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?


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  • 4
    $\begingroup$ Have you tried to learn any category theory? $\endgroup$ – Rob Arthan May 23 '18 at 20:46
  • $\begingroup$ @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. $\endgroup$ – samerivertwice May 23 '18 at 21:33
  • $\begingroup$ 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. $\endgroup$ – Malice Vidrine May 23 '18 at 22:12
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    $\begingroup$ 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. $\endgroup$ – hardmath May 24 '18 at 18:55
  • $\begingroup$ @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. $\endgroup$ – samerivertwice May 28 '18 at 9:20

I believe you are being mislead by Dr. Cheng's motto.

As everyother motto it should not be taken to formally.

What Dr. Cheng is trying to say is that mathematics amounts to the study of logical structres and transformations and relations between them, these data (these structures and relations) are of course logical structures in their own way and have their own transformations and relations, hence they are object of study of mathematics itself. Category theory is the mathematical study if this structures of structures.

Of course here the word logical is not to be interpreted in the formal way (it does not have to do with propositions, connectives and quantification, although category theory can be applied to the study of these very specific structures). Here the word logical is used in an informal way to specify that the objects considered are abstract entities that live in the human mind.

So in short, no category theory does not quantify over propositions (well actually it does when you are working in a category whose objects or morphisms are logical formulas but that is another story for another day).

  • $\begingroup$ Thanks. Her own book blurb says "professor" but her own website says "Dr." so I'm going to go along with that and revert to Dr. unless you have any objection? $\endgroup$ – samerivertwice May 24 '18 at 9:48

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