Why mysticism? What is Formula? Do formulas rule the world?

So it turns out, that our education, even university, primarily aimed at accumulating a certain amount of knowledge and methods, how this knowledge should be used. For most, for those who, in one way or another, are involved in their life with the application of the acquired knowledge of this, basically, quite enough. But for those people, who wants to expand the scope of already existing knowledge, this creates a certain problem. Because, to expand the scope, it is usually not enough just to add something new, previously unknown to existing stock. The greatest advance is achieved more often then, when the very basics of knowledge are revised. And here are some points, forming these very foundations, remain unclear, and even worse, mythologized. I mean here the role and place of mathematics in the physical picture of the world. Very often, the successes of mathematical methods in physics are given noticeably mystical significance.. And this is done not only by people far from the cutting edge of science., but also outstanding physicists and mathematicians. As an example of the quintessence of such a view, I can point to the well-known article Nobel Prize Laureate in Physics 1963 of the year J.P. Wigner, “The Unexplained Effectiveness of Mathematics in Natural Sciences“.

The emergence of such an idea of ​​the role of mathematics goes back to the hoary antiquity. Very many, probably, heard, that after he discovered incommensurable numbers, Pythagoras sacrificed a hundred bulls to the gods. Maybe this is a legend, maybe not, but in itself the awareness of the presence of one of the deepest gaps in the foundations of the mathematical building deserved, to be celebrated.

The language of mathematics has become so sophisticated today, which has ceased to be understood by most people. And few people remember and are aware of the fact, that it was formed as part of our everyday language. And here I want to somehow steal mathematics from that Olympus, which she climbed, peel off the veil of mysticism from her, with which she hid from the idle glances of most people. I want to say, what mathematics has remained a part of the human language, somewhat specific, but no more different from everything else, What is the difference, eg, English from French.

Today's mathematics prefers to emphasize, what does she say about perfect things, seemingly completely stripped of the husk of reality, surrounding us. Is it so? Not at all. This is quite obvious in the case of the most basic concepts of mathematics. – eg, let's take the concept “number”. Yes, this is a perfect concept. But we know well where it came from.. And we use it in everyday language in the same way, like mathematicians. Applied to our practical problems, this concept does not carry any mysticism. The formula 1 + 1 = 2 is clear to everyone. And it's understandable, why is she correct… truth? Is it always? 1(one) sheep plus 1(alone) gate equals 2(two) old ladies at the gate? Funny? Funny. Why? Because, what formula, dealing with numbers (subject of mathematics) can be applied to the real world (in other words, in physics, chemistry, etc. and so on. ) only in a certain way, and not as God puts it on your soul. And there is nothing strange about it. This is how any language works. Pure mathematicians forgot about it? So it's just funny, but what is inexplicable here?

Numbers by themselves, as an idea, have certain properties. We have the right to forget everything in the world and deal only with these properties.. And in mathematics there is such a direction. We've played enough with these properties, learned something special (gave some property a special name, introduced a new concept into the language). Can we affirm, that such a property will take place, if we count the rams? We can! To the same extent, in which it will be applicable, if we count cows, stones, etc. It already exists, this property, in the real world, whether we knew about him or not, whether we named it or not. Concept of the multitude formalized by mathematicians in the nineteenth century. But this concept itself has existed in everyday languages ​​since time immemorial.. What important did mathematicians do?? Gave a definition of the concept itself, and how it can and should be handled. Have identified some idea, which works when talking about any sets. Formalized. Slightly modified idea of ​​the set, slightly differently formalized, endowed with some specific properties, got the name “group“. I want to focus on this particular mathematical concept.. because, that it was precisely the application of group theory to physics that gave rise to Wigner's admiration, bordering on mysticism. Stimulated to write books and articles on this topic. Yes, mathematicians very diligently understood many of the properties, which ideas may have (ideal concepts), called groups. Created group theory, special section of mathematics. They often did it just for the love of art., but not always. Many details of group theory have been clarified by examples of very practical, for example, in the physical theory of the structure of crystals. And here's the question, so what, that it was group theory that later proved to be so effective in quantum theory, and in other branches of physics? Just very, very many phenomena in the real world can be described in appropriate terms. The theory has grown from an example of a limited range of phenomena, but as formalization some very general properties, after distraction from all others, unimportant details (unimportant from the point of view of this theory itself). Why group theory is so pervasive in physics? Yes, if only because, what We all, speaking, which means communicating, transmitting information to each other, and thus describing our world, we make up exactly a group in the strictest mathematical sense. It was a small excursion about too much enthusiasm for the incomprehensible effectiveness of mathematics.. And now a little more serious about formulas, method of formalizing statements in mathematics, physics, and in all others, any exact sciences.

formula. What it is? Any formula can be represented as statement, that something is equal to something. A = B. Sometimes this is a very private statement about, that this something, under certain conditions, takes on some meaning. Such a statement is of little interest to us precisely because of its particular. We are more interested in general statements, which are always executed. We call such statements the laws of nature. And marvel at their beauty. And trying to find all such laws. We are also trying to understand, how is it, why did nature have laws? Why are they like that, and not others? Where did they come from, who imposed them on nature? And reproduce, in this way, religion even in science.

But what is the general formula? After all this the same identity, simple tautology! If A = B everywhere (and always), doesn't that mean, what we called two different names for the same thing?! Discovered this and… wrote down another law of nature. A slightly unexpected and seemingly strange answer to some of the questions, listed just above. But the answer is quite clear. And exhaustive, leaving no more freedom for the flight of fantasy and myth-making. What questions are removed by this answer? These and others like them: Why did nature have laws? Where did they come from, who imposed them on nature? Answer: some of the laws we formulated ourselves, by entering different names for the same phenomena, not realizing their identity.

Why then some the laws, not all? Are there more equals among equals?? there is. And that's why. I have deliberately simplified the general formulas, to highlight one of their most important properties – be identities, tautologies. And slyly kept silent about that, what, describing the real world, we have produced a huge variety of names. And by no means always through negligence and misunderstanding. On the contrary, striving for a better understanding, we clarify the meaning of our definitions of objects and phenomena, observed in the real world (mathematics is the refinement method!) and break down some concepts into components, or, on the contrary, integrate them into the whole. How do we write the definitions themselves? Of course, in the form of formulas. Usually, these are formulas of the form F(…..)=0. Some combination of verbal mathematical objects vanishes. This is also a general formula. AND, when she talks about fairly general properties of the real world, it can also be called a law of nature. And by a much greater right. Although we wrote it, but they didn't just write, and wrote down some ratio, some property of the real world. There are two sides here, and both are inseparable. One side – we have isolated something, we named him. Maybe you can isolate a little differently, and certainly can be called differently. On the other hand – then, what is isolated and named, there is an existing property of the real world. May be, not reflecting the world in its entirety, maybe even distorting it a little, but – at least partially – appropriate to this world. Other methods of dismemberment and names – another description of the world. Also eligible. In this way, among the laws of nature we have discovered, there are also definitions. Definitions, descriptions of some general properties of the world. Want an example? Newton's laws.

Are there any other laws of nature, which do not directly qualify as tautologies and definitions? In a sense there is, although they can be largely attributed to the definitions. But it is better to separate them into a separate type of defining conditions. Since these laws of nature and formulas, their recording, it is more convenient to consider not as identities, but as equations, guaranteeing the fulfillment of certain conditions. That is, we are aware of the fact that, what, generally speaking, the situation in the real world is described by some more general definitions, but here in this particular (possibly very significant) its part has additional specificity, which is written by the corresponding equations, not obliged to be satisfied always and everywhere.

Can you answer the question now “do formulas rule the world?“

My answer – The world is ruled by formulas.

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