Philosophy of Mathematics: Structure and OntologyOxford University Press, 1997. aug. 7. - 296 oldal 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. |
Tartalomjegyzék
3 | |
Mathematics and Its Philosophy | 21 |
A Realist Manifesto | 36 |
Structure | 71 |
Epistemology and Reference | 109 |
How We Got Here | 143 |
Construction Modality Logic | 181 |
Modality Structure Ontology | 216 |
Structure and Reality | 243 |
263 | |
273 | |
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abstract accept According analysis antirealist applied argues arithmetic assertions axioms called chapter characterize claim classical coherent collection concepts concerning consequence consists construction constructor course defined definition determine discourse domain dynamic elements equivalence example exemplify existence fact Field finite follows formal framework Frege function geometry given Hilbert hold idea ideal identity independent infinite interpretation intuition knowledge language least logic mathe mathematical objects mathematicians mathematics matter meaning modal model-theoretic natural numbers natural-number structure noted notion objects ontology ordinary particular pattern perhaps perspective philosophical physical places possible practice present principle problem proof properties provides question range realism reason reference relations result role satisfiability semantics sense sentence sequence set theory simple sort speak statements struc structuralist suggest thesis things tion translations true truth ture understand universe variables