**Mathematics is a way of studying our world**

Mathematics studies our world in its maximum abstraction. Physicists are used to working with manifestations, but mathematicians work with abstraction: they look for those properties of our world purified from manifestations, which can be studied and then applied. There is a terrific article by our great mathematician Manin, entitled “Mathematics as a Language for Describing Possibilities. At the end of his article Yury Manin gives the example of a shaman, who didn’t decide what to do with the tribe, but gave advice to the leader of the tribe on the best way to proceed. Yuri concludes with the following words: “Mathematics describes the phase space of the real world, the space of possibilities. It studies the laws that determine possible trajectories in this phase space, as well as the conditions, the set of information that is required to choose a particular phase trajectory. So it basically tells society that if you do this, you will go this way; if you do that, you will evolve that way. And this is such a higher view of mathematics.

[The Russian mathematician] Alexei Nikolayevich Krylov compares mathematics with a workshop: mathematicians prepare all kinds of tools for various needs. And when mankind has to solve some problem, professors of mathematics – scientists-experts of these tools – give mankind the tool it needs to solve the next problem: sometimes a rough file, sometimes a fine fileā¦

New fields are constantly emerging; mathematics expands a little, and from the new frontiers one can see further fields that can continue to be explored, which will then result in benefits for all of humanity. Here we can give a very simple example – the article “From Lobachevsky’s “crazy” geometry to GPS-navigators”. Here Lobachevsky invented his geometry. He died without even seeing a single working model. And of course, he could not think that later Riemannian geometry would be created. And later, on the basis of Riemannian geometry, Einstein would create his theories of relativity, special and general relativity. And now we, every day using a GPS navigator, use including the general and special theories of relativity. Because if we did not take into account the effects that they give, the error in determining the coordinates on the ground would be huge and the GPS would not be needed. The power of mathematics is that always one way or another these seemingly abstract studies then find their application to mankind.

Does the world only seem mathematical or is it a property of nature?

It is, of course, a property of nature. In general, all mathematicians are Platonists. There’s a little book called “Proof from the Book” – it’s about the fact that somewhere there are written proofs, and we humans can get close to them. And indeed, sometimes you take a proof, it can be very short, but it proves something very important. Actually, the book Proof from the Book itself begins with Euclid’s proof of the existence of an infinite number of prime numbers. A proof that has endured for centuries, and yet is really very beautiful, powerful, and interesting! And sometimes you look and see that there’s still something we don’t know and that’s why the proof is so complicated.

Mathematics dispels the fog – it explores what we haven’t yet explored in our world. It reveals the essence by formalizing and abstracting from something that is already attuned to that essence. And then it remains to apply these laws either to the subject or to the problem, depending on what humanity needs.

Math is all around us

Let’s talk about those manifestations of mathematics in our surrounding world which are clear to everyone, and on the other hand, reveal a mathematical approach, a mathematical component.

For example, the cycloidal curve made it possible to create the first isochronous pendulum clock, in which the period of oscillation did not depend on the amplitude. These were the first clocks. Obviously, there’s a lot of math in engineering. Here at school, everyone went over the parabola. But there is an optical property of a parabola, namely, that rays of light passing parallel to the axis of the parabola, after being reflected from it, fall into focus. Parabolic dishes, satellite dishes that look at a satellite, work according to this principle. And here’s a clear, simple example that has to do with high school math.

Or let’s take colors. The way computers provide us with color, how it adds up, it’s all based on mathematics. Just recently we were celebrating the 50th anniversary of the moon landing, the delivery of the moon rover to the moon. And in 1970, our Soviet lunar program began, and there was a device that we all know, which is a cataphot, an angle reflector. Three planes are mutually perpendicular to each other, and if you shine a beam on them, then after reflection from all three mirrors, the beam goes parallel to the direction from which it came. And it does not matter where you shine from, the beam will come back to you.

Exactly for this reason out of these little corners, mutually perpendicular to the three planes, make a cataphot on the