Maths. What is it really? Reflections

We have been taught this since childhood, it is called the queen of sciences, it is considered the foundation for the study of other sciences, but what does it look like? Is she correct? What is the future of this science? I tried to reflect on these questions and this is what came of it.

About axioms

Everyone knows that mathematics begins with axioms. For example, there are Peano axioms that allow us to manipulate natural numbers. In general, all that we now have in mathematics came out of some obvious axioms that someone once invented. Something like: “But let it be so!” And others picked up: “Suppose it is. Then .............. " And these sentences after “then” reached such things as mathematical analysis, discrete mathematics, etc.

It doesn’t seem surprising to you that out of the now obvious things that someone had already invented we created such a powerful system that makes flying at the speed of sound, looking at other universes, etc., possible? By system, I mean the axioms and what was created on them.

So, can we invent our own axioms? Build on them theories with which you can also create a plane? Will the whole system be deceitful? Maybe there is a set of axioms that will make the formulas of mathematical analysis superfluous and with the help of them the plane will be able to design any student? Maybe the paradoxes associated with quantum mechanics after its revision will become solvable and other unprecedented possibilities will be available to the village! Suddenly, the mathematical path that was chosen by mankind has already (or will) lead us into a dead end and we will not be able to create anything new? After all, every year the discoveries of scientists are becoming more distant from real life and their benefits are felt less and less. And there are more paradoxes.
')

About paradoxes

And what are the paradoxes? After all, paradoxes, this is the first sign that the system is not perfect. Why do people, having run into a paradox, let it pass by the eye and continue to develop a seemingly not completely correct mathematical system? It doesn't even occur to people that the current mathematics should be reconsidered from the very beginning. Probably because of the extraordinary technical progress achieved with the help of the current mathematics, and the reluctance to admit that a tremendous amount of time people created a dead-end system.

In the previous paragraph, I wrote: “the mathematical system is not completely correct,” but this is wrong. Still, can we consider our current mathematics to be true? Of course we can, since I am writing text on a computer now, and not manually; because I don't have to wash things in the river; since at any moment I can call home while being in the other side of the globe, etc. But we do not have the right to say that there are no other true mathematical systems.

About the future of mathematics

Have you ever thought about the future of mathematics? I think she will ever lead us to a dead end. New scientific discoveries in the field of mathematics will be made only at such a high level that they cannot be used in real life. Or there will be discoveries that improve existing theories. (For example, finding some simpler formulas that make calculations faster). Already in our time there is a tendency to such discoveries. Although some scientists are trying to find new approaches to theories. For example, Mikhail Gromov is trying to make a revolution in geometry and for his work he was awarded the Abel Prize this year, but it is limited to the theories already created, which are believed to be correct. That is, if you created a theory that contradicts with some recognized true theory, then your theory is wrong and that is what limits us in creating a new one. How can you say what is true and what is not?

About restrictions

Speaking of restrictions. Where do you think such abstracts come from: “Time travel is impossible! This contradicts many laws of physics and mathematics! ” But how do we know that only current physics and mathematics are the only things that can describe a phenomenon? Maybe if we fundamentally revise all of mathematics, and then rebuild physics under it, time travel will seem not so surprising.
HG Wells:

- This is the seed of my great discovery. You make the mistake of saying that you cannot move in Time. If I, for example, very vividly recall an event, then I return to the time it takes place and, as it were, mentally absent. I make a leap into the past for a moment. Of course, we cannot remain in the past on any particle of Time, just as a savage or an animal cannot hang in the air at least six feet from the ground. In this respect, a civilized person has an advantage over a savage. He, in spite of the force of the force, can rise up in a balloon. Why, then, can one not hope that in the end he will also be able to stop or accelerate his movement along Time or even turn in the opposite direction?
“It's absolutely impossible ...” began Philby.
- Why not? The Time Traveler asked.
“This is contrary to reason,” answered Philby.
- What reason? Said the Time Traveler.
“Of course, you can prove that black is white,” said Philby, “but you will never convince me of that.”

Philby could not explain why this seemed impossible to him. And this is not surprising. Nowadays these things seem crazy. But once stories about flights to Mars were from the realm of fantasy.

Do you think that humanity is doing the right thing by winding more and more turns on this huge “coil” with a weak base? Or is it worth reviewing the base and rewinding the coil? I apologize for the metaphor.

By the way, most likely the only ones who can give us new ideas on how to rebuild the foundation are very young children. Yes, children of 5-6 years old, who are not yet completely clear that 7 * 9 = 63, that through any two points it is possible to draw a straight line, that V = abc, etc. And all because the more you do the math, the harder it is for you to imagine the obvious things are not obvious.

All of the above is just my personal reflections. If there is a factual error somewhere, then please excuse me and correct it. Thank.