Saturday, December 13, 2014

Mathematical "existence"
In 1976, when I delivered the John Locke Lectures at Oxford, I often spent time with Peter Strawson, and one day at lunch he made a remark I have never been able to forget. He said, "Surely half the pleasure of life is sardonic comment on the passing show".  This blog is devoted to comments, not all of them sardonic, on the passing philosophical show.
Hilary Putnam

The preceding three posts have been fairly technical and full of details. So that the wood won’t be lost for the trees, I want to give a fairly short summary of the main points. Here they are:

(1) An interpretation of mathematics must be compatible with a scientific realist understanding of physics. From my first publications on the subject in the 1960s I have insisted that 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) must be interpretable in a “realist”, that is to say, a non-verificationist (or not simply “operationalist”) manner.  I believe many proposed interpretations fail that test. (Brouwer’s Intuitionism was my example in “What is Mathematical Truth”).
(2) Both objectualist interpretations (interpretations under which mathematics presupposes the mind-independent existence of sets as “intangible objects”) and potentialist/structuralist interpretations (interpretations under which mathematics only presupposes the possible existence of stuctures with the properties ascribed to sets, such as the “modal logical interpretations”) can meet the foregoing constraint.
(3) But the actual existence of sets as “intangible objects” suffers not only from familiar epistemological problems, but from a generalization of a problem first pointed out by Paul Benacerraf, “Benacerraf’s Paradox”, namely many identities (or proposed indenties) between different categories of mathematical “objects” seem undefined—e.g. are sets a kind of function or are functions a sort of set? For me, this tips the scales decisively in favor or potentialism/structuralism.
(4) In “Mathematics Without Foundations”, where I first proposed the modal logical interpretation), I claimed that conceptualism and potentialism are “equivalent descriptions”. In the three prededing posts I have retracted that claim.  But I don’t agree with Steven Wagner that rejecting objectualism requires one to say that sets, functions, numbers, etc., are fictions, and that the mathematics  student on the street is making a mistake when she says that there is a prime number between 17 and 34.  I now defend the view that potentialism is a rational reconstruction of our talk of “existence” in mathematics. Rational reconstruction does not “deny the existence” of sets (or, to use the example I used in the last post), of “a square root of minus one”; it provides a way of construing such talk that avoids the paradoxes.



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  3. Dear Prof. Putnam,
    (There seem to be two junk replies to this post, unfortunately.) Good for you for rejecting the fictionalism Wagner offers! I am curious about the status of the objects of rational reconstruction: if they are not fictions, what are they? Vague or contradictory concepts? My own impression of Benacerraf's Paradox is that it shows that rival ontologies one gives of sets (be it structures, possible structures or something more general still) are equivalent because their components are denumerable, and we have not yet reached a basic account of the natural numbers. If one says numbers are not objects but concepts (as some neo-Fregeans do) and thus functions, question immediately arises how many functions they are, how one distinguishes between functions, etc. An account like Frege's seems more basic in making equinumerosity prior to counting. But of course the machinery by which he formalized it was flawed. Still, with regard to your question of mathematical "existence", given the choice between elimination and whatever it is reconstruction works upon, don't we need some status like Frege's "being objective" as opposed to existing (being actual)? I know it is mysterious, but so are numbers.

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