why do scientists use mathematical equations at describing reality?

Why is mathematics good at describing reality?

Introduction

One of the most interesting problems in the philosophy of science is the relationship between mathematics and physical reality. Why is mathematics so good at describing what happens in the universe? After all, many areas of mathematics were formed without any involvement of physics, yet they ended up being the basis in describing some physical laws. How can this be explained?

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

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

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Why is mathematics so good at describing natural phenomena?

Of all the ways of expressing thoughts, why does mathematics work best? Why, for example, can’t we predict the exact trajectory of celestial bodies in the language of poetry? Why can’t we express the complexity of Mendeleev’s periodic table in a piece of music? Why does meditation not help much in predicting the result of quantum mechanics experiments?

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 effectiveness of mathematics in the natural sciences is something mystical and there is no rational explanation.

Albert Einstein wrote on the subject:
How can mathematics, a 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 recourse to experience, comprehend the properties of the universe? [Einstein]

Let’s be clear. The problem really arises when we see mathematics and physics as two different, perfectly formed and objective fields. If you look at the situation from this perspective, it is really unclear why these two disciplines work so well together. Why are the discovered laws of physics so well described by (already discovered) mathematics?

This question has been pondered by many people, and they have given many solutions to this problem. Theologians, for example, have proposed an Being who constructs 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 naturalist cousins) believe in the existence of a “world of ideas” which contains all mathematical objects, forms, as well as Truth. That is also where the physical laws are found. The problem with the Platonists is that they introduce another concept of the Platonic world, and now we have to explain the relationship between the three worlds (translator’s note. I never understood why the third world, but left it as it is). This also begs the question of whether non-ideal theorems are ideal forms (objects of the world of ideas). What about disproved physical laws?

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

In two articles (one, two) Makr Seltzer and I (Noson Janowski) formulated a new view of the nature of mathematics (translator’s note). In general, those articles say the same thing as here, but in a much more extended way). We showed that symmetry plays an enormous role in mathematics, just as it does in physics. This view gives a rather original solution to the problem posed.

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What is physics

Before we consider the reason for the effectiveness of mathematics in physics, we must talk about what physical laws are. To say that physical laws describe physical phenomena is a bit frivolous. For starters, we can say that each law describes many phenomena. For example, the law of gravity tells us what happens if I drop my spoon, it also describes the fall of my spoon tomorrow, or what happens if I drop my spoon in a month on Saturn. Laws describe a whole set of different phenomena. You can go at it from the other side as well. One physical phenomenon can be observed in completely different ways. Some will say that an object is stationary, some will say that the object is moving at a constant speed. A physical law must describe both cases in the same way. Also, for example, the theory of gravitation should describe my observation of a falling spoon in a moving car, from my perspective, from the perspective of my friend standing on the road, from the perspective of a guy standing on his head, next to a black hole, etc.

This begs the next question: how do we classify physical phenomena? Which ones are worth grouping together and attributing to one law? Physicists use the concept of symmetry for this purpose. In colloquial speech, the word symmetry is used for physical objects. We say that a room is symmetrical if the left side of it is similar to the right side. In other words, if we swap the 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 a transformation if the law describes the transformed phenomenon in the same way. For example, physical laws are symmetrical with respect to space. That is, a phenomenon observed in Pisa can also be observed in Princeton. Physical laws are also symmetrical in time, i.e., an experiment conducted today should yield 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 correspond. Relativity according to Galileo 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 have generalized the concept of symmetry in different ways: local symmetry, global symmetry, continuous symmetry, discrete symmetry, etc. Victor Stenger grouped the many types of symmetry by what we call point of view invariance. This means that the laws of physics must remain the same, no matter who observes them or how. He showed how many areas of modern physics (but not all) can be reduced to laws satisfying invariance with respect to the observer. This means that phenomena belonging to the same phenomenon are related, even though they may be viewed differently.

The real importance of symmetry came with Einstein’s theory of relativity. Before him, people would first discover some physical law and then find the property of symmetry in it. Einstein, on the other hand, used symmetry to find a law. He postulated that the law must be the same for a stationary observer and for an observer moving at close to the speed of light. With this assumption, he described the equations of the special theory of relativity. This was a revolution in physics. Einstein realized that symmetry is the defining characteristic of the laws of nature. It is not the law that satisfies the symmetry, but the symmetry generates the law.

In 1918, Emmy Nöther showed that symmetry is even more important in physics than previously thought. She proved a theorem linking symmetries to the laws of conservation. The theorem showed that each symmetry gives rise to its law of conservation and vice versa. For example, invariance on displacement in space generates the law of conservation of linear momentum. Time invariance generates the law of conservation of energy. Orientation invariance generates the law of conservation of angular momentum. After that, physicists started looking for new kinds of symmetries to find new laws of physics.

Thus we 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, independent of man. Because they are invariant with respect to place, time, and man’s view of them, it seems that they exist “out there. However, there is another way of looking at it. Instead of saying that we are looking at many different corollaries from external laws, we can say that man has isolated some observable physical phenomena, found something similar in them, and combined them into a law. We notice only what we perceive, call it a law, and leave out the rest. We cannot give up 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. A law of physics must have symmetry of applicability. That is, if a law works with one type of object, 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 of magnitude. On the other hand, the law may not work for macro-objects with low speed. All similar objects are related to the same law. We will need this type of symmetry when we discuss the relationship of mathematics to physics.

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What is mathematics

Let’s take some time to understand the very essence of mathematics. We’ll look at three examples.

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

A more complicated example. One of the most important theorems of algebraic geometry is the Hilbert zero theorem (https://ru.wikipedia.org/wiki/Теорема_Гильберта_о_нулях ). It consists in the fact that for every ideal J in a polynomial ring there exists a corresponding algebraic set V(J) and for every algebraic set S there exists an ideal I(S). The connection of these two operations is expressed as I(V(J)) = \sqrt J, where \sqrt J is the radical of the ideal. If we replace one algebra with another, we get a different ideal. If we replace one ideal with another, we get another algebraic unit.

One of the basic concepts of algebraic topology is the Gourevitch homomorphism. For every topological space X and positive k there exists a group of homomorphisms from a k-homotopy group to a k-homological group. h_{*}: \pi_k(X) \rightarrow H_k(X). This homomorphism has a special property. If space X is replaced by space Y and k is replaced by k’, the homomorphism is different \pi_{k’}(Y) \rightarrow H_{k’}(Y). As in the previous example, some particular case of this statement does not matter much in mathematics. But if we collect all cases, we get a theorem.

In these three examples, we looked at changing the semantics of mathematical expressions. We swapped oranges for apples, we swapped one idea for another, we swapped one topological space for another. The important thing about this is that by making the right substitution, the mathematical statement remains true. We argue that this very property is a basic property of mathematics. So we will call a statement mathematical if we can change what it refers to and still have the statement remain true.

Now to each mathematical statement we will need to attach a scope. When a mathematician says “for every integer n,” “Let’s take a Hausdorff space,” or “let C be a co-commutative, coassociative involutive coalgebra,” he is defining a scope for his statement. If this statement is true for one element in the scope, it is true for every element (provided that this very scope is chosen correctly, note).

This substitution of one element for another can be described as one of the properties of symmetry. We call it symmetry of semantics. We argue that this symmetry is fundamental to both mathematics and physics. In the same way that physicists formulate their laws, mathematicians formulate their mathematical assertions, while simultaneously determining in which domain the assertion retains the symmetry of semantics (in other words, where the assertion works). Let’s go further and say that a mathematical statement is a statement that satisfies the symmetry of semantics.

If there are any logicians among you, the concept of symmetry of semantics will be quite obvious to them, because a logical statement is true if it is true for every interpretation of a logical formula. Here we are saying that a mate statement is true if it is true for every element in the scope.

Some may argue that this definition of mathematics is too broad and that a statement satisfying the symmetry of semantics is simply a statement, not necessarily mathematical. We reply that, first, mathematics is in principle quite broad. Mathematics is not just about talking about numbers, it is about forms, statements, sets, categories, microstates, macrostates, properties, etc. For all of these objects to be mathematical, the definition of mathematics must be broad. Second, there are many statements that do not satisfy the symmetry of semantics. “It’s cold in New York in January,” “Flowers only come in red and green,” “Politicians are honest people.” All of these statements do not satisfy the symmetry of semantics and therefore are not mathematical. If there is a counterexample from the field, the statement automatically ceases to be mathematical.

Mathematical statements also satisfy other symmetries, such as the symmetry of syntax. 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 “Jordanian curve,” and we will mean the same thing. In practice, mathematicians try to use the simplest syntax (6 instead of 5+2-1).

Some symmetric 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 a statement is true, it will also be true tomorrow in another part of the globe. It does not matter whether it is spoken by Mother Teresa or Albert Einstein, or in what language.

Since mathematics satisfies all these types of symmetry, it is easy to see why we think that mathematics (like physics) is objective, timeless and independent of human observation. When mathematical formulas begin to work for completely different problems, discovered independently, sometimes in different centuries, it begins to seem that mathematics exists “out there”. However, the symmetry of semantics (which is exactly what happens) is the fundamental part of mathematics that defines it. Instead of saying that there is one mathematical truth and we have only found a few instances of it, we will say that there are many instances of mathematical facts and the human mind has combined them together to create a mathematical statement.

why do scientists use mathematical equations at describing

physics or Why is mathematics good at describing physics?

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

Our first example is gravity. A description of one phenomenon of gravity might look like “In New York City, Brooklyn, Main Street 5775, on the second floor at 9:17:54 p.m., I saw a two-hundred-pound spoon fall and bang on the floor 1.38 seconds later.” Even if we are that careful with our notes, they won’t help us much in describing all the phenomena of gravity (which is what a physical law is supposed to do). The only good way to write down this law would be to write it down as a mathematical statement, attributing to it all the observed phenomena of gravity. We can do this by writing Newton’s law F = G\frac{m_1 m_2}{d^2}. Substituting mass and distance, we get our particular example of a gravitational phenomenon.

Similarly, in order to find the extremum of motion, we have to apply the Euler-Lagrange formula \frac{\partial L}{\partial q} = \frac{d}{d t} \frac{\partial L}{\partial q’}. All minima and maxima of motion are expressed through this equation and are determined by semantics symmetry. Of course, this formula can also be expressed in other symbols. It can even be written in Esperanto, in general it does not matter what language it is expressed in (the translator could discuss this topic with the author, but it is not so important for the result of the article).

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

In each of the three examples given, physical laws are naturally expressed only through mathematical formulas. All physical phenomena we want to describe are inside a mathematical expression (or more precisely, in particular instances of that expression). In terms of symmetries, we say that the physical symmetry of applicability is a particular case of the mathematical symmetry of semantics. More precisely, it follows from applicability symmetry 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 must be at least as large).

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 Plato’s world and are not central ideas in mathematics. Both physicists and mathematicians choose their statements so that they fit many contexts. It is not strange that the abstract laws of physics originate in the abstract language of mathematics. As is the fact that some mathematical statements were formulated long before the corresponding laws of physics were discovered, because they obey the same symmetries.

We have now completely solved the riddle of the efficiency of mathematics. Although, of course, there are still many unanswered questions. For example, we can ask why people have physics and mathematics at all. Why are we able to notice the symmetries around us? Part of the answer is that to be alive is to exhibit the property of homeostasis, so living beings must defend themselves. The better they understand their environment, the better they survive. Inanimate objects, such as rocks and sticks, do not interact with their surroundings in any way. Plants, on the other hand, turn toward the sun and their roots reach for water. A more complex animal can notice more things in its environment. Humans notice many patterns around them. Chimpanzees or dolphins, for example, cannot. The regularities of our thoughts we call mathematics. Some of these patterns are patterns of physical phenomena around us, and we call these patterns physics.

One might wonder why there are any regularities in physical phenomena at all? Why an experiment conducted in Moscow will yield the same results if it is conducted in St. Petersburg? Why will a released ball fall with the same speed, despite the fact that it was released at a different 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, we would not exist. Life takes advantage of the fact that nature has some predictable phenomena. If the universe were completely random, or like some kind of psychedelic picture, no life, at least intellectual life, could survive. The anthropic principle, generally speaking, does not solve the problem at hand. Questions like “Why does the universe exist,” “Why is there anything,” and “What is going on here in the first place” remain unanswered for now.

Although we have not answered all the questions, we have shown that the presence of structure in the observable universe is quite naturally described in the language of mathematics.

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