Complexity of finite sequences of zeros and ones and the geometry of finite function spaces

Recently, in the American book Murphy’s Laws, I found a clear classification of all sciences: “If it stinks, then it’s chemistry; when nothing works, it’s physics; and if you can’t understand a word, it’s mathematics.”

I have struggled with this notion all my life.

In my opinion, mathematics is just a part of physics, an experimental science that reveals to mankind the most important and simple laws of nature.

The only difference between mathematics and physics is that in physics experiments cost millions or even billions of dollars, while in mathematics they cost a few rubles or kopecks.

Today I intend to show you how, with the help of simple experiments, new and unexpected laws of nature can be discovered.

Following Newton and Poincare, I consider the best of these discoveries to be important milestones in the progress of human civilization. Some modern mathematicians take the opposite view.

For example, Hardy explained the words of Gauss “mathematics is the queen of sciences” by the complete uselessness of both. The director of the Max Planck Institute for Mathematics in Bonn wrote (in an article dedicated to the 2000th anniversary of Jesus Christ) that mathematics is a formalized transfusion from empty to empty, and its contribution to solving the main problem of modern post-industrial humanity is, in his opinion, “in the abstract the best minds from more dangerous pursuits than mathematics.”

“Some idiots think,” he says, “that mathematics is useful for physics and technology” (there is no word “idiots” in his article, but he argued with me, and I ordered this article from him on behalf of the International Mathematical Union, which published by 2000). “The real benefit,” he says, “is that if instead of Fermat’s problem, mathematicians would improve cars or airplanes, then there would be much more harm.”

I’m not going to improve cars, or planes, or even cryptography, to which today’s report relates.

But all my life I have been following the recipe of Dirac, who taught how to create New Physics, in the following words:

“First of all,” Dirac said, “one must discard all so-called ‘physical representations’, for they are nothing but a term for denoting the obsolete prejudices of previous generations.”

To begin, according to him, should be with a beautiful mathematical theory.

“If she is really beautiful,” says Dirac, “then she will certainly turn out to be an excellent model of important physical phenomena. So we need to look for these phenomena, develop applications of a beautiful mathematical theory and interpret them as predictions of new laws of physics, ”this is how, according to Dirac, all new physics, both relativistic and quantum, is built.

By the way, few people know that three years before Hilbert’s problems and ten years before Einstein, Poincaré formulated the main problem left by the 19th century as a legacy to the 20th century in the field of mathematics. In Poincare’s formulation, the main task is as follows: the construction of a mathematical theory for relativistic and quantum phenomena. Well, by the way, he did this for the relativistic case – however, for some reason, in a strange way, Einstein forgot to refer to it until 1945. In 1945, he mentioned that Minkowski advised him to read that ten years before Einstein, Minkowski’s friend Poincaré had published. Here.

It is even less well known that Dirac’s relativistic electronic equations arose from the ancient mathematical theory of braids. Namely: Dirac noticed, based on the topology of a family of elliptic curves in algebraic geometry, that in the group of spherical braids of four threads there is an element of the second order, and he interpreted this discovery as a theory of the electron spin, which has 2 values ​​(this means that in order to , in order for the particle to return to its previous position, it needs to rotate not by 360 degrees, but by 720).

This was not clear to anyone, and therefore they did not believe him. To convince physicists of the validity of the corresponding strange mathematical theorem (stating that the fundamental group of the SO(3) group of rotations of three-dimensional space consists of two elements), Dirac demonstrated the corresponding experiment by physically fabricating his second-order spherical braid.

This braid is made like this: a sphere and another smaller sphere concentric with it are taken and connected with four ropes. Four nails are driven into the outer sphere, four into the inner one, and four ropes connect them, but in such a way that these ropes do not go along the radius, but intertwine with each other. The second, exactly the same, braid (this is called the “spherical braid”) – the second braid, arranged in exactly the same way, connects a smaller sphere with an even smaller one.

And now, the element of the second order is what it is.

This means that if you remove the middle sphere, you get four ropes connecting the largest to the smallest. Now, they turned out to be unentangled, they were entangled between large and medium, entangled between medium and small in the same way. And if the middle one is removed, then between the large and small ones they can be dragged by a continuous transformation to radial non-entangled ones. Floor the trivial braid is learned.

This is the mathematical theorem in question, which Dirac proved. Dirac made these spheres and burned the middle one. The spheres were connected by loose ropes, and physicists believed in the spin theory. So he proved it.

By the way, neither physicists nor mathematicians now know this. Maybe one I read from Dirac how it’s done and how he came up with it. But physicists believe in spin, because it is proclaimed there, they give Nobel prizes for it, which means that everyone already knows this, that this is a famous, great thing. And everybody believes, just because it’s proclaimed to be so.

So. In fact, this discovery of Dirac – the theory of spin – was based on an experiment that proved a mathematical theorem.

Therefore, I will now begin the story of my experiments, which I will now show you, but I will only make one preliminary remark about the report.

Faraday, who learned everything by himself, was the first to organize public scientific lectures for unprepared listeners, and they were wonderful. As a result of his lectures, Faraday came to this conclusion: “A truly instructive lecture can never be popular, and a truly popular one can never achieve real instructiveness.”

I will try to refute this point of view of the great physicist (whose theories formed the basis of modern civilization after Maxwell wrote them mathematically in the form of his famous system of equations).