185.

After Lobachevsky, non-Euclidean geometry developed considerably.

The first was the German mathematician Riemann, based on the ideas of Lobachevsky and others, who established a broader geometry, now known as Riemannian geometry.

Since then, non-Euclidean geometry has been officially recognized and established.

If Euclidean geometry is based on geometry under a classical plane.

Non-Euclidean geometry, then, is a type of geometry that specializes in the study of surface states.

Geometry has been greatly expanded and extended after the establishment of non-Euclidean geometry.

It's like Newton's classical mechanics only holds true at low speeds after the advent of the theory of relativity.

Non-Euclidean geometry reveals the curved nature of space, making the Euclidean geometry of flat space a special case.

In the 19th century, geometry can be understood as a generalized non-Euclidean movement: from three-dimensional to high-dimensional, from straight bending......

In addition, the development of projective geometry also gave Euclidean geometry the final blow, making Euclidean geometry completely fall from its sacred position.

Due to the flourishing development of geometry in the 19th century, many schools of geometry were derived.

Finally, in order to unify geometry, Hilbert, one of the most famous mathematicians of the 19th century, systematically sorted out the original axiomatic system in his 1899 book "Foundations of Geometry".

The axiom of the act is that there is no way to deduce it from other kilometers, but is a basic fact that is self-evident according to human reason and intuition.

This is also the real reason why some people say that mathematics is a subjectively defined discipline by human beings, because axioms cannot be proven, and can only be defined by human intuition.

And human intuitive feeling is subjective, a feeling of the objective laws of the universe.

But is the objective law of the universe as perceived by human intuition necessarily what the axiom describes? Is it really completely unshakable?

Is it true that there is nothing wrong with human intuitive feelings?

Won't the objective laws of the universe be distorted after being intuitively felt by human beings?

These problems are a major problem that has plagued the entire mathematical community and even the scientific community since the beginning of the 20th century.

This is like a two-dimensional being, he will never have a three-dimensional sense, so the world he sees is always two-dimensional, and the objective laws he sees are only a projection of the high-dimensional world on the two-dimensional level, not the whole thing.

Therefore, a certain axiom that two-dimensional beings feel is justified may be a completely different form at the three-dimensional level.

For example, two-dimensional organisms can propose the axiom that an infinitely extending straight line cannot be bypassed.

This axiom can be said to be natural and absolutely correct in the two-dimensional world.

But this correctness is based on the subjective observation of two-dimensional beings in the two-dimensional world.

It is an axiom that is subjectively defined by two-dimensional organisms, and then two-dimensional organisms can develop a set of two-dimensional mathematics based on this axiom.

However, if we look at it from the perspective of three-dimensional biology, we find that this axiom is completely untenable.

Therefore, it can even be said to the extreme that modern mathematics and physics, as well as other sciences, are based on man's observation of the universe, and develop a kind of subjective and objective mixed disciplines, because we will be constrained by our own perceptual organs.

So much so that after entering the 21st century, some of the more radical scientists are wondering: "We don't even know whether the starry sky we see is real or not." ”

The development of non-Euclidean geometry has profoundly revealed this cruel reality.

That's human intuition, and it's not reliable.

This unreliability became even more evident in the 20th century with the rapid development of science.

The theory of relativity and quantum mechanics both illustrate how unreliable human intuition is.

Geometry, in the final analysis, is an edifice built on axioms. A theorem is deduced from the axioms, which eventually forms the entirety of geometry.

Therefore, once there is a problem with the axiom itself, the foundation of the entire mathematical edifice will also be shaken.

But one of the greatest implications of non-Euclidean geometry is that it reveals the possibility that humans can use mathematics to describe the high-dimensional world.

In other words, although human thinking is subjective, we can still find how to describe the world as objectively as possible.

And this process is necessarily not correct from the beginning, from Euclid to non-Euclidean geometry, from Newtonian mechanics to relativity and quantum mechanics.

These theories discovered by mankind are all limited, that is, they need to add some prerequisites to make sense.

Euclid, for example, is only true on a straight plane.

Newtonian mechanics is established only at low speeds.

The theory of relativity holds true only on a macroscopic scale.

Quantum mechanics is only true at the microscopic scale.

These most important scientific theories of mankind must be established only with the addition of some prerequisites.

Pursue the ultimate axiom of the universe that is completely universal, has no preconditions at all, and can be true in any case.

It is the ultimate axiom of the universe that can be objectively and eternally unchanging under any circumstances, without being influenced by any observer, without any subjective influence.

This is the ultimate ideal pursued by all scientists, countless generations, one after another.

According to Shutong, that is the only truth of the universe and the only root of the Three Thousand Great Roads.

Although the subjective intuition of human beings is not reliable.

But luckily, we still have math.

Although there are some concepts in mathematics, and there are axioms that are subjectively defined, no one knows whether they are correct or not in the end.

However, after rigorous logical deduction, mathematics is indeed the only and most objective tool that human beings can use. Even if the basis of this objectivity is somewhat subjective.

But it doesn't matter, just like the development from Euclidean geometry to non-Euclidean geometry.

Even if mathematics has some subjectivity, as long as human beings continue to doubt and create, then mathematics can become more objective and become the sharpest weapon for human beings to objectively explore the universe.

The greatest creativity of human beings lies in self-doubt, and this is why they can continue to improve.

As Descartes once said, "If we want to pursue the truth, we must doubt everything as much as we can once in our lives." ”

Of course, such skepticism must be based on an attitude of scientific inquiry, not blindly denying everything.

Scientific skepticism is based on thinking and argumentation, and every doubt is to make one's theory more objective.

If you want to be more objective, you need a rigorous logical deduction and a detailed argumentation process.

Instead of doubting and denying everything by casually YY.

This is also the biggest difference between professional scientists and some "civil sciences".

As long as we have this scientific skeptical spirit, even if the starry sky we see is fake, then sooner or later, we will be able to find an objective way to observe the real starry sky.

That's science.

It is also a scientific concept that Cheng Li has always believed in.

Cheng Li, in the process of struggling to move forward in such a sea of question banks, unconsciously sprouted this scientific concept that has always been hidden in his heart.

Behind these questions, he felt the countless generations of people on the earth, the advance of that era, the history of mathematics and even the history of science that spiraled forward alternately in doubt and affirmation.

He felt the scientific ideas behind these problems, as well as the progress and discoveries in the fields of physics, chemistry and biology corresponding to each problem.

This made Cheng Li gradually have a trace of understanding, and gave him a deeper understanding and understanding of the definition of "mathematics".

Under such an understanding, Cheng Li's mysterious information in the sea began to become active again.

Subsequently, when facing the difficult questions that he could not answer, Cheng Li's mind began to flash continuously, and one after another flashed out of his mind.

Originally, he couldn't figure it out at all, but when he thought about it, he suddenly understood that he had figured it out, and at that moment, Cheng Li had a feeling of refreshment that was all over his body, and it was a feeling that was more joyful than any emotion.

In this way, Cheng Li advanced by leaps and bounds in the arithmetic monument, and after 10 hours of hard fighting, he finally came to the 2501st floor.

Began the mathematical journey of the 20th century.

=====

(This chapter can be regarded as an exposition of some of the rabbit's personal views on science over the years.)

I know that some readers don't like to read such content with a scientific tendency.

But rabbits really like it!

Since I was in the fourth grade of elementary school, I have always liked to read all kinds of popular science books.

Every summer vacation, when others go out to play, I like to squat in the bookstore to read popular science books.

Therefore, the content of this recent part of the history of mathematics can be regarded as a little capricious creation of my personal creation.

I like to watch this kind of content myself, and I'm sure some people will like to watch it.

As for readers who don't like this content, please forgive me.

This part of the content is almost over, and the next thing is the battle of Qingling Island.

It is also the last and biggest climax of the first volume, and I will try to write it well, so stay tuned!)