Vaughan Pratt has rewritten the Wikipedia article on mathematical object, and the result is a great improvement.
I still think that a suitable approach would be to pick out mathematical objects among all abstract objects by specific properties they have.
- A mathematical object is always an abstract object that satisfies certain axioms specific to the object.
- A mathematical object is inert and unchanging.
- A mathematical object is defined crisply, no fuzzy allowed.
- And so on.
I wrote about some of those points here, but not the part about axioms. Watch this space!
These ideas are not settled. As one commenter said, Wikipedia articles should not be the product of current research. What Vaughan has written is about right for now.
Specific comments:
1) Above, I reworded my comment about mathematical objects satisfying axioms in response to an objection by a reader. For example, a model of untyped lambda calculus is an object S for which the function space S -> S is isomorphic to S. Such a thing does not exist in the category of sets, but it does in the category of topological semigroups and also in the realizability topos.
2) In a note to the category mailing list, Vaughan also said:
“It might also be worth mentioning coalgebras, and perhaps more importantly dually defined structures such as locales which are understood better in terms of the morphisms from them to a cogenerator rather than those to them from a generator, i.e. dual elements rather than elements. Also toposes as a more general codomain of the forgetful functor than the particular topos Set.”
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