Nikolai Andreev, Mathematician: Sometimes Formulas Are Clearer Than Words

13 October, 16:42

Original post by vm.ru (in Russian)

How to mine ore and harden steel? Why is the plane able to take off? What color is a black hole in space? It is not common for adults to ask such questions — you either know everything, or you just take it as it is. This is an axiom that does not require proof. But sometimes you want to look at highly specialized things differently. Learning the little details is not for the sake of curiosity, but for understanding life. Within the new section “Simple Questions”, we are going to discuss this topic with professionals.

Our first hero is Nikolai Andreev, the Head of the Laboratory of Popularization and Promotion of Mathematics. With a mild jittery I walked into the Steklov Mathematical Institute of the Russian Academy of Sciences — the place where he works. One of my most unpleasant nightmares is the one, where I“m back in high school and haveing a math test. But I don“t remember a single formula. However, the worry fades quickly, since my conversationalist smiles and being friendly. Before starting the conversation, he invites me to go up to the portrait gallery of scientists working at the institute.

— Here is the whole history of Russian mathematics! — emphasizes Nikolai Andreev. — And about each one you can tell a lot, and not all about theorems. Boris Delone is a geometer. He developed a classification of mountain passes that tourists still use. Alexey Krylov is a renowned shipbuilder. Mstislav Keldysh is a mathematician of Sergei Korolev“s space program . Leonid Kantorovich is a Nobel laureate in economics. Alexey Pogorelov is the author of a well-known geometry textbook. These people are highly respected.

The sunset slips through the windows and enlightening the piano.

— Why do you need this here?

— Piano? People play in the evenings...

Who are these “people”, I will learn in a few minutes — young scientists talk in a room. This space is full of shelves, charted books with worn covers, where the leather sofa and cupboard of the XIX century, owned by Paphnutiy Chebyshev, founder of Petersburg mathematical school, are standing. The tone of the conversation is so inspiring that I am eager to join it, but I“m not sure if I can really become a proper part of this conversation. Soon they leave.

— At the popular science lectures, which I give to schoolchildren, I say that if you go into science, one of the perks will be communicating with very interesting people, — says the scientist, noticing my curiosity. — And as you can see, we have many young people here. Mathematicians are really nontrivial, often very well- and wide-educated persons with whom you can talk about almost everything: literature, music, theatre, photography. Although there“s less trivia discussion in this community.

Substantive Approach

The chalkboard, full of formulas, the man with the glasses who flips it over, the mutter and the check-up with instruments — that“s what I was waiting for, but didn“t get. That“s not what math is really about. As a philologist, who seemed to understand “what the author wanted to say in a novel”, I ask what studies mathematics, what kind of problems it has.

— The essence of mathematics is the study of the world. And what normal people face — to divide or multiply numbers, to remember the formula, and so on — is only the initial “gear” of mathematics, — explains Nikolai Andreev.

— The main purpose of this science is to understand the world, in particular, to see, to define what is common in phenomena around us, to study their principle, and then to help people applying the knowledge they have acquired to other fields. While physics and chemistry are based on the realization of this ideal world in our space, mathematics studies its quintessence.

For the reader to experience this, the scientist addresses us to the book “Mathematical Component” (2019), in which the editors from his laboratory united leading Russian mathematicians as authors.

— When you look at the cover of this book, you“ll see a football, chips, sunflowers, a satellite dish, a wheeled pair of railway cars, a map of the Earth, a musical scale. Mathematics is everywhere, — Andreev gives examples. — And if you look closer, you“ll find that the answers to the seemingly unrelated questions — why the octave on the piano is divided into 12 steps, whether the Gregorian calendar is the best designed, what the size of the A4 paper format — come from the same mathematical basics. One of the articles, written by mathematician and philosopher Yuri Manin, is entitled: “Mathematics is a language of possibility description”. For humanity, this science determines the initial conditions that you need to achieve something. In this book, Yuri Manin explains why governors need mathematics as advisers. We also have a section “Book Shelf”, which contains books that may be interesting to read.

But I am still haunted by the notorious “X” and “Y”, infinite symbols. I“m asking him to tell me about mathematical language.

— Formulas are the tool that makes transferring knowledge from one person to another possible, simple, and accurate. And in this sense, it is also a language that makes this process much easier. It is common for people to be a little afraid of formulas. But an actual formula writing is much more transparent, understandable, and clear than the words, — assures Nikolai Andreev.

— When we were writing our book, we tried to keep formulas behind in order not to frighten the reader. And we were arguing a lot! Although we understood all the topics from a mathematical point of view, it was much more difficult to convey meaning to the Russian language than mathematical.

The Focus

I“m eager to learn what“s on the minds of mathematicians these days.

— Classifying problems is an ungrateful business, but there are some guidelines. Every four years there is a congress of mathematicians. In our country, it was only in 1966 in Moscow. Next will be summer 2022 in Saint Petersburg. There will be thousands of leading mathematicians from all over the world. The top mathematics will be awarded the most famous mathematician prize — the Fields Medal. The plenary reports selected by the International Committee will also be read, — says Nikolai Andreev. — All of this demonstrates the direction of mathematics. At the similar congress in 1900 it was a speech by David Hilbert, who formulated 23 problems that influenced 20th-century mathematics.

And at the turn of the 21st century, the Clay Mathematics Institute decided to reuse this idea, and so the seven challenges of the millennium emerged.

My conversationalist explains that these tasks are difficult even in their design. But some of the unresolved ones can be formulated in a way that even a schoolboy can understand.

— For example, there“s a lot that we don“t understand about prime numbers — those that are divisible by themselves and one — 2, 3, 5, and so on. By the way, Pafnuty Chebyshev was the first, after Euclid (who proved that primes are infinite set), to advance solving problems about their distribution. Now, among them are the so-called twins — two prime numbers, the difference between which is two —  11 and 13, 17 and 19, and further, — tells Andreev. — It has not yet been proven whether this is an infinite set or, as far as the natural series goes, there will always be two prime numbers that differ by two.

I“m asking how that applies to the reality that surrounds us.

— Prime numbers are the basis of one of the encryption algorithms that we use every day: it“s the banking, it“s the HTTPS Internet protocol and others. I mean, on the one hand, at school, we“re still learning what a prime number is. On the other, we use it every day, often without even noticing. On the third, there are challenges related to them that have not yet been solved.

I“m asking if there are any discoveries in the mathematical environment.

— Mathematicians are peaceful people, they do not argue, but share their knowledge with each other at conferences, seminars to allow others to continue developing some studies, — smiles Andreev. — That“s what mathematics is good for, it makes it possible not to argue, but logically to convince each other. Besides, mathematics teaches honesty.

Basic Knowledge

Charmed by my conversationalist speech, I understand that I was wrong to think of this science as something through the roof, as many people do. But the popular question is, why teach math at school when people don“t notice how it works around them?

— To see the beauty of mathematics and enjoy it, it is necessary to have a certain level of knowledge. If you, for example, can“t open parentheses, then it“s going to be hard to convert an expression and solve a problem. And such a necessity takes away the love of mathematics from many schoolchildren, — says Andreev. — Some people are starting to think that they are humanitarians and are not given that. But, if they had learned, without analyzing, how to add fractions, to open brackets, and so on, they would have felt the interest of the problem rather than the difficulty of solving it. Then more people would love math. In addition, mathematics teaches to reason logically and to prove one“s position in such a way that the other can test its right. Society cannot live without it. So in any case, the study of mathematics changes a person — improves the ability to reason, to work with certain objects, not to declare: “Snow is red because I want so!”, but to prove everything logically.

When it comes to the idea that mathematics is not needed in real life, for many, the argument is that a calculator can calculate everything.

And this is where I learn a new fact.

— Did you know that the modern computer is based on the Karatsuba algorithm? For centuries, one number by another was multiplied by a column. And mathematician Anatoly Karatsuba, a scientist from our institute, in the 60s of the XX century, wondered if it could be done faster, — Nikolai enlightens me. — And I came up with my own multiplication method, which is now being used in all computers and calculators, which is improving the lives of all of humanity. In general, if you think about what mathematics is embedded in the subjects around us, not only will life be more interesting, but you can do a lot more in life.

Enlightenment

We also met with the Head of the Laboratory for Popularization and Promotion of Mathematics to find out how we can popularize mathematics, what projects are already working, and showing results.

— Our main project “Mathematical Etudes” is available online There are films about solved and unsolved mathematical problems, applications of mathematics in engineering. Another direction — development of visual mathematical models, — Nikolai Andreev shows a puzzle illustrating the Pythagorean theorem, billiards demonstrating the optical property of parabola, which is used in satellite antennas and other models in the laboratory. — I dream that someday these models will become visual aids in schools. But such decisions have to be made at the government level. We are still creating an electronic encyclopedia on our website. In our channel on one of the social media, we launched the project “Mathematical model in school!”: we invite families to make a mathematical model, donate it to the school, and send us a photo. In response, we will send a book “Mathematical component”.

By the way, mathematics that studies the perfect world is hard to imagine as an object of the real world. Developing such models is, therefore, a non-trivial and important exercise. I also want to see the opening of the Mathematics Museum one day. There the beauty of this science will be felt in all possible senses. We are also talking about world-class mathematical centers, which are created within the framework of the national project “Science and universities”.

— Our country already has several regional scientific and educational mathematical centers. For example, the Caucasus Mathematics Center is very vivid as well as the Adygei State University, — shares Andreev. — In particular, there, together with our institute, has been created the first mathematical park in Russia, where mathematical sculptures are placed in the open air. It may interest passers-by, because they also relate to the visualization of mathematics. And, you“re right, four world-class international mathematics centers were created some time ago, one of which is based on our institute. Even with the pandemic, this allowed us to invite young and smart Russian and foreign scientists. And our Laboratory for the Popularization and Promotion of Mathematics, which was created in 2010, has given a new motivation to development.

On The Big Screen

I also want to know how you can look at mathematics in a live-action.

— It is important to recall the unique Soviet experience of live-action educational films.

For the younger ones, it is, for example, the “Arithmetic” episode based on the Eralash series. For teenagers and adult audiences it was the fiction film by Semyon Reitburt “The Mathematician and The Devil” (1972), which was about now proven, and then not yet, Fermat“s theorem. It was the first role of Alexander Kaidanovsky, who is a brilliant Devil. Reitburt has other educational live-action films — it is a very interesting approach to popularization! Some YouTube channels, mostly foreign ones, such as 3Blue1Brown, Mathologer, and others, — adds Andreev. — Our project “Mathematical Etudes” is an example of the use of computer 3D graphics, making mathematics visual.

Nikolai Andreev also tells about scientific documentaries.

— In the representation of science on television, Russia is unfortunately not yet ahead of the world, but sometimes we try to make shows, mainly documentaries. In particular, this year was the 200th anniversary of the great mathematician Pafnuti Chebyshev, — says Andreev. — And we filmed a movie about him in this very laboratory. It was a part of the project “Scientific Russia. Great names”. In fact, the field of science that Pafnuty Lvovich started — the theory of approximation of function — we still use today, in particular when transmitting a television signal.

Nikolay Andreyev — the Head of the Laboratory for Popularization and Promotion of Mathematics of the Steklov Mathematical Institute. Born on February 5, 1975, in Saratov. Graduated from the Mechanical and Mathematical Faculty of Moscow State University. In 2010 he received the President of the Russian Federation Prize for Science and Innovation for Young Scientists, and his works were awarded the Gold Medal of the Russian Academy of Sciences in 2017.