I'll Take the Lead in Surrendering

At this moment, not only did Napoleon's heart skip a beat, and he secretly thought something was wrong, but even Laplace and the others were startled. What? Dean Bonaparte already had a method to prove this strange geometry? But this was normal. If there was anyone in the world who could quickly solve this problem, "Joseph who never makes mistakes" was certainly the most likely candidate.

At that moment, Joseph, standing on the podium, slowly greeted Fourier: "Mr. Fourier, could you please distribute my paper to everyone? I can take a break and have some water. Once they've read it, we can continue our discussion."

After saying this, Joseph leisurely returned to his seat, picked up his teacup, and began to drink. Meanwhile, Fourier distributed a paper to everyone.

A copy of Joseph's paper was also presented to Napoleon. Napoleon lowered his head and saw the title. "An Attempt at an Interpretation of Non-Euclidean Geometry." He opened the paper with a sense of despair, trying to find any flaws. Although he knew that the probability of a paper published by Joseph at this time being flawed was probably even smaller than the probability of him leading 100,000 France troops into open battle against fewer than 1,000 Prussia and being completely annihilated. (After all, there was still the possibility of a few meteorites falling from the sky and hitting them.)

Napoleon's mathematical abilities were actually quite good. While, to be honest, he was still quite a bit short of the level of an academician, he would definitely be considered a top student among ordinary people. Therefore, he wouldn't have had the problem of not being able to understand academic papers.

Napoleon, with a sense of wishful thinking, quickly skimmed through the paper. It was indeed typical of Joseph's style: rigorous in its arguments, leaving no gaps, and, as a bonus, it included the derivation of one or two new mathematical tools.

"Is this using Differential Geometry? The entire proof seems to be working perfectly." Napoleon looked up at Laplace and the others beside him. He saw that they all had their eyes wide open, but none of them seemed inclined to speak.

"Damn it, they probably won't even see the problem. Joseph really did realize such a triangle on a hyperbola. This, this... I'm so stupid, really, I actually came to Joseph's base to challenge him. I thought he really wouldn't retaliate, I forgot that this guy is always petty..."

Laplace and his colleagues finally finished reading the paper, even more carefully than Napoleon, but like Napoleon, they couldn't find any errors in it.

"Joseph who never makes a mistake." This phrase popped into many people's minds, and at the same time, they felt that the mountain weighing on them had become a little heavier.

Joseph had finished his tea and refilled it twice more. Seeing that everyone had basically finished, he put down his teacup and said slowly, "It seems everyone has finished reading? Now, does anyone still have any doubts about Mr. Lucien Evans's paper?"

Everyone remained silent.

Joseph then added, "Actually, besides my method, there is an even more ingenious proof, which was also completed by my friend, Academician Gauss. You can all take a look."

So Fourier then sent Gauss's paper to everyone to see.

Gauss's paper is also titled "An Attempt at an Interpretation of Non-Euclidean Geometry," but his line of reasoning is indeed different from Joseph's. His approach is simpler and more unique. He uses 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 also hold!

This concise derivation and beautiful proof are full of mathematical beauty, and for Laplace and others, nothing could be more stunning.

"I think everyone has no more doubts about Mr. Lucien Evans's paper, which was actually written anonymously," Joseph began. "If that's the case, then I'll announce the results of this hearing. Well, I believe Mr. Fourier made the correct assessment of this paper. Now, who agrees and who disagrees?"

So everyone, including Napoleon, agreed.

"Very good, I am pleased to see that our Academy of Sciences of Sciences is, after all, the Academy of Sciences, and everyone is willing to reason things out. Whether something is right or wrong, everyone is willing to let their papers speak for themselves. Hmm, Mr. Fourier, you made your decision to approve this paper before you saw a complete proof. And we all know that this paper contains many things that go beyond our common sense and are difficult for us to accept. Now, I would like to ask you to explain why you made your decision to approve this paper before you saw a complete proof."

Fourier nodded and walked up to the podium.

"Distinguished academicians, when I first saw this paper, I also found it absurd and unbelievable, and firmly believed that it must contain some kind of error. However, at the time, I felt that although the author of this paper had produced an absurd piece of writing, the mathematical level he demonstrated in the paper was astonishing. I think anyone who truly restrained their aversion and carefully read the paper should have been able to discover this. I thought to myself: even if this paper is indeed wrong and absurd, it is a more profound error and absurdity, like Zeno's paradox (Achilles can never catch a tortoise that is slightly ahead of him). It is obviously absurd, but it may very well be an absurdity with very profound implications. It is an absurdity worthy of serious consideration. Just as the study of Zeno's paradox led to in-depth research on finite and infinite, continuous and discrete mathematics."

So I carefully studied the paper again. To be honest, this kind of research terrified me. My heart told me that it must be wrong; there was no such logic in the world. But my brain told me that, mathematically speaking, the paper was flawless.

This was a truly terrifying thing, because it almost meant that our mathematics contradicted reality. It was very likely that our mathematics was fundamentally flawed. At the time, I was so terrified by this thought that I couldn't even eat.

Even Laplace couldn't help but nod in agreement with this statement. Because it was indeed terrifying. It was just as frightening as the cosmic 3K microwave background radiation suddenly exhibiting isotropic fluctuations with an amplitude between one and five percent, or as if the universe itself were flickering.

"But then, something suddenly came to mind. It was the dean's 'Bonaparte Bright Spot Experiment,' which seemed completely contrary to common sense. Didn't that experiment also sound completely unrealistic? But given the right conditions, it really does happen in reality. So I found some comfort in thinking that perhaps neither mathematics nor reality was wrong, but rather my own understanding of reality was flawed. The real world is so vast, yet the scope we can access is so limited. What right do we have to decide what is realistic and what is unrealistic? Perhaps, under certain special conditions, this strange geometry can really be realized? Just like, given the right conditions, we can actually see a bright spot in the shadow cast by an opaque object."

So, I discussed this paper, along with my ideas, with the dean and Academician Gauss. They all agreed with my ideas and worked with me to find the conditions in reality that would allow this strange geometry, which is different from intuition, to hold true. The final result is the two papers you just saw.

"This matter has deeply moved me." Fourier listened for a moment, then continued, "We should be more cautious about what reality is. Don't assume we truly know what reality is. Often, the real world is not what we imagine. In comparison, I think what is derived from mathematical deduction might 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, our imagination can deceive us, but mathematics cannot. That's my thought, thank you all."

So everyone started clapping.

At this point, Joseph also stood up. As the host of the meeting and the president of the French Academy of Sciences, he was about to deliver his concluding remarks.

"Gentlemen, Mr. Fourier's speech just now greatly inspired me. I suddenly recalled a pagan story. In faraway India, there was a story about a king who brought an elephant to several people who were born blind. He asked them, 'What does an elephant look like?' One blind man who touched the elephant's body said, 'An elephant is like a wall.' Another who touched the elephant's leg said, 'An elephant is like a pillar.' A third who touched the elephant's trunk said, 'An elephant is like a snake.' But we know that they were all wrong."

When we laugh at the blind men and the elephant, have we ever thought about ourselves? The universe is far greater than an elephant; compared to the universe, we are far less than bacteria. The area a blind man can touch represents a far greater proportion of the entire elephant than the entire universe itself, encompassed by all human perception. Our situation is actually more difficult than that of the blind. The blind cannot see light, but we cannot see all light either. Many rays of light and many sounds clearly exist, yet we cannot see or hear them. In this sense, aren't we also blind? We are facing a universe far larger than an elephant. In this context, isn't it equally ridiculous to use our limited sense of touch as the basis for judging reality?

Therefore, we should be humble in the face of nature and the world, and not assume that we truly understand what the real world is. Otherwise, it may at any moment blow our minds away with a phenomenon that we cannot comprehend for the time being.

Therefore, we should try our best to reduce our prejudices and self-imposed limitations, and not presumptuously try to define what the world should be like.

Finally, as Mr. Fourier just mentioned, he felt that mathematical deductions might be more reliable than our sight and hearing. This is indeed reasonable. After all, our eyes are blind to some light, our ears are deaf to some sounds, our sense of touch is unresponsive to vibrations below certain thresholds, and our sense of smell also has limitations. Only mathematics seems to be the universally applicable truth."Joseph paused here, then smiled and said, "But in the end, please allow me to tell one more story to scare everyone a little."

A little chick, through countless observations, discovered a pattern: whenever a peasant woman appeared, delicious grains of rice would fall down, allowing him to eat his fill. He observed this countless times without exception, until he became certain that this could be a foundation for understanding the world, an axiom: the appearance of a peasant woman always meant grains to eat. But one day, the peasant woman appeared again, but instead of grains, she brought a knife. The chick, following the axiom, rushed forward and became chicken soup.

Aren't the axioms of mathematics also based on observations and the discovery of so-called intuitive patterns? Who knows if we might also be that little chick? The real world may be very different from what we imagine. Therefore, we must be cautious, we must be more skeptical, and we must not have too many prejudices. We must judge everything based on the actual reactions of the real world."

Then everyone started clapping.

"Today's hearing was truly inspiring," Monge remarked beside Napoleon. "I think I should tell my students today's story so they too can be educated."

Napoleon pursed his lips, thinking, "Joseph will surely recount this story in detail in the next issue of the journal Mathematics. How could he not publicize something like this? Hmm, there are so many elements in this story. Self-righteous people, bound by old ideas; humble and cautious people who can overcome their prejudices; people who suddenly realize their mistakes and correct them; people who uphold the truth and are not afraid of power... Is there anything that better reflects 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 example in this story. No, my image in this story must be one of respect for science, respect for truth, courage to correct mistakes, and broad-mindedness..."

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