By J. Franklin
Arithmetic is as a lot a technology of the genuine global as biology is. it's the technology of the world's quantitative features (such as ratio) and structural or patterned features (such as symmetry). The e-book develops an entire philosophy of arithmetic that contrasts with the standard Platonist and nominalist ideas.
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Extra resources for An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure
If it were admitted that those truths were literally true of mundane reality, then there would be a large body of Aristotelian mathematical knowledge, in no need of Platonist reinterpretation. If then the world did expand so that the boundary between the instantiated and the uninstantiated blew out infinitely, perhaps to the higher infinities, most of mathematical knowledge might be literally true of the (non-abstract) world. Be that as it may, the ‘problem of uninstantiated universals’ is a genuine one and must be faced.
Small finite structures have plenty to keep the mathematician occupied, and the body of knowledge about them is extensive. If it were admitted that those truths were literally true of mundane reality, then there would be a large body of Aristotelian mathematical knowledge, in no need of Platonist reinterpretation. If then the world did expand so that the boundary between the instantiated and the uninstantiated blew out infinitely, perhaps to the higher infinities, most of mathematical knowledge might be literally true of the (non-abstract) world.
Nor is it sufficient excuse – though it may to some extent explain the lack of communication – that the quantity theory tends to be more Aristotelian and structuralism more Platonist in its realism. The quantity theorists tend to discuss ratios existing between actual lengths, times and so on, and mostly situate their theory in an Aristotelian realist theory of universals such as that of Armstrong. 4 But the gulf between the two sides is not as wide as it seems. 5 On the other hand, quantity theorists admit a need to deal with uninstantiated quantities such as very large numbers, tending to make their approach a semi-Platonist form of Aristotelianism (as described in Chapter 2).