Modal Logic and Mathematical Existence

I agree with Hilary in general terms that when set theorists talk about all sets, it needs to be understood in a relative manner. Relative to what? To the ordinals available to serve as ranks. It is the large cardinals that extend this range. So uncovering a new large cardinal concept augments the universe of sets. As Hellman [2] points out, this conception can already be seen, at least in embryo, in Zermelo’s account in his [5].

But Hilary suggests more: a program to use the formal apparatus of modal logic develop the idea that the class of sets and proper classes generally have only a possible existence.[1] And his student Geoffrey Hellman has made a bravura effort to carry out this program in his [2]. However, to paraphrase a trenchant comment of Poincare:

It is difficult to see that the word possibly acquires when written ?, a virtue it did not possess

when written possibly.[2]

Of course Frege, Russell, and Hilbert had answers for Poincare: Frege and Russell used a conceptual apparatus using э and other symbols to demonstrate that mathematics could be formalized in a formal system, and Hilbert proposed to use that knowledge to overcome the doubts about set-theoretic mathematics. And what came after would have been a surprise to them as well as to Poincare.

Hellman uses second order S5 to prove that a “modalist” need not give up anything that’s available to the mathematical platonist, that, as it were, for mathematical purposes possible existence is an adequate substitute for actual existence. But his formalism is syntactic: to provide meaning to his formalism would land him back in the platonic soup. Moreover his second order formalism includes full comprehension axioms, and as Quine pointed out long ago, this is to admit at least a modicum of set theory. So while I admire Hellman’s heroic effort, I wonder whom it is for. Will it convince mathematical constructivists or predicativists to give up their doubts about set theory? Not the ones I know! Will set theorists who, while making free use of proper classes in their technical work, have qualms about about the concept, be reassured? Again I doubt it. Certainly no one will propose a full modal formalism for the purpose of computer proof verification. So I am skeptical, but I will be very pleased if I am proved wrong, if this remarkable project turns out to yield fruitful results.

Martin Davis, June 13, 2015 Farewell to Hilary Putnam (1926-2016)

I have been very fortunate in having Hilary Putnam in my life as a close friend and a collaborator. Our families lived together in a house in Ithaca, New York in the summer of 1957 where Hilary and I were attending a five week Institute for Logic at Cornell University. We spent the following three summers working together, 1958 and 1959 in Eastern Connecticut, where I was on the faculty of the Connecticut branch of Rensselaer Polytechnic Institute, and 1960 at the University of Colorado Boulder where we attended a conference on physics for mathematicians.

In our time together there was hardly a topic in the full range of human intellectual inquiry into which our conversations did not range. This was in addition to our technical work which certainly includes contributions of which I’m very proud. Also our educations had been sufficiently complementary that we were really able to learn from each other; this included matters remote from our technical work. When I showed Hilary a copy of my first book that had just arrived from the publisher smelling of printer’s ink, he offered to find an error on any page. When I offered the reverse side of the title page, certain that the few lines of text on that page would be free of error, Hilary noticed that the word “permission” was missing its first “i”!

Hilary’s sharp mind, wit, and humane attitude toward life made his company a pleasure and our work together always fun. I miss him very much.

Martin Davis, June 13, 2015

References

[3] [4] [5] [6]

5

  • [1] Recent work by Joel Friedman on modalism [1] should also be mentioned.
  • [2] “It is difficult to see that the word if acquires when written Э, a virtue it did not possess whenwritten if.” [3], p. 156.
  • [3] Friedman, J. (2005). Modal platonism: An easy way to avoid ontological commitment toabstract entities. Journal of Philosophical Logic, 34, 227-273.
  • [4] Hellman, G. (1989). Mathematics without numbers: Toward a modal-structural interpretation.Oxford.
  • [5] Poincare, H. (2012). Science and method, translated from French by Francis Maitland,Thomas Nelson and Sons, London 1914. Facsimile Reprint: Forgotten Books. http://www.forgottenbooks.org.
  • [6] Weyl, H. (1950). David Hilbert and his mathematical work. Bulletin of the American Mathematical Society, 50, 612-654.
 
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