Why mathematics describes reality

One of the most interesting problems of the philosophy of science is the relationship of mathematics and physical reality. Why is math so well describes what is happening in the universe? Because many areas of mathematics were formed without any involvement of physics, however, as the result was, they became the basis of the description of some physical laws. How can this be explained?

Most clearly this paradox can be observed in situations where some physical facilities were first opened mathematically, and then later found evidence of their physical existence. The most famous example is the discovery of Neptune. Urbain Le Verrier made this discovery just calculating the orbit of Uranus and investigating the discrepancies in the predictions with the real picture. Other examples are the Dirac's prediction of the existence of positrons, and Maxwell's assumption that the variations in electric or magnetic field should give rise to waves.

Even more surprisingly, some of the mathematics existed long before physicists realised that they are suitable to explain some aspects of the universe. Of a conic, Apollonius studied even in ancient Greece, was used by Kepler in the early 17th century to describe the orbits of the planets. Complex numbers have been proposed for several centuries before physicists began to use them to describe quantum mechanics. Non-Euclidean geometry was created decades before the theory of relativity.

Why is math so well describes natural phenomena? Why, of all ways of expression, the math works best? Why, for example, it is impossible to predict the exact trajectory of the movement of celestial bodies in the language of poetry? Why can't we translate the complexity of the periodic table of music? Why meditation doesn't help much in predicting the outcome of experiments of quantum mechanics?

Nobel prize winner Eugene Wigner in his article "The unreasonable effectiveness of mathematics in the natural sciences", also asking 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's no rational explanation».

Albert Einstein on this occasion wrote:

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

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Let's be clear. The problem really arises when we perceive math and science as 2 different, perfectly formed and an objective region. If you look at the situation from this side, it really is unclear why these two disciplines work so well together. Why the laws of physics so well describes the (already open) mathematics?

This question is pondered by many people, and they gave many solutions to this problem. Theologians, for example, suggested the Creature, which builds the laws of nature, and thus uses the language of mathematics. However, the introduction of such a Being only complicates things. Platonist (and their cousins naturalists) believe in the existence of a "world of ideas", which contains all mathematical objects, shapes, and Truth.

There are also physical laws. The problem with Plutonite that they introduce another concept of the Platonic world, and now we have to explain the relationship between the three worlds. The same question arises whether the non-ideal theorem ideal forms (objects of the world of ideas). How about disproved physical laws?

The most popular version of solving the problem of effectiveness of mathematics is that we are studying math, watching the physical world. We understand some properties of addition and multiplication counting sheep and stones. We have studied the geometry, observing physical forms. From this point of view, it is not surprising that physics is for math, because math is formed in a careful study of the physical world.

The main problem with this solution is that mathematics is used well in areas far from human perception. Why is the hidden world of subatomic particles is so well described by mathematics, are studied through the estimate of 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, which is formed by the observation of objects moving at normal speed?

 

What is physics

Before considering the reason for the effectiveness of mathematics in physics, we need to talk about what the laws of physics. To say that physical laws describe physical phenomena, a few serious. For starters, you can say that every law describes many phenomena.

For example the law of gravity tells us what will happen if I drop my spoon, he also describes drop my spoon tomorrow, or what will happen if I drop the spoon after a month on Saturn. The laws describe a whole range of different phenomena.

You can go to and from the other side. One physical phenomenon can be observed in completely different ways. Someone will say that the object is stationary, someone that the object is moving with constant speed. Physical law must describe both cases equally. Also, like the theory of gravity should describe my observation of the incident of the spoon in a moving car, from my point of view, from the point of view of my friend standing in the road, from the point of view of the guy standing on his head, next to a black hole, etc.

The next question is: how to classify physical phenomena? What should group together and ascribe the same law? Physics use the concept of symmetry. Colloquially, the word symmetry is used for physical objects. We say that the room is symmetrical if the left part is similar to the right. In other words, if we reverse the parties, then the room will look the same.

Physics slightly expanded this definition and apply it to physical laws. Physical law is symmetrical in relation to the conversion if the converted law describes the phenomenon in the same way. For example, physical laws are symmetric in space. That is, the phenomenon observed in Pisa, as can be observed in Princeton. Physical laws are symmetric in time, i.e., an experiment done today should give the same results as if it was held tomorrow. Another obvious symmetry is the orientation in space.

There are many other types of symmetries, which must conform to physical laws. Relativity for Galieu requires that the physical laws of motion remained unchanged, regardless of stationary or moving with constant speed. Special relativity claims that the laws of motion must remain the same even if the object is moving at a speed close to the speed of light. General 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: the local symmetry, global symmetry continuous symmetry discrete symmetry, etc. Victor Stenger combines many types of symmetry in what we call the invariance relative to the observer (point of view invariance). This means that the laws of physics must remain the same, regardless of who and how they are watching. He showed how many areas of modern physics (but not all) can be reduced to laws satisfying the invariance relative to the observer. This means that phenomena are related to one phenomenon associated, despite the fact that they can be considered in different ways.

Understanding the real importance of symmetry was Einstein's theory of relativity. Before him, people first discovered a physical law, and then found in him a property of symmetry. Einstein used symmetry to find the law. He postulated that the law should be the same for a stationary 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 satisfy the law of symmetry, and symmetry creates the law.

In 1918 Emmy Noether showed that the symmetry is an even more important concept in physics than I thought. She proved a theorem linking symmetries with conservation laws. The theorem showed that every symmetry gives rise to its own conservation law, and Vice versa. For example, the invariance to shift in space produces the law of conservation of linear momentum. 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 looking for new types of symmetries to find new laws of physics.

Thus we have identified what is called physical law. From this point of view it is not surprising that these laws seem to us objective, timeless, independent of the person. Since they are invariant with respect to location, time, and look at them the person the impression that they exist "somewhere". However, it is possible to look the other way. Instead of saying that we look at the many different consequences of external laws, we can say that people have assigned some observable physical phenomena, they found something similar and combined them into law. We notice only what we perceive, call it the law and ignored everything else. We can't abandon the human factor in understanding the laws of nature.

Before we go any further, we must mention one of symmetry, which is so obvious that it is rarely mentioned. The law of physics must have symmetry in the application (symmetry of applicability). That is, if the law works with the object of the same type, then it will work with a different object of the same type. If the law is true for a single positively charged particle moving with a speed close to the speed of light, it will work for another positively charged particle moving with the speed of the same order. On the other hand, the law may not work for macro-objects with low speed. All similar objects are associated with a single law. We need this kind of symmetry when we discuss the relationship of mathematics with physics.

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What is mathematics

Let's spend a little time to understand the essence of mathematics. We will look at 3 examples.

A long time ago, a farmer discovered that if you take nine apples and mix them with four apples, then eventually you'll get thirteen apples. Some time later he found that if nine oranges to connect with four oranges, then get thirteen oranges. This means that if he will exchange each Apple for an orange, the amount of fruit will remain unchanged. In some time, mathematicians have accumulated enough experience in such matters, and brought mathematical expression 9 + 4 = 13. It's a 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. It lies in the fact that for each ideal J in a polynomial ring there is 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 
is a radical ideal. If we replace one of the Sal. mn-in another, we get another ideal. If we replace one ideal with another, we get a different ALG. mn-in.

One of the basic concepts of algebraic topology is the Hurewicz homomorphism. For each topological space X and positive k there is a group of homomorphisms of k-homotopical group in k-homology group.
. This homomorphism has a special property. If the space X is replaced by the space Y, and
replace
then the homomorphism will be different
. As in the previous example is a particular case of this statement is not important to mathematics. But if we collect all the cases, we get the theorem.

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

Now to every mathematical statement, we need to put a scope. When a mathematician says "for each integer" n "will Take the space is Hausdorff", or "let C be cocommutative, coassociative the involutive coalgebra", it determines the applications for its approval. If this assertion is true for one element of the scope, then it is true for each (assuming the correct choice of the applications).

This replacement of one element to another, can be described as one of the properties of symmetry. We call this symmetry of semantics. We argue that this symmetry is fundamental, both for mathematics and for physics. In the same way as physicists formulate their laws of mathematics formulate their mathematical statements simultaneously determining in which applications the true symmetry of semantics (in other words, this statement works). Let's go further and say that a mathematical statement — a statement that satisfy symmetry of semantics.

If among you there are logic, they the concept of symmetry of semantics is quite evident that a logical statement is true if it is true for each interpretation of the logical formula. Here we say that the Mat. the statement is true if it is true for each element of the application.

Someone may object that this definition of mathematics is too broad and that the approval that satisfy symmetry of semantics — just a statement, not necessarily mathematical.

We will answer that first, mathematics is in principle quite broad. Math is not only talking about numbers, it's about the forms, statements, sets, categories, microstates, the microstates, properties, etc. all these objects were mathematics, definition of mathematics needs to be broad. Secondly, there are many claims that do not satisfy symmetry of semantics. "In new York in January is cold, Flowers are only red and green", "Politicians are honest". All these allegations do not satisfy symmetry of semantics, and hence not mathematical. If there is a counterexample from the scope of, the approval automatically ceases to be mathematical.

Mathematical statements also satisfy the other symmetries such as the symmetry of the 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 speak of a "continuous curve smokeparticles", "simple closed curve", "the Jordan curve," and we will have in mind the same thing. In practice, mathematicians try to use the most simple syntax (6 instead of 5+2-1).

Some symmetric properties of mathematics seem so obvious that they did not say. For example, mathematical truth is invariant with respect to time and space. If the statement is true, it will be true also tomorrow in another part of the globe. And no matter who will say mother Teresa or albert Einstein, or what language.

Since mathematics covers all these types of symmetry, it is easy to understand why we think that mathematics (and physics) objective runs out of time and independent of human observations. When mathematical formulas begin to work for completely different tasks, open independently, sometimes in different centuries, it begins to seem that mathematics exists "out there somewhere".

However, symmetry of semantics (which is exactly what happens) is a fundamental part of mathematics that defines it. Instead say that there is one mathematical truth, and we only found few cases we say that there are many cases of mathematical facts and human mind combined them together, creating a mathematical statement.

 

Why is math good description of the physics?

Well, now we can ask the questions why is math so well describes the physics. Let's look at 3 physical law.

• Our first example is gravity. The description of the phenomenon of gravity might look like "In new York city, Brooklyn, 5775 main street, on the second floor in 21.17:54, ounce I saw a spoon that fell and hit the floor after 1.38 seconds." Even if we are so careful in our records, we're not much help in the descriptions of all phenomena of gravity (namely, it is supposed to do the physical act). The only good way to write this law will write its mathematical statement, attributing to him all the observed phenomena of gravity. We can do this by writing Newton's law
  . Substituting mass and distance, we will receive our specific example of gravitational phenomena.
  
  

 

• Similarly, in order to find the extremum of the movement, to apply the formula of Euler-Lagrange
  . All the lows and highs of the movement 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 be written even in Esperanto, in General, no matter what language it is expressed (on this subject, the translator would debate with the author, but the result is not so important).
  
  

 

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

In each of the three examples of physical laws are expressed naturally using only mathematical formulas. All the physical phenomena we want to describe are inside mathematical expressions (more precisely in the particular cases of this expression). In terms of the symmetries we say that the physical symmetry of applicability is a special case of the mathematical symmetry of semantics. More precisely, from the symmetry of applicability, it follows that we can replace one object to another (of the same class). Then a mathematical expression that describes the phenomenon must have the same property (that is, its scope needs to be at least not less).

In other words, we want to say that mathematics works so well in the description of physical phenomena, because physics and mathematics were formed in the same way. The laws of physics are not in the world lovely Platonov and non-Central ideas in mathematics. Physics, and mathematics choose their assertions so that they are suited to many contexts. There is nothing strange, that the abstract laws of physics originate in the abstract language of mathematics. Like the fact that some mathematical statements formulated long before had opened the relevant laws of physics, because they are subject to the same symmetries.

Now we have completely solved the riddle of the effectiveness of mathematics. Although, of course, there are still many questions that have no answers. For example, we can ask why humans generally have physics and mathematics. Why are we able to notice the symmetry around us? Part of the answer to this question is that to be alive means to show the property of homeostasis, therefore living beings should be protected. The better they understand their environment, the better they survive. Inanimate objects such as stones and sticks, does not interact with its environment. Plants, on the other hand, turns to the sun, and their roots are drawn to water. The more complex the animal can notice more things in their environment. People notice a lot of patterns. A chimpanzee or, for example, dolphins can't do it. The patterns of our thoughts we call mathematics. Some of these regularities are the regularities of physical phenomena around us, and we call these laws of physics.

One may wonder why physical phenomena there are patterns? Why the experiment was performed in Moscow will give the same results if performed in Saint-Petersburg? Why you release the ball will fall at the same speed, despite the fact that it was released at another time? Why chemical reaction to proceed equally, even if I look at it different people? To answer these questions we can turn to the anthropic principle.

If in the universe there were no laws, we would not exist. Life enjoys the fact that nature has some predictable effects. If the universe was completely random, or like some kind of psychedelic painting, no life, at least intelligent life, would not be able to survive. The anthropic principle, generally speaking, does not solve the problem. Questions like "Why is the universe", "Why is there something" and "That" remain unanswered.

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

 

 

P. S. And remember, only by changing their consumption — together we change the world! ©

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Source: geektimes.ru/post/270542/