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In The Oxford Handbook of Philosophy of Mathematics and Logic, Stewart Shapiro states in his introductory section:

Quine himself accepts mathematics (as true) only to the extent that it is applied in the sciences. In particular, he does not accept the basic assertions of higher set theory because they do not, at present, have any empirical applications.

I feel as though "higher set theory" can mean either "formally independent results in set theory having to do with infinity, large cardinal axioms, etc." or just "infinite numbers in general," and from the context it isn't clear to me which one exactly he means. If Quine's view of spacetime was that of Einstein's general relativity, he probably would have believed in some sort of continuum of real numbers, so I am compelled to believe it is the first.

Did Quine ever write about the continuum hypothesis, large cardinal axioms, or any other of those topics in "higher set theory" explicitly? Did he ever propose in writing any sort of way to handle questions about the higher infinite in his New Foundations, or the extensions thereof, such as Mathematical Logic? Or was he skeptical of its importance due to the inability for it to be pertinent to science and therefore didn't give those issues any attention?

  • You can see some comments in the final part of Willard Van Orman Quine, Set Theory and its Logic Belknap Press (revised ed 1969). – Mauro ALLEGRANZA Apr 13 '17 at 9:05
  • Quine's point of view is related (the source of ?) the so-called Indispensability Arguments : "(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories. (P2) Mathematical entities are indispensable to our best scientific theories. [Therefore:] (C) We ought to have ontological commitment to mathematical entities." Higher-cardinals have no "scientific applications"; thus, they are dispensable. – Mauro ALLEGRANZA Apr 13 '17 at 12:33
  • @MauroALLEGRANZA I am aware of what the indispensability argument is but my question isn't whether or not Quine or other natrualists/empiricsts think large cardinal axioms are dispensable, it's whether or not he wrote specifically on that topic. Thank you for the link to his book, I'll see if I can find a copy. – Not_Here Apr 13 '17 at 17:48
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Quine did not specifically study "higher set theory", and his positions on the issue are mostly generalities following from his empirical holism (mathematics is the "entrenched" part of the "web of belief" that touches on experience at the observational boundaries) combined with the indispensability argument (what is indispensable in empirical science, e.g. sets and numbers, must be granted ontological rights). His more detailed remarks on the issue came in response to Parsons' critique of his positions in Review of Parsons (1984) and Reply to Parsons (1986):

"So much of mathematics as is wanted for use in empirical science is for me on a par with the rest of science. Transfinite ramifications are on the same footing insofar as they come of a simplificatory rounding out, but anything further is on a par with uninterpreted systems." [Quine 1984, p. 788]

"I recognize indenumerable infinities only because they are forced on me by the simplest known systematizations of more welcome matters. Magnitudes in excess of such demands, e.g., ב_ω [the cardinal number of V_ω(N) and of V_ω+ω] or inaccessible numbers, I look upon only as mathematical recreation and without ontological rights." [Quine 1986, p. 400]

Quine is even more explicit in Pursuit of Truth, where he comes out in support of the axiom of constructibility due to "the considerations of simplicity, economy, and naturalness that contribute to the molding of scientific theories generally", and because "it inactivates the more gratuitous flights of higher set theory, and incidentally it implies the axiom of choice and the continuum hypothesis", thus "rounding out" ZFC. This is followed by a favorable mention of constructivism and predicativism, which of course do an even better job of "inactivating" the higher set theory. Indeed, Field's fictionalist programme of reducing mathematics of natural sciences to a predicative minimum is close to the realization of Quine's own youthful dream (with Goodman) of complete nominalism about mathematics. If successful, it would allow to dispense with the indispensability argument, and platonism even about numbers. See Burgess's Why I Am Not a Nominalist, who calls Quine and Putnam "false friends of numbers" on this score.

As mentioned, Parsons pointed some incongruities in Quine's views of mathematics. A detailed critical analysis of this critique and Quine's responses to it are in Tieszen's book chapter Gödel and Quine on Meaning and Mathematics:

"Parsons notes that, according to Quine, there is no higher necessity than physical or natural necessity. Set theory is supposed to be on par with physics in this respect. The problem is that there is tension in Quine’s own view of this. It conflicts with his view of mathematical existence... Quine is a platonist about set theory. The notion of object in set theory, however, and the structures whose possibility it postulates, are much more general than the notion of physical object and spatiotemporally or physically representable structure. But then how can Quine maintain that these possibilities are ‘natural’ and that the necessity of logic and mathematics is not ‘higher’?

[...] This is related to another difference noted by Parsons: elementary mathematical truths do not seem to be even more rarefied and theoretical than the theoretical hypotheses of natural science. On the contrary, they seem quite obvious. Quine’s view cannot explain the obviousness of elementary mathematics and parts of logic (Parsons 1980, p. 151). In fact, there seem to be very general principles that are universally regarded as obvious, whereas on an empiricist view one would expect them to be bold hypotheses about which a prudent scientist would maintain reserve."

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This is just part of Quine's naturalism, a sort of science-first approach to everything. That is mainly what underlies his suspicion of higher set theory.

Here is Quine discussing the matter, from his book Pursuit of Truth (1990, pp. 94-95):

  1. Truth in mathematics

What now of those parts of mathematics that share no empirical meaning, because of never getting applied in natural science? What of the higher reaches of set theory? We see them as meaningful because they are couched in the same grammar and vocabulary that generate the applied parts of mathematics. We are just sparing ourselves the unnatural gerrymandering of grammar that would be needed to exclude them. On our two-valued approach they then qualify as true or false, albeit inscrutably.

They are not wholly inscrutable. The main axioms of set theory are generalities operative already in the applicable part of the domain. Further sentences such as the con­tinuum hypothesis and the axiom of choice, which are independent of those axioms, can still be submitted to the considerations of simplicity, economy, and naturalness that contribute to the molding of scientific theories generally. Such considerations support Gödel's axiom of construcribil­ity, 'V = L'. It inactivates the more gratuitous flights of higher set theory, and incidentally it implies the axiom of choice and the continuum hypothesis. More sweeping economies have been envisioned by Hermann Weyl, Paul Lorenzen, Errett Bishop, and currently Hao Wang and Sol­omon Feferman, who would establish that all the mathematical needs of science can be supplied on the meager basis of what has come to be known as predicative set theory. Such gains are of a piece with the simplifications and economies that are hailed as progress within natural science itself. It is a matter of tightening and streamlining our global system of the world.

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