My answer to a question on Quora:

An excellent question. Category theory is important on its own, and has important applications in a number of other mathematical theories; however, the crucial and the most fundamental impact of category theory is invisible and under-reported, it is of cultural nature. It can be compared with the influence of set theory: 99% of mathematicians use only a modicum of naive set theory, ignoring deeply penetrating and frequently very hard results of set theory as the live research discipline which continues to develop and flourish.

There is a telling example: the theory of games of chance was created in the 17th century and gave birth to probability theory; the latter was already quite developed by the time when, in the 20th century, the concept of a deterministic game had finally crystallized – in a paper, of all people, by Zermelo, who proved that chess was a deterministic game: for one of the players, there is a strategy, that is, a function from the set of permissible position to the set of moves, which achieves at least a draw. His paper was published in 1913 (see Zermelo’s theorem (game theory) – Wikipedia). Why did this happen so late? The word “set” in the definition of a strategy came to use only in the second half of the 19th century.

The same is happening with category theory: the vast majority of mathematicians use its ideas and terminology in a very rudimentary and naive form, frequently even without realisation that they are doing so. In the work that I am doing, it had happened to be very important to remember that an algebraic group was a functor from the category of unital commutative rings to the category of groups. Some my colleagues who work in the same theory continue to insist that a group is a fixed set with some operations on it. When my co-author and I recently solved a certain problem which was open since 1999, we were able to do that only because for us a group in question was a functor – not much deeper than that. Why it was not solved by someone else earlier? Because for them a group was just a set.

My answer to a question on Quora:

The answer is very simple: Statistics and Mathematics have different sources of funding. From time to time, mathematical disciplines get separated from mathematics because their end users (industry, the military, the government) wish to pay only for the end product, and do not wish to subsidise its mathematics roots. This has happened with computer science, cryptography, fluid dynamics, etc. — and statistics is just one of many examples. There is no any intrinsic reason for all these disciplines to be separate from mathematics.

Below is a fragment from a book written by my university lecturer in mathematical statistics on the bases of lecture notes taken by his PhD students (they were attending for that purpose our undergraduate lectures). This is the very first chapter, the first definition: The notion of a sample.

As you can see from this example, mathematical statistics can be the purest of pure mathematics.