By J. Franklin
Arithmetic is as a lot a technology of the true global as biology is. it's the technological know-how of the world's quantitative points (such as ratio) and structural or patterned facets (such as symmetry). The e-book develops an entire philosophy of arithmetic that contrasts with the standard Platonist and nominalist suggestions.
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Extra resources for An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure
Brent Mundy argues for the reality of uninstantiated universals by asking how a general theory of quantity relates to empirical evidence about quantities. A nominalist theory faces the problem that standard postulates of the theory of (extensive) quantity such as that the sum of two quantities is a quantity are literally false (for example, if mass means, operationally, measurement in a balance, then two large enough masses may be too large to fit together in a balance, though they do fit individually).
Even more so the higher infinities: ‘set theory is committed to the existence of infinite sets that are so huge that they simply dwarf garden variety infinite sets, like the set of all the natural numbers. ’1 Or as Shapiro writes: It seems reasonable to insist that there is some limit to the size of the physical universe. If so, then any branch of mathematics that requires an ontology larger than that of the physical universe must leave the realm of physical objects if these branches are not to be doomed to vacuity.
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’.