The reason for translating the article was that I was looking for a book by the author “The Outer Limits of Reason” . I never managed to save the book, but I came across an article that in a rather concise form 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 does mathematics describe so well in the universe? Indeed, many areas of mathematics were formed without any involvement of physics, however, as it turned out, they became the basis for 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 later were evidence of their physical existence found. The most famous example is the discovery of Neptune. Urbain Le Verrier made this discovery simply by calculating the orbit of Uranus and investigating the discrepancies between the predictions and the real picture. Other examples are Dirac’s prediction of the existence of positrons and Maxwell’s assumption that oscillations in an electric or magnetic field should produce waves.
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Even more surprising, some areas of mathematics existed long before physicists realized that they were suitable for explaining certain aspects of the universe. The conic 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, of all ways of expressing thoughts, does math 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 not we express the complexity of the periodic table of Mendeleev music? Why is meditation not much help in predicting the result of experiments in quantum mechanics?

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

Albert Einstein wrote about this:

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

Let's clarify. 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, proposed a Creature that builds the laws of nature, and in doing so uses the language of mathematics. However, the introduction of such a Being only complicates things. Platonists (and their cousins ​​are naturalists) believe in the existence of a “world of ideas” that contains all mathematical objects, forms, and also Truth. There are also physical laws. The problem with Platonists is that they introduce another concept of the Platonic world, and now we have to explain the relationship between the three worlds ( comment of the translator. I did not understand why the third world, but left it as it is ). It also raises the question whether non-ideal theorems are ideal forms (objects of the world of ideas). How about refuted physical laws?

The most popular version of the solution of the posed problem of the effectiveness of mathematics is that we study mathematics by observing the physical world. We understood some properties of addition and multiplication, 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 by careful study of the physical world. The main problem with this solution is that mathematics is well used in areas far from human perception. Why is the hidden world of subatomic particles so well described by mathematics, studied by counting sheep and stones? Why does the special theory of relativity, which works with objects moving at speeds close to the speed of light, is well described by mathematics, which is formed by observing objects moving at normal speed?

In two articles ( one , two ) Macr Seltzer and I (Noson Janowski) formulated a new view on the nature of mathematics ( note of the translator. In general, in those articles the same is written here, but much more detailed ). We have shown that, just as in physics, symmetry plays a huge role in mathematics. This look gives a rather original solution to the problem.

What is physics

Before considering the cause of the effectiveness of mathematics in physics, we must talk about what physical laws are. To say that physical laws describe physical phenomena is somewhat frivolous. To begin with, it can be said 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 will happen if I drop the 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 completely differently. Someone will say that the object is stationary, someone that the object is moving at a constant speed. Physical law should describe both cases in the same way. Also, for example, the theory of a fall must describe my observation of a falling spoon in a moving car, from my point of view, from the point of view of my friend standing on the road, from the point of view of the guy standing on his head, next to a black hole, etc. .

The following question arises: how to classify physical phenomena? Which 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 looks like the right. In other words, if we swap the sides, then the room will look exactly the same. Physicists have expanded this definition a bit and apply it to physical laws. The physical law is symmetrical with respect to transformation, if the law describes the transformed phenomenon in the same way. For example, physical laws are symmetrical in space. That is, the phenomenon observed in Pisa can also be observed in Princeton. Physical laws are also symmetric in time, i.e. An experiment conducted today should produce the same results as if it were conducted tomorrow. Another obvious symmetry is orientation in space.

There are many other types of symmetries to which physical laws must conform. Galia relativity requires that the physical laws of motion remain unchanged, regardless of whether the object is stationary or moving at a constant speed. Special 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 Stenger has combined many types of symmetry by what we call invariance with respect to the observer (point of view invariance). This means that the laws of physics must remain unchanged, regardless of who and how observes them. 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 connected, despite the fact that they can be treated differently.

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

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

Thus, we have defined what to call a physical law. From this point of view, it is not surprising that these laws seem to us to be objective, timeless, and independent of man. Since they are invariant with respect to a place, a time, and a person’s look at them, it seems that they exist “somewhere out there”. However, this can be viewed 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 united them into a law. We only notice what we perceive, call it the law and skip everything else. We cannot abandon the human factor in understanding the laws of nature.

Before we move on, we need to mention one symmetry, which 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 the same 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 a speed of the same order. On the other hand, the law may not work for macro-objects at low speed. All similar objects are connected 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 trying to understand the very essence of mathematics. We will look at 3 examples.

A long time ago, a 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 nine oranges were combined with four oranges, then there would be thirteen oranges. This means that if he exchanges each apple for an orange, the amount of fruit will remain unchanged. At some time, mathematicians have accumulated enough experience in such cases and derived the mathematical expression 9 + 4 = 13. This small expression summarizes 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 Hilbert's theorem on zeros ( https://ru.wikipedia.org/wiki/Teorema_Gilbert_o_nuli ). It consists in the fact that for every ideal J in the polynomial ring there exists a corresponding algebraic set V (J), and for every algebraic set S there exists an ideal I (S). The relationship of these two operations is expressed as where - radical ideal. If we replace one alg. many things to another, we get another ideal. If we replace one ideal with another, we get another al. mn-in

One of the basic concepts of algebraic topology is the Hurevich homomorphism. For each topological space X and positive k, there exists a group of homomorphisms from a k-homotopy group to a k-homologous group. . This homomorphism has a special property. If space X is replaced by space Y, and replaced by , 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 changed oranges for apples, we changed one idea for another, we replaced one topological space with another. The main thing is that by making the correct substitution, the mathematical statement remains true. We argue 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 at the same time the statement will remain true.

Now for each mathematical statement we will need to attach the scope. When the mathematician says “for every integer n,” “Take Hausdorff space,” or “let C be a cocummutative, co-associative involutive coalgebra,” he defines the scope for his statement. If this statement is true for one element of the scope, then it is true for everyone ( assuming the correct choice of this very scope, approx. Lane ).

This replacement of one element by another can be described as one of the properties of symmetry. We call this symmetry 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 area of ​​application the statement preserves the symmetry of semantics (in other words, where this statement works). Let us go further and say that a mathematical statement is a statement that satisfies the symmetries of semantics.

If there are logic among you, then the concept of symmetry 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 mate. The statement is true if it is true for each element of the application.

Some may argue that such a definition of mathematics is too broad and that a statement satisfying the symmetries of semantics is just a statement, not necessarily a mathematical one. We will answer that, first, 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 is cold in New York in January”, “Flowers are only red and green”, “Politicians are honest people”. All these statements do not satisfy the symmetries of semantics and, consequently, not mathematical ones. If there is a counterexample from the application area, then the statement automatically ceases to be mathematical.

Mathematical statements also satisfy other symmetries, such as 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 speak of a “continuous self-intersecting curve”, a “simple closed curve”, and a “Jordan curve”, and we will mean the same thing. In practice, mathematicians try to use the most simple syntax (6 instead of 5 + 2-1).

Some symmetrical properties of mathematics seem so obvious that they are not spoken of 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 globe. And no matter who says it - mother 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 tasks, opened independently, sometimes in different centuries, it begins to seem that mathematics exists “somewhere out there.” However, the symmetry of semantics (and this is exactly what is happening) is the fundamental part of mathematics that defines it. Instead of saying that there is one mathematical truth and we only found a few of its cases, we say that there are many cases of mathematical facts and the human mind combined them together, creating a mathematical statement.

Why is mathematics good at describing physics?

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

• Our first example is gravity. A description of one phenomenon of gravity might look like "In New York, Brooklyn, Main 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 careful in our records, they will not help us much in describing all the phenomena of gravity (and this is exactly what physical law should do). The only good way to write this law down is to write it with a mathematical statement, attributing to it all the observed phenomena of gravity. We can do this by writing Newton's Law . Substituting the masses and the distance, we get our concrete example of a gravitational phenomenon.

• Similarly, in order to find the extremum of motion, you need to 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 also be expressed in other characters. It can even be written in Esperanto, in general, it does not matter in what language it is expressed ( a translator could discuss this topic with the author, but for the result of the article it is not so important ).

• The only way to describe the relationship between pressure, volume, quantity, and ideal gas temperature is to write 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 physical phenomena that we want to describe are within a 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 the mathematical symmetry of semantics. More precisely, it follows from the symmetry of applicability that we can replace one object with another (of the same class). So a mathematical expression that describes a phenomenon must have the same property (that is, its scope should be at least not less).

In other words, we want to say that mathematics works so well in describing physical phenomena, because physics and mathematics were 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 fit into 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 are formulated long before the relevant laws of physics were discovered, because they obey one symmetry.

Now we have completely solved the mystery of the effectiveness of mathematics. Although, of course, there are many more questions to which there are no answers. For example, we can ask why people even 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 is to show the property of homeostasis, therefore living beings must defend themselves. The better they understand their surroundings, the better they survive. Inanimate objects, such as stones and sticks, do not interact with their surroundings. Plants, on the other hand, turn toward the sun, and their roots are drawn towards the water. A more complex animal may notice more things in its environment. People notice a lot of patterns around themselves. Chimpanzees or, for example, dolphins cannot do this. Patterns of our thoughts we call mathematics. Some of these patterns are patterns of physical phenomena around us, and we call these patterns physics.

One may wonder why there are any regularities in physical phenomena? Why the experiment conducted in Moscow will give the same results if it is conducted in St. Petersburg? Why will the ball be released fall at the same speed, despite the fact that he was released at another time? Why the chemical reaction will proceed the same, even if different people are looking at it? To answer these questions we can refer to the anthropic principle. If there were no laws 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 similar to some psychedelic picture, then no life, at least the intellectual life, could survive. The anthropic principle, generally speaking, does not solve the problem posed. « », « -» « » .

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