186.

The question of the 2500th floor is also sharply iconic.

Perhaps in every era, there is such a landmark question, as a beginning.

Cheng Li saw this problem for the first time, and he had a feeling that it was really so.

This 2500th layer problem is regarded as the beginning of the 20th century in the field of mathematics.

It comes from 1900, when Hilbert gave a famous lecture called "Problems of Mathematics" at the International Congress of Mathematicians.

In this presentation, Hilbert presented 23 famous mathematical problems.

These 23 questions are known as the 23 Hilbert Questions.

Hilbert's 23 questions, which cover many important areas of modern mathematics, are a series of questions raised by Hilbert systematically for the development of mathematics in the next century.

In the 20th century, these questions stimulated the strong research interest of many mathematicians, and even became the development program of mathematics in the 20th century.

It is extremely rare in the history of science for a scientist to ask a whole set of questions so systematically and so intensively, and to influence the development of a science so persistently.

And the question on the 2500th layer is the first question of Hilbert's 23 - the continuum hypothesis.

"Uh...... This 2500th question is the first question asked by Hilbert 23, and the next 23 questions will not come from this, right?" Cheng Li was first worried when he saw this question.

Of Hilbert's 23 questions, only half of them were solved before Cheng Li crossed, and the other half that were not solved also made significant progress, but Cheng Li did not know how to prove and answer.

"Forget it, you can only take one step at a time. ”

Cheng Li also knew that there was no time to dwell on it, so he went straight forward and wrote down the process of proving the answer to Hilbert's 23 questions on the light sand.

The solution and research of Hilbert's 23 questions greatly promoted the development of a series of mathematical branches such as mathematical logic, geometric foundations, Li Qunlun, mathematical physics, probability theory, number theory, function theory, algebraic geometry, ordinary differential equations, partial differential equations, Riemann surface theory, and variational method.

Some questions, such as the second and tenth questions, also contributed to the development of modern computers.

Of course, it was limited by the level of scientific development at that time and the limitations of personal scientific literacy, research interests, and ideological methods.

Hilbert's 23 questions cannot really cover all areas of the development of mathematics in the 20th century, such as topology, differential geometry, and other mathematical problems that became frontier disciplines in the 20th century, but Hilbert's 23 questions do not cover anything.

And in addition to mathematical physics, there is little involved in applied mathematics and so on. The development of mathematics in the 20th century went far beyond what Hilbert's 23 questions predicted.

After answering the first of Hilbert's 23 questions, Cheng Li went straight to the next floor.

On the next floor, when Cheng Li saw the problem, he breathed a sigh of relief.

Because the question on the 2502nd floor is not the content of Hilbert's 23rd question

It seems that the arithmetic monument is not a rigid copy of the problem, but according to the actual difficulty of the problem, to arrange the appropriate questions for each layer. ”

The 2502nd layer question is about the theory of real functions.

The establishment of set theory in the 19th century first caused a revolution in integral science in the 20th century, which led to the establishment of real variable functions.

After some hard answers, Cheng Li finally solved this problem.

Next, he also ran into the problem of functional analysis, as well as the problem of abstract algebra.

Then, he encountered a problem area that gave him quite a headache - topology.

Topology was an important field of mathematics in the 20th century, which was the study of the continuous properties of geometric figures, and finally developed into a basic discipline of mathematics, followed by differential topology and algebraic topology.

After topology, Cheng Li also encountered the problem of probability theory in the subsequent problems, and it was axiomatic probability theory.

In addition, there are problems such as differential geometry, the theory of multiple complex variable functions, and the paradox of set theory, which is also known as Russell's paradox.

Russell's paradox caused the third mathematical crisis on the planet, and its influence can be seen. Cheng Li was almost not stumped on this question, and finally finally got through the danger.

In addition, there are hard bones such as Gödel's incompleteness theorem and recursion theory.

Eventually, after completing these theoretical parts, Cheng Li came to question 2700.

From here, Cheng Li found that the next problems were all related to practical application.

In the 19th and 20th centuries, it was an era of comprehensive application of mathematics.

And after entering the 20th century, mathematics has made unprecedented progress in practical application.

Many of the esoteric mathematical theories that were created in the 18th and 19th centuries were not even known to the founders themselves at the time to have any practical applications, but were only purely mathematical theories.

But in the 20th century, these mathematical theories, which I didn't know what to do with them, came in handy.

One of the most notable examples of this is the birth of the General Theory of Relativity.

Einstein's theory of relativity is the first time that human beings have systematically constructed a cognitive view of time and space.

The space described by Einstein is not uniform, but becomes curved due to gravitational influence.

In order to describe the spatial properties of surface forms, which are difficult to define clearly in words, Einstein needed a powerful mathematical weapon to support it, and he finally found Riemannian geometry.

Riemannian geometry was founded in 1854, but 60 years later, in 1915, Einstein established the theory of relativity.

The mathematical formulation of general relativity revealed the practical significance of non-Euclidean geometry for the first time, and became one of the great examples of applications in the history of mathematics.

For example, in the 20th century, there were two major physics edifices, one was the theory of relativity and the other was quantum mechanics.

Mathematics also played a decisive role in the construction of the edifice of quantum mechanics.

Unlike the theory of relativity, which was created entirely by Einstein, quantum mechanics is a classic example of collective effort.

Planck, Einstein, Bohr and others were all founders of quantum mechanics.

By 1925, matrix mechanics established by Heisenberg and wave dynamics developed by Schrödinger became the two major schools of quantum mechanics.

At that time, the main problem for scientists was how to unify these two schools of quantum mechanics.

In the end, it was mathematics that brought the two together.

In 1927, Hilbert, von Neumann and Nordheim published "On the Basis of Quantum Mechanics", and began to use integral equations and other analytical tools to try to unify quantum mechanics.

In the end, von Neumann used the very abstract Hilbert space theory to extend Hilbert's spectral theory to quantum mechanics, thus laying the mathematical foundation of quantum mechanics.

In 1932, von Neumann published "The Mathematical Foundations of Quantum Mechanics" and completed the mathematical axiomatization of quantum mechanics.

It was later discovered that Hilbert's engineering of integral equations and the resulting theory of infinitely many variables were almost entirely tailored to quantum mechanics.

This is similar to the birth of the electromagnetic field equation at the beginning.

In addition, things like topology have a wide range of applications in condensed matter physics.

And in addition to the fields of physics and chemistry.

In the 20th century, mathematics began to play an active role in the field of biology.

In particular, the discovery of the double helix structure of DNA has brought the knot theory of algebraic topology into use.

As early as the 19th century, Gauss discussed the problem of knots and pointed out that "counting the windings of two closed curves will be an important task of positional geometry, i.e. topology." ”

He had no idea that his prediction would become an important task in the study of DNA structure 100 years later.

In addition, the invention of CT scanning, which is also inseparable from mathematics, is the physicist Cormac published a mathematical formula to calculate the amount of X-ray absorption by different tissues of the human body, which solved the theoretical problem of computed tomography, so that the CT scanner was invented.

In addition, mathematical statistics, differential equations, topology, integral theory, and probability theory are also applied to population theory and population theory, Boolean algebra is applied to neural network description, and Fourier analysis is applied to biological polymer structure analysis...... And so on, are all examples of the application of mathematics to biology.

In addition to the field of biology, mathematical statistics, operations research, and cybernetics are also classic examples of the application of mathematics in various other disciplines.

Among these mathematical applications, there is one that has produced the most profound changes in the 21st century and directly led to the birth of a new era.

That is closely related to Cheng Li's original work, and it is also the field that Cheng Li is most familiar with - electronic computers!