The reason for the translation of the article was that I was looking for a book by the author of “The Outer Limits of Reason” . I couldn’t hide the book, but I came across an article that, in a rather concise way, shows the author’s view of the problem.

Introduction

One of the most interesting problems of the philosophy of science is the connection between mathematics and physical reality. Why is mathematics so good at describing what is happening in the universe? Indeed, many areas of mathematics were formed without any involvement of physics, however, as it turned out, they became the basis in the description of some physical laws. How can this be explained?


Most clearly, this paradox can be observed in situations where some physical objects were first discovered mathematically, and only then evidence was found of their physical existence. The most famous example is the discovery of Neptune. Urbain Le Verrier made this discovery simply by calculating the orbit of Uranus and exploring the discrepancies between the predictions and the real picture. Other examples are Dirac’s prediction of positrons and Maxwell’s suggestion that waves in an electric or magnetic field should generate waves.

Even more surprisingly, some areas of mathematics existed long before physicists realized that they were suitable for explaining certain aspects of the universe. Conical sections studied by Apollonius in ancient Greece were used by Kepler in the early 17th century to describe the orbits of the planets. Complex numbers were proposed several centuries before physicists began to use them to describe quantum mechanics. Non-Euclidean geometry was created decades before the theory of relativity.

Why does mathematics describe natural phenomena so well? Why out of all the ways in which thoughts are expressed, does mathematics work best? Why, for example, it is impossible to predict the exact trajectory of the motion of celestial bodies in the language of poetry? Why can't we express the complexity of the periodic table with a piece of music? Why doesn't meditation help much in predicting the outcome of experiments in quantum mechanics?

Nobel laureate Eugene Wigner, in his article “The unreasonable effectiveness of mathematics in the natural sciences”, also asks these questions. Wigner did not give us any specific answers, he wrote that "the incredible effectiveness of mathematics in the natural sciences is something mystical and there is no rational explanation for this."

Albert Einstein wrote on this subject:

How can mathematics, the product of the human mind, independent of individual experience, be such an appropriate way to describe objects in reality? Can the human mind then, by the power of thought, without resorting to experience, comprehend the properties of the universe? [Einstein]

Let's be clear. The problem really arises when we perceive mathematics and physics as 2 different, perfectly formed and objective areas. If you look at the situation from this perspective, it is really not clear why these two disciplines work so well together. Why are the open laws of physics so well described by (already open) mathematics?

This question was pondered by many people, and they gave many solutions to this problem. Theologians, for example, have proposed a Being who builds the laws of nature, and at the same time uses the language of mathematics. However, the introduction of such a Being only complicates everything. Platonists (and their cousins ​​naturalists) believe in the existence of a "world of ideas" that contains all mathematical objects, forms, as well as Truth. There are also physical laws. The problem with the Platonists is that they introduce another concept of the Platonic world, and now we need to explain the relationship between the three worlds ( translator's note. I still did not understand why the third world, but left it as it is ). The question also arises whether imperfect theorems are ideal forms (objects of the world of ideas). What about disproved physical laws?

The most popular version of solving the posed problem of the effectiveness of mathematics is that we study mathematics by observing the physical world. We understood some of the properties of addition and multiplication by counting sheep and stones. We studied geometry by observing physical forms. From this point of view, it is not surprising that physics follows mathematics, because mathematics is formed with a careful study of the physical world. The main problem with this solution is that mathematics is used well in areas that are far from human perception. Why is the hidden world of subatomic particles so well described by mathematics, studied by counting sheep and stones? why the special theory of relativity, which works with objects moving at speeds close to the speed of light, is well described by mathematics,

In two articles ( one , two ), Macr Zeltser and I (Noson Janowski) formulated a new look at the nature of mathematics ( comment by a translator. On the whole, the same article is written as here, but much more extensively ). We have shown that, as in physics, symmetry plays a huge role in mathematics. Such a view gives a rather original solution to the problem posed.

What is physics

Before considering the reason for the effectiveness of mathematics in physics, we need to talk about what physical laws are. To say that physical laws describe physical phenomena is somewhat frivolous. To begin with, we can say that each law describes many phenomena. For example, the law of gravity tells us what will happen if I drop my spoon, it also describes the fall of my spoon tomorrow, or what happens if I drop a spoon in a month on Saturn. Laws describe a whole range of different phenomena. You can go on the other side. One physical phenomenon can be observed in completely different ways. Someone will say that the object is motionless, someone that the object moves at a constant speed. A physical law should describe both cases identically. Also, for example, the theory of gravity should describe my observation of a falling spoon in a moving car,

The following question arises: how to classify physical phenomena? Which ones should be grouped together and attributed to one law? Physicists use the concept of symmetry for this. In colloquial speech, the word symmetry is used for physical objects. We say that a room is symmetrical if its left side is similar to the right. In other words, if we swap sides, the room will look exactly the same. Physicists have expanded this definition a bit and apply it to physical laws. A physical law is symmetrical with respect to transformation if the law describes the transformed phenomenon in the same way. For example, physical laws are symmetric in space. That is, the phenomenon observed in Pisa can also be observed in Princeton. Physical laws are also symmetric in time, i.e. experiment, carried out today should give the same results as if it had been carried out tomorrow. Another obvious symmetry is spatial orientation.

There are many other types of symmetries that physical laws must conform to. Galilean relativity requires that the physical laws of motion remain unchanged, regardless of whether the object is stationary, or moving at a constant speed. The special theory of relativity states that the laws of motion must remain the same even if the object moves at a speed close to the speed of light. The general theory of relativity says that the laws remain the same even if the object moves with acceleration.

Physicists generalized the concept of symmetry in different ways: local symmetry, global symmetry, continuous symmetry, discrete symmetry, etc. Victor Stanger joined many types of symmetry according to what we call point of view invariance. This means that the laws of physics must remain unchanged, regardless of who observes them and how. He showed how many areas of modern physics (but not all) can be reduced to laws that satisfy invariance with respect to the observer. This means that phenomena related to one phenomenon are related, despite the fact that they can be considered in different ways.

Understanding the true importance of symmetry has gone with Einstein's theory of relativity. Before him, people first discovered some kind of physical law, and then found a symmetry property in it. Einstein used symmetry to find the law. He postulated that the law should be the same for a motionless observer and for an observer moving at a speed close to light. With this assumption, he described the equations of the special theory of relativity. It was a revolution in physics. Einstein realized that symmetry is a defining characteristic of the laws of nature. Not the law satisfies symmetry, but symmetry breeds the law.

In 1918, Emmy Noether showed that symmetry is an even more important concept in physics than previously thought. She proved a theorem linking symmetries with conservation laws. The theorem showed that each symmetry generates its own conservation law, and vice versa. For example, the invariance for displacement in space gives rise to the law of conservation of linear momentum. Time invariance gives rise to the law of conservation of energy. Orientation invariance generates the law of conservation of angular momentum. After that, physicists began to look for new types of symmetries in order to find new laws of physics.

Thus, we determined what to call a physical law. From this point of view, it is not surprising that these laws seem objective, timeless, and independent of man. Since they are invariant with respect to the place, time, and the person’s view of them, it seems that they exist "somewhere there." However, this can be seen in a different way. Instead of saying that we are looking at many different consequences from external laws, we can say that a person singled out some observable physical phenomena, found something similar in them and combined them into a law. We notice only what we perceive, call it law and skip everything else. We cannot refuse the human factor in understanding the laws of nature.

Before we move on, we need to mention one symmetry that is so obvious that it is rarely mentioned. The law of physics must have symmetry of application (symmetry of applicability). That is, if the law works with an object of one type, then it will work with another object of the same type. If the law is true for one positively charged particle moving at a speed close to the speed of light, then it will work for another positively charged particle moving at the same speed. On the other hand, the law may not work for macro objects at low speed. All similar objects are associated with one law. We will need this kind of symmetry when we discuss the connection between mathematics and physics.

What is math

Let's spend some time understanding the very essence of mathematics. We will look at 3 examples.

A long time ago, some farmer discovered that if you take nine apples and combine them with four apples, you will end up with thirteen apples. Some time later, he discovered that if you combine nine oranges with four oranges, you get thirteen oranges. This means that if he exchanges every apple for an orange, then the amount of fruit will remain unchanged. At some time, mathematicians have gained enough experience in such matters and derived the mathematical expression 9 + 4 = 13. This small expression generalizes all possible cases of such combinations. That is, it is true for any discrete objects that can be exchanged for apples.

More complex example. One of the most important theorems of algebraic geometry is the Hilbert Zero Theorem ( https://ru.wikipedia.org/wiki/Hilbert_Zero Theorem ). It consists in the fact that for each ideal J in the polynomial ring there exists a corresponding algebraic set V (J), and for each algebraic set S there exists an ideal I (S). The relationship of these two operations is expressed as , where is the radical of the ideal. If we replace one alg. many to another, we get a different ideal. If we replace one ideal with another, we get another alg. many

One of the basic concepts of algebraic topology is the Gurevich homomorphism. For each topological space X and positive k, there exists a group of homomorphisms from a k-homotopy group to a k-homological group.. This homomorphism has a special property. If the space X is replaced by the space Y and replaced by , then the homomorphism will be different . As in the previous example, a particular case of this statement does not matter much for mathematics. But if we collect all cases, then we get a theorem.

In these three examples, we looked at changing the semantics of mathematical expressions. We exchanged oranges for apples, we exchanged one idea for another, we replaced one topological space for another. The main thing in this is that by making the correct replacement, the mathematical statement remains true. We claim that this property is the main property of mathematics. So we will call the statement mathematical if we can change what it refers to, and the statement remains true.

Now, for each mathematical statement, we will need to attach a scope. When a mathematician says “for every integer n,” “Take the Hausdorff space,” or “let C be a cocummutative, coassociative involutive coalgebra,” he determines the scope for his statement. If this statement is true for one element from the field of application, then it is true for everyone ( provided that this field of application is correctly selected, approx. Per. ).

This replacement of one element by another can be described as one of the symmetry properties. We call it the symmetry of semantics. We argue that this symmetry is fundamental, both for mathematics and for physics. In the same way that physicists formulate their laws, mathematicians formulate their mathematical statements, while determining in which field of application the statement maintains the symmetry of semantics (in other words, where does this statement work). Let us go further and say that a mathematical statement is a statement that satisfies the symmetry of semantics.

If there are logics among you, then the concept of symmetry of semantics will be quite obvious to them, because a logical statement is true if it is true for each interpretation of a logical formula. Here we say that mat. the statement is true if it is true for each element from the scope.

One might argue that such a definition of mathematics is too broad and that a statement satisfying the symmetry of semantics is just a statement, not necessarily a mathematical one. We will answer that, firstly, mathematics is, in principle, quite wide. Mathematics is not only talking about numbers, it is about forms, statements, sets, categories, microstates, macrostates, properties, etc. For all these objects to be mathematical, the definition of mathematics must be broad. Secondly, there are many statements that do not satisfy the symmetry of semantics. “It's cold in New York in January,” “Flowers are only red and green,” “Politicians are honest people.” All these statements do not satisfy the symmetry of semantics and, therefore, are not mathematical. If there is a counterexample from the scope,

Mathematical statements also satisfy other symmetries, for example, syntax symmetries. This means that the same mathematical objects can be represented in different ways. For example, the number 6 can be represented as “2 * 3”, or “2 + 2 + 2”, or “54/9”. We can also talk about a “continuous self-intersecting curve”, a “simple closed curve”, a “Jordan curve”, and we will mean the same thing. In practice, mathematicians try to use the simplest syntax (6 instead of 5 + 2-1).

Some of the symmetrical properties of mathematics seem so obvious that they are not talked about at all. For example, mathematical truth is invariant with respect to time and space. If the statement is true, then it will be true also tomorrow in another part of the world. And it does not matter who pronounces it - the mother of Teresa or Albert Einstein, and in what language.

Since mathematics satisfies all these types of symmetry, it is easy to understand why it seems to us that mathematics (like physics) is objective, works out of time and is independent of human observations. When mathematical formulas begin to work for completely different problems, discovered independently, sometimes in different centuries, it begins to seem that mathematics exists "somewhere there." However, the symmetry of semantics (and this is exactly what happens) is a fundamental part of mathematics that defines it. Instead of saying that there is one mathematical truth and we have only found a few cases of it, we will say that there are many cases of mathematical facts and the human mind combined them together, creating a mathematical statement.

Why is math good at describing physics?

Well, now we can ask why mathematics describes physics so well. Let's take a look at 3 physical laws.

• Our first example is gravity. The description of one phenomenon of gravity may look like "In New York, Brooklyn, Mine Street 5775, on the second floor at 21.17: 54, I saw a two-hundred-gram spoon that fell and hit the floor after 1.38 seconds." Even if we are so accurate in our records, they will not help us much in the descriptions of all the phenomena of gravity (namely, this is what the physical law should do). The only good way to write down this law is to write it down with a mathematical statement, attributing to it all the observed phenomena of gravity. We can do this by writing Newton's law . Substituting masses and distances, we get our specific example of a gravitational phenomenon.

• In the same way, in order to find the extremum of motion, one must apply the Euler-Lagrange formula . All minima and maxima of motion are expressed through this equation and are determined by the symmetry of semantics. Of course, this formula can be expressed by other symbols. It can even be written in Esperanto, in general, it doesn’t matter what language it is expressed in ( the translator could discuss this with the author, but this is not so important for the result of the article ).

• The only way to describe the relationship between the pressure, volume, quantity and temperature of an ideal gas is to write down the law . All instances of phenomena will be described by this law.

In each of the three examples cited, physical laws are naturally expressed only through mathematical formulas. All the physical phenomena that we want to describe are inside the mathematical expression (more precisely, in particular cases of this expression). In terms of symmetries, we say that the physical symmetry of applicability is a special case of mathematical symmetry of semantics. More precisely, from the symmetry of applicability it follows that we can replace one object with another (of the same class). So the mathematical expression that describes the phenomenon must have the same property (that is, its scope should be at least no less).

In other words, we want to say that mathematics works so well in describing physical phenomena, because physics and mathematics formed in the same way. The laws of physics are not in the Platonic world and are not central ideas in mathematics. Both physicists and mathematicians choose their statements in such a way that they suit many contexts. There is nothing strange in this that the abstract laws of physics originate in the abstract language of mathematics. As in the fact that some mathematical statements were formulated long before the corresponding laws of physics were discovered, because they obey the same symmetries.

Now we have completely solved the mystery of the effectiveness of mathematics. Although, of course, there are many more questions that are not answered. For example, we may ask why people generally have physics and mathematics. Why are we able to notice symmetries around us? Part of the answer to this question is that to be alive means to exhibit the property of homeostasis, so living beings must defend themselves. The better they understand their surroundings, the better they survive. Non-living objects, such as stones and sticks, do not interact with their surroundings. Plants, on the other hand, turn toward the sun, and their roots stretch toward the water. A more complex animal can notice more things in its environment. People notice a lot of patterns around them. Chimpanzees or, for example, dolphins cannot do this. The patterns of our thoughts we call mathematics.

One may wonder why in physical phenomena in general there are some regularities? Why will an experiment conducted in Moscow give the same results if carried out in St. Petersburg? Why will the released ball fall at the same speed, despite the fact that it was released at another time? Why will a chemical reaction proceed the same way, even if different people look at it? To answer these questions we can turn to the anthropic principle. If there were no patterns in the universe, then we would not exist. Life takes advantage of the fact that nature has some predictable phenomena. If the universe were completely random, or looked like some kind of psychedelic picture, then no life, at least an intellectual life, could survive. The anthropic principle, generally speaking, does not solve the problem. Questions such as “Why does the universe exist,” “Why is there something,” and “What is happening here at all,” remain unanswered.

Despite the fact that we did not answer all the questions, we showed that the presence of structure in the observable universe is quite naturally described in the language of mathematics.