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Download An Aristotelian Realist Philosophy of Mathematics: by J. Franklin PDF

By J. Franklin

Arithmetic is as a lot a technology of the true global as biology is. it's the technology of the world's quantitative elements (such as ratio) and structural or patterned elements (such as symmetry). The ebook develops an entire philosophy of arithmetic that contrasts with the standard Platonist and nominalist ideas.

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Extra info for An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure

Example text

Nevertheless, any attempted account of sets should either deliver the numbers-as-properties-of-sets theory as a consequence, or explain away the attractiveness of the theory. The present account of numbers and sets does deliver the theory as a consequence – or almost. Let us take the simplest case, that of 1-element sets: the set {Sydney} is (taking the Armstrong version) the state of affairs of there being a unit-making property of the mass of buildings in eastern Australia, such as ‘being a city’.

The second is that its subject matter is structure or pattern. Reasons will be given for taking both of these to be objects of mathematics, and exact definitions of both these (notoriously vague) concepts will be offered. The exactitude of the definitions will be sufficient to permit a demonstration that the concepts are not identical, though closely related. Two realist theories of mathematics: quantity versus structure Quantity is examined in this chapter and structure in the next. It is concluded that both quantity and structure are real properties and are 31 32 An Aristotelian Realist Philosophy of Mathematics studied by mathematics, but are quite distinct properties.

Indeed, this is his objection to what he calls the ‘eliminative structuralist’ Aristotelian alternative to Platonism. He discusses Hellman’s ‘modal realism’, which agrees with Aristotelianism to the extent of regarding mathematics as (at least sometimes) about possible structures (though Hellman does not support this with an Aristotelian theory of universals; Hellman’s theory is considered further in Chapter 7). According to Hellman, an arithmetic claim Φ means that for any logically possible system S, if S exemplifies the naturalnumber structure, then Φ is true of S.

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