By J. Franklin
Arithmetic is as a lot a technological know-how of the genuine international as biology is. it's the technology of the world's quantitative points (such as ratio) and structural or patterned elements (such as symmetry). The ebook 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
Those can be described as (possibly) uninstantiated structures or as (merely) possible structures, but in either case they are complex forms which could be instantiated in reality – forms about which there can be necessary knowledge. They differ from the Forms of classical Platonism which necessarily lie beyond mundane reality and cannot be literally instantiated in it. Aristotelian forms can be instantiated, but it is for the contingencies of historical reality (or the will of God, or whatever decides such matters) to determine which are in fact instantiated.
Structuralism holds that mathematics studies structure or patterns. 2 The structure is ‘exemplified by’ an infinite sequence of distinct moments in time. Number theory studies just the properties of the structure, so that for number theory, there is nothing to the number 2 but its place or ‘office’ near the beginning of the system. Other parts of mathematics study different structures, such as the real number system or abstract groups. The structuralist theory of mathematics has, like the quantity theory, some initial plausibility, in view of the concentration of modern mathematics on structural properties like symmetry and the purely relational aspects of systems both physical and abstract.
Shapiro argues that there is no acceptable view of necessity and possibility that can be relied on here by ‘modal’ Aristotelianism, so that reliance on necessity cannot replace Platonism. 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).