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I can imagine that there is some theorem that says Category theory is a universal modeling language in some sense. For example, it includes all of set theory, s
by j2kun 5y ago
I can imagine that there is some theorem that says Category theory is a universal modeling language in some sense. For example, it includes all of set theory, so it "includes" everything classical math can model.
But the real question is whether it is a _useful_ modeling language for everything. Engineers know everything has a tradeoff, and category theorists would do well to recognize the limitations of their tools or risk turning people (like me) away after years of study.
- wisnesky 5y agoThe trade-off has to do with computability: because (finitely presented) categories can express so much, reasoning about them (for example, the 'word problem' for them) is undecidable. So to use CT as a modeling language requires automated theorem proving techniques, which are computationally expensive, and fast reasoning is a large part of what we're commercializing.
- RacfeelBudkind 5y ago> But the real question is whether it is a _useful_ modeling language for everything. Category theory is definitely not useful for modeling everything. However, it's useful for modeling a whole lot. Categories have few axoims, and those they do have align well with how humans break down complex problems, so a lot of systems can be modeled neatly with categories. And category models are useful for proving existence, uniqueness, and notions of "maximum" and "minimum" for their "arrows", which are often useful things people try to prove about systems.
- j2kun 5y agoI haven't seen any particularly convincing examples of this outside of pure math.