Recommended readings on my philosophy of mathematics (continued):

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

Yesterday’s post, which should be read before this one, began “Although I have posted a number of times about my philosophy of mathematics, it seems desirable to wrap up this series 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”. The general remarks were yesterday’s post. Today’s post is a list of just four readings by the two authors of what I yesterday called “the Hellman-Putnam modal-structural interpretation.”  Besides wanting to provide a short reading list for those who want to know what that interpretation really is, I confess that I also want to call attention to the fact that (after a long wait—I finished writing my Intellectual Autobiography and Replies in 2009!), my volume in The Library of Living Philosophers (titled, of course, The Philosophy of Hilary Putnam) will actually appear next month! Moreover, the first section of the volume is titled “Philosophy and Mathematics”. The fifth chapter of that section, Geoffrey Hellman’s “Infinite Possibilities and Possibilities of Infinity”, is not only an excellent explanation and short history of the modal-structural interpretation (which turns out to go back to Zermelo!—his paper, which Hellman cites, did not get translated into English until the late 1980s, and neither Hellman nor I knew of it), but also an account of open problems and work in progress. So here are the four readings:

By me: (1) “Mathematics without Foundations” (published in the Journal of Philosophy, Jan, 1967, and collected in my Mathematics, Matter of Method, Philosophical Papers  vol. I, Cambridge, 1975, and also collected in Benacerraf and Putnam, Philosophy of Mathematics; Selected Readings, second edition, Cambridge 1983.

By me: (2) “Set Theory: Realism, Replacement, and Modality” in my Philosophy in an Age of Science, Harvard, 2012.

By Hellman: (3) Mathematics without Numbers, Oxford, 1989

By Hellman: (4) “Infinite Possibilities and Possibilities of Infinity” in The Philosophy of Hilary Putnam, The Library of Living Philosophers, Open Court, 2015.