About math
A few years ago, a friend of mine with a liberal arts education said: “Yes, what do you have in mathematics, everything is strict, everything is open, 2 + 2 is always equal to 4, boring”. 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, I’ve been thinking up here, I don’t understand anything; Of course, I understood that all this was not just so, but sometimes, from the abundance of various abstract theorems, it began to seem to me that this was all invented only to load the students.
For people who use mathematics in practice, it is clear that this is not the case. They represent why this or that might be needed. But what to do to others? For example, a lesson in a regular school:
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“Today we will know what is the sine of the angle. Sine is the ratio of the length of the opposite leg to the length of the hypotenuse ... What, Ivanov, is your question? ... Why is this necessary? You see, this is the basis of trigonometry, which is used in particular in analytic geometry ... Ivanov! Do you sleep or what? ”
At this time, Ivanov dreamed a dream in which he was a great mathematician of ancient times:
“Eh, I want to invent something! True, I can not imagine what. So, take, for example, a triangle. By the way, it is a pity that he was already invented. So what to do with it? Fold all sides? No, it will be the perimeter. And what if you divide? One side to the other? Yes, it's just brilliant! No one had thought of that before! I need to give a Nobel Prize! Here, damn it, Nobel is not even born yet. Well, okay. I will call this attitude of the parties "sine"! And the attitude of other parties - "cosine"! That's great how! And students in school will have something to learn, otherwise they will be completely 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 it is necessary, but to a lesser extent) to think, and why it is needed, what it actually is and how it is used in practice. I don’t want to say that other sciences are easier, no. But, in theory, interest arises more easily to more specific things.
In fact, math is a tool. But no one will think that some inventor thought like this:
“Oh, the mood is so good! Should I invent something? Hmm, well, well, here I have a rusty piece of iron, what can I do with it? Well, let's say, I will make a hole in it, it can be fastened on a finger. So, for a start not bad. Oh, and you can also put it on a stick. Well, let's try. And what a cool, cool thing turned out! Still need to call her somehow. Mmmm ... and if so? ..., no, not that ... Oh! And I will call this thing "hammer"! Normally, quite so manly, the very thing! But why is it needed, I can not imagine. Oh well, maybe someone will come up with it later. ”
Mathematics was created to solve specific problems.
I am interested in the question: was mathematics discovered or invented? I used to think it was open. I reasoned like this: one of the basic concepts is “number”. Did it not exist, say, before the advent of mankind? Now I doubt it very much. As it seems to me, “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 consider that a natural series exists in our world, just as there is a law of attraction, then remember that the ancient tribes had only the numbers “one”, “two”, “three” and “many”. Notice, they did not have other numbers, they probably 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 devoid of the right to exist:
So, we use only the numbers 1, 2, 3 and many. You can create addition rules: 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:
There are coins of one ruble and many rubles in circulation in the country. You come to the store, take a thing for 2 rubles, give 2 coins. If a thing costs a lot, then give the coin a lot of rubles. This is where this theory will be useful.
Of course, the case described is a bit delusional, but, I think, the thought is clear. Now a more realistic example of the Lobachevsky geometry.
One of the Euclidean axioms is:
Lobachevsky replaced it with:
Uninitiated people believe that here they say, replaced the obvious axiom with some kind of delusional. For whom the obvious? Only for us. Euclidean geometry is some of the most appropriate correspondence to our familiar world, but not for mathematics. Lobachevsky showed that a new theory is no less a complete theory, although it does not have the usual analogs for us. But it is successfully used in other models. Interested send to Wikipedia .
Bottom line: Mathematics is an opportunity to invent some abstractions and statements and derive consistent consequences from them. This is a tool created by the power of the human mind. Is this why many people like mathematics?
If you liked this article, then I will happily 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 indicative of the example. I will not correct the text, as there are too many comments about this.
How many times, during the preparation for the exam, I grumbled: “Well, Cauchy, damn it, I’ve been thinking up here, I don’t understand anything; Of course, I understood that all this was not just so, but sometimes, from the abundance of various abstract theorems, it began to seem to me that this was all invented only to load the students.
For people who use mathematics in practice, it is clear that this is not the case. They represent why this or that might be needed. But what to do to others? For example, a lesson in a regular school:
')
“Today we will know what is the sine of the angle. Sine is the ratio of the length of the opposite leg to the length of the hypotenuse ... What, Ivanov, is your question? ... Why is this necessary? You see, this is the basis of trigonometry, which is used in particular in analytic geometry ... Ivanov! Do you sleep or what? ”
At this time, Ivanov dreamed a dream in which he was a great mathematician of ancient times:
“Eh, I want to invent something! True, I can not imagine what. So, take, for example, a triangle. By the way, it is a pity that he was already invented. So what to do with it? Fold all sides? No, it will be the perimeter. And what if you divide? One side to the other? Yes, it's just brilliant! No one had thought of that before! I need to give a Nobel Prize! Here, damn it, Nobel is not even born yet. Well, okay. I will call this attitude of the parties "sine"! And the attitude of other parties - "cosine"! That's great how! And students in school will have something to learn, otherwise they will be completely 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 it is necessary, but to a lesser extent) to think, and why it is needed, what it actually is and how it is used in practice. I don’t want to say that other sciences are easier, no. But, in theory, interest arises more easily to more specific things.
In fact, math is a tool. But no one will think that some inventor thought like this:
“Oh, the mood is so good! Should I invent something? Hmm, well, well, here I have a rusty piece of iron, what can I do with it? Well, let's say, I will make a hole in it, it can be fastened on a finger. So, for a start not bad. Oh, and you can also put it on a stick. Well, let's try. And what a cool, cool thing turned out! Still need to call her somehow. Mmmm ... and if so? ..., no, not that ... Oh! And I will call this thing "hammer"! Normally, quite so manly, the very thing! But why is it needed, I can not imagine. Oh well, maybe someone will come up with it later. ”
Mathematics was created to solve specific problems.
I am interested in the question: was mathematics discovered or invented? I used to think it was open. I reasoned like this: one of the basic concepts is “number”. Did it not exist, say, before the advent of mankind? Now I doubt it very much. As it seems to me, “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 consider that a natural series exists in our world, just as there is a law of attraction, then remember that the ancient tribes had only the numbers “one”, “two”, “three” and “many”. Notice, they did not have other numbers, they probably 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 devoid of the right to exist:
So, we use only the numbers 1, 2, 3 and many. You can create addition rules: 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:
There are coins of one ruble and many rubles in circulation in the country. You come to the store, take a thing for 2 rubles, give 2 coins. If a thing costs a lot, then give the coin a lot of rubles. This is where this theory will be useful.
Of course, the case described is a bit delusional, but, I think, the thought is clear. Now a more realistic example of the Lobachevsky geometry.
One of the Euclidean axioms is:
through a point that is not lying on a given straight line, there passes only one straight line lying with this straight line in one plane and not intersecting it.
Lobachevsky replaced it with:
through a point that is not lying on a given line, at least two straight lines lie that lie with this line in the same plane and do not intersect it.
Uninitiated people believe that here they say, replaced the obvious axiom with some kind of delusional. For whom the obvious? Only for us. Euclidean geometry is some of the most appropriate correspondence to our familiar world, but not for mathematics. Lobachevsky showed that a new theory is no less a complete theory, although it does not have the usual analogs for us. But it is successfully used in other models. Interested send to Wikipedia .
Bottom line: Mathematics is an opportunity to invent some abstractions and statements and derive consistent consequences from them. This is a tool created by the power of the human mind. Is this why many people like mathematics?
If you liked this article, then I will happily 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 indicative of the example. I will not correct the text, as there are too many comments about this.
Source: https://habr.com/ru/post/24213/
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