Criticisms
A critique of nominalist reconstructions in mathematics was undertaken by Burgess (1983) and Burgess and Rosen (1997). Burgess distinguished two types of nominalist reconstructions. Thus, hermeneutic nominalism is the hypothesis that science, properly interpreted, already dispenses with mathematical objects (entities) such as numbers and sets. Meanwhile, revolutionary nominalism is the project of replacing current scientific theories by alternatives dispensing with mathematical objects, see (Burgess, 1983, p. 96). A recent study extends the Burgessian critique to three nominalistic reconstructions: the reconstruction of analysis by Georg Cantor, Richard Dedekind, and Karl Weierstrass that dispensed with infinitesimals; the constructivist re-reconstruction of Weiertrassian analysis by Errett Bishop that dispensed with the law of excluded middle; and the hermeneutic reconstruction, by Carl Boyer, Judith Grabiner, and others, of Cauchy's foundational contribution to analysis that dispensed with Cauchy's infinitesimals.
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