By J. Franklin
Arithmetic is as a lot a technology of the true international as biology is. it's the technology of the world's quantitative elements (such as ratio) and structural or patterned points (such as symmetry). The ebook develops an entire philosophy of arithmetic that contrasts with the standard Platonist and nominalist suggestions.
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Additional info for An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure
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. This is an unacceptable circularity. 16 It is a dubious claim that contemporary logic textbooks do regard sets as more basic than logical necessity;17 to the extent that they do, they follow Frege’s Platonism, which will be criticized in Chapter 7.
From the time of Aristotle to the eighteenth century, one philosophy of mathematics dominated the field. Mathematics, it was said, is the ‘science of quantity’. 1 That theory plainly gives an initially reasonable picture of at least elementary mathematics, with its emphasis on counting and measuring, and calculating with the resulting numbers. It promises direct and comprehensible answers to questions about what the object of mathematics is (certain properties of physical and perhaps non-physical things such as their size), and how those properties are known (the same way other natural properties of physical things are known: by perception in simple cases and inference from perception in more complex ones).
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.
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