We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

This paper proposes an approach to the philosophy of mathematics, deductive pluralism, that is designed to satisfy the criteria of inclusiveness of and consistency with mathematical practice. Deductive pluralism views mathematical statements as assertions that a result follows from logical and mathematical foundations and that there are a variety of incompatible foundations such as standard foundations, constructive foundations, or univalent foundations. The advantages of this philosophy include the elimination of ontological problems, epistemological clarity, and objectivity. Possible objections and relations with (...) some other philosophies of mathematics are also considered. (shrink)