  These issues naturally prompt the question of how much mathematics is indispensable (and hence how much mathematics carries ontological commitment). It seems that the indispensability argument only justifies belief in enough mathematics to serve the needs of science. Thus we find Putnam speaking of “the set theoretic ‘needs’ of physics” (Putnam 1979b, p. 346) and Quine claiming that the higher reaches of set theory are “mathematical recreation ... without ontological rights” (Quine 1986, p. 400) since they do not find physical applications. One could take a less restrictive line and claim that the higher reaches of set theory, although without physical applications, do carry ontological commitment by virtue of the fact that they have applications in other parts of mathematics. So long as the chain of applications eventually “bottoms out” in physical science, we could rightfully claim that the whole chain carries ontological commitment. Quine himself justifies some transfinite set theory along these lines (Quine 1984, p. 788), but he sees no reason to go beyond the constructible sets (Quine 1986, p. 400). His reasons for this restriction, however, have little to do with the indispensability argument and so supporters of this argument need not side with Quine on this issue.