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Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1999., Includes bibliographical references., "Ontology and the Foundations of Mathematics" consists of three papers concerned with ontological issues in the foundations of mathematics. Chapter 1, "Numbers and Persons," confronts the problem of the inscrutability of numerical reference and argues that, even if inscrutable, the reference of the numerals, as we ordinarily use them, is determined much more, precisely than up to isomorphism. We argue that the truth conditions of a variety of numerical modal and counterfactual sentences (whose acceptance plays a crucial role in applications) place serious constraints on the sorts of items to which numerals, as we ordinarily use them, can be taken to refer: Numerals cannot be taken to refer to objects that exist contingently such as people, mountains, or rivers, but rather must be taken to refer to objects that exist necessarily such as abstracta. Chapter 2, "Modern Set Theory and Replacement," takes up a challenge to explain the reasons one should accept the axiom of replacement of Zermelo-Fraenkel set theory, when its applications within ordinary mathematics and the rest of science are often described as rare and recondite. We argue that this is not a question one should be interested in; replacement is required to ensure that the element-set relation is well-founded as well as to ensure that the cumulation of sets described by set theory reaches and proceeds beyond the level w of the cumulative hierarchy. A more interesting question is whether we should accept instances of replacement on uncountable sets, for these are indeed rarely used outside higher set theory. We argue that the best case for (uncountable) replacement comes not from direct, intuitive considerations, but from the role replacement plays in the formulation of transfinite recursion and the theory of ordinals, and from the fact that it permits us to express and assert the (first-order) content of the modern cumulative view of the set theoretic universe as arrayed in a cumulative hierarchy of levels., by Gabriel Uzquiano., Ph.D.