the position expressed there is pretty close to what I was suggesting.” That (1975) essay did, in fact, say some things that jibe with what Davis was to write in “Pragmatic Platonism”, and I still believe those things. Does that mean I agree with “Pragmatic Platonism”? It does. Not only does it formulate Davis’s (and my) view that mathematics includes (but, of course, does not solely consist in) what I called “quasi-empirical” (and Davis calls “inductive”) arguments in a remarkably clear way, it argues persuasively that, while this is something Godel 1Martin Davis published “Pragmatic Platonism” online:
http://foundationaladventures.files.wordpress.com/2012/01/platonic.pdf; shortly after I completed the present essay, Davis sent me an expanded (forthcoming) version, “Pragmatic Realism; Mathematics and the Infinite”, in Roy T. Cook and Geoffrey Hellman (eds.), Putnam on Mathematics and Logic (Cham, Switzerland: Springer International Publishing, forthcoming). The online version was read at a conference celebrating Harvey Friedman’s 60th birthday. All the passages from “Pragmatic Platonism” I quote here are retained verbatim in the expanded version.
2“What is Mathematical Truth?” Historia Mathematica 2 (1975): 529-543. Collected in my Mathematics, Matter and Method (Cambridge: Cambridge University Press, 1975), 60-78. The expanded version of Davis’s “Pragmatic Platonism” referred to in the previous note contains a fine discussion of “What is Mathematical Truth”, for which I am grateful.
H. Putnam (B)
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too believed, and while it is a support for what one might call realism with respect to mathematics, one does not have to be a Godelian Platonist to agree with this. ‘Pragmatic Platonism’ is realism enough. This claim is what I shall write about here.
The structure of this essay is as follows: I first describe the form of realism that Martin defends (being prepared, of course, to learn that I misunderstand it), and say something about the sort of realism (“modal realism”) that I defended in a “Mathematics Without Foundation” as well as in “What is Mathematical Truth”, and have subsequently gone on to elaborate and defend (together with Geoffrey Hellman who has brilliantly worked out the details), and discuss the relation between Davis’s view in “Pragmatic Platonism” and the views I defend. I will close by describing an argument I have given in the past for realism with respect to mathematics, an argument that has been misdescribed as the “Quine-Putnam indispensability argument”, and close by discussing a question that occurred to me on reading “Pragmatic Platonism”, 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.
-  “Mathematics without Foundations,” Journal of Philosophy 64.1 (19 January 1967): 5-22. Collected in Mathematics, Matter and Method, 43-59. Repr. In Paul Benacerraf and Hilary Putnam(eds.). Philosophy of Mathematics: Selected Readings, 2nd ed. (Cambridge: Cambridge UniversityPress, 1983), 295-313.
-  In “Set Theory, Replacement, and Modality”, collected in Philosophy in an Age of Science (Cambridge, MA: Harvard University Press, 2012), and “Reply to Steven Wagner”, forthcoming in ThePhilosophy of Hilary Putnam (Chicago: Open Court, 2015).
-  Geoffrey Heilman, Mathematics without Numbers (Oxford: Oxford University Press, 1989).
-  For a description of the argument and its misunderstandings see my “Indispensability Argumentsin the Philosophy of Mathematics”, in Philosophy in an Age of Science, 181-201.