# Does category theory quantify over its own sentences? [closed]

For a while I've imagined that higher orders of logic "quantify over sentences" of lower orders of logic.

In Dr. Eugenia Cheng's lecture in Sydney today, she showed a slide saying that category theory is the mathematics of mathematics:

Category theory is the logical study of

• the logical study of

• how logical things work.

This reminded me of that notion. Is there a connection here? Is this what Category theory is doing? Taking maths to a level of abstraction where the operators are quantifying over, and acting on, themselves, in such a way that no higher level of abstraction is possible? Does category theory effectively quantify over its own sentences?

## closed as unclear what you're asking by Pece, B. Mehta, hardmath, José Carlos Santos, Matthew TowersMay 25 '18 at 13:04

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• Have you tried to learn any category theory? – Rob Arthan May 23 '18 at 20:46
• @RobArthan no, not beyond the overall conceptualisation of a directed graph in which the vertices represent morphisms. But the idea that morphisms express sameness and difference between things is at the heart of how I think about maths. In logic it was said to me previously "you're quantifying over sentences" before as if to do so was sacrilege, but there seems to be a self-referential quality to category theory that perhaps is analogous to permitting this. – samerivertwice May 23 '18 at 21:33
• There's nothing more self-referential about category theory than any other sort of mathematical theory. There are categories in which you can do arithmetic, and could conceivably arithmetize the theory of categories in the usual way. But this isn't anything new. – Malice Vidrine May 23 '18 at 22:12
• Category theory, like many abstract theories studied by mathematicians, can be presented in a first-order logic formalization where the primitive domain is morphisms (or arrows). But category theory, while providing a powerful abstraction, is not a logic per se. We reason with the sentences of category theory the same way we would reason with sentences of any other theory of mathematics. If interested you might attempt to construct a category of quantifiers. Most investments in mathematical research, no matter how abstract, eventually find application. – hardmath May 24 '18 at 18:55
• @hardmath this was interesting to think about. Functors on quantifiers seem to my inexperienced mind to be two kinds: a) end extensions (on ordinals) and their inverses, and b) type conversions. – samerivertwice May 28 '18 at 9:20