Wednesday, April 8, 2015
Recommended readings on my philosophy of mathematics and some general remarks about it
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.
Although I have posted a number of times about my philosophy of mathematics, it seems desirable to wrap up this series of posts on the subject with some recommended readings (especially as I have the impression that some readers are trying to guess what that philosophy is and guessing wrong!). But first a few general remarks.
(i) In “Mathematics without Foundations” I called what I just referred to as “my philosophy of mathematics” “the modal-logical picture”. There I minimized its philosophical importance, referring to my “picture” as simply an alternative version (an “equivalent” description”) of the standard platonist ontology (“the mathematical object picture”). Even then, however, I stressed that the very existence of this “equivalent description” showed that standard mathematics does not in and of itself require “ontological commitment” to what Quine famously called “intangible objects”.
(2) At least since “Mathematics without Foundations”, I have insisted that what realism in philosophy of mathematics requires is that statements of pure mathematics have objective truth values whether or not human beings could or could not prove or disprove them. As Georg Kreisel once said, mathematics needs objectivity not objects. My version of mathematical realism insists that statements to the effect that such-and-such is possible have objective truth values (so the position can be called “potentialism, or “possibleism”‑the latter sounds ugly, however, so I prefer the former), but rejects the idea that taking modality metaphysically seriously in this way requires positing the actual existence of “possible worlds”. It’s just modality all the way down.
(3) In earlier posts in this series I have explained why I now think the modal logical picture and the objectual picture are not “equivalent descriptions”, although the modal logical picture can be regarded as a rational reconstruction of a naïve “mathematical objects” picture.
(4) The technical core of the version I outlined in “Mathematics without Foundations” was a way of correlating each statement of classical set theory (set theory without individuals, in that paper) with a statement that employs modal operators such as “it is possible that”, but whose predicates refer only to possible concrete objects—so the predicates are all nominalistically acceptable, and only nominalistically acceptable possible individuals are talked about. In a forthcoming paper*, Geoffrey Hellman calls these statements Putnam translates of the corresponding set theoretic statements.
Since the proper formalization of all this is something I owe to Hellman, I think the best way to refer to this philosophy of mathematics is “the Hellman-Putnam modal-structural interpretation”.
(to be continued)
*Geoffrey Hellman, "Infinite Possibilities and Possibilities of Infinity", details in next post.