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 facets (such as ratio) and structural or patterned points (such as symmetry). The booklet develops an entire philosophy of arithmetic that contrasts with the standard Platonist and nominalist techniques.
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Additional info for An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure
Even sets of sets of sets ... are not uncommon in ordinary life – if I sign off for a consignment of three truckloads each with four crates of ten packages with five pairs of shoes each, I have dealt with a set of a set of a set of a set of a set of a set of shoes; the example indicates why multiplication is as important in elementary arithmetic as addition. So what are sets, from an Aristotelian point of view? The Aristotelian cannot rest content with the Platonist story that sets are a simple ‘abstract’ object at which questions should stop, and that the membership 40 An Aristotelian Realist Philosophy of Mathematics relation is sui generis.
The division between the two objects of study roughly corresponds to the division between elementary and higher mathematics. It will be explained how to characterize the notions of ‘quantity’ and ‘structure’ precisely enough so that it can be established that the two are not the same. Yet they have a close enough relationship to give some degree of unity to the subject matter of mathematics. From the time of Aristotle to the eighteenth century, one philosophy of mathematics dominated the field.
As Armstrong puts it, ‘in mathematics, we gain knowledge of entities which are merely possible, and indeed, perhaps nomically impossible ... there can be no question of establishing these conclusions a posteriori ... ’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.
An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure by J. Franklin