At this time, not only Napoleon chuckled in his heart and secretly screamed badly, but even Laplace and they were startled. How? Abbot Bonaparte already had a way to prove this strange geometry? But this is also normal, if there is anyone in this world who can solve this problem quickly, "Joseph who never makes mistakes" is certainly the most likely candidate.

At this time, on the podium, Joseph greeted Fourier slowly: "Mr. Fourier, well, you can help me and distribute my paper to everyone." I can also take a break and drink some water. When they're done, we'll move on to our discussion. ”

With that, Joseph slowly returned to his seat and took his teacup and drank his tea. At this time, Fourier also distributed a paper to everyone.

A copy of Joseph's paper was also sent in front of Napoleon, and Napoleon lowered his head and saw such a title as "An Attempt to Explain Non-Euclidean Geometry". He flipped through the papers, somewhat desperately, trying to find out if there were any loopholes in the round. Although he knew that the paper thrown out by Joseph at this time was in a hole in the likelihood that it was even less likely than that he had brought 100,000 present French troops to fight against less than a thousand Prussians in the field, and that the whole army would be destroyed. (After all, there is still the possibility of a few meteorites falling from the sky and hitting them)

Napoleon's mathematics is actually quite good, although to be honest, it is still a lot worse than the level of academicians, but among ordinary people, it is definitely at the level of a scholar. Therefore, he will not have the problem of not being able to read the paper.

Napoleon quickly read the paper with a fluke mentality. The paper is indeed typical of Joseph's style, with rigorous arguments, no gaps, buy one get one free, and come with a deduction of a new mathematical tool or two.

"Is this using differential geometry? The whole argument process seems to be really fine. Napoleon looked up at Laplace beside them. He saw that all of them had wide eyes, but none of them looked like they wanted to speak.

"It's over, and most of the time you don't see a problem. Joseph, this guy, really realized such a triangle on a hyperbolic plane. This, this...... I was so stupid, really, that I actually ran to Joseph's base to say goodbye to him. I thought he really wouldn't retaliate, forgetting that this guy has always been cautious......"

Laplace: They finally read the paper, they read it more carefully than Napoleon, but like Napoleon, they failed to find any errors in the paper.

"O Joseph, who never makes mistakes." Many people have such a sentence in their minds, and at the same time, they feel that the mountain that is pressing on their bodies is a little heavier.

Joseph had already finished the tea in his teacup and refilled two more cups. At this time, seeing that everyone had basically finished drinking, he put down the teacup and said slowly: "Everyone seems to have finished reading?" Now, do you still have any doubts about the thesis of Mr. Lucien Evans? ”

Everyone was silent.

Joseph then added: "In fact, in addition to my method, there is a more ingenious proof, which has also been completed by my friend, Academician Gauss. You can also take a look. ”

So Fourier sent Gauss's paper to everyone to read.

Gauss's paper is also called "An Attempt to Explain Non-Euclidean Geometry", but his argument is indeed different from Joseph's. His thinking is simpler and more special. He used the concept of projection to prove the compatibility of the new geometry with Euclidean geometry on the unit circle. If Euclidean geometry is true, then the new geometry must also be true!

This succinct derivation, the wonderful proof, is full of mathematical beauty, and nothing could have been more striking than Laplace and others.

"I don't think you have any doubts about the paper of Mr. Lucien Evans, who is actually anonymous, right?" "If that is the case," said Joseph, "I am going to announce the results of this hearing, well, I think Mr. Fourier has made the right assessment in his review of this paper." Now, which of you is for it and who is against? ”

So everyone, including Napoleon, agreed.

"Very good, I am very happy to see that our Academy of Sciences is an Academy of Sciences after all, and everyone is willing to be reasonable. True or not, everyone is willing to use papers to speak. Well, Mr. Fourier, you made the judgment that allowed this paper to pass before you saw a perfect proof. And we all know that in this paper, there are many things that go beyond our common sense and make us feel difficult to accept. Now, I'd like to ask you to tell me why you made the judgment that this paper passed without seeing a perfect proof. ”

Fourier nodded, and stepped onto the podium.

"Dear academicians, in fact, when I first saw this paper, I also felt ridiculous, I felt incredulous, and I was convinced that there must be some kind of error in this paper. It's just that at the time, I thought that although the creator of this paper made a ridiculous composition, the mathematical level he showed in the paper was very amazing. I think anyone who is really serious about restraining the disgust in their hearts and reading this paper seriously should be able to see this. I thought at the time: even this paper is really wrong and ridiculous. It is also a higher error and absurdity, like Zeno's paradox (Achilles can never catch up with a turtle a little ahead of him), which is obviously absurd, but it may well be absurd, but it may well be absurd. is absurdity that deserves to be taken seriously. It's as if the study of Zeno's paradox leads to an in-depth study of the finite and the infinite, the continuous and the discrete.

So I carefully studied this paper again. This kind of research – to be honest, scares me a lot. My heart tells me that this thing must be wrong, and there is no such truth in the world. But my brain told me that there was nothing wrong with the paper, mathematically.

This is a terrible thing, because it almost means that our math and reality are at odds. It is quite possible that our math is fundamentally wrong. At that time, I was so frightened by this idea that I couldn't even eat. ”

For this statement, even Laplace couldn't help but nod his head in agreement. Because, it's really scary. It's as frightening as the cosmic 3K microwave background radiation suddenly oscillating in an isotropic amplitude between 1 percent and 5 percent overall, or the universe flickering.

"But, at this point, something suddenly occurred to me. It was the Dean's 'Bonaparte Bright Spot Experiment', which seemed to go completely against common sense. Doesn't that experiment sound completely unrealistic? But as long as the conditions are right, it will really appear in reality. So I got a little comfort, and I thought, maybe it's not that the math is wrong, it's not that the reality is wrong, but that my own understanding of reality is wrong. The real world is so big, but the scope of our access is so limited. On what basis do we decide what is in line with reality and what is not? Perhaps, under some special conditions, this strange geometry can really be realized? It's like if we can really see a bright spot in the middle of the shadow left by an opaque object if the conditions are right.

Therefore, I discussed this paper and my thoughts with the dean and Academician Gauss. They all agreed with me, and together with me they tried to find the conditions in reality that would make this strange, intuitively different geometry work. The end result is the two papers you just saw.

I was very touched by this. Fourier listened and continued, "We would do well to be more cautious about what reality is. Don't think we really know what reality is. Many times, the real world is not what we think it is. In contrast, I think that what comes out of mathematical deduction may be more reliable than what we see in reality. I remember Dean Bonaparte saying before that our eyes can deceive us, our ears will deceive us, our imagination will deceive us, but mathematics will not. That's what I think, thank you. ”

And so everyone applauded.

At this time, Joseph also stood up, and as the presiding officer of the meeting, the president of the French Academy of Sciences, he would make his closing remarks.

"Ladies and gentlemen, Mr. Fourier's speech just now has inspired me greatly, and I am suddenly reminded of a pagan story. In faraway India, there is a story about a king who led an elephant and came to touch it to some people who were born blind. Then ask them, 'What does an elephant look like?' A blind man who had touched the body of an elephant said: 'The elephant is like a wall.' Another man who touched the elephant's leg said: 'The elephant is like a pillar.' A man who touched the trunk of an elephant said: 'An elephant is like a snake.' But we know that they are wrong.

What about us, when we laugh at the blind man who touches the elephant, do we think of ourselves? The universe is much bigger than elephants, and we are far inferior to the universe and even bacteria to us. The proportion of the entire elephant that a blind man can touch with his hands is much higher than the proportion of the universe itself that all humans can see in all our ways. Our situation is actually more difficult than that of the blind. Blind people can't see light, but neither can we see all the light. There's a lot of light, a lot of sound, it's there, but we can't see it or hear it. In that sense. Aren't we also blind? What we are dealing with is a universe much larger than an elephant. In this case, isn't it just as ridiculous that we still use our limited sense of touch as the basis for judging reality?

Therefore, in front of nature, in front of the world, we must be humble and do not think that we really understand what the real world is, otherwise, it may blow up our heads with a bang at any time with a phenomenon that we cannot understand for the time being.

Therefore, we need to reduce our preconceptions as much as possible, reduce those self-identified rules, and not prejudge what the world is like.

Finally, as Mr. Fourier mentioned earlier, he thinks that mathematical deduction may be more reliable than our sense of sight and hearing. And that makes sense. After all, our eyes are blind to some light, our ears are deaf to some sounds, our sense of touch is insensitive to vibrations below certain thresholds, and our sense of smell is also limited. At this point, Joseph paused, then smiled, "But at the end of the day, allow me to tell you one more story to scare everyone."

There was a chick who, through countless observations, discovered a pattern. That is, every time a peasant woman appears, there will be delicious grain falling down for him to eat. He observed it countless times, without exception, so much so that he was sure that this could be used as a basis for understanding the world, an axiom. That is, when the peasant woman appears, there will be grain to eat. As a result, one day the peasant woman appeared again, but instead of the grain, she brought a knife. The chicks that come up according to justice become chicken soup.

Aren't the axioms of our mathematics also the so-called intuitive laws that we have discovered based on repeated observations? Who knows if we'll be the same chicks? The real world may be very different from what we think of us. Therefore, we must be cautious, we must have more doubts, we must not have too many prejudices, and everything is judged by the actual reaction of the real world. ”

So everyone applauded.

"Today's hearing was really inspiring." At Napoleon's side, Monsieur Monges exclaimed, "I feel that I should tell today's story to my students, so that they may also be educated." ”

Napoleon pursed his lips and thought to himself, "Joseph will certainly tell this story in detail in the new issue of Mathematics." How could he not publicize such a thing? Well, there are so many factors in this story. self-righteous, people who are bound by old ideas; A person who is modest and cautious and can overcome his own prejudices; Those who wake up and can change their past wrongs; Hold fast to the truth and not fear the powerful...... Is there a better story than this story that reflects the scientific spirit of the French Academy of Sciences and the Roman Academy of Sciences? The only painful thing is that I'm going to be a villain in this story. No, my image in this story must be a ...... who respects science, respects the truth, has the courage to correct mistakes, and has a big heart."

Thinking of this, Napoleon hurriedly raised his hand and said, "Dean, I have something to say......"