## Philosophy of Mathematics: Structure and OntologyDo numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic.As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences.Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. |

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antirealist arithmetic articulated assertions axiom of choice axioms background ontology cardinal chapter characterize Chihara claim classical logic coherent construction Dedekind defined discourse domain domain of discourse Dummett dynamic epistemic epistemology equivalence Euclidean Euclidean geometry example excluded middle exemplify the natural-number existence fictionalist finite formal framework freestanding Frege function geometry Hellman Heyting Hilbert ideal constructor identity implicit definition impredicative definition infinite interpretation intuition intuitionistic invoked isomorphism language least mathe mathematical objects mathematical structures mathematicians matics metalanguage modal model theory model-theoretic model-theoretic semantics natural numbers natural-number structure nominalistic notion ontology ordinary pattern perspective philosophical realism philosophy of mathematics physical objects places possible principle problem quantifiers question Quine real numbers realism in ontology reference relations rem structuralism role second-order second-order logic sentence sequence set theory set-theoretic hierarchy Shapiro singular terms space-time statements static struc structuralist sublanguage system that exemplifies theorem thesis tion truth truth-value ture variables