Why I Am Not α Nominalist

ions which are infinite and do not lend themselves to a constructivist interpretation-which make classical mathematics effective. One should try to imagine how much help mathematics could have provided twentieth century quantum physics if for the past hundred years it had developed using only abstractions from "constructive objects". Most likely, the standard calculationsions from "constructive objects". Most likely, the standard calculations with infinite dimensional representations of Lie groups which today play an important role in understanding the microworld, would simply never have occurred to anyone. ([11], pp. 172-173) (Mention of quantum mechanics should remind us that it is unclear whether the methods of Chihara and Field are adequate even for present-day science in its entirety. For Chihara the problem is a minor one, and could probably be solved by adopting a somewhat stronger system of predicative analysis than the particular weak system Σω he favors. For Field, the problem is a major one, for he has given us no idea how he proposes to treat quantum theory, which differs radically (owing to its use of infinite-dimensional apparatus and to its statistical character) from the one theory he does treat in detail, Newtonian gravitational theory). But I need not enlarge on the costs for present-day and future physics of a nominalistic revolution. Surely the burden of proof is on the revolutionary, who proposes a drastic departure from our thus far eminently successful policy of ontological tolerance in common sense and scientific theory construction. Until it is shown that nominalism offers physical science some substantive advantages, I for one am prepared to dismiss its revolutionary proposals as motivated only by medieval superstition ("Ockham's razor") and fastidious bigotry (cf. [4], Objection viii). Chihara and Field have gone a long way toward constructing nominalistic alternatives empirically equivalent and pragmatically only slightly inferior to our current scientific theories. Their work suggests that an ontology of abstracta may be one feature of those current theories that is merely conventional, in the best sense of the word (that of David Lewis [8]). This suffices to cast considerable doubt on some more extreme versions of realism. It does not suffice to cast doubt on moderate versions of realism, which merely observe that our current theories seem to invoke abstracta and that we do not yet have reasons to abandon those theories. For to characterize some feature of our present ways of doing things (in scientific theorizing or in driving) as conventional is not in itself to criticize that feature. And Chihara