By J. Franklin
Arithmetic is as a lot a technological know-how of the genuine international as biology is. it's the technological know-how of the world's quantitative facets (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 techniques.
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Extra resources for An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure
12 Thus, suppose there are seven black swans on the lake now. The proposition refers to a part of the world, the black biomass on the lake now, and a structuring property, being a black swan. Both are necessary to determining that the relation between the mass and the property should be ‘seven’: if it were a different mass (for example the black swans on or beside the lake now) or a different unit-making property (for example being a swan organ) then the numerical relation would be different. Therefore numbers are not properties of parts of the world simply, but must be properties of the relation between parts of the world and the unit-making properties that structure them.
According to Hellman, an arithmetic claim Φ means that for any logically possible system S, if S exemplifies the naturalnumber structure, then Φ is true of S. Shapiro objects: Recall that in contemporary logic textbooks and classes, the logical modalities are understood in terms of sets. To say that a sentence is logically possible is to say that there is a certain set that satisfies it. According to the modal option of eliminative structuralism, however, to say that there is a certain set is to say something about every logically possible system that exemplifies the structure of the set-theoretic hierarchy.
The nonadjacency of shades of blue is a necessary fact about the blue spectrum (as Platonism holds), but whether an intermediate shade of blue is instantiated is contingent (contrary to extreme Platonism, which holds that universals cannot be literally instantiated in reality). It is the same with uninstantiated mathematical structures, according to the Aristotelian of Platonist bent: a ratio (say), whether small and instantiated or huge and uninstantiated, is part of a necessary spectrum of ratios (as Platonists think) but an instantiated ratio is literally a relation between two actual (say) lengths (as Aristotelians think) and is thus something found in the physical world.
An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure by J. Franklin