About math

    A few years ago, one of my acquaintances with humanitarian education said: “What is wrong with mathematics? Everything is strict, everything is open, 2 + 2 is always equal to 4, boredom.” Unfortunately, I was still a schoolboy and could not adequately answer.


    How many times, during the preparation for the exam, I grumbled: “Well, Cauchy, damn it, invented it here, it’s not clear, he had nothing to do.” Of course, I understood that all this is not just like that, but sometimes, from the abundance of various abstract theorems, it began to seem to me that this was all invented only to load students.

    People who use mathematics in practice, understand that this is not so. They represent why this or that may be needed. But what should others do? For example, a lesson in a regular school:

    “Today we will find out what the sine of the angle is. Sinus is the ratio of the length of the opposite leg to the length of the hypotenuse ... What, Ivanov, do you have a question? ... Why is this needed? You see, this is the basis of trigonometry, which is used in particular in analytical geometry ... Ivanov! Are you sleeping or something? ”

    At this time, Ivanov had a dream in which he was a great mathematician of ancient times:

    “Oh, I want to invent something! True, I can not imagine what. So, take, for example, a triangle. By the way, it’s a pity that it was already invented. So what to do with it? Fold all sides together? No, it will be the perimeter. But what if divided? One side to the other? Yes, it's just brilliant! Nobody had thought of this before! I need to give the Nobel Prize! Well, damn it, Nobel was not even born yet. Well, okay. Let me call this aspect ratio "sine"! And the attitude of other parties is a "cosine"! That's great how! And the students in the school will have something to learn, otherwise they are quite lazy! ”

    Absurd? Of course! But the one who has an interest in mathematics, this abstract science, is just fine. In other sciences, at least, it is not necessary (or necessary, but to a lesser extent) to think about why this is necessary, what it actually is and how it is applied in practice. By no means do I want to say that other sciences are easier, no. But, in theory, interest more easily arises to more specific things.

    In fact, math is a tool. But no one will have the thought that any inventor thought like this:
    “Oh, the mood is so good! Wouldn't I invent something? Hmm, so-so, here I have a rusty piece of iron, what would I do with it? Well, let's say I make a hole in it, it can be hooked on a finger. So, for a start, not bad. Oh, and you can still put it on a stick. Let's try it. And what, cool, cool little thing turned out! Still need to somehow call her. Mmmm ... and if so? ..., no, not that ... Oh! And I'll call this thing a hammer! Normal, quite so masculine, the most! But why is it needed, I can’t imagine. Oh well, maybe someone will come up with it later. ”

    Mathematics was created to solve specific problems.

    My question is: was mathematics discovered or invented? I used to think it was open. I reasoned this way: one of the basic concepts is “number”. Didn’t it exist, say, before the advent of mankind? Now I doubt it very much. It seems to me that “number” is a concept invented by man, it does not exist by itself.
    The next logical step is the concept of the natural series. If you think that the natural series exists in our world, just as there is a law of attraction, then remember that among the ancient tribes there were only the numbers “one”, “two”, “three” and “many”. Note that they didn’t have other numbers, probably they had enough of these. Theoretically, a mathematical theory can be derived from these numbers. The question is how much it will be in demand. For example, such a theory is not without a right to exist:

    So, we use only numbers 1, 2, 3 and a lot. You can come up with the rules of addition: 1 + 1 = 1, 1 + 2 = 3, 1 + 3 = many, 3 + 3 = many, etc. Is such a theory convenient? In our world, of course, no. And if you imagine this:
    In the country, there are coins in denominations of one ruble and many rubles. You come to the store, take a thing for 2 rubles, give 2 coins. If a thing costs a lot, then give away a coin in many rubles. Here then this theory will be useful.

    Of course, the described case is a little crazy, but, I think, the idea is clear. Now a more realistic example with Lobachevsky geometry.
    One of the Euclidean axioms is:
    through a point that does not lie on a given line, only one line passes, lying with the given line in the same plane and not intersecting it.

    Lobachevsky replaced her with:
    at least two lines pass through a point that does not lie on a given line, lying on the same plane from the given line and not intersecting it.

    Uninitiated people believe that they say that he replaced the obvious axiom with some kind of delusional. For whom is it obvious? Only for us. Euclidean geometry is some of the most suitable correspondence to our familiar world, but not for mathematics. Lobachevsky showed that from the new axioms a no less complete theory is obtained, although it has no familiar analogues for us. But it is successfully used in other models. Interested in sending to Wikipedia .

    Bottom line: Mathematics is the ability to come up with some abstractions and statements and derive consistent consequences from them. This is a tool created by the power of the human mind, is that why many people like mathematics?

    If you liked this article, then I will gladly continue to write about mathematics (more precisely, about the philosophy of mathematics), since there are a lot of thoughts :))

    UPD: I apologize for a typo (meaning 1 + 1 = 2), which slightly reduced the example. I will not correct in the text, as there are too many comments on this subject.

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