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Philosophy of Mathematics: Cognitive Science
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The philosophy of mathematics articulated and defended in this book goes by the name of “structuralism”, and its slogan is that mathematics is the science of structure. The subject matter of arithmetic, for example, is the natural number structure, the pattern common to any countably infinite system of objects with a distinguished initial object and a successor relation that satisfies the induction principle. The essence of each natural number is its relation to the other natural numbers. One way to understand structuralism is to reify structures as ante rem universals. This would be a platonism concerning mathematical objects, which are the places within such structures. Alternatively, one can take an eliminative, in re approach, and understand talk of structures as shorthand for talk of systems of objects or, invoking modality, talk of possible systems of objects.
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Innovations in the philosophy of language during the 20th century renewed interest in the question as to whether mathematics is, as if often said, the language of science. Although most mathematicians and physicists (and many philosophers) would accept the statement "mathematics is a language", linguists believe that the implications of such a statement must be considered. For example, the tools of linguistics are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way than other languages. If mathematics is a language, it is a different type of language than natural languages. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems.
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Towards a Philosophy of Real Mathematics is an impassioned criticism of this state of affairs. It begins with a largely polemical introduction. On the negative side, Corfield attacks what he calls the “foundationalist filterâ€â€”the view that, since all mathematics can be reduced to certain foundational systems, the only really interesting philosophical questions involve the status of those systems (and, perhaps, the status of any meta-mathematics used in the reduction). By applying this filter—often unconsciously—philosophers “fail to detect the pulse of contemporary mathematics†and underestimate the importance of mathematical history (since completed reductions “screen off†the actual historical development of central mathematical concepts). They ... ignore an important lesson from general philosophy of science: that interesting philosophical questions arise at many levels of description. After all, few contemporary philosophers would claim that philosophy of biology is pointless simply because “everything ultimately reduces to particle physics.â€1
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The philosophy of mathematics is the philosophical study of the concepts and methods of mathematics. It is concerned with the nature of numbers, geometric objects, and other mathematical concepts; it is concerned with their cognitive origins and with their application to reality. It addresses the validation of methods of mathematical inference. In particular, it deals with the logical problems associated with mathematical infinitude.
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This position is crystallized by Professor Bar-Hillel in "On a neglected ontology-free philosophy of mathematics," Problems in the Philosophy of Mathematics, T. Lakatos, ed. (Amsterdam, 1997), p. 136. Professor Robinson states his position in "Formalism 64," of Logic, Methodology, and Philosophy of Science, T. Bar-Hillel, ed, (Amsterdam, 1965), pp. 228-246.
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The Department encourages research in feminist topics in epistemology, philosophy of science, moral and political philosophy, and aesthetics. Naomi Scheman teaches courses in feminist philosophy. Other faculty members increasingly use feminist perspectives in their courses in ethics, history of philosophy, philosophy of language, philosophy of science, and other areas. In the Spring of 2005, 9 faculty members participated in a Feminism in Philosophy seminar, where they discussed the contributions feminists have made to their respective fields.
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