New PDF release: An Aristotelian Realist Philosophy of Mathematics:

By J. Franklin

ISBN-10: 1137400730

ISBN-13: 9781137400734

ISBN-10: 1349486183

ISBN-13: 9781349486182

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

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Additional resources for An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure

Sample text

Determinables and determinates The reason we know about uninstantiated universals such as huge numbers is that they occur in structured ranges of universals called determinables. Colour is a determinable, while an exact shade of colour such as Cambridge Blue is a determinate – a precise way of being a colour, among the wide range of possible ways of being a colour. 57 metres a determinate length. The way in which determinables are divided into determinates is unlike the way in which classification works via genus and differentia.

19 Those opposing sources of mathematical objectivity must be compatible despite their apparent tension, since sometimes it happens that pure mathematics discovers structures whose applicability is unsuspected, followed by scientists’ discovery that those very structures describe some aspect of reality. 20) SemiPlatonist Aristotelianism explains the metaphysics underlying these different aspects of the objectivity of mathematics. The same mathematical properties may be instantiated (hence perceptible and measurable) or uninstantiated and merely possible (hence accessible, if at all, by some other, purely intellectual, method).

So, a set refers to the division of a heap into objects in abstraction from which unit-making properties are doing the dividing. Thus the view that numbers are properties of sets is somewhat more abstract than the claim that they are relations between a heap and a unit-making property, but compatible with it. That gives some explanation, too, of why numbers are much older than explicit set-theory: to count ‘two cities’ can be done prior to abstracting from which unit-making property is dividing the heap into two, as is needed to form the set {Sydney, Hong Kong}.

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An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure by J. Franklin


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