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?


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.


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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