4 ms·

> Since it is being gradually replaced by category theory as a more modern, practically better, and a more powerful foundation of mathematics i don't see this.

> Since it is being gradually replaced by category theory as a more modern, practically better, and a more powerful foundation of mathematics i don't see this. set theory is very practical and is a good thing to use to build things up in a granular manner. category theory is useful to connect and relate things. they're at two different ends of the spectrum. category theory isn't "replacing" anything.

Koshkin 6y ago
The truth is, at least since the early 1960s category theory has been a much, much more powerful tool in many branches of modern mathematical research than set theory. (By the way, it can be an enlightening exercise, to try and restate the simple notions about sets purely in terms of maps - notions like 'B is a subset of A', 'C is a union of A and B', 'map f is injective/surjective', 'C is a Cartesian product of A and B', etc. Highly recommended!) Category theory is very deep and easily subsumes much, if not all, of set theory. The difference in power is staggering. Sets in general are now considered just one of many categories and play a subordinate role. I agree that set theory is more useful when studying math at an elementary level, but the sooner one starts getting used to categorical ways of reasoning, the better one's chances to make progress in understanding modern mathematics.
- nikofeyn 6y ago
  you're missing the argument i made. i didn't say category theory isn't useful or powerful. i said it isn't replacing set theory in practice. you can make it through the lion's share of a ph.d. in math without using category theory, but you'll use set theory ubiquitously up until you start doing categorical things. of course category theory is a powerful relator. but it is awkward for when simple sets do just fine. loring tu's manifolds book is a good example of this. set theory is used throughout with categorical concepts sprinkled about to show there's power of relation there. and that's what i said. set theory is helpful as a brick. category theory is helpful as an architect.

Koshkin 6y ago

The truth is, at least since the early 1960s category theory has been a much, much more powerful tool in many branches of modern mathematical research than set theory. (By the way, it can be an enlightening exercise, to try and restate the simple notions about sets purely in terms of maps - notions like 'B is a subset of A', 'C is a union of A and B', 'map f is injective/surjective', 'C is a Cartesian product of A and B', etc. Highly recommended!) Category theory is very deep and easily subsumes much, if not all, of set theory. The difference in power is staggering. Sets in general are now considered just one of many categories and play a subordinate role. I agree that set theory is more useful when studying math at an elementary level, but the sooner one starts getting used to categorical ways of reasoning, the better one's chances to make progress in understanding modern mathematics.

nikofeyn 6y ago
you're missing the argument i made. i didn't say category theory isn't useful or powerful. i said it isn't replacing set theory in practice. you can make it through the lion's share of a ph.d. in math without using category theory, but you'll use set theory ubiquitously up until you start doing categorical things. of course category theory is a powerful relator. but it is awkward for when simple sets do just fine. loring tu's manifolds book is a good example of this. set theory is used throughout with categorical concepts sprinkled about to show there's power of relation there. and that's what i said. set theory is helpful as a brick. category theory is helpful as an architect.