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.
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Notes
- 1.
Martin 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 60\(\mathrm{th}\) 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.
- 3.
“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 University Press, 1983), 295–313.
- 4.
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 The Philosophy of Hilary Putnam (Chicago: Open Court, 2015).
- 5.
Geoffrey Hellman, Mathematics without Numbers (Oxford: Oxford University Press, 1989).
- 6.
For a description of the argument and its misunderstandings see my “Indispensability Arguments in the Philosophy of Mathematics”, in Philosophy in an Age of Science, 181–201.
- 7.
The method of indivisibles was invented by Bonaventura Cavalieri in 1637.
- 8.
For a detailed account, see Kanamori, Akihiro (2004), “Zermelo and set theory”, The Bulletin of Symbolic Logic 10 (4): 487–553, doi:10.2178/bsl/1102083759, ISSN 1079-8986, MR 2136635.
- 9.
David Lewis, On the Plurality of Worlds (Oxford: Blackwell, 1986).
- 10.
Here I am going by Davis’s reference to the use of Grothendieck ’s infinity topoi by Wiles and Taylor in their proof of Fermat’s “Last Theorem”.
- 11.
By “consilience ” I mean that the results are not only consistent, but that they extend one another, often in unexpected directions.
- 12.
The relevant publications are, in addition to the already mentioned “What is Mathematical Truth” and “Mathematics without Foundations”, are “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 The Philosophy of Hilary Putnam (Chicago: Open Court, 2015).
- 13.
Brouwer’s Intuitionism was my example of an interpretation that is incompatible with scientific realism in “What is Mathematical Truth”, 75.
- 14.
Gödel’s Platonism is a prototypical “objectualist” interpretation, but the term “intangible objects” was used by Quine in Theories and Things, (Cambridge, MA: Harvard University Press, 1981), 149.
- 15.
For a fine defense of the claim that a statement can be true but unverifiable, see Tim Maudlin “Confessions of a Hard-Core, Unsophisticated Metaphysical Realist”, forthcoming in The Philosophy of Hilary Putnam. Maudlin rightly criticizes me for giving it up in my “internal realist” period (1976–1990); after I returned to realism sans phrase in 1990 I defended the same claim in a number of places, e.g. “When ‘Evidence Transcendence’ Is Not Malign: A Reply to Crispin Wright,” Journal of Philosophy 98.11 (November 2001), 594–600.
- 16.
Paul Benacerraf (1965), “What Numbers Could Not Be” Philosophical Review Vol. 74, pp. 47–73.
- 17.
Quine is often described as a “reluctant” Platonist because of statements like this one: “I have felt that if I must come to terms with Platonism, the least I can do is keep it extensional”, Theories and Things (Cambridge, MA: Harvard University Press, 1990), 100.
- 18.
Hellman, ibid, second page (the page proofs I have seen do not indicate the forthcoming page numbers).
- 19.
An example of my translation method (from “Mathematics without Foundations”) is this: If the statement 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 point in G, then it is possible 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 \(\Box \) (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 \(G''\)).
- 20.
A model of Zermelo (or Zermelo-Fraenkel) set theory is standard just in case (1) it is well-founded (no infinite descending membership chains), and (2) power sets are maximal.
- 21.
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 to speak of truths being conceptually necessary relative to a conceptual background. I would not be surprised if Martin Davis agreed with this.
- 22.
Menahem Fisch, “The Emergency Which has Arrived: The Problematic History of 19th Century British Algebra—A Programmatic Outline”, The British Journal for the History of Science, 27: 247–276, 1994.
- 23.
Officially, Hellmann and I avoid literal quantification over possible worlds or possibilia, relying entirely on modal operators that officially we avoid literal quantification over possible worlds or possibilia, relying entirely on modal operators.
- 24.
Mark Colyvan , “Indispensability Arguments in the Philosophy of Mathematics,” in E.N. Zalta, ed., The Stanford Encyclopedia of Philosophy (Fall 2004 Edition), http://Plato.stanford.edu/archives/fall2004/entries/mathphil-indis/. Colyvan is also the author of The Indispensability of Mathematics (Oxford: Oxford University Press, 2001).
- 25.
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.: Harvard University Press, 1976), 107–132 and in Paul Benacerraf and Hilary Putnam , eds., Philosophy of Mathematics, 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 Logical Point of View (Cambridge, Mass.: Harvard University Press, 19802), 1–19; W.V. Quine, “Two Dogmas of Empiricism,” Philosophical Review, 60, 1 (January 1951): 20–43; reprinted in his From 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.: Harvard University Press, 1981), 1–23; W.V. Quine, “Success and Limits of Mathematization,” in Theories and Things (Cambridge, Mass.: Harvard University Press, 1981), 148–155.
- 26.
Colyvan is referring to “What is Mathematical Truth” and Hilary Putnam, Philosophy of Logic (New York: Harper and Row, 1971), reprinted in Mathematics, Matter and Method: Philosophical Papers Vol. 1, 2nd edition, (Cambridge: Cambridge University Press, 1979), 323–357.
- 27.
“What is Mathematical Truth?”, 72.
- 28.
“Reflection” here denotes producing a stronger system by adding a consistency statement for a given system. If the systems are indexed by notations for constructive ordinals—that is, elements of a recursive well-ordering—and the ordering is already proved to be a well-ordering, one can continue “reflection” into the transfinite, when one comes to a “limit notation” by adding a suitably formalized statement to the effect that the union of the systems with indexes below the limit notation is a consistent system.
- 29.
Turing , A.M. (1939), ‘Systems of Logic Based on Ordinals’, Proceedings of the London Mathematical Society, Ser. 2 45, pp. 161–228.
- 30.
“What is Mathematical Truth”, 73.
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Putnam, H. (2016). On Davis’s “Pragmatic Platonism”. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_13
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