The Sort of Realism I Defend

I recently (Dec. 2014) described my philosophy of mathematics[1] in three posts on my blog ( In brief, the main points were:

(1) An interpretation of mathematics must be compatible with scientific realism. It is not enough that the theorems of pure mathematics used in physics come out true under one’s interpretation of mathematics—even some antirealist interpretations arguably meet that constraint—the content of the “mixed statements” of science (empirical statements that contain some mathematical terms and some empirical terms) also needs to be interpretable in a realist way. For example, if a theory talks about electrons, according to me it is talking about things we cannot see with the naked eye, and not simply about what measuring instruments would do under certain circumstances, as operationalists and logical positivists maintained. I believe many proposed interpretations fail that test.[2]

  • (2) Both objectualist interpretations (interpretations under which mathematics presupposes the mind-independent existence of sets as “intangible objects”[3] and potentialist/structuralist interpretations (interpretations under which mathematics only presupposes the possible existence of structures that exemplify the structural relations ascribed to sets), may meet the foregoing constraint. For example, under both Godel’s (or Quine’s) Platonist interpretations and Hellman’s and my modal logical interpretation the logical connectives are interpreted classically. In contrast to this, under Brouwer’s interpretation, the logical connectives (including “or” and “not”) are interpreted in terms of (Brouwer’s version of) provability. For example, in Intuitionism, “P or Q” means “There is a proof that either there is a proof of P or there is a proof of Q”. But according to scientific realists, the statement that a physical system either has a property P or has a property Q, does not entail that either disjunct can be proved, or even empirically verified. A statement can be true without being verifiable at all.[4] But if statements of pure mathematics are interpreted intuitionistically, mustn’t statements of physics also be interpreted in terms of the same non-classical understanding of the logical connectives?
  • (3) But, while positing the actual existence of sets as “intangible objects” may justify the use of classical logic, it suffers not only from familiar epistemological problems (not to mention conflicting with naturalism, which is the reason Davis gives for rejecting it), but from a generalization of a problem first pointed out by Paul Benacerraf,[5] a generalization I call “Benacerraf’s Paradox”, namely that too many identities (or proposed identities) between different categories of mathematical “objects” seem undefined on the objectualist picture—e.g. are sets a kind of function or are functions a sort of set? Are the natural numbers sets, and if so which sets are they? etc. For me, the objectualist’s lack of an answer that isn’t completely arbitrary tips the scales decisively in favor or potential- ism/structuralism.
  • (4) Rejecting objectualism (as Martin and I both do) does not requires one to say that sets, functions, numbers, etc., are fictions. (I hope Martin agrees.)

In “Mathematics without Foundations”, where I first proposed the modal logical interpretation), I claimed that objectualism and potentialism are “equivalent descriptions”, which was a mistake. I now defend the view that potentialism is a rational reconstruction of our talk of “existence” in mathematics, rather than an “equivalent” way of talking. Rational reconstruction does not “deny the existence” of sets (or, to change the example), of “a square root of minus one”; it provides a construal of such talk that avoids the paradoxes. In Davis’s language, the mathematician is talking about, for example, entities that play the role of a square root of minus one in certain hypothetical worlds, but unlike Godel she does not suppose that such entities exist in some Platonic realm. (Godel claimed we can perceive them with the aid of a special mental faculty.)

  • [1] The relevant publications are, in addition to the already mentioned “What is Mathematical Truth”and “Mathematics without Foundations”, are “Set Theory, Replacement, and Modality”, collectedin Philosophy in an Age of Science (Cambridge, MA: Harvard University Press, 2012), and “Replyto Steven Wagner”, forthcoming in The Philosophy of Hilary Putnam (Chicago: Open Court, 2015).
  • [2] Brouwer’s Intuitionism was my example of an interpretation that is incompatible with scientificrealism in “What is Mathematical Truth”, 75.
  • [3] Godel’s Platonism is a prototypical “objectualist” interpretation, but the term “intangible objects”wasusedbyQuinein Theories and Things, (Cambridge, MA: Harvard University Press, 1981), 149.
  • [4] 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.
  • [5] Paul Benacerraf (1965), “What Numbers Could Not Be” Philosophical Review Vol. 74, pp. 47-73.
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