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Philosophy of Mathematics: Structure and Ontology
Do 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.
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.
296 pages, Paperback
First published January 1, 1997
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February 11, 2020Math
January 1, 2017
A clearly written and philosophically fecund defense of ante rem structuralism
November 28, 2025
The structuralist approach to the philosophy of math is a form of Platonism. Goes deeper for a short book. I work a little in math, and the experience of working with a lot of college math is more like a rock than something invented. You kick the tires of an equation, and the equations will kick back. I tend to view the structural system like a Lego block system where the blocks are axioms that fit together in a certain way, depending on the nature of the axioms, and build into a variety of real structures. Anyway, at least that is how I picture it in my head.
Update 2/10/2023 explains numbers and math as something like the position of shortstop in baseball or president of the United States; it is a function of a certain structure of a game or constitutional government. Individuals who occupy the position may vary, but the structure that makes up the position is a real thing. Numbers exist in the same way.
Reread 11/28/2025
I read this again. Makes a structural argument for Platonic realism of mathematics. it isn't the numbers themselves that matter, but the structural relations. These structures or relations are real, and they are abstract. He explores some modern debates in this area between realist and antirealist positions on this. He talks about how to relate to the realism of Math after non-Euclidean geometry shook Kantian intuitions about time and space, and the reality of numbers after Russell, Frege, and Godel's disruptions in the foundational crisis of Mathematics. And how structural realism works in the philosophy of modal logic and possible worlds. Good stuff.
Update 2/10/2023 explains numbers and math as something like the position of shortstop in baseball or president of the United States; it is a function of a certain structure of a game or constitutional government. Individuals who occupy the position may vary, but the structure that makes up the position is a real thing. Numbers exist in the same way.
Reread 11/28/2025
I read this again. Makes a structural argument for Platonic realism of mathematics. it isn't the numbers themselves that matter, but the structural relations. These structures or relations are real, and they are abstract. He explores some modern debates in this area between realist and antirealist positions on this. He talks about how to relate to the realism of Math after non-Euclidean geometry shook Kantian intuitions about time and space, and the reality of numbers after Russell, Frege, and Godel's disruptions in the foundational crisis of Mathematics. And how structural realism works in the philosophy of modal logic and possible worlds. Good stuff.
August 7, 2011
Une excellente introduction au structuralisme en philosophie des mathématiques.
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