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二加二等于四。
Two plus two equals four.
在所有地方和所有时间，二加二都等于四。
In all places and for all time, two plus two equals four.
但为什么呢？
But why?
数学告诉我们现实的本质是什么？
What does math tell us about the nature of reality?
大卫·贝尔林斯基、塞尔吉奥·克莱曼恩和斯蒂芬·迈耶将在《非同寻常的知识》节目中讨论，现在开始。
David Berlinski, Sergio Kleinemann, and Stephen Meyer on Uncommon Knowledge, now.
欢迎收看《非同寻常的知识》，今天在奥地利萨尔茨堡录制。
Welcome to Uncommon Knowledge, recording today in Salzburg, Austria.
我是彼得·罗宾逊。
I'm Peter Robinson.
大卫·贝尔林斯基曾在斯坦福大学、罗格斯大学、纽约城市大学和巴黎大学教授数学、哲学和英语。
David Berlinski has taught math, philosophy, and English at universities including Stanford, Rutgers, the City University of New York, and the Universite du Paris.
希望你喜欢这个发音，大卫。
I hope you like that pronunciation, David.
完美。
Perfect.
他是《一、二、三：基础数学》等书籍的作者，即将出版的新作是《永恒的玫瑰》。
He is the author of books including one two three absolutely elementary mathematics and his forthcoming volume, The Perpetual Rose.
塞尔吉奥·克莱曼是罗马尼亚人，现任普林斯顿大学数学教授。
A native of Romania, Sergio Kleinemann is a professor of mathematics at Princeton.
用他自己的话来说，他目前的研究兴趣包括黑洞的数学理论，更具体地说是刚性与稳定性。
In his own words, his current interests include the mathematical theory of black holes, more precisely the rigidity and stability.
我读着这些话，却完全不明白它们是什么意思。
I'm reading these words without having any idea what they mean.
以及捕获面和奇点的动力学形成。
And the dynamic formation of trapped surfaces and singularities.
引号结束。
Close quote.
也许你可以向我们稍微解释一下这部分，塞尔吉奥。
I'll ask you to explain a little bit of that maybe Sergio.
发现研究所科学与文化中心主任斯蒂芬·迈耶，最初的职业生涯是一名地球物理学家。
The director of the Discoveries Institute Center for Science and Culture, Stephen Meyer, started his professional life as a geophysicist.
他重返校园，在剑桥大学获得了科学史与科学哲学博士学位。
He returned to school earning a doctorate from Cambridge in the history and philosophy of science.
他已成为美国智能设计领域最具影响力的思想家之一。
He has established himself as one of America's leading thinkers in intelligent design.
他最近的著作《上帝假说的回归》。
His most recent book, The Return of the God Hypothesis.
大卫、塞尔吉奥、史蒂夫，欢迎你们。
David, Sergio, Steve, welcome.
在《上帝假说的回归》——史蒂夫最新的著作中，他提出，三个相对较新的科学发现表明，科学需要重新回归某种超验观念。
In The Return of the God Hypothesis, Steve's latest book, he argues that three relatively recent developments suggest that science needs to return to some notion of the transcendent.
这三个发现分别是大爆炸、宇宙的精细调谐，以及DNA的发现。
And these three developments are the big bang, the fine tuning of the universe, and the discovery of DNA.
读完史蒂夫的书后，一位非常杰出且知名的数学家把史蒂夫拉到一旁说：你只提到了三个暗示超验心智的发展。
After reading Steve's book, a certain very accomplished well known mathematician took Steve aside and said, you only named three developments that suggest a transcendent mind.
还有第四个。
There's a fourth.
塞吉奥，你这话是什么意思？
Sergio, what did you mean by that?
首先，我得说，史蒂夫谈的是那些发展，但数学是永恒的。
Well, first, I I should say, Steve talked about developments, and mathematics is forever.
我的意思是，它已经存在了几千年。
I mean, been around for thousands of years.
所以这样比较并不完全公平。
So it's not quite fair to compare.
但数学，按其定义，处理的是它自身的一种现实感，我认为这种现实与物理现实一样客观。
But mathematics has, by definition, deals with its own sense of its own reality, which is, I claim, as objective as physical reality.
比如，黑洞就是这样，对吧？
So, for example, black holes are like that, right?
根据定义，黑洞有广义相对论这一数学理论作为预测依据。
A black hole, by definition, we have a mathematical theory of general relativity that predicts black holes.
但根据定义，黑洞是无法被直接观测到的。
But by definition, a black hole cannot be seen.
尽管如此，我们仍能断言它的存在。
So nevertheless, we can assert its existence.
为什么？
Why?
因为广义相对论是一个自洽的理论。
Because general relativity is a consistent theory.
所以，说到黑洞，真把我吓坏了。
So to take this black holes scare the daylights out of me.
我们回头再谈黑洞，我肯定还会提到，但我的头现在都疼了。
We'll come back to black holes, I'm sure, but my mind already hurts.
只是当你谈到你的工作和那种严谨性时，我感觉就是这样。
It's just that when hearing about your work and the rigidity alright.
用通俗的话来说，也就是对我而言，二加二等于四是真实的。
In layman's terms, which is to say for me, two plus two equals four is real.
这并不是幻觉。
That's not a figment.
它不是我们思维、心理过程，或大脑神经中偶然发生的过程的产物。
It's not an artifact of our mind, of mental processes, accidental processes that might be going on in our neurons.
没错。
Correct.
无论我认为二加二等于三还是五，我都是错的。
Whether I think it's two plus two equals three or five, I'm wrong.
二加二确实等于四，这是客观存在的真实。
Two plus two does equal four and that is objectively real.
没错。
Correct.
对。
Right.
因此，存在一种超越我们之外的概念性客观现实。
Therefore, there is a conceptual objective reality that exists in outside us.
对。
Right.
它不是物质的。
It's not material.
它不是物质的。
It's not material.
这实际上非常重要。
And this is actually a big deal.
大卫耸了耸肩。
David shrugs.
是的。
Yeah.
当然，这很重要。
Of course, it's a big deal.
我的意思是，二加二等于四是一个有趣的例子。
I mean, two plus two equals four is an interesting example.
但你可以从更基本的理念中推导出这种生物学推论，这本身就是一个令人兴奋且有趣的事实。
But you can derive that biological inference from still more fundamental ideas, which is an exciting and interesting fact all on its own.
你不必一开始就断言二加二等于四，我站在这里，我别无选择。
You don't have to begin by affirming two plus two equals four, there I stand, I can do no other.
你可以说，我是从更原始的概念元素中推导出这个结论的。
You can say, I've derived that from still more primitive conceptual items.
但当你不断回溯，追问算术系统最初的前提和公理时，你所能提供的唯一辩护，就是整个体系的一致性。
But when you go back and back and back and back and you ask about the initial assumptions, the axioms of the system about arithmetic, there is no additional defense that you can offer beyond the consistency of the whole.
这是一种非常有趣的位置，让人置身其中。
Which is a very interesting position to find oneself.
所以我要引用你书里的一段话。
So I'm going to quote to you from your book.
这里什么都没有。
Nothing here.
一、二、三。
One, two, three.
我想这应该是同一个观点，因为这表明我确实理解了你。
I think this is the I'm hoping this is the same point because that will indicate that I have actually understood you.
引述：数字及其所支持的运算，无法被分析为某种更根本的东西而消解。
Quote, neither the numbers nor the operations they make possible permit an analysis in which they disappear in favor of something more fundamental.
正是数字本身才是根本的。
It is the numbers that are fundamental.
它们或许可以被更好地理解。
They may be better understood.
它们或许可以被更好地描述，但无法被超越。
They may be better described, but they cannot be bettered.
我仍然认为这是对的。
I I still think that's true.
请注意，当你说到二加二等于四时，这是一种断言。
Bear in mind, when you say two plus two equals four, that's an assertion.
是的。
Yes.
我在那段话中所主张的是，当你回到算术的基础，期望或希望摆脱数字时，你会非常失望，因为它们会再次出现。
What I'm arguing for in that particular passage is that when you go back to the foundations of arithmetic, in the expectation or the hope that you can get rid of the numbers, you're going to be very disappointed because they reappear.
好的。
Alright.
我打算
I'm going
再次引用大卫的话。
to quote David once again.
但我把这一点交给你们两位来判断。
But I put this to to the two of you for judgment.
我假设他会同意自己的观点。
I'm assuming he will agree with himself.
尽管在大卫的情况下，这始终是个问题。
Although in David's case, this is always a question.
再次引用他《一二三》一书中的原话：在所有被提出、评估、接受、推迟、延后或驳回的论点中，唯有数学中的论证才具备迫使人们信服的力量。
Again, from his book one two three, quote, Across the vast range of arguments offered, assessed, embraced, deferred, delayed, or defeated, it is only within mathematics that arguments achieve the power to compel allegiance.
没有任何哲学理论能够解释为何会如此。
No philosophical theory has ever shown why this should be so.
这是数学之谜的一部分。
It is a part of the mystery of mathematics.
因此，你从某种源自亚里士多德、看似合理的哲学立场出发，但我仍可以说：你知道，我并不信服。
So you argue from some philosophical point that derives from Aristotle and has seemed straight but I can still say, you know, I'm not persuaded.
但当你对我说‘二加二等于四’时，我当然必须承认，你是对的。
But when you say to me two plus two equals four, I have to of course you're right about that.
这是不是说
Is that the I
我可以从一名从事自然科学和科学哲学工作者的角度来谈谈这一点。
can speak to this from the standpoint of someone who's worked in the natural sciences and as a philosopher of science.
自然科学通过经验观察证据支持其结论，科学家们会通过比较特定理论或假设的解释力或预测力来评估它们，但这些论证的逻辑形式并不能得出演绎上的确定结论。
The natural sciences provide empirical observational evidence in support of conclusions And scientists will evaluate particular theories or hypotheses by comparing their explanatory power or their predictive power, but the logical form of those arguments does not render a deductively certain conclusion.
在最好的情况下，你会推断出一个提供最佳解释的假设。
You, in the best of cases, will make an inference to a hypothesis which provides the best explanation.
等一下。
Hang on one second.
为我们区分一下演绎和推论。
Just distinguish deduction from inference for us.
一个演绎论证会从一个大前提开始。
So a deductive argument will start with a major premise.
所有人都是会死的。
All men are mortal.
一个小前提。
A minor premise.
关于世界的一个事实。
Some fact about the world.
苏格拉底是人。
Socrates is a man.
苏格拉底是人。
Socrates is a man.
然后是结论。
And then a conclusion.
因此，苏格拉底是会死的。
Therefore Socrates is mortal.
对。
Right.
好的。
Okay.
如果前提为真且推理有效，那么结论就可以以某种确定性被确认。
And if the premises are true and the reasoning is valid then the conclusion can be affirmed with some kind of with certainty.
但在自然科学中，你从观察到的关于世界的事实出发，试图从这些事实推断出某种概括（这就是归纳论证），或者推断出某种可能解释你所见现象的因果过程。
But the in the natural sciences you start with facts that you've observed about the world and you want to infer from those facts to either some kind of generalization, that would be an inductive argument, or to some sort of causal process that might explain what you're seeing around you.
这些论证通常被称为溯因推理，就像我们看侦探剧时所享受的那种侦探式推理。
Those arguments are typically characterized as abductive, the kind of detective reasoning that we enjoy when we watch detective shows.
哥伦布
Columbo.
是的
Yeah.
哥伦布或者有人在试图找出凶手是谁。
Columbo or someone's trying to figure out who done it.
因此，当你分析这些溯因推理和归纳推理的形式时，会发现它们并不能带来确定性。
And so those abductive and inductive inferences when you examine the logical forms turn out not to give you certainty.
它们可能带来可能性，或者带来相对可能性——即一个理论远优于另一个，但它们无法像数学和数理逻辑那样提供那种确定性。
They may give you plausibility, they may give you comparative plausibility where one theory is very much better than another, but they don't give you the kind of certainty that mathematics alone and mathematical logic This
这就是为什么一位优秀的科学家永远不会说比‘该理论是x y z，并且在现有最佳证据下暂时成立’更多的内容。
is why a good scientist will never say any more than the theory is x y z and on the best evidence it holds up for now.
而数学家则完全有信心说：我已证明了它。
Whereas a mathematician is feels perfectly confident in saying I've proven it.
证明了它。
Proven it.
我们有一个证明。
We have a proof.
当我说科学中的证明时，数学家们却比我们更胜一筹。
When I say proof in science the mathematicians however are better than we.
当你说到证明某事时，你是认真的。
When you say you prove something you mean it.
是的。
Yes.
好的。
Alright.
明白了。
Okay.
那么，你继续吧。
So this go ahead.
它可能是错的，但会有人向我指出它确实是错的。
It could be wrong, but somebody will show it to me that it's Yeah.
所以
So
如果这让我们谈到塞吉奥在《推论》杂志上的文章，大卫，那是你主编的杂志。
if this brings us to Sergio's article in Inference, a magazine that you edit, David.
这篇文章的标题是《对维格纳一篇论文的反思》。
The article is entitled Reflections on an Essay by Wigner.
现在，尤金·维格纳，我得先介绍一下，是二十世纪的一位数学家和物理学家，于1960年。
Now, Eugene Wigner, I have to set this up, was a twentieth century mathematician and physicist in 1960.
他写了一篇著名的文章，名为《数学在自然科学中不合理的有效性》。
He wrote a famous essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
维格纳注意到，数学毕竟只是存在于我们头脑中的思维活动，却如此有效地用于描述甚至预测物理世界的各个方面，这让他感到惊讶。
Wigner noted his surprise that mathematics, which after all goes on in our minds, should prove so useful in describing and even predicting aspects of the physical world.
好的。
Okay.
你能给我举几个这样的例子吗？
Can you give me a couple of examples of this?
我的意思是，当我自己想，等等，我半夜做了一个梦，醒来后发现它并不真实。
I mean, when I think to myself, wait a minute, so I have a dream in the middle of the night, wake up and it turns out it was untrue.
但如果我做了一个数学方程，醒来后它依然是正确的。
But if I do a mathematical equation and I wake up, it's still true.
嗯，塞吉奥写了一篇精彩的文章，他指出你可以从非常简单的数学出发，逐步推导出越来越复杂的形式。
Is that Well, Sergio wrote a brilliant essay and what he showed was that you can start with very simple mathematics and build up to more and more and more complex forms of math essentially deductively.
而这些复杂的数学形式，比如微积分、微分方程，能够完美地映射到物理世界，描述自然界中实际发生的过程，从而对自然现象提供极其精确的描述。
And then those complex forms of math, take the calculus, take differential equations, they map beautifully onto the physical world to describe actual processes that are taking place in nature so that they provide very precise descriptions of things that are going on in nature.
维格纳所暗示的是，这种现象对许多物理学家而言是一种谜团或难题。
And what Wigner is alluding to is the mystery that this, or the puzzle this induces for a lot of physicists.
为什么我们通过一系列演绎推理发展出来的数学，能够如此完美地对应那些我们甚至尚未观测到的自然过程？
Why should the math that we have developed through a series of deductive steps effectively from our own reasoning map so beautifully to processes that we sometimes haven't even observed yet?
大卫举了几个很好的例子，比如一、二、三种数学结构，它们在被应用于物理学之前很久就已经被发展出来了，但后来却变得至关重要。
And David has a number of great examples in one, two, three of mathematical structures that were developed well before they had any application to physics, but then later were crucial.
也许他应该谈谈这一点。
And maybe he should speak to that.
是的。
Yes.
我的意思是，尤金·维格纳提出了一个非常有趣的观点，人们一直在讨论它。
Well, I mean, it's not Eugene Wigner raised a very interesting point and people have been discussing it.
如果你看看理论物理学，那些伟大的体系——牛顿力学、广义相对论、量子力学——都离不开大量的数学。
If you look at theoretical physics, the great structures, Newtonian mechanics, general relativity, quantum mechanics, can't do it without a lot of mathematics.
你真的需要大量的数学。
You just need a whole lot of mathematics.
问题是，我们知道做量子力学需要数学，但不需要昆虫学。
And ignorance, the question is, you know, we need mathematics to do quantum mechanics, but we don't need entomology.
为什么昆虫不会出现在量子力学中，而数字和复数却会？
How come the bugs don't figure in quantum mechanics, but the numbers and the complex numbers do?
这是一个引人深思且富有启发性的问题，但我们暂时不谈量子力学。
And that is a rewarding and a provocative question, but we don't have turn to quantum mechanics.
这里有一个玻璃杯。
There's a glass here.
这里有一个玻璃杯。
There's a glass here.
一个玻璃杯。
One glass.
一个玻璃杯。
One glass.
你面前有多少个玻璃杯？
How many glasses are in front of you?
两个。
Two.
你凭什么确信你面前有两个玻璃杯？
From where do you derive that assurance that there are two glasses in front of you?
这不是物理观察，因为物理情境本身并不能说明一加一等于二。
It's not a physical observation because nothing in the physics of the situation reveals the fact that one and one are two.
这是一些人会认为是额外的东西。
That is something someone would think that's additional.
现在我们可以将这一切分解为更小的步骤，这正是二十世纪逻辑学家们所做的。
Now we can break that all down into smaller steps, and that's what logicians have done in the twentieth century.
他们向我们展示了证明方法可以分解为非常小的步骤。
They've shown us that the method of proof can be decomposed into very small steps.
事实上，这些步骤小到计算机都能执行。
In fact, so small a computer can execute them.
而初始假设可以变得如此普遍，甚至涵盖整个数学领域，比如集合论或范畴论。
And the initial assumptions can be made so general, in fact so general that they encompass all of mathematics as in set theory, or categories theory for that matter.
但我们在这一切中处于一种相当尴尬的境地。
But we are in all this in a rather an awkward position.
我此刻正望着一片美丽的阿尔卑斯湖，想象一下，我们看到对岸有个人开始赤脚走在水面上，没有任何辅助。
I happen to be looking out at a beautiful alpine lake now, and just imagine we see somebody on the other shore who begins walking across the water without any assistance whatsoever.
他只是一步接一步地行走，正朝我们这边走过湖面。
He's just crossing, walking one step in front of the other, and he's crossing the lake toward us.
而他完全保持干燥。
And he comes completely dry.
他出现在电视摄像机前，我们问：你是怎么做到的？
He appears in front of the television camera, and we say, How did you do that?
他说：嗯，我迈了很小的步子。
And he says, Well, I took very small steps.
我迈了很小的步子。
I took very small steps.
我们自然的反应会是，这很了不起，你确实过来了。
Now, our natural reaction would be that's commendable, and you got across.
但不知怎的，这并没有回答问题。
But somehow or other, it's not the answer to question.
我们所有人都处在这样一种境地：看着某人走过一大瓶水，并用‘看我的脚，小步子’来解释他的成功。
And we are all in the position of watching someone cross a large bottle of water, and explaining his success by saying, look at my feet, small steps.
我们就是处在这样的位置。
That's where we are.
好的。
Okay.
所以塞尔吉奥，等等。
So Sergio well, wait.
让我回到你的论文。
Let me go back to your essay.
我再次引用你的话。
I'm quoting you once again.
维格纳指出的谜题部分源于一个永恒的问题：数学是像主要的物理理论一样，通过探索和发现而发展的科学，还是一种发明，是人类思维的创造？
The mystery Wigner points out arises in part from the perennial question of whether mathematics is a science advanced by exploration and discovery, like the main physical theories, or whether it is an an invention, a creation of the human mind.
我认为数学是通过探索和发现发展起来的。
I argue that mathematics developed through exploration and discovery.
也就是说，它就像地下的矿藏。
That is to say, it is like a nugget in the ground.
你发现了它。
You find it.
对。
Correct.
它有自己的道理。
It has its own okay.
是的。
Yeah.
不。
No.
当然。
Absolutely.
我的意思是，你可以想象一个登山者正努力攀登山顶。
I mean, one image that you could make is that of an alpinist that is is going trying to go on the top of the mountain.
他有一个目标。
He has an idea.
今天在用登山作比喻，不过是的。
Doing alpine metaphors today, but Yeah.
他知道自己想去哪里。
He has an idea where he wants to go.
对吧？
Right?
所以这非常重要。
So that's very important.
这是做数学的一部分。
That's part of doing mathematics.
不仅仅是演绎。
It's not just deduction.
你需要有一种对目标的洞察，这非常关键。
There's sort of a vision of where you want to go, which is very important.
灵感。
Inspiration.
是的。
Yeah.
没错。
Right.
这与灵感有关。
Has something to do with inspiration.
但接着，你知道，你面前有一件非常客观的东西，对吧？
But then, you know, you have something very objective in front of you, right?
对于登山者来说，那块石头就是你需要考虑的东西，因为你不能掉下去。
The stone, in the case of the alpinist, that you have to take into account that you are not going to fall.
你必须触摸那块石头，才能确切知道你要去哪里。
Have to touch the stone to know exactly where you are going.
如果你不这么做，就会陷入麻烦。
And if you don't, you get into trouble.
所以数学也非常相似。
So mathematics is very similar.
人们总觉得一切都是演绎的。
People have the feeling that everything is deductive.
其实不是。
It's not.
我的意思是，在这方面它与自然科学非常相似。
I mean, it is very similar in that respect to physical sciences.
自然科学也是如此，你有一些想法，比如说，有一种预期。
Physical sciences, also, you have some idea of you have, let's say, an expectation.
你提出一个假设。
You make a hypothesis.
对。
Right.
对吧？
Right?
假设的选择并不是一个演绎的过程。
The choice of a hypothesis is not a deductive thing.
这是一种洞察。
It's an It's insight.
然后你试图证明它能与所有其他经验相吻合。
And then you try to show that it fits everything else, all the other experiences.
所以，这就是证明的过程：你有发现的过程，然后是证明的过程。
So And that's the process of proving So something, you have the process of discovery, but then the process of justification.
证明的过程。
The process of justification.
因此，从本质上讲，数学看起来像一连串逻辑推导，计算机也可以做到——我的意思是，一旦你有了这个链条，计算机就能非常快地完成，甚至可以验证，尽管目前还没有计算机能验证大型证明。
So mathematics, at the end of the day, it looks like a chain of logical sequences that the computer can also, I mean once you have the chain, the computer can go very fast to it and maybe even check as though there are yet no computers that can check large.
我的意思是，它们可以验证小型证明，但无法验证大型证明。
I mean they can check small proofs, but not large.
无论如何，这是一个非常好的例子，很好地说明了数学与物理学之间的关系。
In any case, it's sort of a very good example, think that illustrates very well relations between mass and physics.
以几何学为例。
Take geometry.
几何学确实是第一个真正意义上的物理世界理论。
So the geometry was the first, really, theory of the physical world.
对吧？
Right?
我的意思是，它描述的就是这个。
I mean, it's what it describes.
它一直可以追溯到
It's going all the back
欧几里得。
to Euclid.
你是这个意思吗
Is that what you
它确实可以追溯到。
It goes back.
好吧。
All right.
所以欧几里得显然试图从中提炼出一些数学陈述。
So Euclid tried, obviously, tried to make sort of mathematical statements out of.
但这毫无疑问是一种物理理论。
But it's a physical theory, without doubt.
几何学是一种物理理论。
Including geometry is a physical theory.
然后它发展了。
Then it develops.
于是，这里出现了一些数学特有的东西，与物理学不同。
So here comes up something specific to mathematics, different from physics.
数学家可以接受一个理论，然后基于与物理学家完全不同的标准进行发展。
Mathematicians can take a theory and then develop based on very different criteria than a physicist would do.
所以，你知道，他们对某些问题感兴趣，是因为这些问题很优美，或者因为他们觉得这些问题会带来对其他事物的某种理解。
So, you know, they are interested in in problems because they are beautiful or because because they feel that it it will lead to a certain understanding of something else.
这种自由在几何学中确实延续了大约两千年，几乎与物理世界没有任何联系。
And this freedom really, in the case of geometry, went for, I don't know, two thousand years without essentially no connection back to the physical world.
我的意思是，欧几里得几何一开始就在那里。
I mean, Euclidean geometry was there to start with.
但到了十九世纪，出现了高斯、罗巴切夫斯基、黎曼，二十世纪初又出现了闵可夫斯基。
But then by the nineteenth century, have Gauss, you have Robachevsky, you have Gauss, you have Riemann, you have Minkowski at the beginning of the twentieth century.
然后，突然间，所有这些都成了不可或缺的组成部分。
And then all of a sudden, all that stuff becomes an essential ingredient.
我的意思是，这不仅仅是技术性的，它是一种具有特殊生成力的核心要素。
I mean it's not just, you know, it's not just the technical, it's an essential ingredient special generativity.
直接应用于基础物理理论。
Comes directly applicable to fundamental physical theory.
我认为这种情况在历史上发生过很多次，但可能并没有得到充分的认可。
And I think this has happened many, many times in and maybe it's not very well it's not very well acknowledged.
所以，我可以问一下吗，塞鲁特？不用说，我无法独立评估你的工作，因为你完成了两千页的证明，两千页严密的数学推理——我即使活到两千岁，每天读一页，或者每年读一页，也肯定还是无法理解，对吧？
So can I ask, when you, Serjut, needless to say, I cannot evaluate your work on my own because you have done a 2,000 page proof, 2,000 pages of close mathematical reasoning, I could have I could live to 2,000 and not I could read a page a day, or a page a year, and it still would escape me, I'm sure?
这是关于爱因斯坦方程中某个方面的稳定性。
And this was on the stability of some aspect of that Einstein.
对。
So Right.
我可以问一下吗，你当时觉得自己是在创作一件艺术品吗？
May I ask, did you think you were doing a work of art?
你是想创造一些美丽的东西，还是觉得自己在探究现实？
Creating something beautiful, or did you think you were interrogating reality?
我想让你明白，我正试图理解你作为一名数学家，在面对一个深刻而极其困难的问题时，内心的感受和想法。
Do you see that I'm trying to understand what you thought what it felt like to you as a working mathematician engaged on a deep, very difficult problem.
答案是两者都有。
The answer is both.
我的意思是，我之所以接受这个问题，是因为它本身吸引我。
I mean, I would take the problem in the first place.
我是一名数学家，而不是物理学家。
I'm a mathematician and not a physicist.
我对数学感兴趣，我相信最好的数学总以某种复杂的方式与物理学相连。
I'm interested in I believe that the best mathematics is connected somehow with physics in complicated ways.
但物理学中有许多问题，我会选择那些能让我作为一名数学家产生审美满足感的问题。
But there are many problems in physics, and I will pick the one that satisfies my aesthetical feeling as a mathematician.
所以这就是
So that's
你的审美感受。
Your aesthetical feeling.
那么解释一下。
So explain that.
你想要一些……一些那种……
You want something Something that's
我觉得非常优美。
that I feel is very beautiful.
它非常深刻。
It's it it it's very profound.
它带来了许多有趣的问题。
It gives lots of very interesting questions.
好的。
Okay.
所以这是其中一个方面。
So that's that's one aspect.
但对我来说，它还必须有第二个方面，那就是它应该对物理世界有所阐述。
But then then it has to be the second, for me at least, there has to be a second aspect, which is that it should say something about the physical world.
在这种情况下，它确实做到了，对吧？
And in this case, it does, right?
我的意思是，这个解法是这样的，对吧？
I mean, the care solution so here is how it goes, right?
你有广义的活动，这在1915年底由爱因斯坦很好地提出了。
You have general activity, which was well formulated by Einstein at the end of 2015.
抱歉，是1915年。
1915, excuse me.
当时这是一种全新的引力理论。
And this was at the time a new theory of gravity.
大质量物体会弯曲所谓的时空。
Massive bodies curve what's called space time.
然后，人们找到了某些解。
And then then certain solutions were found.
施瓦西是第一个在2016年发现所谓施瓦西解的人，那是在爱因斯坦理论提出后紧接着的一年。这个解是一个具有大量对称性的稳态解，你可以从相对论、爱因斯坦场方程中推导出一个精确公式。
Schwarzschild was the first to found in 2016, a year immediately a year after, found the so called Schwarzschild solution, which is a stationary solution with a lot of symmetries that you can actually extract from the theory of relativity, from the Anschelen equation, as exact formula.
而这可能会引发许多问题，因为它存在奇点。
And that can lead to lots of issues, because it has a singularity.
这后来与彭罗斯奇点定理相关联，他因此获得了诺贝尔奖。
This is connected later on with the Penrose Singularity theorem for which he actually got a Nobel Prize.
他是唯一一位获得诺贝尔物理学奖的数学家，尽管并没有其他人获得过。
He's the only mathematician to have gotten a Nobel Prize in physics, despite there's nobody else.
部分原因是，数学如此完美地应用于了广义相对论。
Part, guess the math applied so beautifully to the Right.
或者说是应用于一个曾经被提出的问题。
Or to a question that was was
只是一个问题。
just one question.
这非常重要。
This was very important.
然后，克尔提出了第二个重大进展。
And then there was a second major development by Kerr.
这是在1963年，克尔解被发现。
This was 1963, where the Kerr solutions was.
好的。
Okay.
现在你有了一个包含施瓦西解的克尔家族。
So now you have a Kerr family which includes Schwarzschild.
这是一个依赖于两个参数的大家族。
It's a large family depending on two parameters.
这些是爱因斯坦方程的精确解，对吧？
So these are exact solutions of the Einstein equation, right?
我的意思是，从数学角度来看，它们是真实的，因为对我而言，数学上的真实意味着这些是你可以明确写出的方程的解，这是一种客观事实。
I mean, you know, from a mathematical from a mathematical point of view, they are real because for me, reality, mathematical reality has to do with objective fact that these are solutions of an equation which you can write down.
这在概念上
It's it's conceptually
我可以打断一下吗？
And may I interrupt for just a moment?
所以，如果我理解得没错的话，爱因斯坦的一个了不起之处在于——当然，请纠正我。
So if if I understand one of the remarkable things about Einstein by the way, of course, correct me.
请说。
Jump in.
我在这里说得太浅显了。
I'm doing baby talk here.
因为说到这个话题，这已经是我能讲得最深入的水平了。
Because that is the top of my form when it comes to this material.
爱因斯坦在1915年提出了广义相对论，而我们现在是2025年，这一个多世纪以来，随着新卫星和新技术的出现，不断有新的实验进行。
Einstein comes up with General Relativity in 1915, and here we are in 2025, and there's still experiments there have been experiments that have been done over the course of the succeeding century as new satellite new technology makes new experiments.
而且每一次都——
And every single time,
是广义相对论
it's the General theory
广义相对论得到了验证。
of is general is proven proven out.
是的。
Yeah.
也就是说，爱因斯坦当年在黑板上提出的这一理论，经过一个世纪的实验，结果证明它能够准确描述并预测现实。
That is to say, this theory that Einstein came up with on a chalkboard for a century of experiments now, it turns out to correspond with and predict reality.
如果你的工作能够通过某种方式设计出针对黑洞的实验，那么你的理论也会得到验证。
And your work, if we could somehow devise experiments on black holes, your work would prove out.
到这个程度，它是真实的吗？
It's real to that extent?
所以我喜欢称之为
So I like to call it
对现实的检验。
a test of reality.
因此，克尔解的稳定性。
So the fact that the Kerr solution is stable.
对吧？
Right?
这是一个数学表述，但蕴含着丰富的物理内涵。
It's a mathematical statement, but with a lot of physical content.
因为假设它不稳定，它虽然是爱因斯坦方程的一个正确解，且从特定初始条件开始。
Because let's say if it was not stable, so it's a solution, a correct solution of the Einstein equation, which starts with specific initial conditions.
对吧？
Right?
所以稳定性的问题在于，你对初始条件做微小扰动。
So the issue of stability is now you make small perturbations of the initial conditions.
然后突然间，我们得到了完全不同的结果，与克尔解毫无关系。
And all of a sudden we get something entirely different which has nothing to do with the solution, the care solution.
这种情形被称为不稳定。
That type would be called instability.
对吧？
Right?
所以，如果这个解是不稳定的，那就意味着它没有任何物理意义。
So if the care solution would be unstable, it means it doesn't have any physical meaning.
对吧？
Right?
因为你知道，它并不对应于自然界中任何你能识别为与之相符的事物。
Because you know, it doesn't correspond to anything that you can recognize in nature as corresponding to that.
对吧？
Right?
所以，稳定性问题是一个根本性的问题，在
So the stability, the issue of stability is a fundamental issue in
这是一个测试。
It's a test.
这是一个标志，你用于
It's a marker You for
可以说。
can say it.
所以我称之为现实的数学检验。
So I call it the mathematical test of reality.
让我们来看看这里发生了什么，因为是的，请说。
Let us what's going on here because Yes, please.
这太
It's so
为我解释清楚。
Explicate for me.
从哲学的角度来看，有一个深刻的假设：那些在数学上一致、连贯、稳定的体系，能够为我们提供通向物理现实的指引，仿佛物理世界中内置了一种理性，这种理性与我们进行这种高级数学研究时所运用的理性相契合。
Well, a philosophical point of view here is that there's a deep assumption that that which is mathematically consistent, coherent, stable is going to give us a guide to physical reality as if there's a rationality built into the physics that somehow matches the rationality that's at work when we're doing this type of advanced mathematics.
这就是维格纳之谜。
And so that's the Wigner mystery.
为什么我们内心的理性与外部自然的理性能够相互匹配？
Why does the reason within match the rationality of nature external to us, the reason without?
好的。
Alright.
展开剩余字幕（还有 480 条）
现在我们进入了一个新的领域。
So now we move into territory.
我已经深陷其中，但我还是继续游下去。
I'm already in over my head but I continue swimming.
水越来越深了。
It gets deeper.
我可以提出一个简单的观点来帮助理解吗？因为我们已经谈到了场方程、广义相对论及其解，但塞尔吉奥最初是关于几何学展开讨论的。
May I offer a simple thing that might help just with because we got into the field equations and general relativity and the solutions and but Sergio started initially talking about geometry.
是的。
Yes.
数学对象具有稳定属性这一理念。
And just the idea that mathematical objects have stable properties.
这就是数学家们认为它们是真实的原因。
This is why mathematicians regard them as real.
你知道，圆具有某些基本属性。
You know, a circle has certain basic properties.
它有周长和面积，我们可以计算这些量。
It's got a circumference, an area, and we can calculate these things.
这些属性对所有思考圆的人而言都是成立的。
And those properties are true for all people who think about circles.
几何对象具有某些稳定的性质，我们可以用数学方式描述它们，而这些性质独立于我们的思维。
It's there are stable properties that that geometric object has that we can describe mathematically that's independent of our minds.
对。
Right.
然而，正如萨罗什在我们最近的会议上所解释的，这些性质的稳定性是现实的一种体现，是独立于心智的客观现实的体现。
And yet it and yet the stability of those properties is a token as as Saroshu has explained in our recent conference, it's a token of reality, of a mind objective reality or a mind independent objective reality.
因此，数学家们并不认为自己是在创造新的数学公式。
And that's why mathematicians don't think that they're inventing new mathematical formulas.
他们几乎普遍认为自己是在发现某种东西，而不是发明。
They think almost universally feel that they're discovering something, not inventing.
不过，也有些人不这么认为。
Well, there are some who don't.
但有趣的是，物理学家总是把数学称为人类思维的发明。
But what's interesting is that physicists always refer to mathematics as being an invention of the human mind.
这就是维格纳所说的，抱歉，爱因斯坦说的，是人类思维的发明，而维格纳也表达了非常相似的观点。
That's what Wigner Sorry, Einstein says an invention of the human mind, but Wigner says something very similar.
所以我总是对这一点感到惊讶。
So I'm always surprised to see that
爱因斯坦认为他发明了广义相对论。
Einstein felt he invented the general
不是的。
No.
不是的。
No.
爱因斯坦认为数学家是在发明东西。
Einstein feels that mathematicians invent things.
对吧？
Right?
所以他实际上称之为人类思维的自由创造。
So He actually called it a free creation of the human mind.
自由？他指的是什么？
Free By which he meant what?
这并不太清楚，因为如果数学是人类思维的自由创造，为什么数学命题却如此必然呢？
It's not really clear because if it's a free creation of the human mind, why are mathematical propositions so dreadfully necessary?
嗯。
Mhmm.
我对二加二等于四没什么选择余地，我猜你也没有。
I didn't have much choice about two plus two equals four, and I presume you didn't either.
但这里有一个
But there's a
这与自发地创造出加法这样的概念有些矛盾。
I It's kind of at odds with the notion of spontaneous spontaneously reaching an invention like addition.
这根本不像是一种发明。
Doesn't seem to be an invention at all.
但我有理由解释为什么物理学家会这么做。
But I have a reason why physicists do that.
因为，我怀疑牛顿不会这么说。
Because, and I doubt that Newton would have said that.
不会。
No.
这是现代物理学家的观念，他们是唯物主义者。
It's a modern it's a modern physicist who are materialists.
他们相信世界上只有物质，包括心灵在内的一切都必须以某种方式被决定。
Do believe that there is just matter and everything, the mind including, has to be determined somehow.
但他们不可能是对的。
But they can't be right about it.
好，注意这里辩证法的要点。
Okay, here notice what's in dialectic here.
做数学的数学家们在发展数学时，通常相信他们是在发现某种真实且独立于他们心灵的东西，而不是像发明内燃机那样进行创造。
The mathematicians who are doing the math, are developing the math, typically believe that they are discovering something that is real and independent of their minds, not inventing something like an internal combustion engine or Exactly.
我的意思是，任何发明都必须有一个起点。
I mean an invention has to have a starting point.
对吧？
Right?
这意味着在那个起点之前，那个数学事实并不存在。
So it means that before that starting point, that mathematical fact did not exist.
对吧？
Right?
在毕达哥拉斯发现它之前，毕达哥拉斯定理并不成立。
Pythagoras theorem was not true before Pythagoras discovered it.
这有点荒谬。
It's kind of ridiculous.
但一个怎样的人怎么可能……这里一定有复杂之处。
But how can anybody who who how can there must be complexities here.
我确信其中存在复杂性。
I'm sure there are complexities.
我不太理解。
I'm not grasping.
但二加二等于四，对我来说是成立的。
But two plus two equals four is true for me.
对你来说也是成立的。
It's true for you.
对大卫来说也是如此。
It's true for David.
无论大卫在任何时刻感觉多么反常，这仍然是真的。
No matter how perverse David may be feeling at any given moment, it's still true.
这是真的，并且一直以来都是真的。
It's true, and it has been true for all time.
因此，是不是有什么东西……不，这直接给了唯物主义一记重击？
Therefore, there is something isn't that doesn't that just put us a stake in the heart of materialism right there?
有些东西是存在于我们之外的。
Something exists outside us.
我们可以称之为理性，也可以称之为理念世界，好吧。
Something we can call it reason, could call it platea Okay.
那么我们来谈谈柏拉图。
So let's go to Plato.
如果我理解得没错的话，在《理想国》中，柏拉图区分了可知世界和可感世界；可感世界是我们能看到和触摸到的。
If I understand this much, in the Republic Plato draws a distinction between the intelligible world, The sensible world is what we can see and touch.
而可知世界是我们能够通过理性理解的事物，但
And the intelligible world is that which we can is intelligible to us, but
对理智的把握。
Access to the intellect.
对理智的把握。
Access to the intellect.
好的。
Okay.
因此，他把他的理念形式放在了那里。
And so that's where he places his ideal forms.
那里有一个圆。
There is a circle out there.
那里有一个三角形。
There is a triangle out there.
柏拉图说它们具有独立的存在，但他似乎暗示存在着一个理想形式的领域， somewhere out there。
Plato says it does have an independent existence but he seems to suggest that there's a there's a realm of ideal forms someplace out there.
阿奎那在一千五百年后出现，认为理念存在于心灵之中。
Aquinas comes along fifteen hundred years later and says, ideas exist in minds.
对我们所有人和所有时间都成立的、可理解但非物质的东西，存在于上帝的心灵中。
And something that is true for all of us and for all time that is intelligible but immaterial exists in the mind of God.
大卫？
David?
是的。
Yeah.
也许吧。
Maybe.
大卫在那一刻比任何时候都更像他自己。
David was never more David than in that very moment.
到目前为止，我完全听不懂这个讨论。
I can't make any sense of the discussion so far.
这是我的问题。
That's my problem.
魏格纳所关注的是一个真正的问题：如果数学对每一个物理理论都是必不可少的，那么除非这是一种 trivial 的解释，否则不可能存在一个物理理论能够从物理上解释数学。
There is a problem to which Wigner was calling attention, which is a real philosophical that is if mathematics is essential for every physical theory, it cannot be the case, unless it's a trivial explanation, that there is a physical theory that physically explains mathematics.
我
I
意思是，如果一个人用鲤鱼当饵去钓鲤鱼，那他就是在进行一种无意义的追求。
mean if a man proposes to catch a carp by baiting his hook with a carp, he's engaged in a trivial pursuit.
他已经有鲤鱼了。
He has the carp.
如果我们需要一个包含数学的物理理论来解释数学，那我们就不再拥有一个真正的物理理论了。
If we need a physical theory that includes mathematics to explain mathematics, we no longer have a physical theory.
我们有一个物理理论和一个数学理论。
We have a physical and a mathematical theory.
这就是困境，我认为这是惠格纳所关注的深层困境。
And that's the dilemma, I think the deep dilemma to which Wigner is calling attention.
我们希望坚持某些原则，作为文化使命的一部分。
There are certain principles we'd like to hold on to as part of the cultural imperative.
我们希望坚持这一基本观念：世界是物理的。
We'd like to hold on to this fundamental idea the world is physical.
我们生活在一个物理世界中。
We live in a physical world.
我不是说这个观念值得称赞，我说它是一种文化使命，因为它看起来令人安心。
I'm not saying this is a commendable idea, I'm saying it's a cultural imperative because it seems so reassuring.
听好了，我们面对的是无法衡量的问题。
Look, we're faced with imponderables.
基本事实是，事物是物质的或物理的物体。
The basic fact, the thing is a material or a physical object.
从文化和智力上来说，得出这样的结论非常不便：要理解这个物理对象，我们需要大量非物理的事实；而要解释关于数学的大量非物理事实，没有数学的物理理论根本无法做到解释。
It is very inconvenient culturally and intellectually to come to the conclusion that in order to understand that physical object, we need a whole lot of non physical facts about And in order to explain a whole lot of non physical facts about mathematics, there is no conception of a physical theory without mathematics that can do the explaining.
因此，我们陷入了一个困境：如果数学真的像谢尔盖和史蒂夫所说的那样有用——他们在这方面完全正确——数学在日常生活中如此有用，那么我们关于世界是一个物理系统的观念就一定存在根本性错误。
So we're left in the position that if mathematics is as useful as Sergei and Steve says, and they're absolutely right about that, it's useful in daily life, there is something fundamentally wrong with our idea of the world as a physical system.
正是如此。
Exactly.
这不可能是对的。
It's something cannot be right.
必须有所取舍。
Something has to give.
要么我们发展出不依赖数学的物理理论。
Either we develop physics physical theories with no mathematics.
费曼曾推测过类似的观点。
Feynman conjectures something of this sort.
要么我们就承认唯物主义根本不可能正确。
Or we agree that materialism simply cannot be right.
二者选其一，但总得有一个让步。
One of the two, but something has to go.
但物理学家们至今至少还没有放弃物理性的理念，这就是问题所在。
But physicists who, until now at least, have not given up on the idea of physicality or So that's that's that's the issue.
但他们没有正视大卫刚刚描述的困境，
But they're not reckoning with the dilemma that David just described,
我觉得。
I think.
不是我。
Not me.
我的意思是，这个困境在文献中至少已经存在五十年、六十年，甚至一百年了。
I mean, that dilemma has been in the literature for at least fifty years or sixty a hundred
我不明白。
Don't understand.
你们这些聪明人，如果某件事在文献中已经存在了半个世纪，这本质上就是一个停止标志，意味着：等等，等等，等等。
All of you bright people, if something's been in the literature for half a century, which essentially is a stop sign, saying wait a moment, wait a moment, wait a moment.
你在这里必须对现实的本质做出一个基本决定。
You have a basic decision to make here about the nature of reality.
如果真是这样，那么学术界、你们这些学者怎么能忽视它呢？
Either it it Well, then how can the academy, how can you academics just ignore it?
好吧，嗯。
Well, so, okay.
所以我不明白为什么。
So I I don't know why.
而且就像
And and like
大卫，我把责任归于你，施特格纳。
David I hold you responsible, Sturgeon.
我不想讨论存在、理念、柏拉图式理念的问题。
I don't want to talk about issues of existence, ideas, platonic ideas.
也许这一步走得太远了，这可能就是我们的共识。
Maybe it's a step too far, and that may be what we agree.
但我对现实有一个操作性的定义。
But I have an operational definition of reality.
因此，现实就是对某个特定对象的一致性表征。
So reality is consistency of representations of a particular object.
这在物理学中成立，在数学中也成立。
That's true in physics and that's true in mathematics.
数学对象是真实的。
Mathematical objects are real.
一位数学家在解决数学问题时，用这种方式计算，用那种方式计算，总是得到相同的结果。
A mathematician who works on a mathematical problem calculates in this way, calculates in that other way, always gets the same result.
我的意思是，我们所做的事情显然具有某种客观性，对吧？
I mean, there's something so obviously objective about what we do, right?
因此，说这些只是人类思维的发明，在我看来是荒谬的。
That somehow to claim that these are just inventions of the human mind is ridiculous to me.
我的意思是，
I mean,
在哲学中，总有一种方式可以为任何立场辩护。
there is a way to defend any position in philosophy.
我的意思是，我们完全可以争论说，一切都是人类思维的发明，因为人类思维才是唯一真实存在的东西。
I mean, we could argue that there are inventions of the human mind because the human mind is the only thing that really exists.
毕竟，从贝克莱开始就有一条非常崇高的传统，正是持这种观点。
After all, there's a very noble tradition going right back to Berkeley, which makes exactly that case.
存在就是被感知。
To be is to be perceived.
但有一些因素阻止我们回归贝克莱和唯心主义。
But there is something that inhibits a return to Berkeley and idealism.
因为说唯一真实存在的就是心灵，听起来似乎荒谬可笑。
In that, it sounds vaguely preposterous to say the only thing that exists is a mind.
这与我们所发展的物理理论的宏伟性不符。
It doesn't comport with the magnificence of the physical theories we've developed.
确实不符。
It just doesn't.
这不是一个论点，而是一个观察。
That's not an argument, it's an observation.
但话虽如此，我们确实正面临被压缩到越来越狭窄的冰流中的危险。
But having said that, we are really in danger of being reduced to an ever narrowing ice flow.
正如谢尔盖刚刚提到的，冰流指的是表征的一致性。
The ice flow, as Sergey just mentioned, is the consistency of representations.
嗯，'表征'这个术语有点晦涩。
Well, representations is kind of an obscure term.
我们不如回到最基本的问题上？
Why don't we get down to basics?
它指的是我们理论的一致性。
It's the consistency of our theories.
那么，什么是理论呢？
Well, what is a theory?
我们是可以对此给出答案的。
Well, we can provide an answer to that.
理论是一大组句子的集合。
A theory is kind of a large group of sentences.
那么句子是什么呢？
And what are sentences?
它们是能够被判断为真或假的某种断言，如果这些断言具有一致性，那在理论的可信度和可取性方面，我们就已经做得很好了。
They're things that make certain kinds of assertions that can be true or false, and if they're consistent, that is about as good as we can get in terms of the credibility and commendability of a theory.
因此，我们现在站在冰流上说：我们对物理世界有一个表征或理论，并且它是自洽的。
So we're reduced now on our ice floater saying, well, we have a representation or a theory about the physical world and it's consistent.
但当我们加以审视时，却发现它并不是一个物理对象。
But when we examine it, we find out it is not a physical object.
它涉及了非物理的实体，比如数学对象。
It invokes non physical substances like mathematical objects.
我们不必非得做出决定。
We don't have to say, we need not make a decision.
它们存在于思维中，还是存在于外部世界？
Do they exist in the mind or they exist in the external world?
它们是存在的。
They exist.
这我们就足够了。
That's all we need.
数字二它是存在的。
The number two exists.
我不必告诉你它存在于你的思维中，还是存在于我的思维中。
I don't have to tell you whether it exists in your mind, exists in my mind.
这无关紧要。
That's irrelevant.
它是存在的。
It exists.
这才是决定性的陈述。
That's the determinative statement.
只要它存在，而且我们承认它不是物理的，那么我们就不得不面对一个问题：为什么我们对物理世界最深刻的理解却包含了非物理的东西？
And as long as it exists, and we acknowledge it's not physical, then we're left with a position of saying, how come our best view of the physical world incorporates things that are not physical?
为什么呢？
Why is that?
存在是一个本体论问题。
Existence is an ontological question.
我更倾向于把现实当作我可以处理的东西。
I prefer reality as something that I can work with.
至于存在，我不确定。
While existence, I don't know.
谈到存在这个问题时，我感到迷茫，对吧？
It comes to the issue of existence, I feel lost, right?
不知道。
Don't know
像你一样
like You
感到迷茫？
feel lost?
我的意思是，我不知道什么存在，什么不存在
I mean, I don't know what exists and what does not
在这一切中让我感到着迷的是，这些数学对象——无论是二次方程、圆还是更高级的数学形式——都具有稳定的特性。
What intrigues me in all this is the idea that those mathematical objects, whether it's the quadratic equation or a circle or more advanced forms of mathematics, have stable properties.
你所说的稳定性是指这个吗？
Is the consistency you're talking about?
我在谈论现实。
I talk about reality.
是的。
Yeah.
我更愿意把现实称为客观性。
I prefer to call reality as being objectivity.
一致性。
Consistency.
是的。
Yeah.
塞尔吉奥的意思是，这些数学结构或对象具有一种独立于我是否认可这些属性的现实性。
What Sergio was saying is that these mathematical structures or objects have a reality that's independent of whether or not I affirm those properties myself.
它们是独立于心智的，但本质上又是概念性的。
They're mind independent, and yet they're conceptual, essentially.
它们不是物质的。
They're not material.
它们是概念性的。
They're conceptual.
这就引出了一个问题：它们存在于哪里？
So it does raise the question, where do they reside?
如果有概念存在，而维基，你正是通过引入柏拉图在朝这个方向推进。
If there are concepts, and this is where, Vicki, you were moving in this direction by bringing Plato in.
柏拉图认为，存在某种概念性的现实，它们存在于某种天界之中，但阿奎那对这一观点的批评是，这与我们的经验不一致。
Plato had the idea that there are conceptual realities that exist in some sort of heavenly realm, but Aquinas's critique of that was that doesn't make any that's not consistent with our experience.
思想存在于心智中，因此，如果这些物质和数学结构的客观属性独立于我们的心智，而这些数学结构本身又是概念性的，那就意味着它们并非漂浮在某处，而是源于或根本存在于某种超越性的心智之中。
Ideas exist in minds, and so if there are these objective properties of material, of mathematical structures that are independent of our minds, and if these mathematical structures are themselves conceptual, it implies not that they're floating around someplace, but they originate or reside fundamentally in some transcendent mind.
这就是对数学实在论的有神论观点。
That's that's the theistic take on the mathematical realism.
史蒂夫说，数学存在于上帝的心中。
Steve says, math exists in the mind of God.
塞尔吉奥说，别跟我提这个。
And Sergio says, no, don't bother me with that.
我是一名执业数学家。
I'm a working mathematician.
我只需要一支粉笔和一块黑板。
All I need is a piece of chalk and a blackboard.
这是一种对现实的操作性定义。
Well, an operational definition of reality.
对吧？
Right?
我认为，那种认为所有真实的东西都是物理的观念是说不通的，因为数学对象也是真实的。
And the notion that somehow everything that's real, it's physical, doesn't make sense to me because mathematical objects are also real.
我认为，无论我们如何切割这个问题，无论你持哪种观点，
And I think we're all saying whichever, you know, where you slice this whether you're a pointless
是的，我认为这完全正确。
Yeah, I think that's absolutely right.
我的意思是，即使我们采纳贝克莱的观点——存在即被感知，我们最终还是会得出同样的结论：一个彻底一致且连贯的宇宙观不可能是纯粹物理的。
I mean, even if we adopt Berkeley's position that to be is to be perceived, we wind up at the same position that a thoroughly consistent and coherent coherent view of the universe simply can't be physical.
对。
Yeah.
它根本不可能是。
It simply can't
好吧。
be Okay.
实际上，这甚至让我这个平凡的大脑也意识到，这真是一个极其深刻的发现，我不会称之为洞见，而是一种发现。
Actually actually strikes even my little mind as a really quite profound, I'm not going to call it an insight, it's a discovery.
它是一种真实存在的、现实本身的真正层面。
It's something, it's a real aspect of reality itself.
我可以回到之前对话中的某个话题吗？
Can I revert back to something earlier in the conversation?
当然可以。
Sure.
这只是来自大卫的书里的一件事，他曾经在美国就这个话题做过一次演讲，当时让我觉得特别有趣。
It's just something that from David's book, he gave it to, he gave a talk in The US on this one time, and it just was so intriguing to me.
我想应该是你书里的第一、二、三章，你在路上谈到这个的时候，他展示了复变量、复数的概念。
It was I think it was from your book one, two, three when you were talking about it on the road, and it was he showed how I think it was the complex variables, the complex numbers.
还记得数学里的负一的平方根吗？
Remember the square root of negative one from math?
围绕复数发展出了一整套数学体系，看起来它似乎与现实毫无关联，但我读研时上过一门复变函数的课，大卫指出，这套体系是大约在什么时候被发明出来的？
There's this whole mathematical apparatus that's been developed around complex numbers, and it seems like it has absolutely nothing to do with anything, but there's a whole body of mathematics I on took a course in grad school on complex variables, and David showed that this was invented something like what was it?
是一百四十年左右吗？
Hundred and forty or developed?
一百四十年。
Hundred forty.
早在一百四十年前。
A hundred and forty years before.
也许是两百年。
Maybe two hundred.
也许是两百年。
Maybe two hundred.
结果发现，它对于量子力学至关重要，而量子力学是我们最基础的物理理论之一。
And then lo and behold it's absolutely crucial for doing quantum mechanics, which is our most fundamental physical theory, or one of them.
是的，如果我可以的话
Yeah, if I can
对，请纠正
add Yeah, please correct
关于虚数 i 的发明历史，令人着迷的是，它最早出现在14到15世纪的意大利。
What is fascinating about the history of the invention of the number I is that, well, it came up in Italy in the fourteenth century, fourteen and fifteen.
是十六世纪。
Sixteenth century.
不，再早一点。
No, little earlier.
1450年，对，好吧。
50, yeah, Okay.
1450年，可能再早一点。
Fourteen fifty, a bit earlier maybe.
但无论如何，这源于他们求解方程的愿望。
But in any case, it was based on their desire to solve equations.
所以他们想先求解二次方程。
So they wanted to solve first the quadratic equation.
当时已经有一个公式了。
There was a formula already.
阿拉伯人似乎也早就知道了。
The Arabs apparently also knew it already.
无论如何，他们进一步研究了三次方程。
In any case, they went to the third order equation.
他们发现，引入这个符号——负一的平方根——虽然毫无意义，但却很有用。
And they found that it pays to introduce this symbol, square root of minus one, which makes no sense.
你不能取负一的平方根，但他们还是把它放了进去。
You cannot take the square But root of minus one, they just put it there.
他们做出了一个惊人的发现：你可以使用这个数，最终却能得到一个三次方程的所有实数解。
And they made this incredible observation that you can use this number, and at the end you are getting a solution of a cubic equation which are all reals.
这些复数与他们原本要解决的问题毫无关系。
They had nothing to do with those complex numbers.
但它们却出现在了公式中，这真是非常不可思议，对吧？
But they enter into the formula for so this was quite an amazing thing, right?
于是，他们逐渐习惯了使用负一的平方根。
So then little by little they got used to this using square root of minus one.
到了十九世纪的某个时候，这种做法甚至被更加形式化了。
And at some point in the nineteenth century, it was even made more formal.
负一的平方根被赋予了几何定义，于是产生了复数和复变函数，随之而来的是海量的应用。
Was given a geometric definition of scores of minus one, and you have complex numbers and complex functions, and then enormous amount of applications came out of it.
所以负一的平方根显然是真实的。
So scores of minus one is obviously real.
它在被发现之前就已经存在了。
It existed before it was discovered by
非常有趣。
very interesting.
我正在
I'm
利用长期
making use of long
之后。
after.
等级。
Grade.
当他们引入负数时，我就不再关注了。
When they introduced negative numbers, I stopped paying attention.
负数与数字无关。
Negative numbers have No.
无关。
No.
与现实没有任何关系。
Nothing to do with reality.
但我那时错了。
But I was wrong then.
彼得，这不是负数。
Peter, it's not negative.
这些是复数。
These are complex numbers.
负一的平方根。
The square root of minus one.
负数就是负一、负二。
Negative numbers are minus one, minus two.
想想债务。
Think of debts.
我们现在不是在谈债务。
We're not talking about debts now.
我们在谈复数。
We're talking about a complex number.
这里有一个极其有趣的观点。
Here's the extraordinarily interesting point.
十五世纪有一位古怪的意大利数学家，他发现如果我引入符号 i 等于负一的平方根，就能构建一个严密的推理链条，得出正确的答案。
Got this weirdo Italian mathematician of the fifteenth century who figures out that if I introduce the symbol I equals the square root of minus one, I can solidify a chain of inference and come out with the right answer.
简直太惊人了。
Just absolutely amazing.
然而，当数学家们开始深入思考时，完全可以用两个实数和一组操作规则来替代负一的平方根。
However, when mathematicians start to think about it, it's entirely possible to get rid of the square root of minus one in favor of two real numbers and a set of rules for manipulating them.
突然之间，负一的平方根就消失了。
All of a sudden, the square root of minus one is gone.
你最终留下的，就是最初的东西：数字三和七，按某种顺序并遵循某些规则。
You're left with what you began with, the real numbers three and seven, in a certain order and obeying certain rules.
一种乘法规则，这是至关重要的。
A multiplication rule, which is essential.
这就是复数。
That's the complex number.
复数。
The complex number.
没错。
Exactly.
这是一种新的乘法规则，它在所谓的实数领域中原本并不存在。
It's a new multiplication rule, which was not did not exist in the realm of of real, what is called real But natural
这引入了一个非常有趣的分析观点：本体论可以被简化为一套规则和规范。
that that introduces a very interesting analytical point that ontology can be reduced in favor of a system of rules and regulations.
你可以减轻数学的本体论负担。
You can reduce the ontological burden of mathematics.
你可以说，我要抛弃数字，转而使用集合。
You can say, I'm going to get rid of the numbers in favor of the sets.
我要抛弃复数，转而使用实数的有序对。
I'm going to get rid of the complex numbers in favor of ordered pairs of real numbers.
我要抛弃实数，转而使用收敛序列。
I'm going to get rid of the real numbers in favor of convergent sequences.
我要抛弃这么多东西。
I'm going to get rid of so much.
但当你减轻本体论的负担时，你却增加了规则的负担。
But as you reduce the burden of ontology, you increase the burden of your regulations.
因此，你永远无法达到数学从无中产生的那个点。
So you actually never get to the point where mathematics appears from nothing.
你永远达不到那个点。
You never get to that point.
生物学只是对真正真实事物的肯定，某种实质性的实在。
Biology just being the affirmation of what's really true, something substantially real.
我的意思是，生物学家喜欢说生命只能来自生命，这是十九世纪的真知。
I mean biologists love to say life comes only from life, nineteenth century true.
生命只能来自生命，而它如何可能不来自生命则是一个彻底的谜。
Life comes only from life and how it might not come from life is an utter mystery.
但同样真实的是，语言只能来自语言，而且数学也只能来自数学。
But it's also true that language comes only from language, and it's additionally true that mathematics only comes from mathematics.
这些似乎都是我们有丰富经验的过程，它们没有明确的起源点。
These seem to be processes with which we have a good deal of experience, which have no point of origin.
数学没有任何起源之处。
There is no place in which mathematics originates.
语言也没有任何起源之处。
There's no place in which language originates.
生命同样没有任何起源之处。
And there is no place in which life originates.
它们很可能是宇宙本身的基本特征。
They may well be fundamental features of the universe itself.
但即便如此，数学是有历史的。
But nevertheless, mathematics has a history.
在这篇文章中
In this article
但好吧。
But Alright.
它在过去是无限的。
It's infinite in the past.
对。
Right.
我的意思是，这是人类的。
I mean, it's a human.
所以，肌肉身体中有一些人类的特质。
It's a so there there is something human about muscle bodies.
我们对这些现实的发现是有历史的，但这些现实本身并没有历史。
There's a history of our discovery, but not a history to the realities.
没错。
Exactly.
发现。
The the discover.
方式。
Way
对。
Right.
好的。
Okay.
大卫，这是你最近的一次对话，我想，是在剑桥和史蒂夫的对话。
David, this is you in a recent this is a conversation you had, I think, with Steve in Cambridge.
我要引用一段话。
I'm gonna quote a passage.
稍等一下。
Hold up a finger.
这个手指能是不同的颜色吗？
Could this finger be a different color?
是的。
Yes.
它能再长一点吗？
Could it be slightly longer?
是的。
Yes.
它能是弯曲的吗？
Could it be crooked?
是的。
Yes.
但它能始终不是一根手指吗？
But could it ever be anything other than one finger?
不能。
No.
这个数字是必需的。
The number is obligatory.
数字是手指本质上所具有的东西。
The number the number is something the finger essentially has.
引号结束。
Close quote.
好的。
Alright.
所以我们现在进入了亚里士多德的领域，讨论
So now we're in the realm of Aristotle and the difference between
本质属性与非本质属性之间的区别
essence Essential properties and nonessential
和偶然属性。
And accidents.
本质属性和偶然属性。
Essential properties and accidents.
解释一下这个。
Explain this.
嗯，我觉得我像在小心翼翼地走路，不过你做得不错，罗宾逊。
Well, I feel as though I'm tiptoeing You're doing pretty well, Robinson.
好的。
Alright.
数字代表了现实的一个本质方面。
So numbers represent an essential aspect of reality.
这很重要。
That's a big deal.
这只是一个非常笼统的陈述。
Well, it's a very general statement.
我更倾向于——上帝保佑，原谅我提起海德格尔——我更喜欢他的表述方式。
I much prefer, God forbid me, forgive me for introducing Heidegger, I much prefer his formulation.
海德格尔的著作中有一些非常有趣的段落，我承认。
Heidegger, they're very interesting passages in his work, I admit it.
他说，当我们观察物体时，无法将这个玻璃杯的单一性与其本身分离。
And he says, look, when we look at objects, we cannot separate the oneness of this glass from the object itself.
但我们可以改变它的颜色，改变它的形状，它仍可能是同一个物体。
But we can change the color, we can change the shape of It the could be the same object.
我们不能说这个物体原本可以是两个。
We can't say this one object could have been two.
这根本说不通。
That just doesn't go.
因此，当我们谈论物理对象时，我们现在只讨论物理上可实现的对象。
So when we talk about physical objects, we're only talking now about physically realizable objects.
它们的数学属性是其本质所在。
Their mathematical aspects are essential to them.
适用于数字的，同样也适用于形状。
What holds for numbers also holds for shape.
我们不必像尤金·维格纳那样说：看看量子力学，以及希尔伯特空间对复数的必然要求。
We don't have to do what Eugene Wigner did and say, look at quantum mechanics and the remarkable fact that Hilbert spaces require of complex numbers.
现在，看看这个玻璃杯。
Now, just look at the glass.
这个玻璃杯需要引入一个自然数。
The glass requires the introduction of a natural number.
两个玻璃杯需要引入两个自然数。
Two glasses require the introduction of two natural numbers.
这和在量子场论中引入复数一样神秘。
That is every bit as mysterious as the invocation of complex numbers in quantum field theory.
同样神秘。
Bit as mysterious.
我们还不太清楚为什么。
And we don't know quite why.
好的。
Okay.
所以你们三位都愿意同意，你们三位都坚持认为，数学的存在，数学以一种奇特的方式与现实对应并帮助我们探究现实——我反复使用‘现实’这个词——证明了现实并非纯粹局限于我们五感所能触及的范围。
So all three of you are willing to agree, all three of you are willing to insist that mathematics, the existence of mathematics, the weird way in which mathematics seems to correspond with and help us to investigate reality in a way that is real, I'm using reality over and over again, proves that reality is not purely, not limited to what we can access by our five senses.
这是一个线索。
It's a hint.
这并不能证明。
It doesn't prove.
这是一个线索。
It's a hint.
哦，连这一点我都已经跟不上了。
Oh, now I've lost deep ground even from that.
哦，所以我们只剩下这个线索了。
Oh, so all we have is a hint.
那为什么你不觉得史蒂夫愿意接受呢？
So why aren't you I think Steve is willing to go.
现在，我把话放到了史蒂夫嘴里。
Now, here I am putting words into Steve's mouth.
史蒂夫愿意说，我们这里讨论的是心智
Steve is willing to say, we're dealing here with the mind of
我认为大卫将会
I think what David's gonna
无论我怎么逼你，大卫都不会那样做
David won't do that no matter how I could pin you down and
我不会给你一个类似这样的例子
you I'll wouldn't give you sort of an example of something like this.
所以，你知道，在牛顿之后不久，就有了牛顿力学
So, you know, immediately after Newton, there was Newtonian mechanics.
当时的想法是，牛顿力学必须解释一切
The idea was that Newtonian mechanics has to explain everything.
然后出现了麦克斯韦，也就是麦克斯韦方程组
Then came Maxwell, the Maxwell equations.
为了使麦克斯韦方程组与牛顿力学相协调，他们需要以太这个概念
And in order to adjust the Maxwell equations to Newtonian mechanics, they needed this notion of ether.
这里的以太是指，我想说的是，爱因斯坦的伟大洞见在于，我们根本不需要它
Ether here is that there, at some point I mean, Einstein's great insight was we don't need it.
对吧？
Right?
所以我认为，情况是一样的。
So it's the same thing, I believe.
我们不需要这种唯物主义的世界观。
We don't need this materialistic representation of the world.
干脆忘掉它吧。
Just forget about it.
现实意味着比那更广泛的东西。
Reality means something broader than that.
这与数学本身最明显的表达方式相矛盾。
It's inconsistent with the most obvious presentation of mathematics itself.
很明显，数学属性不是物质的。
It's obvious that mathematical properties are not material.
你可以构建一个形而上学体系来解释这一点。
You can invent a metaphysical system that explains that away.
这就是为什么这不算一个证明，但是
That's why it's not a proof, but
所以没人证明过进食者确实存在。
So nobody proved the eater does Yeah.
我们只是把它摒弃掉。
Not We just get rid of it.
是的。
Yeah.
我认为我们对此就该这么做。
And I think that's what we should do about it.
对。
Yeah.
我觉得我可以同意这一点。
Think I could agree with that.
但回到你刚才的评论，彼得，我认为更简单的方式是说，这个谜团就是数学本身的存在。
But going back to your last remark, Peter, I think it's just much simpler to say that the mystery is just the existence of mathematics.
就是这样。
It's just that.
因为它是最基本的。
Because it's fundamental.
我们完全可以这么说，当然哲学家们也早就说过，我们可以摒弃物理世界。
We could well say, and of course philosophers have well said, we can get rid of the physical world.
从形而上学的角度来看，这并不是问题。
Metaphysically, that's not a problem.
贝克莱展示了所有事物都只是知觉或观念。
Berkeley showed how everything is a perception or an idea.
外部世界就此消失。
External world just disappears.
但我们无法摒弃数学世界。
But we can't get rid of the mathematical world.
这是无法消除的。
That's ineliminable.
它的存在是一个深刻的谜题。
And its existence is a profound mystery.
它为什么会在那里？
What is it doing there?
为什么我们用数学的方式来理解事物？
Why do we see things in mathematical terms?
我提出这个问题，并不是因为我有一个秘密答案要透露。
Now, I'm not asking this question because I have a secret answer I'm prepared to vouchsafe.
我本以为你会用
Was hoping you'd wrap up the conversation with the
答案来结束这场对话。
answer.
我觉得这真是一个巨大的谜题。
I find it a great mystery.
数学的纯粹存在令人深深困惑。
The sheer existence of mathematics is deeply puzzling.
你会同意所说的每一个字。
You will agree with every word of that.
是的。
Yeah.
当然。
Absolutely.
你会同意，但你能更进一步吗？
And you'll agree, but can you take it farther?
我只是对这种论点感到着迷，我之前在对话中已经复述过：数学对象具有稳定的属性，因此它们具有独立于我们心智的客观性，但它们又是概念性的，根据我们的经验，它们不可能漂浮在某种柏拉图式的天堂中，相反，我认为更合理的看法是，它们最终源自上帝的心智。
Well, I just am intrigued with this kind of argument that I recapitulated earlier in the conversation that mathematical objects have stable properties, therefore they have an objectivity that is independent of our minds, and yet they are conceptual which by our experience that they must not be floating around somewhere in the Platonic heavens, but rather it makes more sense to me to think that they ultimately issue from the mind of God.
而这正是数学之所以能神秘地适用于物理世界的根本原因。
And that that is the deep reason for the mysterious applicability of mathematics to the physical world.
请注意，史蒂夫，你正在接近贝克莱的观点。
Bear in mind, Steve, that you're reaching a position very close to Berkeley's position.
谁？
To who?
谁？
To?
贝克莱。
Berkeley.
巴思主教。
Bishop Barth.
对。
Right.
我的意思是，如果你说存在就是被感知。
I mean, if you say that to be is to be perceived.
巴思主教，17世纪英国、18世纪英格兰的教士和哲学家，最著名的是出现在鲍斯韦尔的《约翰逊传》中，当时鲍斯韦尔提到约翰逊曾这样反驳贝克莱。
Bishop Barthew, seventeenth century British eighteenth century English churchman and philosopher who appears possibly most famously in Boswell's Life of Johnson when Boswell I rebuke says Berkeley thus.
于是，约翰逊踢了一块石头。
Thus, Johnson kicking a rock.
但关键是，你
But the point is you're
面对一个显而易见的问题：如果你不看月亮，爱因斯坦也讨论过这个问题，那么月亮是否依然存在？
face the obvious question, if you're not looking at the moon, Einstein discusses this too, does the moon continue to exist?
巴克莱的回答是：是的，它作为上帝心中的一个思想而存在。
And Barckley's response was, yes, it exists as a thought in the mind of God.
这与史蒂夫刚才所论证的观点非常接近。
Which is very close to what Steve was just arguing.
尽管他当然不是
Although he's certainly not
我不是阿瑟林。
I'm not Artholin.
我不认为物理世界只是我心中的产物，我认为它具有独立于我们心智的特性，是的。
I don't think the physical world is as a I think it has a mind, an independence of our mind Yeah.
它是上帝创造的，源于上帝的心智。
Of the mind of God who created it.
但我觉得数学现实的终极源头很可能就是上帝的心智。
But it's But I think the ultimate source of mathematical reality may well be the mind of God.
但有趣的是，你愿意走得比我认为许多当代分析哲学家更远，朝向巴克莱式的分析方向。
But it's interesting that you are prepared to go further than I think many contemporary analytic philosophers are prepared to go in the direction of a Barclian kind of analysis.
我认为在这种情况下，
Which I think is, in this case,
唯一的
the only
至少对于数学而言，这是唯一合理的分析。
At analysis that makes least for math.
是的。
Yeah.
至少对于数学而言。
At least for math.
因为这太神秘了。
Because it is so mysterious.
各位，我打算提最后一个问题。
Boys, I'm going to attempt for last question here.
我打算引入一个新概念，那就是美。
I'm going to attempt to introduce one new concept, and that is the concept of beauty.
我不知道这会怎样，但这里有一段来自即将上映的纪录片《万物的故事》的片段。
I have no idea how this will go, but here's an excerpt from the forthcoming documentary, The Story of Everything.
科学中有一个叫做美的原则的说法，认为真正的理论往往体现出数学上的美感或结构上的和谐。
There's something in science called the beauty principle that says true theories often convey a mathematical beauty or structural harmony.
弗朗西斯·克里克在看到DNA分子模型时曾被引用说：‘这太美了。’
Upon looking at their model of the DNA molecule, Francis Crick was quoted as saying, it's so beautiful.
它一定是正确的。
It's gotta be right.
解释一下。
Explain that.
为什么美要介入其中？
Why should Beauty enter into this?
我们其实并不清楚，但它在科学中通常被称为一种启发式指南，一种发现的指引。
We don't really know, but it tends to be what's called a heuristic guide in science, a guide to discovery.
在影片接下来的部分中，几位物理学家发表了更为深刻的评论，指出他们的数学美感认知常常成为发现的指引。
And in the section of the film that follows, there were perhaps even more trenchant comments from a couple of the physicists who were saying how often their perception of mathematical beauty had been a guide to discovery.
我认为是保罗·狄拉克最先说过，理论的优美比与数据一致更重要，因为最终我们可能会误判对数据的感知，但我们默认现实本身具有某种数学上的美感。
I think it was Paul Dirac who first said that it's more important for the theories to be beautiful than to have them consistent with the data because eventually, well, because we can be mistaken about how we're perceiving the data, but there is an assumption that there's something mathematically beautiful about about reality itself.
可理解性。
Intelligibility.
狄拉克对美着迷。
Dirac was mesmerized by Yeah.
抱歉。
Sorry.
谁？
Who was?
狄拉克。
Dirac.
狄拉克对美着迷于
Dirac was mesmerized by by
你刚才提到在选择要做的项目时，我想你用了‘优雅’这个词，对吗？
And you said a moment ago when you were choosing projects on which to work, you I think you used the word, is it elegant?
它是美的吗？
Is it beautiful?
审美的。
Aesthetic.
当然。
Of course.
为什么呢？为什么是这样？
Why why is that why?
这个解决方案是一个令人难以置信的、了不起的成果。
The care solution is an incredible incredible object.
我的意思是，真的非常美。
I mean, very really beautiful.
我的意思是，是的。
I mean, yeah.
我们又在探讨另一个谜题吗？一个美学上的谜题？
Are we on another mystery An aesthetic mystery?
美当然在我们选择问题的方式中扮演着根本性的角色，也引导着我们走向真理。
Beauty plays a fundamental role, of course, in the way we choose problems, but also in the way we are guiding ourselves towards the truth.
我的意思是，走向问题的解决方案。
I mean, towards the solution of a problem.
不知怎的，我们会拒绝那些牵强附会、不够优美的论证。
Somehow we reject arguments which are contrived, which are not beautiful.
我们不会称它们为优美的。
We don't call them beautiful.
是的，我的意思是，这神奇地是真的。
Yeah, I mean, it's mysteriously true.
物理学中当然充满了这样的例子。
And physics, of course, is full of such examples.
麦克斯韦，你知道，麦克斯韦方程组的发现，最初是法拉第通过实验已经发现了电磁学的三大定律。
Maxwell, know, the way the Maxwell equations were discovered, it was first Faraday who had the three laws of electromagnetism already discovered experimentally.
那是
That's
对的。
right.
但他根本不是数学家，所以只是把那些定律摆在那里，没有进一步发展。
And then, but he was not a mathematician at all, so he just left it there, just stated the laws.
正是麦克斯韦意识到，如果把这些表述纳入数学框架，就会发现某种不对称性，这引导他推导出一个新方程，最终促成了电磁理论。
And it was Maxwell who realized that if you put those statements within mathematics, there is a lack of symmetry sort of guided him towards a force one which led to electromagnetism.
所以，所有
So And all
这个现代世界的科技。
of this technology of the modern world.
美？
Beauty?
麦克斯韦。
Maxwell.
通过麦克斯韦。
With Maxwell.
你是在接近美吗？
You're coming upon beauty?
是的。
Yeah.
我想听听这些人的看法，因为是的。
I'd like to hear from these guys because Yeah.
我通常把这件事留给我的裁缝。
I kind of reserve that for my tailors.
你知道，作为一名职业数学家，数学在人们所做的一切中都起着根本性的作用。
You know, can tell you that as a working mathematician, it plays absolutely a fundamental role in everything people do.
真的吗？
Really?
几乎没有数学家会说，我做这个是因为它太丑了，对吧？
There are very few mathematicians who would say, I work on this because it's just ugly, right?
我的意思是，你知道，他们选择问题或方向是基于
I mean, you know, they choose the problems or directions based on
这只是表面说法，谢尔盖。
That's just the party lines, Sergei.
我们其实藏着各种各样的数学糟粕。
There are all sorts of mathematical drabs that we keep hidden.
没人会告诉我湍流是一个优美的课题。
Nobody's going to tell me turbulence is a beautiful subject.
哦，这是一个极棒的课题。
Oh, it's fantastic subject.
很棒，但不是……美观，有点笨拙，但我们的预期是，我们会通过
Fantastic, but not Well, beautiful, clumsy, but the expectation is that we'll find the beauty in turbulence by
预期是很容易被买来的。
Expectations are easily purchased.
这可是饥荒价。
Comes at famine prices.
好吧。
Alright.
他现在这么反常，踢他一脚。
Kick him because he's being perverse now.
他是在捣蛋。
He's being mischievous.
最后一个问题。
Last question.
艾萨克·牛顿，这位为我们带来了天文学、流体动力学和微积分数学的人。
Isaac Newton, the man who gave us mathematics that on astronomy fluid dynamics and Calculus.
牛顿解释了很多东西。
And Newton explains a lot.
这是牛顿的一段话。
Here's a quotation from Newton.
一位天上的主宰作为宇宙的君主治理着整个世界。
A heavenly master governs all the world as sovereign of the universe.
引号结束。
Close quote.
对超越我们自身的神圣或智慧的认可，几千年来一直被视为理所当然，即使在牛顿时代仍是如此，但随后却被逐出了知识界和学术界。
A recognition of the divine, or a mind that transcends our own, is taken for granted for thousands of years, as recently as Newton, and then it gets kicked out of intellectual life and kicked out of the academy.
数学的奥秘是否暗示，唯物主义的错误是一种偏差，现在或许正在得到纠正？
Does the mystery of mathematics suggest that the materialist error was an aberration that ought to and may be may be ending now.
史蒂夫，你愿意走到这一步吗？
Are you willing to go that far, Steve?
是的。
Yeah.
当然。
Of course.
我认为这完全正确。
I think that's exactly right.
牛顿如此美妙地阐明了我们一直讨论的这种可理解性原则。
And what Newton illustrates so beautifully is that this principle of intelligibility that we've been talking about.
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