I am not a mathematician but I’ve been studying category theory for about a year now. From what I’ve learned so far it seems that it’s main benefit within pure mathematics is that it gives a way of translating between different domains of mathematical discourse. On the face of it, even if you’ve provided a common set-theoretic foundation for all areas of math, it isn’t obvious how higher level constructions in say, geometry, can be translated into the language of algebra or topology, or vice versa. So category theory was invented to facilitate this process of sharing mathematical insights across mathematical sub-disciplines. (I think specifically the context in which it originated was algebraic topology, which as the name implies uses techniques from abstract algebra to study topology.)
Later, computer scientists realized that category theory was useful for thinking about the structure of programs (e.g., data types and functions). For example, the concept of a Monad in functional programming which allows the simulation of side effects in a pure functional programming language comes directly from category theory. Bartosz Milewski is the person to look to if you are interested in learning about this aspect of things.
Even more recently (the last 10 years or so) people have started applying category theory to science more generally. Two books by David Spivak explore this here and here. I think much of this work in applied category is too recent to expect to see much in the way of big practical discoveries or breakthroughs. It remains to be seen if it will produce major innovations, but I think it is very promising. The hope is that category theory will provide scientists a way to model model more of the structure of both their research domain and the research process itself in a unified formalism. It also shows promise for modelling natural language concepts and argumentation, which could lead to better methods of computer knowledge representation.
On a more philosophical level, some have argued that category theory provides support for structuralism in the philosophy of mathematics. This view argues that mathematical entities are essentially structures, which is to say patterns of relationship. In category theory, what an object is is entirely determined by the pattern of relationships (morphisms) with other objects, within a given context (category). This contrasts with set theory, where sets are described in terms of their internal structure of elements and subsets. In practice, this means that set theory starts from the bottom (the empty set) and builds up to the whole mathematical universe, while category theory starts from the top (the category of categories) and then defines everything else in terms of universal properties.
Essentially, category theory validates the intuition that the number 5 isn’t some specific object floating out in Platonic heaven, nor is it just a made up meaningless symbol. It is a structure that is defined by it’s properties, and those properties are all determined by its relations to everything else. Without actually studying category theory it is difficult to see how this idea could be cashed out in a rigorous non-hand wavy way.
I am not a mathematician but I’ve been studying category theory for about a year now. From what I’ve learned so far it seems that it’s main benefit within pure mathematics is that it gives a way of translating between different domains of mathematical discourse. On the face of it, even if you’ve provided a common set-theoretic foundation for all areas of math, it isn’t obvious how higher level constructions in say, geometry, can be translated into the language of algebra or topology, or vice versa. So category theory was invented to facilitate this process of sharing mathematical insights across mathematical sub-disciplines. (I think specifically the context in which it originated was algebraic topology, which as the name implies uses techniques from abstract algebra to study topology.)
Later, computer scientists realized that category theory was useful for thinking about the structure of programs (e.g., data types and functions). For example, the concept of a Monad in functional programming which allows the simulation of side effects in a pure functional programming language comes directly from category theory. Bartosz Milewski is the person to look to if you are interested in learning about this aspect of things.
Even more recently (the last 10 years or so) people have started applying category theory to science more generally. Two books by David Spivak explore this here and here. I think much of this work in applied category is too recent to expect to see much in the way of big practical discoveries or breakthroughs. It remains to be seen if it will produce major innovations, but I think it is very promising. The hope is that category theory will provide scientists a way to model model more of the structure of both their research domain and the research process itself in a unified formalism. It also shows promise for modelling natural language concepts and argumentation, which could lead to better methods of computer knowledge representation.
On a more philosophical level, some have argued that category theory provides support for structuralism in the philosophy of mathematics. This view argues that mathematical entities are essentially structures, which is to say patterns of relationship. In category theory, what an object is is entirely determined by the pattern of relationships (morphisms) with other objects, within a given context (category). This contrasts with set theory, where sets are described in terms of their internal structure of elements and subsets. In practice, this means that set theory starts from the bottom (the empty set) and builds up to the whole mathematical universe, while category theory starts from the top (the category of categories) and then defines everything else in terms of universal properties.
Essentially, category theory validates the intuition that the number 5 isn’t some specific object floating out in Platonic heaven, nor is it just a made up meaningless symbol. It is a structure that is defined by it’s properties, and those properties are all determined by its relations to everything else. Without actually studying category theory it is difficult to see how this idea could be cashed out in a rigorous non-hand wavy way.