Fox of France

At this time, not only did Napoleon feel a little bit of a thump in his heart, secretly screaming that it was not good, even Laplace and the others were startled. how? Dean Bonaparte already has a way to prove this weird geometry? But this is also normal. If there is anyone in this world who can quickly solve this problem, "Joseph who never makes mistakes" is of course the most likely candidate.

At this time, on the podium, Joseph greeted Fourier slowly: "Mr. Fourier, um, please help me to distribute my paper to everyone. I can just take a break, Drink some water. When they finish watching, we can continue our discussion."

After saying this, Joseph slowly returned to his seat, picked up the teacup and drank tea. At this time, Fourier also distributed a paper to everyone.

A paper by Joseph was also sent in front of Napoleon. Napoleon lowered his head and saw a title like "An Attempt to Interpret Non-Euclidean Geometry". He flipped through the paper in desperation, trying to find if there were any loopholes in this round of papers. Although he knew that the thesis that Joseph threw out at this time, the possibility of being in a loophole was more likely than the probability that he would bring 100,000 of the current French army to fight against less than 1,000 Prussians in the field, and the entire army would be wiped out Possibly even smaller. (After all, there is still the possibility that a few meteorites fell from the sky and just hit them)

Napoleon's mathematics is actually not bad. Although to be honest, it is still far from the level of an academician, but among ordinary people, it is definitely at the level of a master. Therefore, he would not have the problem of not being able to understand the paper.

Napoleon quickly read the paper with a fluke mentality. This paper is indeed a typical Joseph style, with rigorous arguments and no gaps, and buy one get one free, and it also comes with a deduction of one or two new mathematical tools.

"This is differential geometry? The whole argument process seems to be really okay." Napoleon raised his head and looked at Laplace and the others beside him. He saw that their eyes were all wide open, but none of them seemed to speak.

"It's over, I probably can't see the problem. Joseph, this guy, really realized such a triangle on a hyperbolic surface. This, this... I'm so stupid, really, I actually ran to Joseph's base Lilai and him don't show any signs. I thought he really wouldn't retaliate, forgetting that this guy has always been narrow-minded..."

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

"Joseph who never makes a mistake." This sentence came to many people's minds, and at the same time, they felt that the mountain that was pressing on them was a little bit heavier.

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

Everyone is silent.

Joseph then said: "Actually, apart from my method, there is another more ingenious proof, which was also completed by my friend, Academician Gauss. You can also take a look."

So Fourier sent Gauss's paper to everyone.

This paper by Gauss 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 holds, then the new geometry must hold too!

This concise derivation, wonderful proof, full of mathematical beauty, is nothing more shocking to Laplace and others.

"I think everyone should have no doubts about the thesis written by Mr. Lucien Evans, who is actually an anonymous person?" Joseph said, "If so, I will announce this hearing The result of the meeting, well, I think Mr. Fourier made a correct evaluation in the review of this paper. Now, who of you agrees and who opposes?"

So everyone, including Napoleon, agreed together.

"Very well, I am very pleased to see that our Academy of Sciences is an Academy of Sciences after all, and everyone is willing to reason. Whether it is true or not, everyone is willing to use papers to speak. Well, Mr. Fourier, you are seeing the perfect Before the proof, we made the judgment to pass this paper. And we all know that in this paper, there are many things that go beyond our common sense and make us feel unacceptable. Now, I would like to invite Let me tell you why you made the decision to pass this paper before you saw the perfect proof."

Fourier nodded and stepped onto the podium.

"Dear academicians, in fact, when I first saw this paper, I also felt absurd and unbelievable, and I was convinced that there must be some kind of error in this paper. It's just that I thought at the time that the paper's Although the creator made an absurd composition, the level of mathematics he showed in the paper is amazing. I think anyone who really restrains his disgust in his heart and reads this paper seriously Everyone should be able to discover this. I thought at the time: Even if this paper is really wrong and absurd, it is also a higher level of error and absurdity, just like Zeno’s paradox (Achilles always pursues not a tortoise a little ahead of him) is obviously absurd, but it may well be an absurdity with very deep connotations. An absurdity that deserves to be taken seriously. It is as if the study of Zeno's paradox led to An in-depth study of finite and infinite, continuous and discrete alike.

So I carefully studied this paper carefully. Research like this -- frankly, terrifies me. My heart told me that this thing must be wrong, there is no such reason in the world. But my brain told me that there was nothing wrong with this paper mathematically.

This is really a scary thing, because it almost means that our mathematics and reality are contradictory. Chances are our math is fundamentally wrong. At that time, I was so scared by my own thoughts that I couldn't even eat. "

Even Laplace couldn't help but nod in agreement with this statement. Because, this is really too scary. It is just as frightening as the cosmic 3K microwave background radiation suddenly has an isotropic fluctuation with an amplitude between 1% and 5% as a whole, or the universe flickers.

"However, at this moment, I suddenly remembered one thing. It was the dean's 'Bonaparte bright spot experiment' that seemed to go 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, I thought, maybe neither the math is wrong, nor the reality is wrong, but my own understanding of reality is wrong. Reality The world is so vast, but the range we can touch is so limited. How can we decide what is in line with reality and what is not? Maybe, under certain special conditions, this Can this strange geometry really be realized? Just as 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 agree with my idea, and together with me try to find out the condition that can make this kind of strange geometry different from intuition. The final result is the two papers you just saw.

This incident touched me very much. Fourier listened and continued, "We'd better be more cautious about what is reality." 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 the things derived from mathematical deduction may be more reliable than the reality we see. I remember Dean Bonaparte once said that our eyes can deceive us, our ears can deceive us, and our imagination can deceive us, but mathematics does not. That's what I thought, thank you all. "

So everyone applauded.

At this time, Joseph also stood up. As the host of the meeting, the president of the French Academy of Sciences, he will deliver a concluding speech.

"Gentlemen, Mr. Fourier's speech just now gave me a lot of inspiration. I suddenly remembered a pagan story. In far away India, there is such a story. It is said that there was a king who led a big Come and touch a few people who are blind from birth. Then ask them: 'What does an elephant look like?' A blind man who touched the elephant's body said: 'An elephant is like a wall.' Another The one who touched the elephant's leg said, 'An elephant is like a pillar.' The one who touched the elephant's trunk said, 'An elephant is like a snake.' But we know they were all wrong .

And we, when we laugh at the blind man who touches the elephant, do we think of ourselves? The universe is bigger than an elephant, and we are far inferior to the universe, even bacteria. The proportion of the range that blind people can touch with their hands in the whole elephant is much higher than the proportion of the range that all human beings can see in all our ways and the universe itself. Our situation is actually more difficult than that of the blind. Blind people can't see light, but we can't see all light either. A lot of light, a lot of sound, obviously exist, but we can't see or hear. In this sense. Aren't we blind too? What we have to face is a universe much larger than an elephant. In this case, we still use our limited sense of touch as the basis for judging reality. Isn't it just as ridiculous?

Therefore, in front of nature and the world, we must be humble, and don't think that we really understand what the real world is, otherwise, it may use a phenomenon that we cannot understand at any time to blow our heads off with a bang .

Therefore, we should reduce our prejudices as much as possible, reduce those self-identified rules and regulations, and don't self-righteously define what the world looks like.

Finally, Mr. Fourier mentioned just now that he felt that mathematical deduction may be more reliable than our vision and hearing. This also does make sense. After all, our eyes are blind to some lights, our ears are deaf to some sounds, our sense of touch is insensitive to vibrations below a certain threshold, and our sense of smell is similarly limited. universal truth. At this point, Joseph paused, then smiled and said, "But at the end, please allow me to tell another story to scare everyone."

There was a little chicken who discovered a pattern through countless observations. That is, whenever a peasant woman appears, delicious millet will fall to make him full. He has observed countless times without exception, so that he is sure that this can be used as a basis for understanding the world, an axiom. That is, when the peasant woman appears, there will be millet to eat. As a result, one day the peasant woman appeared again, but she did not bring millet, but a knife. According to the axiom, the chicken that greets it becomes chicken soup.

Aren't the axioms of our mathematics also the so-called intuitive laws discovered based on repeated observations? Who knows if we might be that chick too? The real world may be very different from our minds. Therefore, we must be cautious, we must have more doubts, we must not have too many prejudices, and everything must be judged by the actual reaction in the real world. "

So everyone applauded.

"Today's hearing was really inspiring." Beside Napoleon, Mr. Monge sighed, "I think I should share today's story with my students so that they can also be educated."

Napoleon curled his lips, thinking: "Joseph will definitely tell this story in detail in the new issue of "Mathematics". How could he not publicize such a thing? Well, the various aspects of this story There are really too many factors. People who are self-righteous and bound by old ideas; people who are humble and prudent and able to overcome their own prejudices; Can this story better reflect the scientific spirit of the French Academy of Sciences and the Roman Academy of Sciences? The only painful thing is that I have to be a negative foil in this story. No, my image in this story must be to respect science , respect the truth, have the courage to correct mistakes, and have a broad mind..."

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