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## Unity of Mathematics

### Perspectives in Analysis, Geometry, and Topology May 19-25, 2008, Stockholm

I want to use this opportunity, first, to thank the organizers of this beautiful conference, and I want to thank specifically professor Lennart Carleson for his very precise and great reaction to the events that were mentioned. I don't want to discuss the events, I don't want to use this my experience. It will be for another occasion. I just want to discuss the subject of unity of mathematics.

First of all let me ask why Mathematics has this property, why this unity is really what we see. I think that the main cause for this is that in all the branches of Mathematics we do study one thing, and this one thing is excellent, it is not often mentioned. It is what I would like to call {\it Mathematical Reality\/} or {\it Mathematical Universe.\/} It is not just the Universe that we live in, it's not parallel, it's not quite similar to it, but anyway all Mathematicians know what it is. When we work with mathematical objects, we work with objects of this universe, and this universe is quite connected. I mean that it has very nice topology: everything is related with everything.

It is very real: it is very resistant to an attempt to do anything arbitrarily. Try to prove a wrong theorem, you will see it is not easy. Try to work with bad definitions which are not well thought and you will see that it is not a good idea.

It is not an easily accessible place. We do not have originally senses which would allow us to get into it. We need to learn these things, various mathematical stuff, to get skills. So, it is not available to everybody, sort of partially closed.

We need to develop specific senses for this. One of them is, of course, the sense of proof: we really can check that something is correct. This is a language which sounds like the usual language that people speak, but it is not the same language. It is more precise.

This universe is somehow subdivided, but the subdivision is very vague, and often you can get to a quite unexpected area of mathematics just by one step from what you are doing. The division is induced mainly by division of mathematicians which was mentioned by professor Carleson in his speech, but I think that the main cause of this division of mathematicians is not in the new information technology. The information technology just help us to orient in this world. The cause is rather shrinking of the education period: the longer you study mathematics before you start to do mathematics the more you know about this universe.

I would suggest as a remedy to fight with this subdivision just make it longer and richer. In particular I think that prospective mathematicians should start to study mathematics earlier and this should be somehow more elite education as it happens somewhere. I think this is in interests of our community to have more people how know better this universe.

The picture of the mathematical universe or mathematical reality may be very attractive for newcomers. I want to share my own experience. When I was a teenager I was very excited by flights to the space to study other planets and stars, and, well, I even did something for that, some studies and training. But I was happy to stop at a good time, because the mathematical universe provides much more beautiful opportunity to study remote worlds. And you need very little to get into these new worlds, just your brain, you need to study new things, you need to think a little bit, you need to learn. I think the consistent picture of mathematical reality may attract young people just because it is attractive.

Of course, the main attraction is a success. How do mathematicians become mathematicians? Usually some sort of success: olympiad or exam, or class, and some beauty of mathematics, and this is already a mathematician.

I don't mean to discuss the nature of this mathematical reality, I think that it is real and there are doubts: quite often mathematicians do not agree that this is something real. Probably, historically the first discussion of this things can be traced to Plato's World of Ideas. Some people are allergic to ideas, and then we can speak, instead of ideas or the world of ideas, about a sort of interface between the physical world and our brain. You cannot solve problems without reformulation of them, without models and this is done usually with mathematics. But I don't think that mathematicians do think about mathematics as about this universal interface. We think about our mathematical objects as about very real objects. They can be models of something, it does not matter. They do belong to this mathematical reality.

Another point of view is that the mathematical world is a sort of toolset for study of physics or other things. We do not think in this way.

There is a problem of foundations and quite often this is an objection against speaking about this strange subject that mathematics is real or not real, maybe there is some contradiction and so on. We know pretty well that there is no contradiction. We do believe this.

If it would be some contradiction between some crazy axioms like it was at the very beginning... well, it was not really a contradiction, but it was an over optimistic hope that everything can be combined in some huge axiomatic system (may be, the shorter the better) from which everything will follow. You know, this Hilbert program... It was not realized because it was too optimistic, but it doesn't make sense to consider this as an obstruction to think about our domain as about something independent.

Really, mathematics now, well, at least publicly, has no object of study. Without objects it looks sort of pointless. We may speak about studying numbers, figures, integrals, but it's not clear why it worths to study. And this is a part of great problem which mathematics faces now.

It used to be much more respected by most of educated people, because they were very excited, in general, by mathematics at school. At school it was a great Mathematics, it was Geometry. In Geometry they learned how to argue, how to think. And now this Geometry in most western countries has gone, and this was done by mathematicians. Well, not by me personally, but by mathematicians of the previous generations, who did not understand what they were doing.

So what is instead? Instead is Calculus! In general education Geometry doesn't exist any more. It is not attractive, it is rather painful. This painful experience forces many people with power to cut our financing. It's really a difficult problem, I do not know how can we solve it. But if the solution at all exists, it should be looked for not in the general education, not in pump, how it is described in American discussions on this, because we have no teachers.

I mean that the hight school teachers are not competent for this, in general. There should be some good teachers.

It would be great if there were some positions for teachers at high school which would be paid as professor's positions and would be given to some professors. I would love to work at such school! And this school should attract really good students. I mean, this should be elitist process. I do not see how it can be done otherwise.

There is a difficult question about applied mathematics. To tell the truth, I do not know what is applied mathematics. There is mathematics which studies this universe of ideas, a mathematical universe, and there are application of these things. What is applied mathematics, how should it be qualified? If you take some formula and use it to calculate something, is it applied mathematics or not? So where is the borderline?

I think that as a community, mathematicians should accept existence of applied mathematics we shouldn't argue against this, let it be... but there should be one important thing: they, people who call themselves applied mathematicians, have different needs, they have different tastes. So we have to have a right to separate from them, to distance from them. If they like to get help, it's great, we can help them! When they like to stay at the same departments, quite often it happens that they stay and we don't. This is what happened in Uppsala, although it was presented in a different way.

I want to mention also a relation to Physics. Physics developed in a closed connection to mathematics, it was basically the same subject for centuries, it was the same culture. In 20th century it was separated somehow. And nonetheless it is still the same, physicists still think about the same set of notions, it's just another domain like algebra for geometers. The legendary effectiveness of physics in mathematics and mathematics in physics, I think, is not much greater than in general, if you look in a large scale, than the effectiveness, say, of geometric vision in function theory or algebra in geometry. I think, they accumulated huge experience and very nice collection of images. We have to work together, cooperate with them, as long as this is theoretical physics, and it should be really interesting, of course. They have different values sometimes. They do not like proofs sometimes, or we like the proofs too much, but these are not that important differences.

I don't want to take your attention too long. It is really late now. I have mentioned most of the things that I wanted to mention and I would be very grateful if you ask me something or tell me something about this. Thank you.

1. I think I will make a philosophical question, but I think this a philosophical talk.

- Do you really think these people who are not very sure about the direction or importance of certain topics in high school are mathematicians? For me it is not that if you have a degree in mathematics, that you are a mathematician. It is more than that. OK, I am just a student, not a mathematician yet, but in some other sense I could say, OK, I have a master degree, I am a mathematician. But I am not. I think you are a mathematician when you do mathematics, when you work in it. And when you work in it, you have clearness in mind that certain topics are extremely important and can be not cut off from the education.

-[Answer] The things which are cut out from the education usually are things which are proclaimed to be not useful. This is a completely wrong point of view. It led to disasters. The useful things which should be kept in the education are those which develop your mind. I think this is the major principle which should be formulated as much as possible.

-[The question continued:] To my opinion, those people who cut off certain topics in education are not mathematicians. They may have some degrees.

-[Answer.] Oh no, they were mathematicians. It started in France with the reform of geometry. Deudonne wrote a book about elementary geometry and linear algebra. He suggested to replace this "useless" elementary geometry with linear algebra. Trigonometry was claimed to be useful only for 4-5 nice categories of people like teachers of mathematics and geodesists. Something like that.

No, this is a common point of view. When I discussed these things with mathematicians they quite often say ok, let us skip in our education, say, even in mathematical education something, I don't know, say, Lebesgue integrals, because it is useless for most of people who study this. Well, you may skip everything, and what is left?

In general, the value of education is what is left after the student forgot everything specific. I think, that this is a correct principle.

-[Question] I'd like to point out that there is a huge difference between different countries. As myself, I can compare Germany and France quite well, and mathematics has just much higher importance in France, in the whole society, than in Germany. In France, for somebody to become a teacher in high school, in mathematics, the diploma they have to pass actually is quite exigent, whilst in Germany, those to become a high school teacher are rather the worst students. This also something which is certainly influenceable, and I am really sure that mathematical education in high school in France is much better than in Germany.

- [Answer] Of course, there are national differences and national specifics. France with its system of Grand Ecoles, when all the elite of the country passed through this Classe Prepertoire, is different from other countries where the political elite did not pass through this school. It's obvious. So, there is some immunity somewhere, say, in France. On the other hand, France is more formal in education than other countries. What is the main point? The tendency is the same. The trend of development, if you trace this over the number of years (I am 60 now). I can compare what happened and there are people who are elder than me and I think all would agree.

I did not want to say anything polemic, I just want to express our common opinion. If I am wrong I would be happy to hear objections. What I try to formulate, maybe I was not very successful in this, is more or less common feeling of mathematicians.

I saw, for example, as for applied mathematics, recently I have applied to many universities at math departments. It is natural in my situation... I visited them, I talk to them and I know that this is quite common problem, because of difference in needs, in directions, in the ways how they want to use resources. Quite often one person appears in math department and is very welcome, and then all the new hirings are in the same domain... This is also what I could see.

-[Question] How do you suggest making a public face of mathematics reflect this point of view?

-[Answer] I do not suggest anything as an organizer. I am just a person who may speak up about this, so, that's what I am doing.

-Sure...

- If you like this, you may do something that you... like. On some occasion you can use these ideas, or you may say that this is all wrong, but I think that at least attraction of attention to these problems is useful, and occasionally it may happen that some of the attempts will be successful. As for this elitist education, I think this is really important. and if at some place it would be possible to realize, it should be done, but done by right people.

-[Question] Is it true that two line version of your talk can be formulated as follows. Let us try to return geometry to school.

-[Answer] If it was realistic I would be happy to say that this is so. I do not think that it is realistic in a short term, even in Russia where this subject still exists at school, but is not included into exams, which is very important. But first, it should be included at universities, on the first course, I tried to do this in Uppsala. I gave several times lectures in elementary geometry.

-[Question (continued)] Is it true that this is the main point of your talk?

-[Answer] No, the main point is that there exists a mathematical reality. Period.

There exists the second thesis: it should distance from applied mathematicians and interact more with physicists.

-[Answer] No, I did not say this. I said exactly that we should have a right to distance from applied mathematics when it's getting really difficult. We should have a right, we have a moral right to say: well, this is not mathematics, this is not mathematics. It may be very useful for making whatever presentation about, say, the life of ants, but it is not what I want to have the next door in this department, not what I want to be taught to mathematical students. It is disorienting to them.

-[Question] There is another point about applied mathematics, which is that one shouldn't be labeling some brand of activity as applied mathematics as much as lots of mathematics is potentially applicable which is not endows domain as applied mathematics.

-[Answer] Yes, that's right. What is called applied mathematics is each time specific. Say, when I try to ask my colleges in Russia who are applied mathematicians: what is applied mathematics? Quite often they said: it is something about optimal control and things like that. And, well, today it is this, tomorrow it will be some computer algebra and I have to say that some of my own mathematics is used now by applied mathematicians. I am happy not to do this, but just in case I will join them, it's not a problem. I don't know what is applied mathematics.

-[Question] That is good, that is good, yea. Of course in Russia, in Soviet Union we had traditional good mathematicians working on applied problems. Gelfand, Kolmogorov.

-[Answer] And from time to time they were picked up, in Stalin time, and put somewhere to do this applied mathematics. They could do this. It was useful, why not?

-[Question] What do you think about the role of recreational mathematics? People who are not necessarily mathematicians, but were attracted by mathematical reality?

-[Answer] That's a good tool to infect with mathematics!

-[Question] Infect with what?

-[Answer] Well, you know, a mathematician is a person who cannot not to do mathematics. Let me say that in this way. So, this is a sort of sick person. To make them sick you have to present something attractive, and this is one of the ways to do this.

-[In Russian] Zarazit'

- To poison with mathematics. On its own, it may be nice. This is a part of mathematical universe, this is an easy way to invite people there. And we have to use this.

-[Question] It would promote mathematics in society.

-[Answer] Well, when we promote mathematics in society, the best way would be just to teach the prospective elite in a nice way. So that they would appreciate really this mathematics. When we made them suffer with uninteresting problems and do not show them anything good, we prepare for the next generation what we experience now.

-[Question] Is not this a contradiction, I mean, from the point of view of an average student or an average politician, the best way of being good is not being taught at all?

-[Answer] That's why. Because they have no good experience. You know, Napoleon was proud to prove theorems, and it was published, it was well known. It was another situation in the society, when popular magazines were happy to publish some problems. It is another situation, it is very difficult to get back to it, and probably impossible at one step.

-[Question] People's popular experience is curiously split. You have thousands of people working on, for example, sudoku process, which is low level mathematical problems, and very few people being caught up in periodicals by interesting mathematical problems, which could be much better then sudoku.

-[Answer] Yes, suggest something much better then sudoku and that's it.

- Trigonometry...

-[Answer] Probably, not that attractive. Well, trigonometry was necessary just to be able to do all these problems in elementary calculus where all the functions are polynomials, rational functions and trigonometric functions. Then we can say that this is analysis, but analysis of elementary functions. By the way, if you just take old textbooks on analysis and compare with modern books, you will see the great difference. Interesting things were washed out. Just get them back!

Well, this is just a general call! I want to use this opportunity, I do not want to organize whatever! I am not an organizer! Disclaimer: I am not this, I am not that! I am just a mathematician at this bad situation of being sixty.

-[Question] Another comment about the mathematical reality is that it is separate from us. It is what we do, that is what we up.

-[Answer] Yes, but I wouldn't even try to discuss if it is inside of us or outside. It's just we have, what we experience. This is very important thing in our life and it should be understood in this way by mathematicians. I think, this is a good point of view that what we are doing is some unity. This is the unity of mathematics, the unity of ideas which developed for quite a while.

-[Question] I want to stress one nice definition of applied mathematics that emerged in discussion there: it was "group of mathematicians working on applied problems". I think that should be the definition of applied mathematics. So, I mean whenever somebody is casing in a hiring program part and there is a pretense that this is applied mathematics, one should measure this against high standards. Gelfand was working on some applied problems. And if the candidate doesn't quite match...

-[Answer] I had these arguments, but it did not work. What is another problem which we really faced with Burglind at Uppsala University, is coming of new kind of administration, which are sort of managers from business. This is really a new problem and I cannot say much about this. Of course, personally, if someone suggested me money for just getting to a new, more interesting job, what can I say against this? But for the community it is terrible, because we have this tenure institution and it started to disappear. But I am not ready for final formulation.