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Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. Mathematical objects are exhaustively defined by their place in such structures. Consequently, structuralism maintains that mathematical objects do not possess any intrinsic properties but are defined by their external relations in a system. For instance, structuralism holds that the integer 1 is exhaustively defined by being the successor of 0 in the structure of the theory of natural numbers. By generalization of this example, any integer is defined by their respective place in this structure of the number line. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra. Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what ''kind'' of entity a mathematical object is, not to what kind of ''existence'' mathematical objects or structures have (not, in other words, to their ontology). The kind of existence mathematical objects have would clearly be dependent on that of the structures in which they are embedded; different sub-varieties of structuralism make different ontological claims in this regard. Structuralism in the philosophy of mathematics is particularly associated with Paul Benacerraf, Michael Resnik and Stewart Shapiro. ==Historical motivation== The historical motivation for the development of structuralism derives from a fundamental problem of ontology. Since Medieval times, philosophers have argued as to whether the ontology of mathematics contains abstract objects. In the philosophy of mathematics, an abstract object is traditionally defined as an entity that: (1) exists independent of the mind; (2) exists independent of the empirical world; and (3) has eternal, unchangeable properties. Traditional mathematical Platonism maintains that some set of mathematical elements–natural numbers, real numbers, functions, relations, systems–are such abstract objects. Contrarily, mathematical nominalism denies the existence of any such abstract objects in the ontology of mathematics. In the late 19th and early 20th century, a number of anti-Platonist programs gained in popularity. These included intuitionism, formalism, and predicativism. By the mid-20th century, however, these anti-Platonist theories had a number of their own issues. This subsequently resulted in a resurgence of interest in Platonism. It was in this historic context that the motivations for structuralism developed. In 1965, Paul Benacerraf published a paradigm changing article entitled "What Numbers Could Not Be".〔Paul Benacerraf (1965), “What Numbers Could Not Be”, ''Philosophical Review'' Vol. 74, pp. 47-73.〕 Benacerraf concluded, on two principle arguments, that set-theoretic Platonism cannot succeed as a philosophical theory of mathematics. Firstly, Benacerraf argued that Platonic approaches do not pass the ontological test. He developed an argument against the ontology of set-theoretic Platonism, which is now historically referred to as Benacerraf's identification problem. Benacerraf noted that there are elementarily equivalent, set-theoretic ways of relating natural numbers to pure sets. However, if someone asks for the "true" identity statements for relating natural numbers to pure sets, then different set-theoretic methods yield contradictory identity statements when these elementarily equivalent sets are related together. This generates a set-theoretic falsehood. Consequently, Benacerraf inferred that this set-theoretic falsehood demonstrates it is impossible for there to be any Platonic method of reducing numbers to sets that reveals any abstract objects. Secondly, Benacerraf argued that Platonic approaches do not pass the epistemological test. Benacerraf contended that there does not exist an empirical or rational method for accessing abstract objects. If mathematical objects are not spatial or temporal, then Benacerraf infers that such objects are not accessible through the causal theory of knowledge.〔Paul Benacerraf (1973). "Mathematical Truth", in Benacerraf & Putnam ''Philosophy of Mathematics: Selected Readings'', Cambridge: Cambridge University Press, 2nd edition. 1983, pp. 403–420.〕 The fundamental epistemological problem thus arises for the Platonist to offer a plausible account of how a mathematician with a limited, empirical mind is capable of accurately accessing mind-independent, world-independent, eternal truths. It was from these considerations, the ontological argument and the epistemological argument, that Benacerraf's anti-Platonic critiques motivated the development of structuralism in the philosophy of mathematics. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Structuralism (philosophy of mathematics)」の詳細全文を読む スポンサード リンク
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