3 ms·
Category theory would help them. Yoneda's lemma tells us that elements of collections are uniquely isomorphic to the transformations which create those elements
by myWindoonn 5y ago
Category theory would help them. Yoneda's lemma tells us that elements of collections are uniquely isomorphic to the transformations which create those elements. This explains the butterfly effect where seemingly-unrelated parts of logical statements are causally connected.
This post would have had bite 70 years ago, before we had "formally formal" theories of categories which can subsume any possible set theory. But now the door is closed. Interestingly, the transformations seem to be more important than the elements; whether a logic is e.g. reversible is a property of transformations and not collections.
- AnimalMuppet 5y agoIsn't that a bit recursive? Don't you use formal logic in creating category theory? Then using category theory to create formal logic is circular.
- myWindoonn 5y agoSets are 0-categories, so this critique would also suggest that 0-category theories, set theories, are circular too. But we actually place formal logic before sets, and then use sets to reflectively study formal logic without infinite regress. This is a version of the general pattern that there are no informal justifications for formal reasoning. Once a person tastes formality, then they can judge for themselves whether to adopt it, but there's no self-sustaining proof statement.