Comments on Hilary Putnam’s Remarks on “Pragmatic Platonism”

I was delighted to learn that Hilary and I agree about so much concerning the nature of mathematical knowledge. Here I will concern myself with a few aspects where his remarks suggest that our views may differ, as well as to see say a little more about certain topics than I did in my original essay.

Mathematics and Natural Science

To a mathematician it is certainly gratifying that our field is so richly applicable in science, if only for the economic advantages that accrue even to those of us whose work is remote from applications. And of course there are important and difficult philosophical problems in understanding this relationship. As was indicated in a famous essay by Eugene Wigner, it all seems almost too good to be true. But I don’t see that this relationship sheds any light on the question with which my essay deals: how is that we can obtain objective knowledge about infinite entities.

Hilary suggests that this connection helps to show that intuitionism is unsatisfactory as a foundation of mathematics. I am more persuaded by a semi-facetious remark that Hilary himself made to me in conversation many years ago: “Do the intuitionists intend to put people who use non-intuitionisitic methods in jail?” Mathematicians will just use whatever methods seem to work and when faced with methodological difficulties will not long retreat to “safe methods” but learn how best to work around the difficulties. The evolution of the ideas of the great mathematician Hermann Weyl illustrates this well. Convinced that full-blooded mathematical analysis was methodologically unsustainable, even remarking that it was a “house built on sand”, he became a disciple of Brouwer, writing (much to the chagrin of his teacher Hilbert) “Brouwer, Das ist die Revoultion!” Many years later, writing Hilbert’s obituary, still admiring Brouwer, Weyl wrote that trying to develop mathematics in an intuitionistic setting leads to “an almost unbearable awkwardness” [4]. Weyl was well acquainted with mathematical physics, especially with relativity, but never referred to that as a reason to abandon intuitionism.

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