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In 1965, Benacerraf

published a paradigm-changing article What numbers could not be which stimulated structuralism in the philosophy of mathematics.

The article argued that it wasn't possible to reduce natural numbers to set theory; since there are many definitions of the natural numbers; for example, Von Neumann's where 0={}, 1={0}={{}}, 2={1}={{{}}}, ... etc; amongst others.

Thus Benacerraf argued that the relationship between set theory and numbers is not ontological in the platonic sense; in that given the specific differences between different set-theoretic definitions of the natural numbers; there are certain questions which will be dependent on the definition chosen; this goes against the platonic spirit.

Does category theory, as an alternative foundation of mathematics, solve this problem? And if it does, how and at what cost?

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McLarty's "Numbers Can Be Just What They Have To" (http://www.cwru.edu/artsci/phil/NumbersCanBeJustWhattheyHaveTo.pdf) was written to answer precisely this question.

In particular, he argues that categorical set theory meets the demand that Benacerraf required of a structuralist account of mathematics.

It is a short, nice paper and I don't hope to summarize it adequately, so I will not, and instead just leave you with the citation.

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Yes it does. Category theory defines natural numbers internally. If C is a category, then the natural number object (if it exists) is the initial diagram of shape 1 -> N -> N in C, where 1 is terminal and N is any object. This basically means that it is the biggest object containing a distinguishable element (a 0, here encoded as an arrow 1 -> N), and a successor unary function (here encoded as N -> N).

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