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 facets (such as ratio) and structural or patterned points (such as symmetry). The e-book develops a whole philosophy of arithmetic that contrasts with the standard Platonist and nominalist thoughts.
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Additional resources for An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure
To be the heap a + b + ... regarded as having a, b ... as parts, and a realist version of this would be to take the set to be the ‘trope’ or state of affairs of the heap a + b + ... ’s having a, b, ... 26 Simons suggests that the same formal properties can be obtained by allowing the singleton of x to be the state of affairs of x’s being identical to itself27 (the unit-making property is implicit here, in that without such a property there would not be an x to pick out). The essence of all these suggestions is that at the basic philosophical level necessary in this question, we cannot help ourselves naively to the notion of ‘object’.
What those properties are and how they make up the larger ones is something to be considered later (Chapters 3 and 4). Semi-Platonist Aristotelianism At this point it may be wondered whether it is not a very Platonist form of Aristotelianism that is being defended here. It has a structured space of universals, not all instantiated, into whose necessary interconnections the soul has insights. That is so. But there are three, not two, distinct positions covered by the names Platonism and Aristotelianism: 1.
Even philosophers could manage the concept to some degree: when Plato is recorded as being forced to add to the ‘featherless biped’ definition of ‘man’ the characteristic ‘with broad flat nails’ (in response to Diogenes’ plucked chicken),23 there is plainly an understanding involved of the extension of a predicate. 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.
An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure by J. Franklin