By J. Franklin
Arithmetic is as a lot a technological know-how of the genuine global as biology is. it's the technology of the world's quantitative elements (such as ratio) and structural or patterned features (such as symmetry). The e-book develops a whole philosophy of arithmetic that contrasts with the standard Platonist and nominalist concepts.
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Additional resources for An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure
There have been two main suggestions from realists about the object of mathematics, as to what that object is. The first theory, the one that dominated the field from Aristotle to Kant and that has been revived by a few recent authors, is that mathematics is the ‘science of quantity’. 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.
Semi-Platonist Aristotelianism makes sense of two conflicting intuitions about the objectivity of mathematics, which create difficulties for other theories. On the one hand, its Aristotelian aspect allows it to connect the objectivity of mathematics with the usual objectivity of science arising from perception and measurement: the symmetry of a physical object, for example, can be perceived, quantities can be counted and measured. That is because symmetry and quantitative properties like length are genuinely instantiated in reality and can cause perceptual and measurable knowledge of themselves in the ordinary way of science.
They express ‘concomitant variation’, in Mill’s phrase, or ‘generic relations between the quantitative properties of things’, that is, relations between ranges of, for example, depth and pressure. 8 The lack of instantiation of some values does not tell against the reality of the determinable in general. Uninstantiated Universals and ‘Semi-Platonist’ Aristotelianism 25 It is the same with mathematical structures such as the continuum, Euclidean geometry or infinite numbers (on which more in Chapter 8) and idealizations such as perfect spheres (on which more in Chapters 5 and 14).