My answer on Quora: How does category theory relate to other branches of mathematics?

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:  Is it natural for a mathematician to forget specifics of some parts of math?

I think it is natural. I love to surprise my students by saying that I hardly remember any trigonometric formula beyond \(\sin^2x+\cos^2x=1\)– but I can deduce most standard trigonometric formulae on the spot. On a number of occasions I offered my students a game: give me a wrong formula, say, for \(\sin(x+y)\) — and I will instantly explain you why it is wrong. I have always won. I am not special; I believe that the vast majority of my mathematician colleagues can recover statements of L’Hopital theorems even if they do not remember them exactly.

And there is one more aspect of mathematical memory. The great mathematician Andrew Wiles said in a recent interview Andrew Wiles: what does it feel like to do maths?:

I really think it’s bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it’s a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that’s what actually enables you to get round [the problem].

So if you remembered all the failed attempts before, you wouldn’t try them again. But because I have a slightly bad memory I’ll probably try essentially the same thing again and then I realise I was just missing this one little thing I needed to do.

My answer to a question on Quora: Is it normal that one doesn’t understand a math procedure unless demonstrated by a teacher? Like for example, I cannot grasp what the textbook is trying to say but if I watch a video tutorial on how a procedure is done, I can understand right away.

Today I attended a seminar of mathematics education experts. They are seeing inability of many students to adequately understand written/printed texts as a serious and growing problem facing school mathematics education. Mathematical texts are difficult. What are other (that is, non-mathematical) difficult texts that you have read in your life? For example, have you ever tried to read fine print in an insurance policy? By complexity, this could easily beat maths textbooks. Ability to read and understand a difficult text is a useful skill — a skill for life.

My answer to a question on Quora:  Why do I need to learn conics in math?

This is the second best kept secret of mathematics education: it does not matter what students are taught, what matters is how deep their learning, and how efficient is the network of connections between mathematical facts that grows in their minds in their minds: in how many steps they can get form Fact A to Fact B? Conics could be included in high school courses of mathematics, and could be omitted, it does not really matter. What matters is whether students develop specific mental skills of mathematical thinking.

In one of the best mathematics high schools in the world, conics are used as a training ground of mathematics problem solving, that is, solving problems not seen by students ever before — simply because the theory of conics, if taken seriously, is rich, and because it is really hard to find in the literature good books on advanced level but elementary theory of conics. Students, I was told, really have to work from scratch.

I can offer you one problem; I think it was used at one of the International Mathematical Olympiads of yesteryear.

Assume you are given a sheet of paper with a parabola printed on it. Using only straightedge and compasses, construct the Cartesian coordinate system in which this parabola has equation y=x^2.

People who can solve this problem have reasonably deep understanding of conics.

My answer to a question on Quora: How do I start studying mathematics from the beginning until I get the gold medal in the Mathematics Olympiad?

I know a number of my fellow mathematicians who won gold medals at the International Mathematical Olympiad ( I think you mean that Olympiad). I doubt that any of them got this medal after starting studying mathematics specifically with this aim. The only fruitful way to learn mathematics is because you are interested in mathematics, because you wish to understand mathematics, because you wish to learn to recognise that specific feeling of joy which comes from understanding mathematics.

I once met a young girl who told me that she loved mathematics because she felt that mathematics biought her closer to  the God. This is a good reason for studying mathematics, but it is essentially the same as the one just described by me, but expressed from a different viewpoint. But let us think a bit: does winning a gold medal brings you closer to the God? I doubt that. Medal is from people, not the God.

An answer to a question on Quora: Is pure math hard? What type of people would study pure math?

My answer to a question in Quora: Why do pure mathematicians keep proving new theorems but don’t know their applications in real life?

I am a pure mathematician, but I have some experience of solving raw engineering problems, that is, problems where engineers have no vaguest idea what kind of mathematics could be applied. In one peculiar case I was using in my solution the classification of finite simple groups, the one of the most notorious achievements of the 20th century algebra — its original solution was spread over 100+ journal papers of about 10,000 pages in total (and contained gaps and holes which required additional hundreds of pages of proofs). It is hard to imagine anything more pure and remote from the so-called “real life” then the classification of finite simple groups.

I already said somewhere on Quora that

mathematics can be useful, but what makes it useful is not the same as what makes it mathematics.

Mathematics is a living organism which has to meet its own needs just to stay alive. It can be usefully compared with a cow. Cow is useful, for example, she gives us milk. Her udder definitely belongs to applied mathematics. But let us accept that the whole remarkably useful cow is applied mathematics.

The cow needs food and water, and air for breathing, etc. For production of milk, once in a year cow needs a rendezvous with a bull. We may perhaps compare pure mathematics with this bull.

Then I had commented a someone else’s answer:

Someone’s answer: There was a mathematician who believed math to be irrelevant if it had real world applications, so he devised a branch of mathematics using only on/off states. His name was George Boole, and the Boolean math he created is the basis of most of the electronic devices in use today. You don’t know what will become useful, but the math will be there for it.

My comment:

I disagree with your assessment of Boole. He designed what is now known as Boolean algebra as a way of checking and resolving logical arguments by some kind of arithmetic with True and False being values used in place of numbers. Prior to 19th century (when Boole lived) the use of logic was confined to Theology and Law, and Aristotelian logic taught to students (as a rule, of privileged classes) who were supposed to become priests or lawyers. As it is clear from his writing, Boole had aim to create logic for the masses, mental tools assisted by calculation on paper, for ordinary working people to understand, for example, court proceedings. He was, in nowadays terminology, a social activists: fought for improvement of working conditions for shopworkers, founded schools, credit unions, etc. The purpose of logical calculus that he invented was very pragmatic.

My answer to a question on Quora: Have you ever gotten answers by intuition in math but don’t know how to explain it?

In mathematics, an answer without explanation (proof) is not an answer. What you call “answer” or “result” (while this result is still not proven) is called a “conjecture”.

• Yes, mathematicians systematically produce conjectures in the process of work.
• Yes, they frequently do that using their intuition.
• These conjectures could be proved (sometimes much later) , or get refuted and die.

Refutation is no less important than proof. Without refutation, mathematics is a car without brakes.

In my life, I produced hundreds of conjectures. Most of them died, some instantly. Some are still open – that is, not proven and not refuted.

My answer to a question on Quora: If you are an autodidact, do you think you were born that way, or did you get so curious because of the environment you grew up in?

If you are an autodidact, what is the “actionable information” in an answer to this question? What will you change in your behaviour after receiving a definite answer “this was all nature” or “this was all nurture”?

But if you are a young parent of a child, I think it is my duty to you to say that I think that all children are born instinctive autodidacts, but this trait is usually destroyed in them by their upbringing in the family, school, etc.

In a parallel development, all children are born with natural abilities for mathematics, but by the age of 16 this gift is usually completely destroyed in them by the criminally bad mathematics education — this is why it is considered to be so rare and so precious.