As we all know, one cannot set a flag.
The cow that you blow out on a whim will soon turn into a slap and slap you in the face again.
In the history of science written in Chinese, the behavior of scientists setting up flags is called "building a building."
Among the two most famous ones, one occurred in 1900 and the other also occurred in 1900.
The protagonist for the first time was Lord Kelvin, who had just helped Chen Muwu and provided him with a fund to develop a particle accelerator.
It is said that in his New Year's address to the Royal Society in 1900, he made some prospects for the development of physics in the coming new century, and then said the famous sentence:
"The edifice of physics has been completed, leaving only some cosmetic work. There are only two small dark clouds floating in the beautiful and clear sky."
Mr. Lu Xun once said more than once: "I never said such a thing."
But this time Lord Kelvin also had to say: "I said no such thing either."
In fact, Kelvin did mention the concept of dark clouds back then, but he never mentioned buildings.
The location of the speech was not the Royal Society, but the Royal Institution, and the time was not the first day of the new year, but April 27th.
Kelvin's lecture that day was titled "The Dark Clouds of the Nineteenth Century Over the Dynamic Theory of Heat and Optics."
What he said in his speech was not as bland as the Chinese expression: "Dynamic theory asserts that heat and optics are forms of motion. Now the beauty and clarity of this theory are obscured by two dark clouds. It’s been eclipsed.”
The word eclipsed reflects the seriousness of these two dark clouds. It is not at all as bland as the first expression, as if the two small dark clouds are insignificant.
Chen Muwu always feels that the word "building" is used in Chinese to describe a sense of crisis where the foundation is not solid and is crumbling.
Then two fierce men descended from the sky, Planck and Einstein, who "supported the building before it collapsed" and opened up two new paths for the development of physics, quantum theory and relativity.
As for the second fictitious building, it happened in 1900 at the Second International Congress of Mathematicians in Paris, the capital of France.
I don’t know whether it was during the opening or closing ceremony. The convener of the conference, French mathematician Henri Poincaré, is said to have said this: “…with the help of the concepts of set theory, we can build the entire mathematical edifice… Today, we can say that absolute mathematical rigor has been achieved!"
Poincaré said that he never said the above paragraph. It only appeared in the speech about the building in the history of Chinese mathematics, which is somewhat doubtful.
It's just that the mathematics building at that time was just as shaky as the physics building.
After that, mathematicians came up with a bunch of paradoxes, among which the "Russell Paradox" proposed by Russell, the philosopher who recruited Chen Muwu into the Cambridge Apostolic Society, is the most famous.
In some popular science books, Russell's Paradox is simplified into the Barber's Paradox.
In a city, there is a barber.
He declared that he would shave the faces of all the people in the city who did not shave themselves, and that he would only shave their faces.
One day, the barber saw in the mirror that his beard had grown. He subconsciously picked up the razor, but before doing so, he suddenly thought of what he had said.
If he doesn't shave himself, then he belongs to the "people who don't shave themselves in the city", so he has to shave himself.
But if he shaves himself, he is a "self-shaving person", so he should not shave himself.
In addition to the Barber's Paradox, Russell's Paradox has another popular science form that is easy to understand.
A library compiles a book title dictionary that contains all the books in the library that do not list their own names.
It doesn't matter whether your name is listed in this dictionary or not. The principle is similar to the barber's paradox above.
The introduction of Russell's paradox severely slapped the faces of those mathematicians who said that "all mathematical results can be based on set theory."
A German logician, Gottlob Frege, wrote a book on the basic theory of sets.
When the book was about to be delivered to the printer, Frege received a letter from Russell about Russell's paradox.
He immediately found that his book was messed up by Russell's paradox, and he could only add this sentence at the end of the book: "The most unfortunate thing that happens to a scientist is to end his work just when he is about to complete it." The foundation of what was being done was found to have collapsed.”
After Russell's paradox was published, a series of paradoxes followed: Richard's paradox, Perry's paradox, Grayling's and Nelson's paradox...
These paradoxes, known as semantic paradoxes, shook the foundations of the mathematical edifice and triggered the third mathematical crisis.
The first two mathematical crises occurred in the ancient Greek period.
Hippasus, a student of Pythagoras, discovered that the length of the diagonal of a square with side length one is neither an integer nor the ratio of two integers.
The ancient Greek mathematicians at that time did not know the root number two, let alone the existence of such things as irrational numbers in the world.
Unable to solve the problem, they finally chose to solve the problem:
They threw Hippasos into the Aegean Sea and fed it to the sharks.
The second mathematical crisis, Zeno's paradox that originated in ancient Greece, can Achilles catch up with the tortoise, and does the moving arrow move or not?
The ancient Greeks were first exposed to the problems caused by infinitesimals, and the mathematical crisis really broke out in the era of Newton and Leibniz.
The two of them invented calculus, which is very convenient to use. There is just one question: is the infinitesimal quantity in calculus actually zero?
An infinitesimal quantity may appear in the denominator, so it should not be zero.
But if the infinitesimal quantity is regarded as zero and the terms containing it are removed, the obtained formula can be proved to be correct in mechanics and geometry.
At that time, some people criticized calculus as "the devil's trick" and "using double errors to accidentally obtain scientific but incorrect results."
This crisis was not finally resolved until the nineteenth century, when mathematicians led by Cauchy perfected the specific concept of limits.
As for the third mathematical crisis caused by paradoxes, it was the fastest one to solve.
German mathematicians Ernst Zermelo and Abraham Frankl proposed two sets of theories in 1908 and 1922 respectively. The two sets of theories added together became Z (ermelo, Zermelo)- F (raenkel, Frankl) axiomatic system.
This axiom system axiomizes the construction of sets to eliminate the existence of sets like Russell's paradox, which can be regarded as solving this mathematical crisis.
It was in the same year that Hilbert thought of finding a universal solution to the mathematical crisis that had broken out three times.
He came up with an idea called the Hilbert Plan, which proposed basing all existing theories on a limited set of complete axioms and giving proof that these axioms were consistent.
Hilbert hoped that mathematics would be complete and decidable. He hoped that mathematics would be based on rigorous logic and be the most impeccable truth in the world.
There is such a clause in Hilbert's plan, which is the so-called completeness. People can deduce all theorems starting from the axioms.
If you can't deduce it, it's not a problem with the completeness of the above statement, but a problem with your personal ability.
Axioms are basic mathematical knowledge that people have summarized in long-term practice and serve as the basis for determining the truth or falsehood of other propositions. They cannot be proven and do not need to be proven.
Theorems are true propositions obtained by reasoning based on axioms.
Hilbert is the greatest mathematician in the world today. His words are consistent and very appealing.
Since he proposed this plan, mathematicians have always believed that this plan is correct, and they have been trying to prove that it is correct.
It's just that many years have passed, and no mathematician has been able to obtain this proof.
In the original space and time, it was not until 1931 that Gödel proved another point. In an axiom system, there is always at least one proposition that cannot be proven true or false. If you want to prove or falsify these propositions, you must use the system. New axioms outside.
This is Gödel's first incompleteness theorem. The emergence of this theorem completely negated Hilbert's plan and shattered the dreams of Hilbert and all mathematicians.
Hilbert's original intention was to completely solve the mathematical crisis, but unexpectedly he almost knocked down the foundation of the mathematical building.
This Gödel was the one who solved Einstein's gravitational field equations and solved the Gödel universe that supports time and space travel.
Chen Muwu knew this man because of Gödel's universe, so he naturally knew the two incompleteness theorems he proposed.
When he heard the incompleteness mentioned in Bohr's words, he thought of this theorem and the mathematician Hilbert.
Chen Muwu didn't have much prejudice against Hilbert, but he clearly remembered Hilbert once said, "Physics is really too difficult for physicists."
His original intention was to say that although modern physics relies heavily on advanced mathematics, it has always been used loosely.
But when this sentence comes from the mouth of a mathematician, it is still very uncomfortable for a physics student to hear it.
Anyway, the particle accelerator is now being manufactured step by step, and Chen Muwu has nothing else to do except supervise the work.
So now that you have thought of this, why not come up with this incompleteness theorem, which can be regarded as a small shock to Sir Hilbert from a physicist.
After "denounced" the Germans, Chen Muwu once again fell into a long period of distraction.
Rutherford had long been accustomed to his disciple having such out-of-body experiences from time to time, so he simply took his other disciple Bohr to discuss and teach him the management experience of laboratories and research institutes.
After a long time, Chen Muwu finally woke up from his trance, opened his mouth and blinked his eyes.
"You finally came to life. Did you come up with any good ideas just now?"
A wise disciple is better than a master, Rutherford asked with a smile.
Chen Muwu rubbed his head in embarrassment: "I did have some immature ideas just now."
"What aspect, particle...experiment?"
Bohr followed Rutherford: "Or quantum mechanics?"
"Uh, no, it's just that after Professor Bohr reminded me, I suddenly had some ideas about mathematics."
He scratched his head more frequently.
Although Rutherford was used to Chen Muwu changing his research directions at any time, he still could not imagine that a good student would study mathematics.
He subconsciously reached for the pipe on his desk, and then remembered that Chen Muwu didn't like the smell of tobacco.
There was even a disappointed expression on Bohr's face.
"However, Professor Bohr, I also have some new ideas in quantum mechanics. I may write one or two papers in the near future. I will ask for your advice then."
"It's easy to talk about. I can't talk about giving advice. It's just a mutual discussion between us."
The office began to be filled with happy air again.
After the day's chat was over, the guests and hosts had a great time.
Rutherford acquiesced to the purpose of Chen Muwu's trip and did not recruit this student to the Cavendish Laboratory.
Bohr also got a fairly satisfactory result from Chen Muwu. Since he wanted to study theoretical knowledge and said he wanted to consult with himself, there was inevitably an exchange of letters and telegrams.
Maybe over time, the relationship between the two people will become closer, so that we can slowly figure it out. We must dig him out, oh no, please go to Copenhagen.
Bohr stayed at Cambridge University for a few more days, then went north to Manchester, continued to visit relatives, friends and old friends, and finally boarded the ship from Norwich and returned to Copenhagen, Denmark.
Chen Muwu came out of the leisurely state he had maintained for more than a year and started writing liver papers again.
Every other line is like a mountain. Although I know the law of incompleteness and once understood how Gödel proved it, it is not easy to reproduce that paper.
Fortunately, Chen Muwu not only has cheats for himself, but he also has humanoid cheats at Cambridge University.
Many of the Cambridge Apostolic Society are mathematicians and logicians. They are all students of Russell, and even Russell himself.
There is an essential connection between the incompleteness theorem and Russell's paradox. Both involve negating self-reference and the diagonal method.
It would be a fool not to embrace such a ready-made thigh but to study it on your own, so I took advantage of the opportunity of the Apostolic Society meeting every Saturday night.
The other young masters were all chatting with their glasses of wine. Chen Muwu was eating the expensive ingredients prepared on the dining table, while lowering his profile to ask others about math and logic problems.
He also took the opportunity to go to Russell's office several times to ask for advice, just to write this paper on the incompleteness theorem and get recognition from the mathematics community.
There are rumors in Cambridge University that Dr. Chen from Trinity College became indifferent in physics after winning the Nobel Prize in Physics.
He has become very close to Russell recently and may be developing in a philosophical direction.
(End of chapter)
