This seems rather intuitive to me, and maybe that’s because I trained as a mathematician and I did that because I just happen to have the sort of mind that finds additional layers of abstraction useful on their own, but for the sake of being explicit I’ll tell you why I think it exists and why we would invent it now if it hadn’t already been.
Mathematics is (maybe) fundamentally about finding patterns in the world and reasoning about those patterns. When we see enough patterns, we can find patterns in those patterns, and then when we see enough meta-patterns and we find patterns in those patterns, and so on and on until we can’t find any more patterns. For example, geometry is, at least originally about the patterns we find when drawing stuff on a flat surface; arithmetic is about the patterns we find when we combine countable things; regular algebra is about the patterns we find in arithmetic; abstract algebra is about the patterns we find in regular algebra, geometry, and some other fields of mathematics with particular structures. Category theory is an extension of this pattern finding to the level of finding patterns across broad swaths of otherwise disconnected parts of mathematics.
It’s useful for several reasons, some internal to the theory itself, but I think largely because it gives us more general ways to reason about more concrete things. In addition to mathematics I also trained and work as a programmer, so my view is perhaps a bit biased here, but I find general abstractions useful because they let us deal with many concrete things that we would otherwise have to handle as special cases. With category theory we no longer have a bunch of mathematical silos that require the redevelopment of various concepts, and instead we have a general field that can at least give us for free theorems and structures and relationships for any part of mathematics that it adequately covers, thus I can take a result in category theory and use it to find similar results when applied to various fields.
Category theory also helps, much as abstract algebra did before it, to identify shared patterns across different fields of mathematics to set up correspondences that allow the transmutation of, say, a problem about graphs into a problem about complex variables without relying on a bunch of one-off proofs of shared structure because you can appeal to the categories to show how they relate. Yes, there is always stuff that doesn’t translate between fields because the fields have their own unique parts that are different because they are trying to model different things, but category theory at least lets us abstract away what we can from the noise and notice what’s going on in common.
It’s been a while since I did much academic math so I’m a bit fuzzy on specific results to point to, but I hope that gives a general sense of why category theory seems valuable and important to me.
This seems rather intuitive to me, and maybe that’s because I trained as a mathematician and I did that because I just happen to have the sort of mind that finds additional layers of abstraction useful on their own, but for the sake of being explicit I’ll tell you why I think it exists and why we would invent it now if it hadn’t already been.
Mathematics is (maybe) fundamentally about finding patterns in the world and reasoning about those patterns. When we see enough patterns, we can find patterns in those patterns, and then when we see enough meta-patterns and we find patterns in those patterns, and so on and on until we can’t find any more patterns. For example, geometry is, at least originally about the patterns we find when drawing stuff on a flat surface; arithmetic is about the patterns we find when we combine countable things; regular algebra is about the patterns we find in arithmetic; abstract algebra is about the patterns we find in regular algebra, geometry, and some other fields of mathematics with particular structures. Category theory is an extension of this pattern finding to the level of finding patterns across broad swaths of otherwise disconnected parts of mathematics.
It’s useful for several reasons, some internal to the theory itself, but I think largely because it gives us more general ways to reason about more concrete things. In addition to mathematics I also trained and work as a programmer, so my view is perhaps a bit biased here, but I find general abstractions useful because they let us deal with many concrete things that we would otherwise have to handle as special cases. With category theory we no longer have a bunch of mathematical silos that require the redevelopment of various concepts, and instead we have a general field that can at least give us for free theorems and structures and relationships for any part of mathematics that it adequately covers, thus I can take a result in category theory and use it to find similar results when applied to various fields.
Category theory also helps, much as abstract algebra did before it, to identify shared patterns across different fields of mathematics to set up correspondences that allow the transmutation of, say, a problem about graphs into a problem about complex variables without relying on a bunch of one-off proofs of shared structure because you can appeal to the categories to show how they relate. Yes, there is always stuff that doesn’t translate between fields because the fields have their own unique parts that are different because they are trying to model different things, but category theory at least lets us abstract away what we can from the noise and notice what’s going on in common.
It’s been a while since I did much academic math so I’m a bit fuzzy on specific results to point to, but I hope that gives a general sense of why category theory seems valuable and important to me.