👏🏿 🍞 🧤 The world and n worlds, or mathematics for the humanities ⛹🏻 🔪 🐀

Foreword

In this article, you will learn how mathematics is actually applied in real life. I’ll ask you to forget immediately everything that you were taught at school: mathematics are not dry formulas and endless arithmetic operations. First of all, mathematics is us and what is around us. Before we begin, in my judgment I will admit the following: I will apply mathematical concepts, which I will immediately explain in simple language; after all, this article, for the most part, was started only with the aim of reconciling the humanitarian with the real world.

I am not an advanced mathematician, I am not the son of a math teacher. analysis, however, I am the person who relatively recently understood the essence of the science that I will present here. Since a large number of sources are written by people who suggest that the reader is familiar with the terminology used in advance, I myself encountered enormous difficulties. And since in my experience that part of my life is still fresh where I asked initial questions and did not understand where to step, I will answer the questions of any novice techie right here, and my clumsy pictures will accompany you. And so, our world is ...

Idea of man

Man is accustomed to know. If not for this feature, I would not have written this article, and you would not have read it. And not the fact that we could, in principle, read. And what is a man? For a more detailed understanding of the article, we compose a human model.

We denote some living creature with certain external features and behavior. Let's call him "man." Since man is a creature, he needs to eat food. Let our man know and analyze in order to find the best options for survival. It will be common for a person to systematize his knowledge in order to further assimilate various hypotheses and theories. It is human nature to prove every hypothesis and theory through analysis and previous experience. And so, we have a model of man, a model of ourselves. This model, without taking into account various errors, the so-called "ideal" model, reflects the essence of man - to learn in order to survive. In real life, each person is very individual, we cannot find two identical people, so we will use just such models,which came from mathematics - they allow us to simplify our understanding of the world.

But where does our “model” person live?

A bit of algebra

I will introduce the primitive concepts of vector, vector space and unit vector.

A vector is a segment that has a direction. This concept will help us determine how we see our world.

Vector space is the space of many vectors.

A unit vector is a unit length vector whose origin is a reference point in vector space.

In our case, for simplicity, I will make reasoning based on the two-dimensional space, which is formed by two vectors, the beginnings of which come from one point, called the reference point (Fig. 1).

(Fig. 1, Vector 3 is composed as follows: we set the length of vector 1, draw a straight line from the end of vector 1 parallel to vector 2, then set the length of vector 2 and draw a straight line from its end parallel to vector 1; from the intersection of the drawn lines we draw vector 3 Using this method, you can compose an infinite number of vectors that will lie in one plane.)

By changing the length of vectors 1 and 2, we derive countless new vectors (3, 4, 5, ..., n), which are built on the basis of our two.

And so, let’s try to understand how mathematics will help us in solving the real problem - to understand how the world works. We studied hard algebra and geometry at school, but for what purpose? They shoved us with endless practice, which in reality is not what we need. We were told to consider equations incomprehensible to us by given algorithms - is it mathematics? No. And those who teach it in this way and apparently have no idea what real mathematics is, because it is much simpler than tons of obscure equations. My theory says: it does not matter what to teach a person - it is only important to interest him, and he will learn himself. And this theory works successfully. And now I’m trying to show you how mathematics is applied in the real world.

The first sentence with which mathematics begins: “What if ...?”. But what if our world can be represented by the same model as man? Let's arrange the objects. From the school course of physics, we have all successfully learned that we ~~live in a matrix,~~ all bodies are composed of molecules, and those of even smaller objects. Drawing an analogy with computers (so, again, easier), imagine that every smallest particle is a pixel. It has a set of personal characteristics: color, location. Let's return to our two-dimensional space of vectors. If we introduce such a rule: the beginning of each new vector must lie at a reference point; then we can characterize each vector with two numbers - this is its coordinate along the axes on which the basic vectors lie (the basic vectors are those vectors that formed the space above).

Let the basis vector be a unit vector, and all other vectors be combinations of basis vectors (Fig. 2).

(Fig. 2. The green vector consists of two vectors i and two vectors j. The vectors i and j are the unit vectors of our system, are indicated for convenience.)

Based on the previous, we represent our space around in this way: each smallest particle is the end vector. The length of the vector is the distance from the reference point to the particle. Imagine our vision. Now you are reading this text, and let each letter is the same particle. The distance from your eyes to this letter is the length of the vector. And this vector itself is the direction of your look at the letter. And in this way absolutely everything in our world is characterized.

We live in a matrix

A person cannot understand what a matrix is until he sees it.

(Fig. 3. In parentheses, this is the matrix. It is formed as follows: the coordinates of the basis vector i are written in the left column, the coordinates of the basis vector j in the right column. We have designated them unit for simplicity, respectively, i has a coordinate along the X 1 axis, along the Y axis 0, and j - vice versa.)

The matrix characterizes the space with which we work. In this case, we see a primitive space where everything is described thanks to two straight lines clear to us from school. If we do manipulations with the matrix, then we will change the objects (vectors) that make up this space. We’ll postpone this venture for now.

Now imagine, we have our vector space, where each object is represented by a set of pixels, that is, a set of endpoints of vectors. We can draw in this way absolutely any two-dimensional object. Imagine that this two-dimensional space is our world (for simplicity, I do not introduce the third dimension). Our matrix ... it is everywhere. It is she who describes how we see this world and how we interact with objects. I will draw a red line, which is actually the set of endpoints of the vectors (Fig. 4, line 1).

(Fig. 4. Blue vectors make up the red line)

Next, I have the right to take it and move it to the right, to the place of line 2. This will be another line, because the blue vectors that make up it have different coordinates. But these vectors are still dependent on each other, that is, they have some kind of relationship to each other. Further, I have the right to bend this line, getting line 3, breaking the initial relationship between the vectors. They will still be dependent on each other, but in a "different format." It is possible to break the connection between vectors by dividing this line in half. Then its two parts will already be independent.

Now imagine instead of this line a leaf of budmagi in our space. I can do the same things with him. I can move it from the edge of the table to the other edge, then collapse it, and then tear it apart. Thus linear algebra characterizes our space. And if we can draw analogies between our 2D model and the world, then we can go further.

Our 2D space is actually a plane. That is, looking at this space from the side, we will see only a straight line. Good. We have our model of the world, we can roughly imagine that the human model is inside the material reference point (why the material point? Because we neglect the size and put the model exactly at this point for convenience). Each time a person moves in either direction, he actually draws the coordinates of the vectors to himself and moves them away from himself.

A little more mate. analysis and everything, I promise

There is such a thing as a “limit”. In practice, it is written like this: lim; is short for limit. Now I will explain why it is so complicated. The limit allows you to characterize the sequence or function. Suppose we have a sequence of numbers 1, 2, 3, ..., n. So, if we are talking about natural numbers, then this n will be infinite, because no matter what number you come up with, I can always add another digit to it. Take the school function (1 / x). If we take the variable “x” from the numbers of a sequence of natural numbers, then we can make this “x” infinitely large. But what happens to this function if the “x” becomes infinitely large? She will endlessly tend to zero, but she will never reach it. It will be infinitely small, and an infinite amount of time will continue to decrease.And still can not reach zero. For general awareness, this phenomenon is written as follows:

(Fig. 5. It reads as follows: the limit of the function (1 / x) as x tends to infinity, that is, when x is taken infinitely more and more)

What now to do with this? Why do I need to know? This is the base required for a novice philosopher to have his starter pack. These tools allow you to think about the universe deeper, going into an accurate calculation, simulating various situations and other heaps of interesting things.

Denouement. What if…?

Is there a parallel world? Its existence is possible, if only because smart people have long proved this by mathematics. How did they come to this? Ancient mathematicians spent their whole lives thinking: what if you take a ball and throw it down? but what if in the fall to cut this ball? what if …? And now we ourselves will ask ourselves this question: what if a parallel world exists? How to model it? Remember that we are working with two-dimensional space now? So: imagine this parallel world as the second exactly the same two-dimensional space. But here's the thing we’ll add here: let these two spaces endlessly strive for each other. That is, the limit of one space will be the limit of another and vice versa. Now take the third space and add it to this heap. And the fourth. And the fifth.And they all mutually strive for each other. Why is this impossible? Describing such things in three-dimensional space is more difficult, so it will continue to fantasize with 2D. Here's how our model looks on the side:

(Fig. 6. Ex. - short for “space." All 3 spaces tend to each other)

What if one of the spaces intersects with the other? How will this look in real life? We will get a hole that is simultaneously in our, and in a parallel world. And things falling into this hole will disappear. And they will also appear. But what if these intersections of spaces are black holes that suck in everything that enters them? In the framework of this article, we will not provide evidence of the falsity or truthfulness of these statements. They only serve as an example of how mathematics works in real life: these are not only formulas, but also unreal imagination, comrades.

But then again, we have given very primitive models of two-dimensional space, and string theory says that in our world there is far from three-dimensional space. Calculations with the addition of each new dimension will become much more complicated and, in fact, will not be representable by the human brain. And given that we, people, live in a completely different world, unlike the smallest particles, which probably do not even have a concept of time, we can at an amateur level dream up how to fit into our model. Earlier we talked about matrices. So, this matrix, as we said, is in our head. We see the world as it is laid with us. And those creatures that come into this world also come here by default with this matrix. People, as it were, agree among themselves that they should have such a view of the world: that tree that I see, you see.

Just remember how the server connects to online services. Each user works according to a given list of protocols about connecting to the server. The same thing in life. We are born connecting to a real-world server, and take in a set of protocols that allow us to interact with this world so as not to cause connection errors for other users. That the tree that I see, you saw, comrades. What if there is no real world at all? Suddenly, only WE, living beings, apparently his just that. What if there are entities that carry a different matrix with a different set of numbers, and then our space is distorted for them? What if these entities with a different kind of matrix are the smallest particles that exist in completely different scenarios? So many questions and so few answers ...

Conclusion

What did I want to say? Mathematics is not tedious definitions, rules and formulas. This is the essence of philosophy, which was earlier transmitted by the mythopoetic view. Today, a philosopher, in the first place, is a mathematician who has an analytical mindset, but at the same time combines the part of the humanities that allows him to create incredible things, receiving the power that many ordinary people don’t even dream about. And finally: do matan, friends, not for grades at school, but for your personality, strive to be stronger.

The world and n worlds, or mathematics for the humanities