陶哲轩：数学与物理中最难的问题，以及 AI 的未来

The following is a conversation with Terence Tao, widely considered to be one of the greatest mathematicians in history, often referred to as The Mozart of Math.

以下是与陶哲轩的对话，陶哲轩被广泛认为是历史上最伟大的数学家之一，通常被称为数学中的莫扎特。

00:00 He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics.

他获得了菲尔兹奖和数学突破奖，并为数学和物理领域的一系列真正令人惊叹的领域做出了开创性的工作。

This was a huge honor for me for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions.

这对我来说是一个巨大的荣誉，原因有很多，包括特里在我们所有的互动中向我表现出的谦逊和友善。

It means the world.

它意味着世界。

This is the Lex Fridman Podcast.

这是莱克斯·弗里德曼播客。

To support it, please check out our sponsors in the description or at LexFridman.com/sponsors.

如需支持，请在说明中或访问 LexFridman.com/sponsors 查看我们的赞助商。

And now, dear friends, here’s Terence Tao.

现在，亲爱的朋友们，这里是陶哲轩。

00:49 What was the first really difficult research-level math problem that you encountered, one that gave you pause maybe?

您遇到的第一个真正困难的研究级数学问题是什么？这个问题可能让您犹豫不决？

Well, in your undergraduate education you learn about the really hard impossible problems like the Riemann Hypothesis, the Twin-Primes Conjecture.

嗯，在你的本科教育中，你会学到一些真正困难的不可能的问题，比如黎曼猜想、双素数猜想。

00:57 You can make problems arbitrarily difficult.

你可以让问题变得任意困难。

That’s not really a problem.

这并不是什么问题。

In fact, there’s even problems that we know to be unsolvable.

事实上，甚至有一些我们知道无法解决的问题。

What’s really interesting are the problems just on the boundary between what we can do rather easily and what are hopeless, but what are problems where existing techniques can do 90% of the job and then you just need that remaining 10%.

真正有趣的是那些我们可以轻松完成的事情和毫无希望的事情之间的边界问题，但是哪些问题是现有技术可以完成 90% 的工作，而你只需要剩下的 10%。

I think as a PhD student, the Kakeya Problem certainly caught my eye.

我认为作为一名博士生，挂谷问题确实引起了我的注意。

And it just got solved actually.

实际上它刚刚得到解决。

It’s a problem I’ve worked on a lot in my early research.

这是我在早期研究中大量研究的一个问题。

Historically, it came from a little puzzle by the Japanese mathematician Soichi Kakeya in 1918 or so.

历史上，它源于日本数学家挂谷宗一在 1918 年左右的一个小谜题。

So, the puzzle is that you have a needle on the plane or think like driving on a road something, and you want it to execute a U-turn, you want to turn the needle around, but you want to do it in as little space as possible.

所以，难题是，你在飞机上有一根针，或者像在路上开车一样思考，你希望它执行掉头，你想让针掉头，但你想在尽可能小的空间内完成它。

So, you want to use this little area in order to turn it around, but the needle is infinitely maneuverable.

所以，你想利用这个小区域来转动它，但针的可操作性是无限的。

So, you can imagine just spinning it around.

所以，你可以想象只是旋转它。

As the unit needle, you can spin it around its center, and I think that gives you a disc of area, I think pi over four.

作为单位针，你可以将它绕其中心旋转，我认为这会给你一个面积圆盘，我认为圆周率大于四。

Or you can do a three-point U-turn, which is what we teach people in their driving schools to do.

或者，您也可以进行三点掉头，这就是我们在驾驶学校教人们做的事情。

And that actually takes area of pi over eight, so it’s a little bit more efficient than a rotation.

这实际上需要 pi 的面积超过 8，所以它比旋转更有效。

And so for a while people thought that was the most efficient way to turn things around, but Besicovitch showed that in fact you could actually turn the needle around using as little area as you wanted.

因此，有一段时间，人们认为这是扭转局面的最有效方法，但贝西科维奇表明，事实上，你可以使用尽可能小的面积来扭转局面。

So, 0.01, there was some really fancy multi back and forth U-turn thing that you could do that you could turn a needle around and in so doing it would pass through every intermediate direction.

所以，0.01，有一些非常奇特的多来回 U 形转弯的东西，你可以做你可以转动一根针，这样它就会穿过每个中间方向。

Is

是

This in the two-dimensional plane?

这是在二维平面上吗？

This is in the two-dimensional plane.

这是在二维平面中。

02:52 So, we understand everything in two dimensions.

所以，我们从二维角度理解一切。

So, the next question is: what happens in three dimensions?

那么，下一个问题是：三维空间中会发生什么？

So, suppose the Hubble space Telescope is tube in space, and you want to observe every single star in the universe, so you want to rotate the telescope to reach every single direction.

因此，假设哈勃太空望远镜是太空中的管子，并且您想要观察宇宙中的每一颗恒星，因此您想要旋转望远镜以到达每个方向。

And here’s unrealistic part, suppose that space is at a premium, which totally is not, you want to occupy as little volume as possible in order to rotate your needle around, in order to see every single star in the sky.

这是不切实际的部分，假设空间非常宝贵，但事实并非如此，你想要占据尽可能小的体积，以便旋转你的针，以便看到天空中的每一颗星星。

How small a volume do you need to do that?

您需要多小的体积才能做到这一点？

And so you can modify Besicovitch’s construction.

所以你可以修改贝西科维奇的构造。

And so if your telescope has zero thickness, then you can use as little volume as you need.

因此，如果您的望远镜厚度为零，那么您可以根据需要使用尽可能小的体积。

That’s a simple modification of the two-dimensional construction.

这是二维结构的简单修改。

But the question is that if your telescope is not zero thickness, but just very, very thin, some thickness delta, what is the minimum volume needed to be able to see every single direction as a function of delta?

但问题是，如果你的望远镜不是零厚度，而是非常非常薄，有一定的厚度增量，那么能够看到每个方向作为增量函数的最小体积是多少？

03:45 So, as delta gets smaller, as the needle gets thinner, the volume should go down.

因此，随着 Delta 变小，针变细，成交量就会下降。

But how fast does it go down?

但它下降的速度有多快？

And the conjecture was that it goes down very, very slowly like logarithmically roughly speaking, and that was proved after a lot of work.

我们的猜想是，它下降得非常非常慢，就像对数一样，粗略地说，这是经过大量工作证明的。

So, this seems like a puzzle.

所以，这看起来像是一个谜题。

Why is it interesting?

为什么有趣？

So, it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, combinatorics.

因此，事实证明它与偏微分方程、数论、几何、组合学中的许多问题有着令人惊讶的联系。

For example, in wave propagation, you splash some water around, you create water waves and they travel in various directions, but waves exhibit both particle and wave-type behavior.

例如，在波传播中，您向周围泼洒一些水，产生水波，并且它们向各个方向传播，但波同时表现出粒子和波型行为。

So, you can have what’s called a wave packet, which is a very localized wave that is localized in space and moving a certain direction in time.

因此，你可以得到所谓的波包，它是一种非常局域化的波，它在空间中局域化，并在时间上向某个方向移动。

And so if you plot it in both space and time, it occupies a region which looks like a tube.

因此，如果你在空间和时间上绘制它，它会占据一个看起来像管子的区域。

What can happen is that you can have a wave which initially is very dispersed, but it all focuses at a single point later in time.

可能发生的情况是，你可能会出现一个最初非常分散的波，但后来它全部集中在一个点上。

You can imagine dropping a pebble into a pond and the ripples spread out, but then if you time-reverse that scenario, and the equations of wave motion are time-reversible, you can imagine ripples that are converging to a single point and then a big splash occurs, maybe even a singularity.

你可以想象将一颗鹅卵石扔进池塘，涟漪扩散开来，但是如果你对这种情况进行时间反转，并且波动方程是时间可逆的，你可以想象涟漪会聚成一个点，然后发生大水花，甚至可能是奇点。

And so it’s possible to do that.

所以这是可能做到的。

And geometrically what’s going on is that there’s also light rays, so if this wave represents light, for example, you can imagine this wave as a superposition of photons all traveling at the speed of light.

从几何上讲，发生的事情是还有光线，所以如果这个波代表光，例如，你可以把这个波想象成所有以光速传播的光子的叠加。

05:15 They all travel on these light rays and they’re all focusing at this one point.

它们都沿着这些光线传播，并且都聚焦在这一点。

So, you can have a very dispersed wave focus into a very concentrated wave at one point in space and time, but then it de-focuses again, it separates.

因此，你可以在空间和时间的某一点将非常分散的波焦点变成非常集中的波，但随后它再次散焦，分离。

But potentially if the conjecture had a negative solution, so what that meant is that there’s a very efficient way to pack tubes pointing different directions to a very, very narrow region of a very narrow volume.

但如果猜想有一个负解，那么这意味着有一种非常有效的方法可以将指向不同方向的管子包装到非常狭窄的体积的非常非常狭窄的区域。

Then you would also be able to create waves that start out some…

然后你还可以创造一些波浪来开始......

There’ll be some arrangement of waves that start out very, very dispersed, but they would concentrate, not just at a single point, but there’ll be a lot of concentrations in space and time.

会有一些波的排列，一开始非常非常分散，但它们会集中，不仅仅是在一个点上，而且在空间和时间上会有很多集中。

And you could create what’s called a blowup, where these waves amplitude becomes so great that the laws of physics that they’re governed by are no longer wave equations, but something more complicated and nonlinear.

你可以创造所谓的爆炸，其中这些波的振幅变得如此之大，以至于它们所遵循的物理定律不再是波动方程，而是更复杂和非线性的东西。

06:08 And so in mathematical physics, we care a lot about whether certain equations and wave equations are stable or not, whether they can create these singularities.

所以在数学物理中，我们非常关心某些方程和波动方程是否稳定，是否可以产生这些奇点。

There’s a famous unsolved problem called the Navier-Stokes regularity problem.

有一个著名的未解决问题，称为纳维-斯托克斯正则问题。

So, the Navier-Stokes equations, equations that govern the fluid flow for incompressible fluids like water.

因此，纳维-斯托克斯方程是控制不可压缩流体（如水）的流体流动的方程。

The question asks: if you start with a smooth velocity field of water, can it ever concentrate so much that the velocity becomes infinite at some point?

问题是：如果从平滑的水速度场开始，它是否会集中得如此之大，以致速度在某个点变得无限大？

That’s called a singularity.

这就是所谓的奇点。

We don’t see that in real life.

我们在现实生活中看不到这一点。

If you splash around water in the bathtub, it won’t explode on you or have water leaving at the speed of light or anything, but potentially it is possible.

如果你把水溅到浴缸里，它不会在你身上爆炸，或者水以光速或其他任何速度离开，但有可能。

06:49 And in fact, in recent years, the consensus has drifted towards the belief that, in fact, for certain very special initial configurations of, say, water, that singularities can form, but people have not yet been able to actually establish this.

事实上，近年来，共识已经转向这样的信念：事实上，对于某些非常特殊的初始构型，例如水，可以形成奇点，但人们尚未能够真正确定这一点。

The Clay Foundation has these seven Millennium Prize Problems as a $1 million prize for solving one of these problems, and this is one of them.

克莱基金会将这七个千年奖问题作为解决其中一个问题的 100 万美元奖金，这就是其中之一。

Of these of these seven, only one of them has been solved, at the Poincare Conjecture [inaudible 00:07:18].

在这七个问题中，只有一个已得到解决，即庞加莱猜想 [听不清 00:07:18]。

So, the Kakeya Conjecture is not directly directly related to the Navier-Stokes Problem, but understanding it would help us understand some aspects of things like wave concentration, which would indirectly probably help us understand the Navier-Stokes Problem better.

因此，挂谷猜想与纳维-斯托克斯问题没有直接关系，但理解它可以帮助我们理解波集中等事物的某些方面，这可能会间接帮助我们更好地理解纳维-斯托克斯问题。

Can you speak to the Navier-Stokes?

你能和纳维-斯托克斯通话吗？

07:32 So, the existence of smoothness, like you said, Millennium Prize Problem, You’ve made a lot of progress on this one.

所以，平滑性的存在，就像你说的，千年奖问题，你在这个问题上取得了很大的进展。

In 2016, you published a paper, Finite Time Blowup For An Average Three-Dimensional Navier-Stokes Equation.

2016 年，您发表了一篇论文，Finite Time Blowup For An Average Three-Dimensional Navier-Stokes Equation。

So, we’re trying to figure out if this thing…

所以，我们试图弄清楚这件事是否……

Usually it doesn’t blow up, but can we say for sure it never blows up?

通常它不会爆炸，但我们能肯定地说它永远不会爆炸吗？

Right, yeah.

对，是的。

07:56 So yeah, that is literally the $1 million question.

是的，这确实是 100 万美元的问题。

So, this is what distinguishes mathematicians from pretty much everybody else.

所以，这就是数学家与其他人的不同之处。

If something holds 99.99% of the time, that’s good enough for most things.

如果某件事在 99.99% 的情况下成立，那么对于大多数事情来说就足够了。

But mathematicians are one of the few people who really care about whether really 100% of all situations are covered by it.

但数学家是少数真正关心它是否真的 100% 涵盖所有情况的人之一。

So, most fluid, most of the time water does not blow up, but could you design a very special initial state that does this?

所以，大多数流体，大多数时候水不会爆炸，但是你能设计一个非常特殊的初始状态来做到这一点吗？

And maybe we should say that this is a set of equations that govern in the field of fluid dynamics, trying to understand how fluid behaves.

也许我们应该说这是流体动力学领域的一组方程，试图理解流体的行为。

08:29 And it’s actually turns out to be a really…

事实证明，这真的是……

Fluid is extremely complicated thing to try to model.

流体建模是极其复杂的事情。

Yeah, so it has practical importance.

是的，所以它具有实际意义。

08:43 So this Clay Prize problem concerns what’s called the Incompressible Navier-Stokes, which governs things like water.

因此，这个克莱奖问题涉及所谓的不可压缩纳维-斯托克斯问题，它控制着水等物质。

There’s something called the Compressible Navier-Stokes, which governs things like air, and that’s particularly important for weather prediction.

有一种叫做可压缩纳维斯托克斯的东西，它控制着空气等物质，这对于天气预报尤其重要。

Weather prediction, it does a lot of computational fluid dynamics.

天气预报，它做了很多计算流体动力学。

A lot of it’s actually just trying to solve the Navier-Stokes equations as best they can.

其中很多实际上只是试图尽可能地求解纳维-斯托克斯方程。

Also gathering a lot of data, so that they can initialize the equation.

还收集了大量数据，以便他们可以初始化方程。

There’s a lot of moving parts, so it’s very important from practically.

有很多活动部件，因此从实践来看它非常重要。

Why is it difficult to prove general things about the set of equations like it not not blowing up?

为什么很难证明关于方程组的一般事情，比如它没有爆炸？

Short answer is Maxwell’s Demon.

简短的回答是麦克斯韦妖。

09:17 So, Maxwell’s Demon is a concept in thermodynamics.

所以，麦克斯韦妖是热力学中的一个概念。

If you have a box of two gases in oxygen and nitrogen, and maybe you start with all the oxygen on one side and nitrogen on the other side, but there’s no barrier between them.

如果你有一盒氧气和氮气两种气体，也许一开始所有氧气都在一侧，氮气在另一侧，但它们之间没有障碍。

Then they will mix and they should stay mixed.

然后它们会混合并且应该保持混合状态。

There’s no reason why they should un-mix.

他们没有理由不混合。

But in principle, because of all the collisions between them, there could be some sort of weird conspiracy that maybe there’s a microscopic demon called Maxwell’s Demon that will…

但原则上，由于它们之间的所有碰撞，可能存在某种奇怪的阴谋，也许有一个叫做麦克斯韦恶魔的微观恶魔会……

every time an oxygen and nitrogen atom collide, they’ll bounce off in such a way that the oxygen sort of drifts onto one side and then nitrogen goes to the other.

每当氧原子和氮原子碰撞时，它们都会以这样的方式弹开：氧原子会漂移到一侧，然后氮原子会漂移到另一侧。

And you could have an extremely improbable configuration emerge, which we never see, and which statistically it’s extremely unlikely, but mathematically it’s possible that this can happen and we can’t rule that out.

你可能会出现一种极其不可能的配置，这是我们从未见过的，而且从统计上看这是极不可能的，但从数学上讲，这种情况有可能发生，我们不能排除这种可能性。

10:06 And this is a situation that shows up a lot in mathematics.

这种情况在数学中经常出现。

A basic example is the digits of pi 3.14159 and so forth.

一个基本示例是 pi 3.14159 等数字。

The digits look like they have no pattern, and we believe they have no pattern.

这些数字看起来没有模式，我们也相信它们没有模式。

On the long-term, you should see as many ones and twos and threes as fours and fives and sixes, there should be no preference in the digits of pi to favor, let’s say seven over eight.

从长远来看，你应该看到与四、五和六一样多的一、二、三，在圆周率的数字上不应该有任何偏好，比如说七比八。

But maybe there’s some demon in the digits of pi that every time you compute more and more digits, it biases one digit to another.

但也许圆周率的数字中有一些恶魔，每次你计算越来越多的数字时，它就会使一个数字偏向另一个数字。

And this is a conspiracy that should not happen.

而这是一个不应该发生的阴谋。

There’s no reason it should happen, but there’s no way to prove it with our current technology.

没有理由会发生这种情况，但用我们现有的技术无法证明这一点。

So, getting back to Navier-Stokes, a fluid has a certain amount of energy, and because the fluid is in motion, the energy gets transported around.

因此，回到纳维-斯托克斯定律，流体具有一定的能量，并且由于流体在运动，因此能量会四处传输。

10:53 And water is also viscous, so if the energy is spread out over many different locations, the natural viscosity of the fluid will just damp out the energy and will go to zero.

水也是粘性的，因此如果能量分散在许多不同的位置，流体的自然粘度就会减弱能量并趋于零。

And this is what happens when we actually experiment with water.

这就是我们实际对水进行实验时所发生的情况。

You splash around, there’s some turbulence and waves and so forth, but eventually it settles down and the lower the amplitude, the smaller velocity, the more calm it gets.

你溅起水花，有一些湍流和波浪等等，但最终它会稳定下来，振幅越低，速度越小，就越平静。

But potentially there is some sort of demon that keeps pushing the energy of the fluid into a smaller and smaller scale, and it’ll move faster and faster.

但可能存在某种恶魔，不断将流体的能量推向越来越小的范围，并且它会移动得越来越快。

And at faster speeds, the effect of viscosity is relatively less.

并且在较快的速度下，粘度的影响相对较小。

And so it could happen that it creates some sort of what’s called a self-similar blob scenario where the energy of the fluid starts off at some large scale and then it all sort of transfers energy into a smaller region of the fluid, which then at a much faster rate moves into an even smaller region and so forth.

因此，它可能会产生某种所谓的自相似斑点场景，其中流体的能量以一定的规模开始，然后将能量转移到流体的较小区域，然后以更快的速度移动到更小的区域，依此类推。

11:55 And each time it does this, it takes maybe half as long as the previous one, and then you could actually converge to all the energy concentrating in one point in a finite amount of time.

每一次这样做，花费的时间可能是前一次的一半，然后你实际上可以在有限的时间内将所有能量集中在一个点上。

And that’s scenario is called finite time blowup.

这种情况被称为有限时间爆炸。

So, in practice, this doesn’t happen.

所以，在实践中，这种情况不会发生。

So, water is what’s called turbulent.

所以，水就是所谓的湍流。

So, it is true that if you have a big eddy of water, it will tend to break up into smaller eddies, but it won’t transfer all energy from one big eddy into one smaller eddy.

因此，如果有一个大的水涡流，它确实会分裂成较小的涡流，但它不会将所有能量从一个大涡流转移到一个较小的涡流中。

It will transfer into maybe three or four, and then those ones split up into maybe three or four small eddies of their own.

它可能会变成三到四个，然后这些漩涡可能会分裂成三到四个自己的小漩涡。

So the energy gets dispersed to the point where the viscosity can then keep everything under control.

因此，能量会分散到粘度可以控制一切的程度。

But if it can somehow concentrate all the energy, keep it all together, and do it fast enough that the viscous effects don’t have enough time to calm everything down, then this blowup can occur.

但如果它能以某种方式集中所有能量，将所有能量集中在一起，并且速度足够快，以至于粘性效应没有足够的时间让一切平静下来，那么这种爆炸就会发生。

12:51 So, there were papers who had claimed that, “Oh, you just need to take into account conservation of energy and just carefully use the viscosity and you can keep everything under control for not just the Navier-Stokes, but for many, many types of equations like this.” And so in the past there have been many attempts to try to obtain what’s called global regularity for Navier-Stokes, which is the opposite of finite time blowup, that velocity stays smooth.

因此，有论文声称，“哦，你只需要考虑能量守恒，并仔细使用粘度，你就可以控制一切，不仅适用于纳维-斯托克斯方程，而且适用于很多很多类型的方程。”因此，过去有很多尝试试图获得所谓的纳维-斯托克斯全局正则性，这与有限时间爆炸相反，即速度保持平滑。

And it all failed.

但这一切都失败了。

There was always some sign error or some subtle mistake and it couldn’t be salvaged.

总是存在一些符号错误或一些微妙的错误，而且无法挽救。

13:17 So, what I was interested in doing was trying to explain why we were not able to disprove finite time blowup.

所以，我感兴趣的是试图解释为什么我们无法反驳有限时间爆炸。

I couldn’t do it for the actual equations of fluids, which are too complicated, but if I could average the equations of motion of Navier-Stokes, basically if I could turn off certain types of ways in which water interacts and only keep the ones that I want.

我无法对实际的流体方程做到这一点，因为它太复杂了，但如果我可以平均纳维-斯托克斯运动方程，基本上如果我可以关闭某些类型的水相互作用方式并只保留我想要的方式。

So, in particular, if there’s a fluid and it could transfer as energy from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this one and direct it only into this smaller eddy while still preserving the lower conservation energy.

因此，特别是，如果有一种流体，它可以将能量从一个大涡流转移到这个小涡流或另一个小涡流中，我会关闭将能量转移到这个涡流的能量通道，并将其仅引导到这个较小的涡流中，同时仍然保留较低的守恒能量。

So, you’re trying to make a blowup?

所以，你想制造爆炸吗？

Yeah, yeah.

是啊是啊。

14:00 So, I basically engineer a blowup by changing rules of physics, which is one thing that mathematicians are allowed to do.

所以，我基本上通过改变物理规则来设计爆炸，这是数学家可以做的一件事。

We can change the equation.

我们可以改变方程式。

How does that help you get closer to the proof of something?

这如何帮助你更接近某事的证明？

Right.

正确的。

14:11 So, it provides what’s called an obstruction in mathematics.

因此，它提供了所谓的数学障碍。

So, what I did was that basically if I turned off the certain parts of the equation, which usually when you turn off certain interactions, make it less nonlinear, it makes it more regular and less likely to blow up.

所以，我所做的基本上是，如果我关闭方程的某些部分，通常当你关闭某些相互作用时，它会减少非线性，它会使其更加规则并且不太可能爆炸。

But I find that by turning off a very well-designed set of interactions, I could force all the energy to blow up in finite time.

但我发现，通过关闭一组精心设计的交互，我可以迫使所有能量在有限的时间内爆炸。

So, what that means is that if you wanted to prove the regularity for Navier-Stokes for the actual equation, you must use some feature of the true equation, which my artificial equation does not satisfy.

所以，这意味着，如果你想证明实际方程的纳维-斯托克斯正则性，你必须使用真实方程的某些特征，而我的人工方程不满足这些特征。

So, it rules out certain approaches.

因此，它排除了某些方法。

14:55 So, the thing about math, it’s not just about taking a technique that is going to work and applying it, but you need to not take the techniques that don’t work.

所以，关于数学，不仅仅是采用一种可行的技术并应用它，而且你不需要采用那些不起作用的技术。

And for the problems that are really hard, often though are dozens of ways that you might think might apply to solve the problem, but it’s only after a lot of experience that you realize there’s no way that these methods are going to work.

对于真正困难的问题，通常你可能认为有几十种方法可以解决问题，但只有在大量的经验之后你才会意识到这些方法根本行不通。

So, having these counterexamples for nearby problems rules out…

因此，拥有这些附近问题的反例就排除了……

it saves you a lot of time because you’re not wasting energy on things that you now know cannot possibly ever work.

它可以为你节省大量时间，因为你不会把精力浪费在你现在知道不可能成功的事情上。

How deeply connected is it to that specific problem of fluid dynamics or is this some more general intuition you build up about mathematics?

它与特定的流体动力学问题的联系有多深，或者这是您对数学建立的一些更普遍的直觉？

Right.

正确的。

15:38 Yeah.

是的。

So, the key phenomenon that my technique exploits is what’s called super-criticality.

所以，我的技术利用的关键现象就是所谓的超临界性。

So, in partial [inaudible 00:15:46] equations, often these equations are like a tug of war between different forces.

因此，在部分 [听不清 00:15:46] 方程中，这些方程通常就像不同力量之间的拉锯战。

So, in Navier-Stokes, there’s the dissipation force coming from viscosity, and it’s very well understood.

因此，在纳维-斯托克斯中，存在来自粘度的耗散力，这是很好理解的。

It’s linear, it calms things down.

它是线性的，它能让事情平静下来。

If viscosity was all there was, then nothing bad would ever happen, but there’s also transport that energy from…

如果粘度就是全部，那么就不会发生任何不好的事情，但也可以从……传输能量。

in one location of space can get transported because the fluid is in motion to other locations.

由于流体正在运动到其他位置，所以空间的一个位置可以被运输。

And that’s a nonlinear effect, and that causes all the problems.

这是一种非线性效应，导致了所有问题。

So, there are these two competing terms in the Navier-Stokes Equation, the dissipation term and the transport term.

因此，纳维-斯托克斯方程中有两个相互竞争的项：耗散项和输运项。

If the dissipation term dominates, if it’s large, then basically you get regularity.

如果耗散项占主导地位，如果它很大，那么基本上你就会得到规律性。

And if the transport term dominates, then we don’t know what’s going on.

如果运输术语占主导地位，那么我们就不知道发生了什么。

It’s a very nonlinear situation, it’s unpredictable, it’s turbulent.

这是一个非常非线性的情况，它是不可预测的，它是动荡的。

16:32 So, sometimes these forces are in balance at small scales but not in balance at large scales or vice versa.

因此，有时这些力在小尺度上保持平衡，但在大尺度上不平衡，反之亦然。

Navier-Stokes is what’s called supercritical.

纳维-斯托克斯就是所谓的超临界。

So at smaller and smaller scales, the transport terms are much stronger than the viscosity terms.

因此，在越来越小的尺度上，输运项比粘度项强得多。

So, the viscosity terms are things that calm things down.

所以，粘度项可以让事情平静下来。

And so this is why the problem is hard.

这就是为什么这个问题很难。

In two dimensions, so the Soviet mathematician Ladyzhenskaya, she in the ’60s shows in two dimensions there was no blowup.

在二维中，苏联数学家 Ladyzhenskaya 在 60 年代表明，在二维中没有爆炸。

And in two dimensions, the Navier-Stokes Equation is what’s called critical, the effect of transport and the effect of viscosity about the same strength even at very, very small scales.

在二维中，纳维-斯托克斯方程就是所谓的临界方程，即使在非常非常小的尺度上，传输效应和粘度效应也具有相同的强度。

And we have a lot of technology to handle critical and also subcritical equations and prove regularity.

我们拥有大量技术来处理临界方程和亚临界方程并证明规律性。

But for supercritical equations, it was not clear what was going on, and I did a lot of work, and then there’s been a lot of follow up showing that for many other types of supercritical equations, you can create all kinds of blowup examples.

但对于超临界方程，还不清楚发生了什么，我做了很多工作，然后有很多后续工作表明，对于许多其他类型的超临界方程，您可以创建各种爆炸示例。

17:27 Once the nonlinear effects dominate the linear effects at small scales, you can have all kinds of bad things happen.

一旦非线性效应在小尺度上主导了线性效应，就会发生各种不好的事情。

So, this is sort of one of the main insights of this line of work is that super-criticality versus criticality and subcriticality, this makes a big difference.

因此，这一系列工作的主要见解之一是超临界性与临界性和亚临界性，这会产生很大的差异。

That’s a key qualitative feature that distinguishes some equations for being sort of nice and predictable and…

这是一个关键的定性特征，它区分了一些方程是否良好且可预测......

Like planetary motion, there’s certain equations that you can predict for millions of years or thousands at least.

就像行星运动一样，有些方程可以预测数百万年或至少数千年。

Again, it’s not really a problem, but there’s a reason why we can’t predict the weather past two weeks into the future because it’s a supercritical equation.

再说一次，这并不是一个真正的问题，但我们无法预测未来两周的天气是有原因的，因为这是一个超临界方程。

Lots of really strange things are going on at very fine scales.

许多非常奇怪的事情正在非常精细的尺度上发生。

So, whenever there is some huge source of nonlinearity, that can create a huge problem for predicting what’s going to happen?

那么，只要存在一些巨大的非线性源，就会给预测将要发生的事情带来巨大的问题？

Yeah.

是的。

18:13 And if non-linearity is somehow more and more featured and interesting at small scales.

如果非线性在小尺度上变得越来越有特色和有趣。

There’s many equations that are nonlinear, but in many equations you can approximate things by the bulk.

有许多方程是非线性的，但在许多方程中，您可以对事物进行整体近似。

So, for example, planetary motion, if you want to understand the orbit of the Moon or Mars or something, you don’t really need the microstructure of the seismology of the Moon or exactly how the mass is distributed.

因此，举例来说，行星运动，如果你想了解月球或火星或其他东西的轨道，你实际上并不需要月球地震学的微观结构或质量的确切分布。

Basically, you can almost approximate these planets by point masses, and it’s just the aggregate behavior is important.

基本上，你几乎可以通过点质量来近似这些行星，只是聚合行为很重要。

But if you want to model a fluid, like the weather, you can’t just say, “In Los Angeles the temperature is this, the wind speed is this.” For supercritical equations, the fine scale information is really important.

但如果你想对流体（例如天气）进行建模，你不能只是说：“洛杉矶的温度是这样，风速是这样。”对于超临界方程，精细尺度信息非常重要。

If we can just linger on the Navier-Stokes Equations a little bit.

如果我们能稍微停留在纳维-斯托克斯方程上。

18:54 So, you’ve suggested, maybe you can describe it, that one of the ways to ways solve it or to negatively resolve it would be to construct a kind of liquid computer, and then show that the halting problem from computation theory has consequences for fluid dynamics, so show it in that way.

所以，你建议，也许你可以描述它，解决它或消极解决它的方法之一是构建一种液体计算机，然后证明计算理论中的停止问题对流体动力学有影响，所以用这种方式来展示它。

Can you describe this idea?

你能描述一下这个想法吗？

Right, yeah.

对，是的。

19:22 So, this came out of this work of constructing this average equation that blew up.

所以，这是从构建这个爆炸的平均方程的工作中产生的。

So, as part of how I had to do this, so there’s this naive way to do it, you just keep pushing.

所以，作为我必须这样做的一部分，所以有一种天真的方法可以做到这一点，你只需继续推动即可。

Every time you get one scale, you push it immediately to the next scale as fast as possible.

每次你获得一个等级时，你都会立即将其尽快推到下一个等级。

This is sort of the naive way to force blowup.

这是一种强制爆炸的幼稚方式。

It turns out in five and higher dimensions, this works, but in three dimensions there was this funny phenomenon that I discovered, that if you change laws of physics, you just always keep trying to push the energy into smaller and smaller scales, what happens is that the energy starts getting spread out into many scales at once, so that you have energy at one scale.

事实证明，在五个维度和更高的维度中，这是可行的，但在三个维度中，我发现了一个有趣的现象，如果你改变物理定律，你总是试图将能量推入越来越小的尺度，所发生的事情是能量开始立即分散到多个尺度，这样你就拥有一个尺度的能量。

You’re pushing it into the next scale, and then as soon as it enters that scale, you also push it to the next scale, but there’s still some energy left over from the previous scale.

你把它推到下一个尺度，然后一旦它进入那个尺度，你也把它推到下一个尺度，但是上一个尺度还剩下一些能量。

20:16 You’re trying to do everything at once, and this spreads out the energy too much.

你试图同时做所有事情，这会过度分散能量。

And then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything.

然后事实证明，它很容易受到粘度的影响，实际上只是削弱了一切。

So, it turns out this direct abortion doesn’t actually work.

所以，事实证明这种直接堕胎实际上是行不通的。

There was a separate paper by some other authors that actually showed this in three dimensions.

其他一些作者发表了一篇单独的论文，实际上从三个维度展示了这一点。

So, what I needed was to program a delay, so kind of like airlocks.

所以，我需要的是编程一个延迟，就像气闸一样。

So, I needed an equation which would start with a fluid doing something at one scale, it would push this energy into the next scale, but it would stay there until all the energy from the larger scale got transferred.

所以，我需要一个方程，它从流体在一个尺度上做某事开始，它将把这种能量推入下一个尺度，但它会停留在那里，直到来自更大尺度的所有能量都被转移。

And only after you pushed all the energy in, then you open the next gate and then you push that in as well.

只有当你把所有的能量都推进去之后，你才能打开下一扇门，然后你也把它推进去。

21:01 So, by doing that, the energy inches forward, scale by scale in such a way that it’s always localized at one scale at a time, and then it can resist the effects of viscosity because it’s not dispersed.

因此，通过这样做，能量会逐个尺度地向前移动，每次总是局限于一个尺度，然后它可以抵抗粘度的影响，因为它没有分散。

So, in order to make that happen, I had to construct a rather complicated nonlinearity.

因此，为了实现这一点，我必须构建一个相当复杂的非线性。

And it was basically…

这基本上是……

It was constructed like an electronic circuit.

它的构造就像一个电子电路。

So, I actually thank my wife for this because she was trained as an electrical engineer, and she talked about she had to design circuits and so forth.

所以，我实际上为此感谢我的妻子，因为她接受过电气工程师培训，她谈到她必须设计电路等等。

And if you want a circuit that does a certain thing, maybe have a light that flashes on and then turns off and then on and off.

如果你想要一个能做某件事的电路，也许有一个灯会闪烁，然后关闭，然后打开和关闭。

You can build it from more primitive components, capacitors and resistors and so forth, and you have to build a diagram.

您可以使用更原始的组件、电容器和电阻器等来构建它，并且您必须构建一个图表。

21:47 And these diagrams, you can sort of follow up your eyeballs and say, “Oh yeah, the current will build up here and it will stop, and then it will do that.” So, I knew how to build analog of basic electronic components, like resistors and capacitors and so forth.

这些图表，你可以跟踪你的眼球并说，“哦，是的，电流会在这里积聚，它会停止，然后它就会这样做。”所以，我知道如何构建基本电子元件的模拟，如电阻器和电容器等。

And I would stack them together in such a way that I would create something that would open one gate.

我会把它们堆叠在一起，这样我就能创造出能打开一扇门的东西。

And then there’d be a clock, and then once the clock hits a certain threshold, it would close it.

然后会有一个时钟，一旦时钟达到某个阈值，它就会关闭它。

It would become a Rube Goldberg type machine, but described mathematically.

它将成为鲁布·戈德堡型机器，但以数学方式描述。

And this ended up working.

这最终奏效了。

So, what I realized is that if you could pull the same thing off for the actual equations, so if the equations of water support a computation…

所以，我意识到，如果你能对实际方程做出同样的事情，那么如果水方程支持计算……

So, you can imagine a steampunk, but it’s really water-punk type of thing where…

所以，你可以想象蒸汽朋克，但它确实是水朋克类型的东西……

So, modern computers are electronic, they’re powered by electrons passing through very tiny wires and interacting with other electrons and so forth.

因此，现代计算机是电子的，它们由穿过非常细的电线并与其他电子相互作用的电子提供动力。

22:39 But instead of electrons, you can imagine these pulses of water moving a certain velocity.

但你可以想象这些以一定速度移动的水脉冲，而不是电子。

And maybe there are two different configurations corresponding to a bit being up or down.

也许有两种不同的配置对应于一位向上或向下。

Probably that if you had two of these moving bodies of water collide, they would come out with some new configuration, which would be something like an AND gate or OR gate, that the output would depend in a very predictable way on the inputs.

可能如果两个移动的水体发生碰撞，它们会产生一些新的配置，类似于“与”门或“或”门，输出将以一种非常可预测的方式取决于输入。

And you could chain these together and maybe create a Turing machine.

你可以将它们链接在一起，也许可以创建一个图灵机。

And then you have computers which are made completely out of water.

然后你就有了完全由水制成的计算机。

And if you have computers, then maybe you can do robotics, so hydraulics and so forth.

如果你有计算机，那么也许你可以研究机器人技术、液压技术等等。

And so you could create some machine which is basically a fluid analog, what’s called a von Neumann machine.

因此，您可以创建一些基本上是流体模拟的机器，即所谓的冯诺依曼机器。

23:26 So, von Neumann proposed if you want to colonize Mars, the sheer cost of transporting people in machines to Mars is just ridiculous, but if you could transport one machine to Mars, and this machine had the ability to mine the planet, create some more materials, smelt them and build more copies of the same machine, then you could colonize a whole planet over time.

因此，冯·诺依曼提出，如果你想殖民火星，用机器运送人到火星的绝对成本是荒谬的，但如果你可以运送一台机器到火星，并且这台机器有能力开采地球，创造更多材料，熔炼它们并建造更多同一台机器的副本，那么随着时间的推移，你可以殖民整个星球。

So, if you could build a fluid machine, which yeah, so it’s a fluid robot.

所以，如果你能建造一台流体机器，是的，那么它就是一个流体机器人。

And what it would do, its purpose in life, it’s programmed so that it would create a smaller version of itself in some sort of cold state.

它会做什么，它的人生目的，它被编程，以便它能够在某种冷状态下创建一个较小的自身版本。

It wouldn’t start just yet.

事情还没有开始。

Once it’s ready, the big robot configuration of water would transfer all its energy into the smaller configuration and then power down.

一旦准备就绪，大型水机器人配置会将其所有能量转移到较小的配置中，然后断电。

And then they clean itself up, and then what’s left is this newest state which would then turn on and do the same thing, but smaller and faster.

然后它们会自我清理，剩下的就是这个最新的状态，然后它会打开并做同样的事情，但更小、更快。

24:19 And then the equation has a certain scaling symmetry.

那么方程具有一定的标度对称性。

Once you do that, it can just keep iterating.

一旦你这样做了，它就可以继续迭代。

So, this, in principle, would create a blowup for the actual Navier-Stokes.

因此，原则上，这会给实际的纳维-斯托克斯造成爆炸。

And this is what I managed to accomplish for this average Navier-Stokes.

这就是我为这个普通的纳维-斯托克斯所做的努力。

So, it provided this sort of roadmap to solve the problem.

因此，它提供了这种解决问题的路线图。

Now, this is a pipe dream because there are so many things that are missing for this to actually be a reality.

现在，这只是一个白日梦，因为要真正成为现实还缺少很多东西。

So, I can’t create these basic logic gates.

所以，我无法创建这些基本的逻辑门。

I don’t have these special configurations of water.

我没有这些特殊配置的水。

There’s candidates, these include vortex rings that might possibly work.

有一些候选者，其中包括可能有用的涡环。

But also analog computing is really nasty compared to digital computing because there’s always errors.

但与数字计算相比，模拟计算确实很糟糕，因为总是存在错误。

You have to do a lot of error correction along the way.

一路上你必须做很多错误纠正。

25:05 I don’t know how to completely power down the big machine, so it doesn’t interfere the writing of the smaller machine, but everything in principle can happen.

我不知道如何完全关闭大机器的电源，这样就不会干扰小机器的写入，但原则上一切都可能发生。

It doesn’t contradict any of the laws of physics, so it’s sort of evidence that this thing is possible.

它并不违反任何物理定律，所以这是这件事是可能的证据。

There are other groups who are now pursuing ways to make Navier-Stokes blow up, which are nowhere near as ridiculously complicated as this.

现在还有其他团体正在寻求让纳维-斯托克斯爆炸的方法，而这些方法远没有这么复杂得可笑。

They actually are pursuing much closer to the direct self-similar model, which can…

他们实际上追求更接近直接自相似模型，这可以......

It doesn’t quite work as is, but there could be some simpler scheme they want to just describe to make this work.

它并不能按原样工作，但他们可能想描述一些更简单的方案来使其工作。

There is a real leap of genius here to go from Navier-Stokes to this Turing machine.

从纳维-斯托克斯到图灵机，这是真正的天才飞跃。

25:40 So, it goes from what the self-similar blob scenario that you’re trying to get the smaller and smaller blob to now having a liquid Turing machine gets smaller and smaller and smaller, and somehow seeing how that could be used to say something about a blowup.

因此，它从你试图获得越来越小的斑点的自相似斑点场景到现在拥有一个液体图灵机变得越来越小，并以某种方式看到如何用它来表达关于爆炸的事情。

That’s a big leap.

这是一个很大的飞跃。

So, there’s precedent.

所以，这是有先例的。

26:08 So, the thing about mathematics is that it’s really good at spotting connections between what you might think of as completely different problems, but if the mathematical form is the same, you can draw a connection.

所以，数学的特点是，它非常擅长发现你可能认为完全不同的问题之间的联系，但如果数学形式相同，你就可以画出联系。

So, there’s a lot of previously on what called cellular automata, the most famous of which is Conway’s Game of Life.

因此，之前有很多关于元胞自动机的研究，其中最著名的是康威的生命游戏。

There’s this infinite discrete grid, and at any given time, the grid is either occupied by a cell or it’s empty.

有一个无限离散的网格，在任何给定时间，网格要么被单元格占据，要么是空的。

And there’s a very simple rule that tells you how these cells evolve.

有一个非常简单的规则可以告诉你这些细胞是如何进化的。

So, sometimes cells live and sometimes they die.

因此，细胞有时存活，有时死亡。

And when I was a student, it was a very popular screen saver to actually just have these animations go on, and they look very chaotic.

当我还是个学生的时候，这是一个非常流行的屏幕保护程序，实际上只是让这些动画继续下去，而且它们看起来非常混乱。

In fact, they look a little bit like turbulent flow sometimes, but at some point people discovered more and more interesting structures within this Game of Life.

事实上，它们有时看起来有点像湍流，但在某些时候，人们在这个生命游戏中发现了越来越多有趣的结构。

So, for example, they discovered this thing called glider.

例如，他们发现了一种叫做滑翔机的东西。

27:00 So, a glider is a very tiny configuration of four or five selves which evolves and it just moves at a certain direction.

因此，滑翔机是一种由四到五个自我组成的非常微小的结构，它会不断进化，并且只会朝某个方向移动。

And that’s like this vortex rings [inaudible 00:27:09].

就像这个漩涡环一样 [听不清 00:27:09]。

Yeah, so this is an analogy, the Game of Life is a discrete equation, and the fluid Navier-Stokes is a continuous equation, but mathematically they have some similar features.

是的，所以这是一个类比，生命游戏是一个离散方程，而流体纳维-斯托克斯是一个连续方程，但在数学上它们有一些相似的特征。

And so over time people discovered more and more interesting things that you could build within the Game of Life.

随着时间的推移，人们发现了越来越多有趣的东西，可以在生命游戏中构建。

The Game of Life is a very simple system.

生命游戏是一个非常简单的系统。

It only has like three or four rules to do it, but you can design all kinds of interesting configurations inside it.

它只有三四个规则来完成它，但你可以在里面设计各种有趣的配置。

There’s some called a glider gun that does nothing that spit out gliders one at a time.

有一种叫做滑翔机枪的枪，它什么也不做，只能一次吐出一个滑翔机。

And then after a lot of effort, people managed to create AND gates and OR gates for gliders.

经过大量努力，人们成功地为滑翔机创建了“与”门和“或”门。

27:48 There’s this massive ridiculous structure, which if you have a stream of gliders coming in here and a stream of gliders coming in here, then you may produce extreme gliders coming out.

有一个巨大的荒谬的结构，如果你有一队滑翔机进入这里，又有一队滑翔机进入这里，那么你可能会产生极端的滑翔机出来。

Maybe if both of the streams have gliders, then there’ll be an output stream, but if only one of them does, then nothing comes out.

也许如果两个流都有滑翔机，那么就会有一个输出流，但如果只有一个流有滑翔机，那么什么也不会输出。

So, they could build something like that.

所以，他们可以建造类似的东西。

And once you could build these basic gates, then just from software engineering, you can build almost anything.

一旦你可以构建这些基本的门，那么仅仅通过软件工程，你就可以构建几乎任何东西。

You can build a Turing machine.

您可以构建一台图灵机。

It’s enormous steampunk type things.

这是巨大的蒸汽朋克类型的东西。

They look ridiculous.

他们看起来很可笑。

But then people also generated self-replicating objects in the Game of Life, a massive machine, a [inaudible 00:28:31] machine, which over a huge period of time and always look like glider guns inside doing these very steampunk calculations.

但后来人们也在生命游戏中生成了自我复制的物体，一台巨大的机器，一台[听不清的00:28:31]机器，在很长一段时间内，它总是看起来像滑翔机枪，里面做着这些非常蒸汽朋克的计算。

It would create another version of itself which could replicate.

它将创建自己的另一个可以复制的版本。

That’s so incredible.

这太不可思议了。

A lot of this was like community crowdsourced by amateur mathematicians actually.

实际上，其中很多就像业余数学家众包的社区。

28:42 So, I knew about that work.

所以，我知道那项工作。

And so that is part of what inspired me to propose the same thing with Navier-Stokes.

这也是我向纳维-斯托克斯提出同样想法的部分原因。

Seriously, analog is much worse than digital.

说实话，模拟比数字差得多。

It’s going to be…

这将会是……

You can’t just directly take deconstructions in the Game of Life and plunk them in.

你不能直接在生命游戏中进行解构并把它们放进去。

But again, it shows it’s possible.

但这再次表明这是可能的。

There’s a kind of emergence that happens with these cellular automata local rules…

这些元胞自动机局部规则会出现一种情况……

29:06 maybe it’s similar to fluids, I don’t know, but local rules operating at scale can create these incredibly complex dynamic structures.

也许它类似于流体，我不知道，但大规模运行的局部规则可以创建这些极其复杂的动态结构。

Do you think any of that is amenable to mathematical analysis?

您认为其中任何一个都适合数学分析吗？

Do we have the tools to say something profound about that?

我们是否有工具来表达一些深刻的内容？

The thing is, you can get these emergent very complicated structures, but only with very carefully prepared initial conditions.

问题是，你可以获得这些新兴的非常复杂的结构，但前提是需要非常仔细地准备初始条件。

29:34 So, these glider guns and gates and self-propelled machines, if you just plunk on randomly some cells and you unlink them, you will not see any of these.

因此，这些滑翔机枪、大门和自行式机器，如果你只是随机放置一些单元格并断开它们的链接，你将看不到任何这些。

And that’s the analogous situation with Navier-Stokes again, that with typical initial conditions, you will not have any of this weird computation going on.

这又是纳维-斯托克斯的类似情况，在典型的初始条件下，你不会进行任何这种奇怪的计算。

But basically through engineering, by specially designing things in a very special way, you can make clever constructions.

但基本上通过工程，通过以非常特殊的方式专门设计事物，你可以做出巧妙的构造。

I wonder if it’s possible to prove the negative of…

我想知道是否有可能证明...的否定

30:07 basically prove that only through engineering can you ever create something interesting.

基本上证明只有通过工程才能创造出有趣的东西。

Yeah.

是的。

30:16 This is a recurring challenge in mathematics that I call the dichotomy between structure and randomness, that most objects that you can generate in mathematics are random.

这是数学中反复出现的挑战，我称之为结构与随机性之间的二分法，即在数学中可以生成的大多数对象都是随机的。

They look like random, like the digital supply, well, we believe is a good example.

它们看起来像是随机的，就像数字供应一样，我们认为这是一个很好的例子。

But there’s a very small number of things that have patterns.

但只有极少数事物具有模式。

But now, you can prove something has a pattern by just constructing…

但现在，你可以通过构造来证明某些东西具有模式......

If something has a simple pattern and you have a proof that it does something like repeat itself every so often, you can do that and you can prove that…

如果某件事有一个简单的模式，并且你有证据证明它会经常重复自己，那么你可以做到这一点，并且可以证明......

For example, you can prove that most sequences of digits have no pattern.

例如，您可以证明大多数数字序列没有模式。

So, if you just pick digits randomly, there’s something called low-large numbers.

所以，如果你只是随机选择数字，就会有一种叫做小-大数字的东西。

It tells you you’re going to get as many ones as twos in the long run.

它告诉你，从长远来看，你会得到多达两个的。

But we have a lot fewer tools to…

但我们的工具却少得多……

31:01 If I give you a specific pattern like the digits of pi, how can I show that this doesn’t have some weird pattern to it?

如果我给你一个特定的模式，比如圆周率的数字，我怎样才能证明它没有一些奇怪的模式呢？

Some other work that I spent a lot of time on is to prove what are called structure theorems or inverse theorems that give tests for when something is very structured.

我花了很多时间做的其他一些工作是证明所谓的结构定理或逆定理，这些定理可以测试某些东西是否非常结构化。

So, some functions are what’s called additive.

所以，有些函数就是所谓的加法函数。

If you have a function of natural numbers of the natural numbers, so maybe two maps to four, three maps to six and so forth, some functions are what’s called additive, which means that if you add two inputs together, the output gets added as well.

如果你有一个自然数的自然数函数，那么可能两个映射到四，三个映射到六等等，有些函数就是所谓的加法，这意味着如果你将两个输入加在一起，输出也会被添加。

For example, a multiply by constant.

例如，乘以常数。

If you multiply a number by 10…

如果将一个数字乘以 10...

If you multiply A plus B by 10, that’s the same as multiplying A by 10 and B by 10, and then adding them together.

如果将 A 加 B 乘以 10，则相当于将 A 乘以 10，将 B 乘以 10，然后将它们相加。

So, some functions are additive, some functions are kind of additive but not completely additive.

因此，有些函数是可加的，有些函数是某种可加的但不是完全可加的。

31:47 So, for example, if I take a number, and I multiply by the square of two and I take the integer part of that, so 10 by square route of two is like 14 point something, so 10 up to 14, 20 or up to 28.

举例来说，如果我取一个数字，乘以二的平方，然后取其中的整数部分，那么 10 乘以二的平方就像 14 点一样，所以 10 到 14、20 或到 28。

So, in that case, additivity is true then, so 10 plus 10 is 20 and 14 plus 14 is 28.

因此，在这种情况下，可加性成立，因此 10 加 10 等于 20，14 加 14 等于 28。

But because of this rounding, sometimes there’s round-up errors, and sometimes when you add A plus A, this function doesn’t quite give you the sum of the two individual outputs, but the sum plus/minus one.

但由于这种舍入，有时会出现舍入错误，有时当您将 A 加 A 时，此函数并不能完全给出两个单独输出的总和，而是给出加/减 1 的总和。

So, it’s almost additive, but not quite additive.

所以，它几乎是可加的，但又不完全是可加的。

32:21 So, there’s a lot of useful results in mathematics, and I’ve worked a lot on developing things like this, to the effect that if a function exhibits some structure like this, then it’s basically there’s a reason for why it’s true.

所以，数学中有很多有用的结果，我在开发这样的东西上做了很多工作，大意是，如果一个函数表现出这样的结构，那么它基本上就有一个理由说明为什么它是正确的。

And the reason is because there’s some other nearby function, which is actually completely structured, which is explaining this sort of partial pattern that you have.

原因是因为附近还有其他一些函数，它们实际上是完全结构化的，这解释了您所拥有的这种部分模式。

And so if you have these inverse theorems, it creates this dichotomy that either the objects that you study are either have no structure at all or they are somehow related to something kind of structured.

因此，如果你有这些逆定理，就会产生这种二分法，即你研究的对象要么根本没有结构，要么以某种方式与某种结构化相关。

And in either way, in either case, you can make progress.

无论哪种方式，无论哪种情况，你都可以取得进步。

A good example of this is that there’s this old theorem in mathematics-

一个很好的例子是数学中有一个古老的定理——

33:01 A good example of this is that there’s this old theorem in mathematics called Szemerédi’s Theorem, proven in the 1970s.

一个很好的例子是数学中有一个古老的定理，称为 Szemerédi 定理，于 20 世纪 70 年代得到证明。

It concerns trying to find a certain type of pattern in a set of numbers, the patterns of arithmetic progression.

它涉及试图在一组数字中找到某种类型的模式，即算术级数的模式。

Things like three, five, and seven or 10, 15 and 20, and Szemerédi, Endre Szemerédi proved that any set of numbers that are sufficiently big, what’s called positive density, has arithmetic progressions in it of any length you wish.

像三、五、七或 10、15 和 20 之类的东西，以及 Szemerédi、Endre Szemerédi 证明了任何足够大的数字集，即所谓的正密度，都可以具有任意长度的算术级数。

33:28 For example, the odd numbers have a density of one half, and they contain arithmetic progressions of any length.

例如，奇数的密度为二分之一，并且它们包含任意长度的算术级数。

So in that case, it’s obvious, because the odd numbers are really, really structured.

所以在这种情况下，这是显而易见的，因为奇数确实非常有结构。

I can just take 11, 13, 15, 17, I can easily find arithmetic progressions in that set, but Szemerédi’s theorem also applies to random sets.

我可以只取 11、13、15、17，我可以轻松地在该集合中找到算术级数，但 Szemerédi 定理也适用于随机集合。

If I take a set of odd numbers and I flip a coin for each number, and I only keep the numbers for which I got a heads…

如果我取一组奇数，然后为每个数字掷一枚硬币，并且只保留正面朝上的数字……

So I just flip coins, I just randomly take out half the numbers, I keep one half.

所以我只是抛硬币，我只是随机取出一半的数字，我保留一半。

That’s a set that has no patterns at all, but just from random fluctuations, you will still get a lot of arithmetic progressions in that set.

这是一个完全没有模式的集合，但仅从随机波动中，您仍然会在该集合中得到很多算术级数。

Can you prove that there’s arithmetic progressions of arbitrary length within a random-

你能证明在随机数中存在任意长度的算术级数吗？

Yes.

是的。

34:17 Have you heard of the infinite monkey theorem?

你听说过无限猴子定理吗？

Usually, mathematicians give boring names to theorems, but occasionally they give colorful names.

通常，数学家会给定理起一些无聊的名字，但偶尔他们也会给定理起一些有趣的名字。

Yes.

是的。

The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room, each with typewriter, they type out text randomly, almost surely, one of them is going to generate the entire script of Hamlet, or any other finite string of text.

无限猴子定理的流行版本是，如果一个房间里有无数只猴子，每只猴子都有打字机，它们随机地打出文本，几乎可以肯定，其中一只将生成哈姆雷特的整个脚本，或任何其他有限的文本字符串。

34:24 It’ll just take some time, quite a lot of time, actually, but if you have an infinite number, then it happens.

这只是需要一些时间，实际上是相当多的时间，但如果你有无限的数量，那么它就会发生。

34:44 So basically, the theorem is that if you take an infinite string of digits or whatever, eventually any finite pattern you wish will emerge.

所以基本上，定理是，如果你采用无限的数字串或其他任何东西，最终你想要的任何有限模式都会出现。

It may take a long time, but it will eventually happen.

这可能需要很长时间，但最终会发生。

In particular, arithmetic progressions of any length will eventually happen, but you need an extremely long random sequence for this to happen.

特别是，任何长度的算术级数最终都会发生，但你需要一个极长的随机序列才能发生。

I suppose that’s intuitive.

我想这很直观。

35:04 It’s just infinity.

这只是无穷大。

Yeah, infinity absorbs a lot of sins.

是的，无限吸收了很多罪恶。

Yeah.

是的。

35:11 How we humans supposed to deal with infinity?

我们人类应该如何应对无限？

Well, you can think of infinity as an abstraction of a finite number of which you do not have a bound.

好吧，您可以将无穷大视为无限数量的抽象。

35:15 So nothing in real life is truly infinite, but you can ask yourself questions like, “What if I had as much money as I wanted?”, or, “What if I could go as fast as I wanted?”, and a way in which mathematicians formalize that is mathematics has found a formalism to idealize, instead of something being extremely large or extremely small, to actually be exactly infinite or zero, and often the mathematics becomes a lot cleaner when you do that.

所以现实生活中没有什么是真正无限的，但你可以问自己这样的问题，“如果我有我想要的那么多钱怎么办？”，或者，“如果我能像我想要的那么快走得那么快怎么办？”，数学家形式化的方式就是数学找到了一种形式主义来理想化，而不是非常大或非常小的东西，实际上是无限或零，当你这样做时，数学通常会变得更加清晰。

I mean, in physics, we joke about assuming spherical cows, real world problems have got all kinds of real world effects, but you can idealize, send some things to infinity, send some things to zero, and the mathematics becomes a lot simpler to work within.

我的意思是，在物理学中，我们开玩笑说假设是球形牛，现实世界的问题会产生各种现实世界的影响，但你可以理想化，将一些东西发送到无穷大，将一些东西发送到零，并且数学变得更简单。

I wonder how often using infinity forces us to deviate from the physics of reality.

我想知道有多少次使用无穷大会迫使我们偏离现实的物理学。

So there’s a lot of pitfalls.

所以有很多陷阱。

36:17 So we spend a lot of time in undergraduate math classes teaching analysis, and analysis is often about how to take limits and whether…

因此，我们在本科数学课上花了很多时间教授分析，而分析通常是关于如何接受限制以及是否……

36:28 So for example, A plus B is always B plus A.

例如，A 加 B 始终是 B 加 A。

So when you have a finite number of terms and you add them, you can swap them and there’s no problem, but when you have an infinite number of terms, they’re these sort of show games you can play where you can have a series which converges to one value, but you rearrange it, and it suddenly converges to another value, and so you can make mistakes.

因此，当你有有限数量的项并将它们相加时，你可以交换它们，没有问题，但是当你有无限数量的项时，它们就像你可以玩的表演游戏，你可以有一系列收敛到一个值的序列，但你重新排列它，它突然收敛到另一个值，所以你可能会犯错误。

You have to know what you’re doing when you allow infinity.

当你允许无限​​时，你必须知道你在做什么。

You have to introduce these epsilons and deltas, and there’s a certain type of wave of reasoning that helps you avoid mistakes.

你必须引入这些 epsilon 和 delta，并且有某种类型的推理波可以帮助你避免错误。

36:58 In more recent years, people have started taking results that are true in infinite limits and what’s called finitizing them.

近年来，人们开始得出无限极限下正确的结果，即所谓的有限化。

So you know that something’s true eventually, but you don’t know when.

所以你知道某件事最终是真的，但你不知道什么时候。

Now give me a rate.

现在给我一个价格。

So such…

所以这样的……

If I don’t have an infinite number of monkeys, but a large finite number of monkeys, how long do I have to wait for Hamlet to come out?

如果我没有无限数量的猴子，而是大量有限数量的猴子，我要等多久才能看到哈姆雷特出来？

That’s a more quantitative question, and this is something that you can attack by purely finite methods, and you can use your finite intuition, and in this case, it turns out to be exponential in the length of the text that you’re trying to generate.

这是一个更加定量的问题，你可以通过纯粹有限的方法来解决这个问题，你可以使用你有限的直觉，在这种情况下，它会随着你试图生成的文本的长度呈指数增长。

37:36 So this is why you never see the monkeys create Hamlet.

所以这就是为什么你永远不会看到猴子创造哈姆雷特。

You can maybe see them create a four letter word, but nothing that big, and so I personally find once you finitize an infinite statement, it does come much more intuitive, and it’s no longer so weird.

你也许可以看到他们创建了一个四个字母的单词，但没有那么大，所以我个人发现，一旦你确定了一个无限语句，它确实变得更加直观，并且不再那么奇怪。

So even if you’re working with infinity, it’s good to finitize so that you can have some intuition?

因此，即使您正在处理无穷大，最好进行有限化以便您可以有一些直觉？

Yeah, the downside is that the finitized groups are just much, much messier.

是的，缺点是有限组非常非常混乱。

37:57 So the infinite ones are found first usually, decades earlier, and then later on, people finitize them.

因此，无穷大通常是在几十年前首先被发现的，然后，人们将它们有限化。

So since we mentioned a lot of math and a lot of physics, what is the difference between mathematics and physics as disciplines, as ways of understanding, of seeing the world?

既然我们提到了很多数学和很多物理，那么作为学科、作为理解和看待世界的方式，数学和物理之间有什么区别？

38:07 Maybe we can throw engineering in there, you mentioned your wife is an engineer, give it new perspective on circuits.

也许我们可以把工程学放在那里，你提到你的妻子是一名工程师，给它带来关于电路的新视角。

So this different way of looking at the world, given that you’ve done mathematical physics, so you’ve worn all the hats.

鉴于你已经完成了数学物理，所以你以不同的方式看待世界，所以你已经戴上了所有的帽子。

Right.

正确的。

38:30 So I think science in general is interaction between three things.

所以我认为科学一般是三件事之间的相互作用。

There’s the real world, there’s what we observe of the real world, observations, and then our mental models as to how we think the world works.

有现实世界，有我们对现实世界的观察、观察，然后是我们认为世界如何运作的心智模型。

38:46 We can’t directly access reality.

我们无法直接接触现实。

All we have are the observations, which are incomplete and they have errors, and there are many, many cases where we want to know, for example, what is the weather like tomorrow, and we don’t yet have the observation, but we’d like to.

我们拥有的只是观测数据，这些观测数据是不完整的，而且有错误，而且有很多很多情况我们想知道，例如，明天的天气怎么样，我们还没有观测数据，但我们很想知道。

A prediction.

一个预测。

39:04 Then we have these simplified models, sometimes making unrealistic assumptions, spherical cow type things.

然后我们有这些简化的模型，有时会做出不切实际的假设，球形牛型的东西。

Those are the mathematical models.

这些是数学模型。

39:11 Mathematics is concerned with the models.

数学与模型有关。

Science collects the observations, and it proposes the models that might explain these observations.

科学收集观察结果，并提出可以解释这些观察结果的模型。

What mathematics does, we stay within the model, and we ask what are the consequences of that model?

数学做什么，我们留在模型中，我们问该模型的后果是什么？

What observations, what predictions would the model make of future observations, or past observations?

模型会对未来的观察或过去的观察做出什么观察、预测？

Does it fit?

合适吗？

Observe data?

观察数据？

39:35 So there’s definitely a symbiosis.

所以肯定存在共生关系。

I guess mathematics is unusual among other disciplines is that we start from hypotheses, like the axioms of a model, and ask what conclusions come up from that model.

我想数学在其他学科中的不同寻常之处在于，我们从假设开始，比如模型的公理，然后询问从该模型中得出什么结论。

In almost any other discipline, you start with the conclusions.

在几乎所有其他学科中，你都是从结论开始的。

“I want to do this.

“我想做这个。

I want to build a bridge, I want to make money, I want to do this,” and then you find the paths to get there.

我想建一座桥，我想赚钱，我想这样做，”然后你找到到达那里的路径。

There’s a lot less sort of speculation about, “Suppose I did this, what would happen?”.

关于“假设我这样做，会发生什么？”的猜测要少得多。

Planning and modeling.

规划和建模。

Speculative fiction maybe is one other place, but that’s about it, actually.

推理小说也许是另一回事，但实际上仅此而已。

Most of the things we do in life is conclusions driven, including physics and science.

我们生活中所做的大多数事情都是结论驱动的，包括物理和科学。

I mean, they want to know, “Where is this asteroid going to go?

我的意思是，他们想知道，“这颗小行星要去哪里？

What is the weather going to be tomorrow?”, but mathematics also has this other direction of going from the axioms.

明天的天气怎么样？”，但是数学也有从公理出发的另一个方向。

What do you think…

你觉得怎么样……

40:32 There is this tension in physics between theory and experiment.

物理学中理论与实验之间存在着这种张力。

What do you think is the more powerful way of discovering truly novel ideas about reality?

您认为发现有关现实的真正新颖想法的更有效方法是什么？

Well, you need both, top down and bottom up.

嗯，你需要自上而下和自下而上。

40:42 It’s really an interaction between all these…

这确实是所有这些之间的相互作用……

So over time, the observations and the theory and the modeling should both get closer to reality, but initially, and this is always the case out there, they’re always far apart to begin with, but you need one to figure out where to push the other.

因此，随着时间的推移，观察结果、理论和模型都应该越来越接近现实，但一开始，情况总是如此，它们一开始总是相距甚远，但你需要一个来弄清楚将另一个推到哪里。

41:04 So if your model is predicting anomalies that are not predicted by experiment, that tells experimenters where to look to find more data to refine the models.

因此，如果您的模型预测了实验未预测到的异常，那么这会告诉实验人员在哪里寻找更多数据来完善模型。

So it goes back and forth.

所以它来回。

41:21 Within mathematics itself, there’s also a theory and experimental component.

在数学本身中，也有理论和实验的组成部分。

It’s just that until very recently, theory has dominated almost completely.

只是直到最近，理论几乎完全占据主导地位。

99% of mathematics is theoretical mathematics, and there’s a very tiny amount of experimental mathematics.

99%的数学都是理论数学，只有极少量的实验数学。

People do do it.

人们确实这么做了。

If they want to study prime numbers or whatever, they can just generate large data sets.

如果他们想研究素数或其他什么，他们可以生成大型数据集。

41:41 So once we had the computers, we had to do it a little bit.

因此，一旦我们拥有了计算机，我们就必须做一些事情。

Although even before…

尽管甚至之前……

Well, like Gauss for example, he discovered a reconjection, the most basic theorem in number theory, called the prime number theorem, which predicts how many primes up to a million, up to a trillion.

嗯，就像高斯一样，他发现了一个重猜想，这是数论中最基本的定理，称为素数定理，它可以预测一百万、一万亿以内的素数有多少个。

It’s not an obvious question, and basically what he did was that he computed, mostly by himself, but also hired human computers, people whose professional job it was to do arithmetic, to compute the first hundred thousand primes or something, and made tables and made a prediction.

这不是一个显而易见的问题，基本上他所做的就是计算，主要是他自己计算，但也雇佣了人类计算机，这些人的专业工作是算术，计算前十万个素数之类的东西，并制作表格并做出预测。

That was an early example of experimental mathematics, but until very recently, it was not…

这是实验数学的一个早期例子，但直到最近，它还不是……

42:22 I mean, theoretical mathematics was just much more successful.

我的意思是，理论数学要成功得多。

Of course, doing complicated mathematical computations was just not feasible until very recently, and even nowadays, even though we have powerful computers, only some mathematical things can be explored numerically.

当然，直到最近，进行复杂的数学计算才变得可行，即使在今天，即使我们拥有强大的计算机，也只能对一些数学事物进行数值探索。

42:37 There’s something called the combinatorial explosion.

有一种东西叫做组合爆炸。

If you want us to study, for example, Szemerédi’s theorem, you want to study all possible subsets of numbers one to a thousand.

例如，如果你想让我们研究 Szemerédi 定理，你想研究数字一到一千的所有可能的子集。

There’s only 1000 numbers.

只有1000个号码。

How bad could it be?

有多糟糕？

It turns out the number of different subsets of one to a thousand is two to the power of 1000, which is way bigger than any computer can currently enumerate.

事实证明，1 到 1000 的不同子集的数量是 2 的 1000 次方，这比当前任何计算机可以枚举的都要大得多。

42:59 So there are certain math problems that very quickly become just intractable to attack by direct brute force computation.

因此，某些数学问题很快就变得难以通过直接强力计算来解决。

Chess is another famous example.

国际象棋是另一个著名的例子。

The number of chess positions, we can’t get a computer to fully explore, but now we have AI, we have tools to explore this space, not with 100% guarantees of success, but with experiment.

国际象棋棋局的数量，我们无法让计算机来充分探索，但现在我们有了人工智能，我们有了工具来探索这个空间，不是100%保证成功，而是通过实验。

So we can empirically solve chess now.

所以我们现在可以凭经验解决国际象棋问题。

For example, we have very, very good AIs that don’t explore every single position in the game tree, but they have found some very good approximation, and people are using actually these chess engines to do experimental chess.

例如，我们有非常非常好的人工智能，它们不会探索游戏树中的每个位置，但它们已经找到了一些非常好的近似值，并且人们实际上正在使用这些国际象棋引擎来进行实验性国际象棋。

They’re revisiting old chess theories about, “Oh, when you do this type of opening…

他们正在重新审视古老的国际象棋理论，“哦，当你做这种开局时......

This is a good type of move, this is not,” and they can use these chess engines to actually refine, and in some cases, overturn conventional wisdom about chess, and I do hope that that mathematics will have a larger experimental component in the future, perhaps powered by AI.

这是一种很好的走法，这不是，”他们可以使用这些国际象棋引擎来实际改进，在某些情况下，颠覆有关国际象棋的传统智慧，我确实希望数学在未来会有更大的实验成分，也许由人工智能提供支持。

We’ll, of course, talk about that, but in the case of chess, and there’s a similar thing in mathematics, I don’t believe it’s providing a kind of formal explanation of the different positions.

当然，我们会讨论这一点，但就国际象棋而言，数学中也有类似的事情，我不认为它为不同的立场提供了一种正式的解释。

44:05 It’s just saying which position is better or not that you can intuit as a human being, and then from that, we humans can construct a theory of the matter.

只是说哪种立场更好或哪种立场更好，作为一个人，你可以凭直觉来判断，然后我们人类可以据此构建一个关于事物的理论。

44:27 You’ve mentioned the Plato’s cave allegory.

你提到了柏拉图的洞穴寓言。

In case people don’t know, it’s where people are observing shadows of reality, not reality itself, and they believe what they’re observing to be reality.

如果人们不知道的话，那就是人们正在观察现实的影子，而不是现实本身，并且他们相信他们所观察到的就是现实。

Is that, in some sense, what mathematicians and maybe all humans are doing, is looking at shadows of reality?

从某种意义上说，这是否是数学家甚至全人类正在做的事情，是在观察现实的影子？

Is it possible for us to truly access reality?

我们有可能真正接触到现实吗？

Well, there are these three ontological things.

嗯，存在这三个本体论的东西。

44:55 There’s actual reality, there’s observations and our models, and technically they are distinct, and I think they will always be distinct, but they can get closer over time, and the process of getting closer often means that you have to discard your initial intuitions.

有实际的现实，有观察和我们的模型，从技术上来说它们是不同的，我认为它们将永远是不同的，但随着时间的推移它们可以变得更接近，而接近的过程通常意味着你必须放弃你最初的直觉。

So astronomy provides great examples, like an initial model of the world is flat because it looks flat and it’s big, and the rest of the universe, the skies, is not.

天文学提供了很好的例子，就像世界的初始模型是平坦的，因为它看起来平坦而且很大，而宇宙的其余部分，即天空，却不是。

The sun, for example, looks really tiny.

例如，太阳看起来非常小。

45:38 So you start off with a model, which is actually really far from reality, but it fits the observations that you have.

因此，您从一个模型开始，该模型实际上与现实相去甚远，但它符合您的观察结果。

So things look good, but over time, as you make more and more observations, bring it closer to reality, the model gets dragged along with it, and so over time, we had to realize that the earth was round, that it spins, it goes around the solar system, solar system goes around the galaxy, and so on and so forth, and the universe was expanding.

所以事情看起来不错，但随着时间的推移，当你进行越来越多的观察，使其更接近现实时，模型就会随之被拖拽，所以随着时间的推移，我们必须意识到地球是圆的，它旋转，它绕着太阳系旋转，太阳系绕着银河系旋转，等等，宇宙正在膨胀。

Expansions is self-expanding, accelerating, and in fact, very recently this year…

扩张是自我扩张、加速的，事实上，就在今年……

So even the acceleration of the universe itself, this evidence now is non-constant.

因此，即使是宇宙本身的加速，这个证据现在也不是恒定的。

The explanation behind why that is…

其背后的解释是……

It’s catching up.

它正在迎头赶上。

It’s catching up.

它正在迎头赶上。

46:18 I mean, it’s still the dark matter, dark energy, this kind of thing.

我的意思是，它仍然是暗物质、暗能量之类的东西。

We have a model that explains, that fits the data really well.

我们有一个可以解释的模型，它非常适合数据。

46:23 It just has a few parameters that you have to specify.

它只有一些您必须指定的参数。

So people say, “Oh, that’s fudge factors.

所以人们会说，“哦，那是捏造的因素。

With enough fudge factors, you can explain anything,” but the mathematical point over the model is that you want to have fewer parameters in your model and data points in your observational set.

有了足够的捏造因素，你就可以解释任何事情，”但模型的数学要点是你希望模型中的参数和观测集中的数据点更少。

46:43 So if you have a model with 10 parameters that explains 10 observations, that is a completely useless model, its what’s called overfitted, but if you have a model with two parameters and it explains a trillion observations, which is basically the dark matter model, I think it has 14 parameters, and it explains petabytes of data that the astronomers have.

因此，如果你有一个具有 10 个参数的模型，可以解释 10 个观测值，那么这是一个完全无用的模型，这就是所谓的过度拟合，但如果你有一个具有两个参数的模型，并且它解释了一万亿个观测值，这基本上就是暗物质模型，我认为它有 14 个参数，并且它解释了天文学家拥有的 PB 数据。

47:06 You can think of a theory.

你可以想出一个理论。

One way to think about a physical mathematical theory is it’s a compression of the universe, and a data compression.

思考物理数学理论的一种方式是它是宇宙的压缩和数据的压缩。

So you have these petabytes of observations, you like to compress it to a model which you can describe in five pages and specify a certain number of parameters, and if it can fit, to reasonable accuracy, almost all of your observations, the more compression that you make, the better your theory.

因此，您拥有这些 PB 的观察结果，您希望将其压缩为一个模型，您可以在五页中描述该模型并指定一定数量的参数，如果它能够以合理的精度拟合几乎所有的观察结果，则压缩得越多，您的理论就越好。

In fact, one of the great surprises of our universe and of everything in it is that it’s compressible at all.

事实上，我们的宇宙和其中的一切的最大惊喜之一就是它是可压缩的。

47:32 That’s the unreasonable effectiveness of mathematics

这就是数学不合理的有效性

Yeah, Einstein had a quote like that.

是的，爱因斯坦曾说过这样一句话。

47:40 “The most incomprehensible thing about the universe is that it is comprehensible.”

“宇宙最不可理解的事情就是它是可以理解的。”

Right, and not just comprehensible.

是的，而且不仅仅是可以理解的。

47:45 You can do an equation like e=MC2.

你可以做一个像 e=MC2 这样的方程。

There is actually some possible explanation for that.

实际上对此有一些可能的解释。

47:49 So there’s this phenomenon in mathematics called universality.

所以数学中有一种现象叫做普遍性。

So, many complex systems at the macro scale are coming out of lots of tiny interactions at the macro scale, and normally, because of the commutative explosion, you would think that the macro scale equations must be infinitely, exponentially more complicated than the macro scale ones, and they are, if you want to solve them completely exactly.

因此，许多宏观尺度上的复杂系统都是由宏观尺度上的许多微小相互作用产生的，通常，由于交换爆炸，你会认为宏观尺度方程一定比宏观尺度方程无限、指数地复杂，如果你想完全精确地求解它们，它们确实是复杂的。

If you want to model all the atoms in a box of air…

如果你想模拟一盒空气中的所有原子......

48:21 Like Avogadro’s number is humongous.

就像阿伏伽德罗的数字是巨大的。

There’s a huge number of particles.

有大量的粒子。

If you actually tried to track each one, it’ll be ridiculous, but certain laws emerge at the microscopic scale that almost don’t depend on what’s going on at the macro scale, or only depend on a very small number of parameters.

如果你真的试图追踪每一个，那将是荒谬的，但某些法则出现在微观尺度上，几乎不依赖于宏观尺度上发生的事情，或者只依赖于极少数的参数。

48:35 So if you want to model a gas of a quintillion particles in a box, you just need to know is temperature and pressure and volume, and a few parameters, like five or six, and it models almost everything you need to know about these 10 to 23 or whatever particles.

因此，如果你想对盒子中的 500 亿个粒子的气体进行建模，你只需要知道温度、压力和体积，以及一些参数，例如 5 个或 6 个参数，它就可以对你需要了解的关于这 10 到 23 个或任何粒子的几乎所有内容进行建模。

So we don’t understand universality anywhere near as we would like mathematically, but there are much simpler toy models where we do have a good understanding of why universality occurs.

因此，我们对普遍性的理解并不像我们在数学上所希望的那样，但有更简单的玩具模型，我们确实可以很好地理解普遍性发生的原因。

The most basic one is the central limit theorem that explains why the bell curve shows up everywhere in nature, that so many things are distributed by what’s called a Gaussian distribution, famous bell curve.

最基本的一个是中心极限定理，它解释了为什么钟形曲线在自然界中随处可见，以至于许多事物都遵循所谓的高斯分布，即著名的钟形曲线。

There’s now even a meme with this curve.

现在甚至有一个带有这条曲线的模因。

And even the meme applies broadly.

甚至模因也广泛适用。

49:18 The universality to the meme.

模因的普遍性。

Yes, you can go meta if you like, but there are many, many processes.

是的，如果你愿意的话，你可以使用元，但是有很多很多的过程。

49:22 For example, you can take lots of independent random variables and average them together in various ways.

例如，您可以采用许多独立的随机变量并以各种方式将它们平均在一起。

You can take a simple average or more complicated average, and we can prove in various cases that these bell curves, these Gaussians, emerge, and it is a satisfying explanation.

你可以采用简单的平均值或更复杂的平均值，我们可以在各种情况下证明这些钟形曲线、这些高斯曲线的出现，这是一个令人满意的解释。

49:44 Sometimes they don’t.

有时他们不这样做。

So if you have many different inputs and they’re all correlated in some systemic way, then you can get something very far from a bell curve to show up, and this is also important to know when [inaudible 00:49:55] fails.

因此，如果您有许多不同的输入，并且它们都以某种系统方式相关，那么您可能会得到一些与钟形曲线相距甚远的东西，这对于了解 [听不清 00:49:55] 何时失败也很重要。

So universality is not a 100% reliable thing to rely on.

因此，通用性并不是 100% 可靠的东西。

The global financial crisis was a famous example of this.

全球金融危机就是一个著名的例子。

People thought that mortgage defaults had this sort of Gaussian type behavior, that if a population of a hundred thousand Americans with mortgages ask what proportion of them would default on their mortgages, if everything was de-correlated, it would be an asset bell curve, and you can manage risk of options and derivatives and so forth, and there’s a very beautiful theory, but if there are systemic shocks in the economy that can push everybody to default at the same time, that’s very non-Gaussian behavior, and this wasn’t fully accounted for in 2008.

人们认为抵押贷款违约具有这种高斯型行为，如果十万有抵押贷款的美国人问他们中有多少比例会拖欠抵押贷款，如果一切都去相关，这将是一条资产钟形曲线，你可以管理期权和衍生品等的风险，这是一个非常漂亮的理论，但如果经济中存在系统性冲击可以迫使每个人同时违约，那就是非常非高斯行为，而这不是2008年已全部核算完毕。

50:45 Now I think there’s some more awareness that this systemic risk is actually a much bigger issue, and just because the model is pretty and nice, it may not match reality.

现在我认为人们更多地认识到这种系统性风险实际上是一个更大的问题，仅仅因为模型很漂亮，它可能与现实不符。

So the mathematics of working out what models do is really important, but also the science of validating when the models fit reality and when they don’t…

因此，计算模型的作用的数学非常重要，而且验证模型何时适合现实、何时不适合的科学也很重要……

You need both, but mathematics can help, because for example, these central limit theorems, it tells you that if you have certain axioms like non-correlation, that if all the inputs were not correlated to each other, then you have this Gaussian behavior and things are fine.

你两者都需要，但数学可以提供帮助，因为例如，这些中心极限定理，它告诉你，如果你有某些公理，比如非相关性，如果所有输入彼此不相关，那么你就有这种高斯行为，一切都很好。

It tells you where to look for weaknesses in the model.

它告诉您在哪里寻找模型中的弱点。

51:25 So if you have a mathematical understanding of Szemerédi’s theorem, and someone proposes to use these Gaussian [inaudible 00:51:32] or whatever to model default risk, if you’re mathematically trained, you would say, “Okay, but what are the systemic correlation between all your inputs?”, and so then you can ask the economist, “How much of a risk is that?”, and then you can go look for that.

因此，如果你对 Szemerédi 定理有数学上的理解，并且有人建议使用这些高斯[听不清 00:51:32] 或其他什么来建模违约风险，如果你受过数学训练，你会说，“好吧，但是你所有输入之间的系统相关性是什么？”，然后你可以问经济学家，“风险有多大？”，然后你可以去寻找它。

So there’s always this synergy between science and mathematics.

所以科学和数学之间总是存在这种协同作用。

A little bit on the topic of universality, you’re known and celebrated for working across an incredible breadth of mathematics, reminiscent of Hilbert a century ago.

稍微谈一下普适性的话题，你因在令人难以置信的数学领域的研究而闻名并受到赞誉，让人想起一个世纪前的希尔伯特。

51:52 In fact, the great Fields Medal winning mathematician Tim Gowers has said that you are the closest thing we get to Hilbert.

事实上，伟大的菲尔兹奖获得者数学家蒂姆·高尔斯曾说过，你是我们最接近希尔伯特的人。

He’s a colleague of yours.

他是你的同事。

Oh yeah, good friend.

哦，是的，好朋友。

But anyway, so you are known for this ability to go both deep and broad in mathematics.

但无论如何，你因在数学方面既深入又广泛的能力而闻名。

52:16 So you’re the perfect person to ask.

所以你是最适合提问的人。

Do you think there are threads that connect all the disparate areas of mathematics?

您认为数学的所有不同领域之间是否存在连接的线索？

Is there a kind of a deep, underlying structure to all of mathematics?

所有数学是否都存在一种深层的基础结构？

There’s certainly a lot of connecting threads, and a lot of the progress of mathematics can be represented by taking…

当然有很多连接线，数学的很多进步可以通过……来表示。

52:36 By stories of two fields of mathematics that were previously not connected, and finding connections.

通过两个以前没有联系的数学领域的故事，并找到联系。

52:50 An ancient example is geometry and number theory.

一个古老的例子是几何和数论。

So in the times of the ancient Greeks, these were considered different subjects.

所以在古希腊时代，这些被认为是不同的主题。

I mean, mathematicians worked on both.

我的意思是，数学家对这两个问题都进行了研究。

Euclid worked both on geometry, most famously, but also on numbers, but they were not really considered related.

欧几里得不仅研究最著名的几何学，而且还研究数字，但人们并不认为它们真正相关。

I mean, a little bit, like you could say that this length was five times this length because you could take five copies of this length and so forth, but it wasn’t until Descartes, who developed analytical geometry, that you can parameterize the plane, a geometric object, by two real numbers.

我的意思是，有点像你可以说这个长度是这个长度的五倍，因为你可以复制这个长度的五个副本等等，但直到笛卡尔发展了解析几何，你才可以通过两个实数来参数化平面，一个几何对象。

So geometric problems can be turned into problems about numbers.

所以几何问题可以转化为数字问题。

53:35 Today this feels almost trivial.

今天，这感觉几乎是微不足道的。

There’s no content to this.

没有这方面的内容。

Of course, a plane is X and Y, because that’s what we teach and it’s internalized, but it was an important development that these two fields were unified, and this process has just gone on throughout mathematics over and over again.

当然，平面是X和Y，因为那是我们所教授的内容，并且它已经内化了，但是这两个领域的统一是一个重要的发展，这个过程在数学中一遍又一遍地进行着。

Algebra and geometry were separated, and now we have this fluid, algebraic geometry that connects them, and over and over again, and that’s certainly the type of mathematics that I enjoy the most.

代数和几何是分开的，现在我们有这种流动的代数几何将它们连接起来，一遍又一遍，这当然是我最喜欢的数学类型。

54:06 I think there’s sort of different styles to being a mathematician.

我认为成为一名数学家有不同的风格。

I think hedgehogs and fox…

我认为刺猬和狐狸...

A fox knows many things a little bit, but a hedgehog knows one thing very, very well, and in mathematics, there’s definitely both hedgehogs and foxes, and then there’s people who can play both roles, and I think ideal collaboration, British mathematicians involves very…

狐狸知道很多事情，但刺猬知道一件事非常非常清楚，在数学中，肯定有刺猬和狐狸，然后有人可以扮演这两个角色，我认为理想的合作，英国数学家涉及非常......

You need some diversity, like a fox working with many hedgehogs or vice versa, but I identify mostly as a fox, certainly.

你需要一些多样性，就像一只狐狸与许多刺猬一起工作，反之亦然，但我当然主要认为自己是一只狐狸。

I like arbitrage, somehow.

不知怎的，我喜欢套利。

Learning how one field works, learning the tricks of that wheel, and then going to another field which people don’t think is related, but I can adapt the tricks.

学习一个领域是如何运作的，学习那个轮子的技巧，然后去另一个人们认为不相关的领域，但我可以适应这些技巧。

So see the connections between the fields.

所以看看字段之间的联系。

Yeah.

是的。

54:52 So there are other mathematicians who are far deeper than I am.

所以还有其他数学家比我深刻得多。

They’re really hedgehogs.

他们真的是刺猬。

They know everything about one field, and they’re much faster and more effective in that field, but I can give them these extra tools.

他们了解某个领域的一切，并且在该领域更快、更有效，但我可以为他们提供这些额外的工具。

I mean, you’ve said that you can be both a hedgehog and the fox, depending on the context, depending on the collaboration.

我的意思是，你说过你既可以是刺猬，也可以是狐狸，这取决于具体情况，取决于合作。

55:05 So can you, if it’s at all possible, speak to the difference between those two ways of thinking about a problem?

那么，如果可能的话，您能谈谈这两种思考问题的方式之间的区别吗？

Say you’re encountering a new problem, searching for the connections versus very singular focus.

假设你遇到了一个新问题，正在寻找联系而不是非常单一的焦点。

I’m much more comfortable with the fox paradigm.

我对狐狸范式更加满意。

55:26 Yeah.

是的。

So yeah, I like looking for analogies, narratives.

所以是的，我喜欢寻找类比和叙述。

I spend a lot of time…

我花了很多时间……

If there’s a result, I see it in one field, and I like the result, it’s a cool result, but I don’t like the proof, it uses types of mathematics that I’m not super familiar with, I often try to re-prove it myself using the tools that I favor.

如果有一个结果，我在一个领域看到它，我喜欢这个结果，这是一个很酷的结果，但我不喜欢证明，它使用我不太熟悉的数学类型，我经常尝试使用我喜欢的工具自己重新证明它。

55:53 Often, my proof is worse, but by the exercise they’re doing, so I can say, “Oh, now I can see what the other proof was trying to do,” and from that, I can get some understanding of the tools that are used in that field.

通常，我的证明更糟糕，但是通过他们正在做的练习，所以我可以说，“哦，现在我可以看到另一个证明试图做什么”，从中，我可以对该领域使用的工具有一些了解。

So it’s very exploratory, very…

所以这是非常具有探索性的，非常……

Doing crazy things in crazy fields and reinventing the wheel a lot, whereas the hedgehog style is, I think, much more scholarly.

在疯狂的领域做疯狂的事情，经常重新发明轮子，而我认为刺猬风格更具学术性。

You’re very knowledge-based.

你很有知识。

You stay up to speed on all the developments in this field, you know all the history, you have a very good understanding of exactly the strengths and weaknesses of each particular technique.

您可以了解该领域的所有发展，了解所有历史，并且对每种特定技术的优点和缺点有很好的了解。

I think you rely a lot more on calculation than sort of trying to find narratives.

我认为你更多地依赖于计算，而不是试图寻找叙述。

So yeah, I can do that too, but other people are extremely good at that.

是的，我也能做到，但其他人在这方面都非常擅长。

Let’s step back and maybe look at a bit of a romanticized version of mathematics.

让我们退后一步，也许看一下数学的浪漫化版本。

56:44 So I think you’ve said that early on in your life, math was more like a puzzle-solving activity when you were young.

所以我想你说过，在你生命的早期，数学更像是你年轻时的一种解谜活动。

When did you first encounter a problem or proof where you realized math can have a kind of elegance and beauty to it?

你什么时候第一次遇到一个问题或证明，让你意识到数学可以有一种优雅和美丽？

That’s a good question.

这是一个好问题。

57:11 When I came to graduate school in Princeton, so John Conway was there at the time, he passed away a few years ago, but I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof.

当我来到普林斯顿研究生院时，约翰·康威当时也在场，他几年前去世了，但我记得我参加的第一个研究讲座是康威关于他所谓的极端证明的演讲。

57:28 So Conway just had this amazing way of thinking about all kinds of things in a way that you wouldn’t normally think of.

所以康威以一种你通常不会想到的方式思考各种事情的惊人方式。

So he thought proofs themselves as occupying some sort of space.

所以他认为证明本身占据了某种空间。

So if you want to prove something, let’s say that there’s infinitely many primes, you have all different proofs, but you could rank them in different axes.

因此，如果你想证明某件事，假设有无限多个素数，你有所有不同的证明，但你可以将它们排列在不同的轴上。

Some proofs are elegant, some proofs are long, some proofs are elementary and so forth, and so there’s this cloud, so the space of all proofs itself has some sort of shape, and so he was interested in extreme points of this shape.

有些证明很优雅，有些证明很长，有些证明很基本等等，所以就有了这个云，所以所有证明的空间本身都有某种形状，所以他对这种形状的极值点感兴趣。

Out of all these proofs, what is one of those, the shortest, at the expense of everything else, or the most elementary or whatever?

在所有这些证明中，哪一个是最短的、以牺牲其他一切为代价的，还是最基本的或其他什么？

58:09 So he gave some examples of well-known theorems, and then he would give what he thought was the extreme proof in these different aspects.

所以他举了一些众所周知的定理的例子，然后他会在这些不同的方面给出他认为是极端的证明。

I just found that really eye-opening, that it’s not just getting a proof for a result that was interesting, but once you have that proof, trying to optimize it in various ways, that proofing itself had some craftsmanship to it.

我发现这真的令人大开眼界，这不仅仅是获得一个有趣的结果的证明，而且一旦你有了这个证明，尝试以各种方式优化它，证明本身就有一些技巧。

58:40 It’s certainly informed my writing style, like when you do your math assignments and as you’re an undergraduate, your homework and so forth, you’re sort of encouraged to just write down any proof that works and hand it in, and as long as it gets a tick mark, you move on, but if you want your results to actually be influential and be read by people, it can’t just be correct.

这当然影响了我的写作风格，就像当你做数学作业时，当你是一名本科生时，你的家庭作业等等，我们会鼓励你写下任何有效的证明并将其提交，只要它得到一个勾号，你就继续前进，但如果你希望你的结果真正有影响力并被人们阅读，它就不能只是正确的。

It should also be a pleasure to read, motivated, be adaptable to generalize to other things.

它还应该是一种阅读的乐趣，有动力，能够适应概括到其他事物。

It’s the same in many other disciplines, like coding.

在许多其他学科中也是如此，比如编码。

There’s a lot of analogies between math and coding.

数学和编码之间有很多相似之处。

I like analogies, if you haven’t noticed.

如果你没有注意到的话，我喜欢类比。

You can code something, spaghetti code, that works for a certain task, and it’s quick and dirty and it works, but there’s lots of good principles for writing code well so that other people can use it, build upon it so it has fewer bugs and whatever, and there’s similar things with mathematics.

你可以编写一些东西，意大利面条代码，适用于特定的任务，它又快又脏，而且有效，但是有很多好的原则可以很好地编写代码，以便其他人可以使用它，在它的基础上进行构建，从而减少错误等等，数学方面也有类似的事情。

Yeah, first of all, there’s so many beautiful things there, and [inaudible 00:59:42] is one of the great minds in mathematics ever, and computer science, just even considering the space of proofs and saying, “Okay, what does this space look like, and what are the extremes?”

是的，首先，那里有很多美丽的东西，[听不清 00:59:42] 是数学和计算机科学领域有史以来最伟大的思想家之一，甚至考虑到证明的空间并说，“好吧，这个空间是什么样的，极端是什么？”

59:56 Like you mentioned, coding as an analogy is interesting, because there’s also this activity called the code golf, which I also find beautiful and fun, where people use different programming languages to try to write the shortest possible program that accomplishes a particular task, and I believe there’s even competitions on this, and it’s also a nice way to stress test not just the programs, or in this case, the proofs, but also the different languages.

就像你提到的，编码作为类比很有趣，因为还有一种叫做“代码高尔夫”的活动，我也觉得它美丽又有趣，人们使用不同的编程语言来尝试编写完成特定任务的最短程序，我相信这方面甚至存在竞争，而且这也是一种很好的方法，不仅可以对程序进行压力测试，或者在这种情况下，对证明进行压力测试，还可以对不同的语言进行压力测试。

Maybe that’s a different notation or whatever to use to accomplish a different task.

也许这是一个不同的符号或任何用来完成不同任务的东西。

Yeah, you learn a lot.

是的，你学到了很多东西。

1:00:31 I mean, it may seem like a frivolous exercise, but it can generate all these insights, which, if you didn’t have this artificial objective to pursue, you might not see…

我的意思是，这可能看起来像是一个无聊的练习，但它可以产生所有这些见解，如果你没有这个人为的目标来追求，你可能看不到……

What, to you, is the most beautiful or elegant equation in mathematics?

对你来说，数学中最美丽或最优雅的方程是什么？

1:00:43 I mean, one of the things that people often look to in beauty is the simplicity.

我的意思是，人们在美丽中经常关注的事情之一就是简单。

So if you look at e=MC2…

所以如果你看看 e=MC2...

So when a few concepts come together, that’s why the Euler identity is often considered the most beautiful equation in mathematics.

因此，当几个概念结合在一起时，欧拉恒等式通常被认为是数学中最美丽的方程。

Do you find beauty in that one, in the Euler identity?

您在欧拉恒等式中发现美了吗？

Yeah.

是的。

1:01:08 Well, as I said, what I find most appealing is connections between different things that…

嗯，正如我所说，我发现最吸引人的是不同事物之间的联系……

So if you…

所以如果你...

Pi equals minus one.

Pi 等于负一。

So yeah, people use all the fundamental constants.

所以，是的，人们使用所有基本常数。

Okay.

好的。

I mean, that’s cute, but to me…

我的意思是，这很可爱，但对我来说……

1:01:24 So the exponential function, which is by Euler, was to measure exponential growth.

因此，欧拉提出的指数函数是用来衡量指数增长的。

So compound interest or decay, anything which is continuously growing, continuously decreasing, growth and decay, or dilation or contraction, is modeled by the exponential function, whereas pi comes around from circles and rotation, right?

因此，复利或衰减，任何持续增长、持续下降、增长和衰减、膨胀或收缩的事物，都是由指数函数建模的，而 pi 来自圆和旋转，对吧？

If you want to rotate a needle, for example, a hundred degrees, you need rotate by pi radians, and i, complex numbers, represents the swapping imaginary axes of a 90 degree rotation.

如果你想旋转一根针，例如一百度，你需要旋转pi弧度，而i，复数，代表90度旋转的交换虚轴。

So a change in direction.

所以改变方向。

1:01:53 So the exponential function represents growth and decay in the direction that you already are.

所以指数函数代表了你已经所处的方向的增长和衰退。

When you stick an i in the exponential, now instead of motion in the same direction as your current position, the motion as a right angles to your current position.

当您在指数中添加 i 时，现在不再以与当前位置相同的方向运动，而是以与当前位置成直角的方向运动。

So rotation, and then, so E to the pi i equals minus one tells you that if you rotate for a time pi, you end up at the other direction.

所以旋转，然后，所以 E 到 pi i 等于负一告诉你，如果你旋转一段时间 pi，你最终会到达另一个方向。

So it unifies geometry through dilation and exponential growth or dynamics through this act of complexification, rotation by pi i.

因此，它通过膨胀和指数增长统一了几何学，或者通过这种复杂化行为（按 pi i 旋转）统一了动力学。

So it connects together all these two as mathematics, dynamics, geometry and complex numbers.

因此它将数学、动力学、几何和复数这两者连接在一起。

They’re all considered almost…

他们都被认为几乎……

They were all next-door neighbors in mathematics because of this identity.

因为这个身份，他们都是数学界的邻居。

Do you think the thing you mentioned as Q, the collision of notations from these disparate fields, is just a frivolous side effect, or do you think there is legitimate value in when notation…

你认为你提到的 Q 的东西，来自这些不同领域的符号的碰撞，只是一个无聊的副作用，或者你认为当符号……有合法的价值吗？

1:02:37 Although our old friends come together in the night?

虽然我们的老朋友在夜里聚在一起？

Well, it’s confirmation that you have the right concepts.

嗯，这证明你有正确的概念。

1:02:54 So when you first study anything, you have to measure things, and give them names, and initially sometimes, because your model is, again, too far off from reality, you give the wrong things the best names, and you only find out later what’s really important.

因此，当你第一次研究任何东西时，你必须测量事物，并给它们命名，最初有时，因为你的模型又离现实太远，你给错误的东西起了最好的名字，然后你才会发现真正重要的东西。

Physicists can do this sometimes, but it turns out okay.

物理学家有时可以做到这一点，但事实证明还可以。

So actually, physics [inaudible 01:03:19] e=MC2.

所以实际上，物理学 [听不清 01:03:19] e=MC2。

1:03:18 So one of the big things was the E, right?

所以最重要的事情之一就是 E，对吧？

So when Aristotle first came up with his laws of motion, and then Galileo and Newton and so forth, they saw the things they could measure, they could measure mass and acceleration and force and so forth, and so Newtonian mechanics, for example, F=ma, was the famous Newton’s second law of motion.

所以当亚里士多德首先提出他的运动定律时，然后是伽利略和牛顿等等，他们看到了他们可以测量的东西，他们可以测量质量、加速度和力等等，所以牛顿力学，例如F=ma，就是著名的牛顿第二运动定律。

So those were the primary objects.

这些是主要对象。

So they gave them the central billing in the theory.

所以他们给了他们理论上的中心计费。

1:03:44 It was only later after people started analyzing these equations that there always seemed to be these quantities that were conserved.

直到后来人们开始分析这些方程之后，才发现这些量似乎总是守恒的。

So in particular, momentum and energy, and it’s not obvious that things have an energy.

特别是动量和能量，而且事物具有能量并不明显。

It’s not something you can directly measure the same way you can measure mass and velocity, so both, but over time, people realized that this was actually a really fundamental concept.

它不能像测量质量和速度一样直接测量，但随着时间的推移，人们意识到这实际上是一个非常基本的概念。

1:04:05 Hamilton, eventually in the 19th century, reformulated Newton’s laws of physics into what’s called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object.

汉密尔顿最终在 19 世纪将牛顿物理定律重新表述为所谓的哈密顿力学，其中能量（现在称为哈密顿量）是主导对象。

Once you know how to measure the Hamiltonian of any system, you can describe completely the dynamics like what happens to all the states.

一旦您知道如何测量任何系统的哈密顿量，您就可以完整地描述动力学，例如所有状态所发生的情况。

It really was a central actor, which was not obvious initially, and this change of perspective really helped when quantum mechanics came along, because the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their Newtonian thinking, because everything was a particle and so forth, to quantum mechanics, because everything was a wave, but it just looked really, really weird.

它确实是一个核心角色，最初并不明显，当量子力学出现时，这种视角的改变确实有所帮助，因为研究量子力学的早期物理学家，他们在尝试将牛顿思维（因为一切都是粒子等等）适应量子力学时遇到了很多麻烦，因为一切都是波，但它看起来真的非常奇怪。

1:04:51 You ask, “What is the quantum version of F=ma?”, and it’s really, really hard to give an answer to that, but it turns out that the Hamiltonian, which was so secretly behind the scenes in classical mechanics, also is the key object in quantum mechanics, that there’s also an object called a Hamiltonian.

你问，“F=ma 的量子版本是什么？”，这个问题真的很难回答，但事实证明，经典力学中秘密存在的哈密顿量，也是量子力学中的关键对象，还有一个叫做哈密顿量的对象。

It’s a different type of object.

这是一种不同类型的物体。

It’s what’s called an operator rather than a function, but again, once you specify it, you specify the entire dynamics.

它是所谓的运算符而不是函数，但同样，一旦指定它，您就指定了整个动态。

1:05:17 So there’s something called Schrodinger’s equation that tells you exactly how quantum systems evolve once you have a Hamiltonian.

因此，有一个叫做薛定谔方程的东西，可以准确地告诉你一旦有了哈密顿量，量子系统就会如何演化。

So side by side, they look completely different objects.

因此，它们并排放置时，看起来完全是不同的物体。

One involves particles, one involves waves and so forth, but with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics.

一种涉及粒子，一种涉及波等等，但有了这种中心地位，你就可以开始将许多直觉和事实从经典力学转移到量子力学。

So for example, in classical mechanics, there’s this thing called Noether’s theorem.

例如，在经典力学中，有一个叫做诺特定理的东西。

Every time there’s a symmetry in a physical system, there was a conservation law.

每当物理系统中存在对称性时，就会存在守恒定律。

So the laws of physics are translation invariant.

所以物理定律是平移不变的。

Like if I move 10 steps to the left, I experience the same laws of physics as if I was here, and that corresponds to conservation momentum.

就像如果我向左移动 10 步，我会体验到与在这里相同的物理定律，这对应于动量守恒。

If I turn around by some angle, again, I experience the same laws of physics.

如果我再转过某个角度，我就会再次经历同样的物理定律。

This corresponds to the conservation of angular momentum.

这对应于角动量守恒。

If I wait for 10 minutes, I still have the same laws of physics.

如果我等10分钟，我仍然有同样的物理定律。

1:06:00 It If I wait for 10 minutes, I still have the same laws of physics.

如果我等10分钟，我仍然有同样的物理定律。

So there’s time transition invariance.

所以存在时间转换不变性。

This corresponds to the law of the conservation of energy.

这符合能量守恒定律。

So there’s this fundamental connection between symmetry and conservation.

所以对称性和守恒定律之间存在着这种基本的联系。

And that’s also true in quantum mechanics, even though the equations are completely different, but because they’re both coming from the Hamiltonian, the Hamiltonian controls everything, every time the Hamiltonian has a symmetry, the equations will have a conservation wall.

在量子力学中也是如此，尽管方程完全不同，但因为它们都来自哈密顿量，所以哈密顿量控制一切，每当哈密顿量具有对称性时，方程就会有守恒墙。

Once you have the right language, it actually makes things a lot cleaner.

一旦你有了正确的语言，它实际上会让事情变得更加清晰。

1:06:32 One of the problems why we can’t unify quantum mechanics and general relativity, yet we haven’t figured out what the fundamental objects are.

我们无法统一量子力学和广义相对论的问题之一是，我们还没有弄清楚基本物体是什么。

For example, we have to give up the notion of space and time being these almost Euclidean-type spaces, and it has to be, we know that at very tiny scales there’s going to be quantum fluctuations.

例如，我们必须放弃空间和时间的概念，这些空间和时间几乎是欧几里得类型的空间，而且必须如此，我们知道在非常小的尺度上将会存在量子涨落。

There’s space-time foam and trying to use Cartesian coordinates X, Y, Z.

有时空泡沫并尝试使用笛卡尔坐标 X、Y、Z。

It’s a non-starter, but we don’t know what to replace it with.

这是不可能的，但我们不知道用什么来代替它。

We don’t actually have the concepts, the analog Hamiltonian that sort of organized everything.

我们实际上并没有概念，即可以组织一切的模拟哈密顿量。

Does your gut say that there is a theory of everything, so this is even possible to unify, to find this language that unifies general relativity and quantum mechanics?

你的直觉是否认为万物皆有理论，因此甚至有可能统一，找到这种统一广义相对论和量子力学的语言？

I believe so.

我相信是这样。

1:07:19 The history of physics has been out of unification much like mathematics over the years.

多年来，物理学的历史就像数学一样失去了统一性。

[inaudible 01:07:26] magnetism was separate theories and then Maxwell unified them.

[听不清 01:07:26] 磁力是独立的理论，然后麦克斯韦统一了它们。

Newton unified the motions of heavens for the motions of objects on the Earth and so forth.

牛顿将天体的运动统一为地球上物体的运动等等。

So it should happen.

所以它应该发生。

It’s just that, again, to go back to this model of the observations and theory, part of our problem is that physics is a victim of it’s own success.

只是，再次回到观察和理论的模型，我们的部分问题是物理学是其自身成功的受害者。

That our two big theories of physics, general relativity and quantum mechanics are so good now is that together they cover 99.9% of all the observations we can make.

我们的两大物理学理论——广义相对论和量子力学现在之所以如此出色，是因为它们加在一起涵盖了我们能进行的所有观察的 99.9%。

And you have to either go to extremely insane particle accelerations or the early universe or things that are really hard to measure in order to get any deviation from either of these two theories to the point where you can actually figure out how to combine together.

你必须要么去研究极其疯狂的粒子加速，要么去早期宇宙，要么去研究那些很难测量的东西，以便从这两种理论中得到任何偏差，从而真正弄清楚如何将它们结合在一起。

But I have faith that we’ve been doing this for centuries and we’ve made progress before.

但我相信，我们已经这样做了几个世纪，而且我们之前已经取得了进展。

There’s no reason why we should stop.

我们没有理由停下来。

Do you think you’ll be a mathematician that develops a theory of everything?

你认为你会成为一名发展万有理论的数学家吗？

What often happens is that when the physicists need some theory of mathematics, there’s often some precursor that the mathematicians worked out earlier.

经常发生的情况是，当物理学家需要某种数学理论时，往往会有数学家较早提出的一些先兆。

1:08:24 So when Einstein started realizing that space was curved, he went to some mathematician and asked, “Is there some theory of curved space that mathematicians already came up with that could be useful?” And he said, “Oh yeah, I think Riemann came up with something.” And so yeah, Riemann had developed Riemannian geometry, which is precisely a theory of spaces that are curved in various general ways, which turned out to be almost exactly what was needed by Einstein’s theory.

因此，当爱因斯坦开始意识到空间是弯曲的时，他去找一位数学家询问：“数学家们已经提出的一些弯曲空间理论是否有用？”他说：“哦，是的，我认为黎曼想出了一些办法。”是的，黎曼发展了黎曼几何，这正是一种以各种一般方式弯曲的空间理论，事实证明这几乎正是爱因斯坦理论所需要的。

This is going back to weakness and unreasonable effectiveness of mathematics.

这又回到了数学的弱点和不合理的有效性。

I think the theories that work well, that explain the universe, tend to also involve the same mathematical objects that work well to solve mathematical problems.

我认为解释宇宙的有效理论往往也涉及能够很好地解决数学问题的相同数学对象。

Ultimately, they’re just both ways of organizing data in useful ways.

最终，它们只是以有用的方式组织数据的两种方式。

It just feels like you might need to go some weird land that’s very hard to intuit.

只是感觉你可能需要去一些很难直觉的奇怪的地方。

1:09:17 You have string T=theory.

你有字符串 T=理论。

Yeah, that was a leading candidate for many decades.

是的，这是几十年来的主要候选人。

1:09:25 I think it’s slowly pulling out of fashion.

我认为它正在慢慢退出时尚。

It’s not matching experiment.

这不是匹配实验。

So one of the big challenges of course, like you said, is experiment is very tough because of how effective both theories are.

因此，正如您所说，最大的挑战之一当然是实验非常困难，因为这两种理论都很有效。

1:09:33 But the other is just you’re talking about you’re not just deviating from space-time.

但另一个就是你说你不只是偏离了时空。

You’re going into some crazy number of dimensions.

你将进入一些疯狂的维度。

You’re doing all kinds of weird stuff that to us, we’ve gone so far from this flat earth that we started at, like you mentioned, and now it’s very hard to use our limited ape descendants of a cognition to intuit what that reality really is.

你对我们做了各种奇怪的事情，我们已经远离了我们开始时的平坦地球，就像你提到的那样，现在很难利用我们有限的认知能力的猿后代来直觉现实到底是什么。

This is why analogies are so important.

这就是为什么类比如此重要。

1:10:10 So yeah, the round earth is not intuitive because we’re stuck on it.

所以，是的，圆形的地球并不直观，因为我们被困在它上面。

But round objects in general, we have pretty good intuition over and we have interest about light works and so forth.

但总的来说，对于圆形物体，我们有很好的直觉，并且我们对光的作品等感兴趣。

And it’s actually a good exercise to actually work out how eclipses and phases of the sun and the moon and so forth can be really easily explained by round earth and round moon and models.

这实际上是一个很好的练习，可以真正弄清楚如何通过圆形地球和圆形月亮和模型来轻松解释日食和日月相位等。

And you can just take a basketball and a golf ball and a light source and actually do these things yourself.

你可以拿一个篮球、一个高尔夫球和一个光源，然后自己实际做这些事情。

So the intuition is there, but you have to transfer it.

所以直觉是存在的，但你必须转移它。

That is a big leap intellectually for us to go from flat to round earth because our life is mostly lived in flat land.

从平坦的地球到圆形的地球对于我们来说是智力上的一大飞跃，因为我们的生活大部分时间都生活在平坦的土地上。

1:10:47 To load that information and we’re all like, take it for granted.

加载这些信息，我们都认为这是理所当然的。

We take so many things for granted because science has established a lot of evidence for this kind of thing, but we’re on a round rock flying through space.

我们认为很多事情是理所当然的，因为科学已经为这类事情建立了很多证据，但我们是在一块飞过太空的圆形岩石上。

Yeah, that’s a big leap.

是的，这是一个很大的飞跃。

And you have to take a chain of those leaps.

你必须采取一连串的飞跃。

The more and more and more we progress,

我们进步得越来越多，

Right, yeah.

对，是的。

1:11:15 So modern science is maybe, again, a victim of own success is that in order to be more accurate, it has to move further and further away from your initial intuition.

因此，现代科学可能再次成为自身成功的受害者，因为为了更加准确，它必须离你最初的直觉越来越远。

And so for someone who hasn’t gone through the whole process of science education, it looks more suspicious because of that.

因此，对于没有经历过整个科学教育过程的人来说，它看起来更可疑。

So we need more grounding.

所以我们需要更多的接地气。

There are scientists who do excellent outreach, but there’s lots of science things that you can do at home.

有些科学家的外展工作非常出色，但你也可以在家做很多科学事情。

Lots of YouTube videos I did at YouTube video recently, Grant Sanderson, we talked about this earlier, that how the ancient Greeks were able to measure things like the distance of the moon, distance the earth, and using techniques that you could also replicate yourself.

我最近在 YouTube 视频上制作了很多 YouTube 视频，Grant Sanderson，我们之前谈到过，古希腊人如何能够测量诸如月球距离、地球距离之类的东西，以及使用你也可以复制自己的技术。

It doesn’t all have to be fancy space telescopes and very intimidating mathematics.

它并不一定都是奇特的太空望远镜和非常令人生畏的数学。

Yeah, I highly recommend that.

是的，我强烈推荐。

1:12:01 I believe you give a lecture and you also did an incredible video with Grant.

我相信你做了一次演讲，你还和格兰特一起制作了一段令人难以置信的视频。

It’s a beautiful experience to try to put yourself in the mind of a person from that time shrouded in mystery.

尝试将自己置于那个时代笼罩在神秘之中的人的脑海中，这是一次美妙的经历。

You’re on this planet, you don’t know the shape of it, the size of it.

你在这个星球上，你不知道它的形状，它的大小。

You see some stars, you see some things and you try to localize yourself in this world and try to make some kind of general statements about distanced places.

你看到一些星星，你看到一些东西，你尝试将自己定位在这个世界上，并尝试对遥远的地方做出某种一般性的陈述。

Change of perspective is really important.

改变观点确实很重要。

1:12:29 You say travel broadens the mind, this is intellectual travel.

你说旅行开阔心胸，这就是智力旅行。

Put yourself in the mind of the ancient Greeks or person some other time period, make hypotheses, spherical [inaudible 01:12:41], whatever, speculate.

将自己置于古希腊人或其他时期的人的思想中，做出假设，球形[听不清01:12:41]，无论如何，推测。

And this is what mathematicians do and some other, what artists do actually.

这就是数学家所做的事情，还有一些艺术家实际上所做的事情。

It’s just incredible that given the extreme constraints, you could still say very powerful things.

令人难以置信的是，在极端的限制下，你仍然可以说出非常有力的话。

1:12:48 That’s why it’s inspiring.

这就是为什么它很鼓舞人心。

Looking back in history, how much can be figured out when you don’t have much to figure out stuff with.

回顾历史，当你没有太多东西可以弄清楚时，你能弄清楚多少。

If you propose axioms, then the mathematics does.

如果你提出公理，那么数学就会提出公理。

1:13:01 You follow those axioms to their conclusions and sometimes you can get quite a long way from initial hypotheses.

你遵循这些公理得出结论，有时你可能会与最初的假设相去甚远。

If we can stay in the land of the weird.

如果我们能留在怪异之地就好了。

1:13:10 You mentioned general relativity.

你提到了广义相对论。

You’ve contributed to the mathematical understanding, Einstein’s field equations.

您对爱因斯坦场方程的数学理解做出了贡献。

Can you explain this work and from a mathematical standpoint, what aspects of general relativity are intriguing to you?

您能解释一下这项工作吗？从数学的角度来看，广义相对论的哪些方面让您感兴趣？

Challenging to you?

对你来说有挑战性吗？

I have worked on some equations.

我研究过一些方程。

1:13:31 There’s something called the wave maps equation or the Sigma field model, which is not quite the equation of space-time gravity itself, but of certain fields that might exist on top of space-time.

有一种叫做波动图方程或西格玛场模型的东西，它并不完全是时空引力本身的方程，而是可能存在于时空之上的某些场的方程。

So Einstein’s equations of relativity just describe space and time itself.

所以爱因斯坦的相对论方程只是描述空间和时间本身。

But then there’s other fields that live on top of that.

但除此之外还有其他领域。

There’s the electromagnetic field, there’s things called Yang-Mills fields, and there’s this whole hierarchy of different equations of which Einstein’s considered one of the most nonlinear and difficult, but relatively low on the hierarchy was this thing called the wave maps equation.

有电磁场，有称为杨-米尔斯场的东西，还有整个不同方程的层次结构，爱因斯坦认为其中最非线性和最困难的方程之一，但层次结构中相对较低的是称为波图方程的东西。

So it’s a wave which at any given point is fixed to be on a sphere.

所以它是一个在任何给定点都固定在球体上的波。

So I can think of a bunch of arrows in space and time.

所以我可以想到一堆空间和时间的箭头。

Yeah, so it’s pointing in different directions, but they propagate like waves.

是的，所以它指向不同的方向，但它们像波一样传播。

If you wiggle an arrow, it would propagate and make all the arrows move kind of like sheaves of wheat in a wheat field.

如果你摆动一支箭，它会传播并使所有箭移动，就像麦田里的麦捆一样。

1:14:27 And I was interested in the global regularity problem.

我对全球规律性问题很感兴趣。

Again for this question, is it possible for the energy here to collect at a point?

还是这个问题，这里的能量有可能聚集在一个点吗？

So the equation I considered was actually what’s called a critical equation where it’s actually the behavior at all scales is roughly the same.

所以我考虑的方程实际上是所谓的临界方程，它实际上在所有尺度上的行为都大致相同。

And I was able barely to show that you couldn’t actually force a scenario where all the energy concentrated at one point, that the energy had to disperse a little bit at the moment, just a little bit.

我几乎无法证明你实际上无法强迫所有能量都集中在一个点的情况，能量必须在此刻分散一点点，只是一点点。

It would stay regular.

它会保持规律。

Yeah, this was back in 2000.

是的，那是2000年的事了。

That was part of why I got interested in [inaudible 01:14:58] afterwards actually.

这就是我后来对 [听不清 01:14:58] 感兴趣的部分原因。

So I developed some techniques to solve that problem.

所以我开发了一些技术来解决这个问题。

So part of it, this problem is really nonlinear because of the curvature of the sphere.

因此，由于球体的曲率，这个问题实际上是非线性的。

There was a certain nonlinear effect, which was a non-perturbative effect.

存在一定的非线性效应，属于非微扰效应。

It was when you sort looked at it normally it looked larger than the linear effects of the wave equation.

当你正常地观察它时，它看起来比波动方程的线性效应更大。

And so it was hard to keep things under control even when your energy was small.

所以即使你的能量很小，也很难控制住事情。

1:15:23 But I developed what’s called a gauge transformation.

但我开发了所谓的规范变换。

So the equation is kind of like an evolution of sheaves of wheat, and they’re all bending back and forth, so there’s a lot of motion.

所以这个方程有点像麦捆的演变，它们都来回弯曲，所以有很多运动。

But if you imagine stabilizing the flow by attaching little cameras at different points in space, which are trying to move in a way that captures most of the motion, and under this stabilized flow, the flow becomes a lot more linear.

但如果你想象通过在空间的不同点安装小相机来稳定流动，这些相机试图以捕捉大部分运动的方式移动，并且在这种稳定的流动下，流动变得更加线性。

I discovered a way to transform the equation to reduce the amount of nonlinear effects, and then I was able to solve the equation.

我发现了一种方法来变换方程以减少非线性效应，然后我就能解出方程。

I found the transformation while visiting my aunt in Australia, and I was trying to understand the dynamics of all these fields, and I couldn’t do a pen and paper, and I had not enough facility of computers to do any computer simulations.

我在拜访澳大利亚的姑妈时发现了这种转变，我试图了解所有这些领域的动态，但我没有笔和纸，也没有足够的计算机设施来进行任何计算机模拟。

1:16:08 So I ended up closing my eyes being on the floor and just imagining myself to actually be this vector field and rolling around to try to see how to change coordinates in such a way that somehow things in all directions would behave in a reasonably linear fashion.

所以我最终闭上眼睛坐在地板上，想象自己实际上是这个矢量场，然后滚动以尝试了解如何以某种方式改变坐标，以便所有方向上的事物都会以合理的线性方式表现。

And yeah, my aunt walked in on me while I was doing that and she was asking, “Why am I doing this?”

是的，当我这样做的时候，我姨妈走进来，她问我：“我为什么要这样做？”

It’s complicated as the answer.

答案很复杂。

“Yeah, yeah.

“是啊，是啊。

1:16:30 And okay, fine.

好吧，好吧。

You are a young man.

你是个年轻人。

I don’t ask questions.”

我不问问题。”

I have to ask about how do you approach solving difficult problems if it’s possible to go inside your mind when you’re thinking, are you visualizing in your mind the mathematical objects, symbols, maybe what are you visualizing in your mind?

我想问的是，如果你在思考时可以进入你的大脑，你会如何解决困难的问题，你是否在脑海中想象数学对象、符号，也许你在脑海中想象什么？

1:16:34 Usually when you’re thinking?

通常当你思考的时候？

A lot of pen and paper.

很多笔和纸。

1:16:57 One thing you pick up as a mathematician is I call it cheating strategically.

作为一名数学家，你会发现一件事，我称之为策略性作弊。

So the beauty of mathematics is that you get to change the problem and change the rules as you wish.

所以数学的美妙之处就在于你可以随心所欲地改变问题、改变规则。

You don’t get to do this by any other field.

在任何其他领域你都无法做到这一点。

If you’re an engineer and someone says, “Build a bridge over this river,” you can’t say, “I want to build this bridge over here instead,” or, “I want to put it out of paper instead of steel,” but a mathematician, you can do whatever you want on.

如果你是一名工程师，有人说“在这条河上建一座桥”，你不能说“我想在这里建一座桥”，或者“我想用纸而不是钢来建造它”，但作为一名数学家，你可以做任何你想做的事情。

It’s like trying to solve a computer game where there’s unlimited cheat codes available.

这就像尝试解决一个有无限作弊代码的电脑游戏一样。

And so you can set this, there’s a dimension that’s large.

所以你可以设置这个，有一个很大的维度。

I’ve set it to one.

我已将其设置为一。

I’ll solve the one-dimensional problem first.

我先解决一维问题。

So there’s a main term and an error term.

所以有一个主要术语和一个错误术语。

I’m going to make a spherical call assumption [inaudible 01:17:45] term is zero.

我将进行一次球形呼叫假设 [听不清 01:17:45] 项为零。

1:17:45 And so the way you should solve these problems is not in this Iron Man mode where you make things maximally difficult, but actually the way you should approach any reasonable math problem is that if there are 10 things that are making your life difficult, find a version of the problem that turns off nine of the difficulties, but only keeps one of them and solve that.

所以你应该解决这些问题的方法不是在钢铁侠模式中让事情变得最大困难，而是实际上你应该解决任何合理的数学问题的方法是，如果有 10 件事让你的生活变得困难，找到一个解决问题的版本，可以解决其中的 9 个困难，但只保留其中一个并解决它。

And then so you solve nine cheats.

然后你就解决了九个秘籍。

Okay, you solve 10 cheats, then the game is trivial, but you solve nine cheats.

好吧，你解决了 10 个作弊，那么游戏就很简单了，但是你解决了 9 个作弊。

You solve one problem that teaches you how to deal with that particular difficulty.

你解决了一个问题，它教会你如何应对这一特定的困难。

And then you turn that one-off and you turn someone else something else on, and then you solve that one.

然后你关闭那个一次性的东西，然后你打开其他人的其他东西，然后你解决那个问题。

And after you know how to solve the 10 problems, 10 difficulties separately, then you have to start merging them a few at a time.

当你知道如何分别解决这 10 个问题、10 个困难之后，你就必须开始一次将它们合并起来。

1:18:26 As a kid, I watched a lot of these Hong Kong action movies from our culture, and one thing is that every time it’s a fight scene, so maybe the hero gets swarmed by a hundred bad-guy goons or whatever, but it’ll always be choreographed so that he’d always be only fighting one person at a time and it would defeat that person and move on.

小时候，我看了很多我们文化中的香港动作片，有一件事是，每次都是打斗场面，所以也许英雄会被一百个坏人暴徒或其他什么东西包围，但它总是经过精心设计的，这样他总是一次只与一个人战斗，然后打败那个人并继续前进。

And because of that, he could defeat all of them.

正因为如此，他才能击败所有人。

But whereas if they had fought a bit more intelligently and just swarmed the guy at once, it would make for much worse cinema, but they would win.

但是，如果他们战斗得更聪明一点，立刻包围那个家伙，电影的效果就会更糟，但他们会赢。

Are you usually pen and paper?

你平时都是用笔和纸吗？

1:19:02 Are you working with computer and LaTeX?

您使用计算机和 LaTeX 进行工作吗？

Mostly pen and paper actually.

实际上主要是笔和纸。

1:19:08 So in my office I have four giant blackboards and sometimes I just have to write everything I know about the problem on the four blackboards and then sit my couch and just see the whole thing.

因此，在我的办公室里，我有四块巨大的黑板，有时我只需要把我所知道的有关问题的所有内容写在四块黑板上，然后坐在沙发上看整个事情。

Is it all symbols like notation or is there some drawings?

都是符号之类的符号还是有一些图画？

Oh, there’s a lot of drawing and a lot of bespoke doodles that only makes sense to me.

哦，有很多绘画和很多定制涂鸦只对我有意义。

1:19:23 And the beauty of a blackboard is you erase and it’s a very organic thing.

黑板的美妙之处在于你可以擦除它，这是一个非常有机的东西。

I’m beginning to use more and more computers, partly because AI makes it much easier to do simple coding things that if I wanted to plot a function before, which is moderately complicated, has some iteration or something, I’d had to remember how to set up a Python program and how does a full loop work and debug it and it would take two hours and so forth.

我开始使用越来越多的计算机，部分原因是人工智能使简单的编码工作变得更加容易，如果我之前想绘制一个函数，该函数比较复杂，有一些迭代或其他东西，我必须记住如何设置Python程序以及完整的循环如何工作和调试它，这将需要两个小时等等。

And now I can do it in 10, 15 minutes as much.

现在我可以在 10、15 分钟内完成。

I’m using more and more computers to do simple explorations.

我正在使用越来越多的计算机来进行简单的探索。

Let’s talk about AI a little bit if we could.

如果可以的话，我们来谈谈人工智能吧。

1:20:01 So maybe a good entry point is just talking about computer-assisted proofs in general.

因此，也许一个好的切入点只是谈论一般的计算机辅助证明。

Can you describe the Lean formal proof programming language and how it can help as a proof assistant and maybe how you started using it and how it has helped you?

您能否描述一下精益形式证明编程语言以及它如何作为证明助手提供帮助，也许您是如何开始使用它以及它如何帮助您的？

So Lean is a computer language, much like standard languages like Python and C and so forth, except that in most languages the focus is on using executable code.

所以精益是一种计算机语言，很像 Python 和 C 等标准语言，只不过大多数语言的重点是使用可执行代码。

1:20:25 Lines of code do things, they flip bits or they make a robot move or they deliver your text on the internet or something.

代码行可以做一些事情，它们翻转位或者让机器人移动或者在互联网上传递您的文本等等。

So lean is a language that can also do that.

所以精益是一种也可以做到这一点的语言。

It can also be run as a standard traditional language, but it can also produce certificates.

它也可以作为标准传统语言运行，但它也可以生成证书。

So a software language like Python might do a computation and give you that the answer is seven.

因此，像 Python 这样的软件语言可能会进行计算并给出答案为 7。

Okay, does the sum of three plus four equal to seven?

好吧，三加四的和等于七吗？

1:20:59 But Lean can produce not just the answer, but a proof that how it got the answer of seven as three plus four and all the steps involved.

但精益不仅可以给出答案，还可以证明它如何得到 7 等于 3 加 4 的答案以及所涉及的所有步骤。

So it creates these more complicated objects, not just statements, but statements with proofs attached to them.

因此，它创建了这些更复杂的对象，不仅仅是语句，而是带有附加证明的语句。

And every line of code is just a way of piecing together previous statements to create new ones.

每行代码都只是将以前的语句拼凑在一起以创建新语句的一种方式。

So the idea is not new.

所以这个想法并不新鲜。

These things are called proof assistance, and so they provide languages for which you can create quite complicated mathematical proofs.

这些东西被称为证明辅助，因此它们提供了可以创建相当复杂的数学证明的语言。

They produce these certificates that give a 100% guarantee that your arguments are correct if you trust the compiler of Lean, but they made the compiler really small and there are several different compilers available for the Lean.

如果您信任 Lean 的编译器，他们生成的这些证书可以 100% 保证您的论点是正确的，但它们使编译器变得非常小，并且有几种不同的编译器可用于 Lean。

Can you give people some intuition about the difference between writing on pen and paper versus using Lean programming language?

您能否让人们直观地了解用笔和纸书写与使用精益编程语言之间的区别？

1:21:45 How hard is it to formalize statement?

正式声明有多难？

So Lean, a lot of mathematicians were involved in the design of Lean.

所以精益，很多数学家都参与了精益的设计。

1:21:56 So it’s designed so that individual lines of code resemble individual lines of mathematical argument.

因此，它的设计使得单独的代码行类似于单独的数学论证行。

You might want to introduce a variable, you want to prove our contradiction.

你可能想引入一个变量，你想证明我们的矛盾。

There are various standard things that you can do and it’s written.

您可以执行多种标准操作，并且已写明。

So ideally should like a one-to-one correspondence.

所以理想情况下应该是一对一的对应。

In practice, it isn’t because Lean is explaining a proof to extremely pedantic colleague who will point out, “Okay, did you really mean this?

在实践中，这并不是因为精益正在向极其迂腐的同事解释证明，而后者会指出：“好吧，你真的是这个意思吗？

What happens if this is zero?

如果为零会发生什么？

Okay, how do you justify this?” So Lean has a lot of automation in it to try to be less annoying.

好吧，你如何证明这一点？”因此，精益有很多自动化功能，以尽量减少烦人的事情。

So for example, every mathematical object has to come with a type.

例如，每个数学对象都必须有一个类型。

If I talk about X, is X a rule number or a natural number or a function or something?

如果我谈论 X，X 是一个规则数还是一个自然数或一个函数还是什么？

If you write things informally, it’s often if you have context.

如果你以非正式的方式写东西，通常是有背景的。

You say, “Clearly X is equal to let X be the sum of Y and Z and Y and Z were already rule number, so X should also be a rule number.” So Lean can do a lot of that, but every so often it says, wait a minute, can you tell me more about what this object is?

你说，“显然X等于让X是Y和Z之和，并且Y和Z已经是规则号，所以X也应该是规则号。”所以精益可以做很多事情，但它经常说，等一下，你能告诉我更多关于这个对象是什么的信息吗？

What type of object it is?

它是什么类型的物体？

You have to think more at a philosophical level, not just computations that you’re doing, but what each object actually is in some sense.

你必须在哲学层面上更多地思考，不仅仅是你正在做的计算，而是每个对象在某种意义上实际上是什么。

Is he using something like LLMs to do the type inference or you match with the real number?

他是使用法学硕士之类的东西来进行类型推断还是你与真实数字相匹配？

It’s using much more traditional what’s called good old-fashioned AI.

它使用的是更传统的所谓的老式人工智能。

1:23:23 You can represent all these things as trees, and there’s always algorithm to match one tree to another tree.

你可以将所有这些东西表示为树，并且总是有算法将一棵树与另一棵树相匹配。

So it’s actually doable to figure out if something is a real number or a natural number.

所以实际上可以判断某个东西是实数还是自然数。

Every object comes with a history of where it came from, and you can kind of trace it.

每件物品都有其来源的历史，您可以追踪它。

Oh, I see.

我懂了。

Yeah.

是的。

1:23:41 So it’s designed for reliability.

所以它的设计是为了可靠性。

So modern AIs are not used in, it’s a disjoint technology.

所以现代人工智能并没有被使用，这是一种脱节的技术。

People are begin to use AIs on top of lean.

人们开始在精益的基础上使用人工智能。

So when a mathematician tries to program proven in lean, often there’s a step.

因此，当数学家尝试进行经过精益验证的编程时，通常需要一个步骤。

Okay, now I want to use the fundamental thing on calculus, say to do the next step.

好吧，现在我想使用微积分的基本知识，说做下一步。

So the lean developers have built this massive project called Mathlib, a collection of tens of thousands of useful facts about methodical objects.

因此，精益开发人员构建了这个名为 Mathlib 的大型项目，它收集了数万个有关有条理对象的有用事实。

1:24:09 And somewhere in there is the fundamental calculus, but you need to find it.

那里有基本的微积分，但你需要找到它。

So a lot of the bottleneck now is actually lemma search.

所以现在很多瓶颈实际上是引理搜索。

There’s a tool that you know is in there somewhere and you need to find it.

您知道有一个工具就在某个地方，您需要找到它。

And so there are various search engine engines specialized for Mathlib that you can do, but there’s now these large language models that you can say, “I need the fundamental calculus at this point.” And it was like, okay, for example, when I code, I have GitHub Copilot installed as a plugin to my IDE, and it scans my text and it sees what I need.

因此，您可以使用各种专门针对 Mathlib 的搜索引擎，但现在有了这些大型语言模型，您可以说：“我现在需要基本的微积分。”就像，好吧，例如，当我编码时，我将 GitHub Copilot 作为插件安装到我的 IDE 中，它会扫描我的文本并看到我需要的内容。

Says I might even type, now I need to use the fundamental calculus.

说我什至可以打字，现在我需要使用基本的微积分。

And then it might suggest, “Okay, try this,” and maybe 25% of the time it works exactly.

然后它可能会建议，“好吧，试试这个”，也许 25% 的时间它是正确的。

And then another ten-fifty percent of the time it doesn’t quite work, but it’s close enough that I can say, oh yeah, if I just change it here and here, it’ll work.

然后另外百分之十五十的时间它不太起作用，但它足够接近我可以说，哦，是的，如果我在这里和这里改变它，它就会起作用。

And then half the time it gives me complete rubbish.

然后有一半的时间它给我的都是垃圾。

But people are beginning to use AIs a little bit on top, mostly on the level of basically fancy auto-complete that you can type half of one line of a proof and it will find, it’ll tell you.

但人们开始稍微使用人工智能，主要是在基本奇特的自动完成水平上，你可以输入半行证明，它会发现，它会告诉你。

Yeah, but a fancy, especially fancy with the sort of capital letter F, remove some of the friction mathematician might feel when they move from pen and paper to formalizing.

是的，但是一种幻想，尤其是大写字母 F 的幻想，消除了数学家从笔和纸转向形式化时可能感受到的一些摩擦。

Yes.

是的。

1:25:23 Yeah.

是的。

So right now I estimate that the time and effort taken to formalize it, proof is about 10 times the amount taken to write it out.

所以现在我估计将其形式化、证明所花费的时间和精力大约是写出来所花费的时间和精力的 10 倍。

So it’s doable, but it’s annoying.

所以这是可行的，但很烦人。

But doesn’t it kill the whole vibe of being a mathematician?

但这不会扼杀作为一名数学家的整体氛围吗？

1:25:36 Having a pedantic coworker?

有一个迂腐的同事吗？

Right?

正确的？

1:25:42 Yeah, if that was the only aspect of it, but there’s some cases was actually more pleasant to do things formally.

是的，如果这是唯一的方面，但在某些情况下，正式做事实际上更令人愉快。

So there’s a theorem I formalized, and there was a certain constant 12 that came out in the final statement.

因此，我形式化了一个定理，并且在最终陈述中出现了某个常数 12。

And so this 12 had been carried all through the proof and everything had to be checked all these other numbers that had to be consistent with this final number 12.

因此，这个 12 已经通过了整个证明，并且必须检查所有其他数字，这些数字必须与最终的数字 12 一致。

And so we wrote a paper through this theorem with this number 12.

所以我们用这个数字12通过这个定理写了一篇论文。

And then a few weeks later someone said, “Oh, we can actually improve this 12 to an 11 by reworking some of these steps.” And when this happens with pen and paper, every time you change your parameter, you have to check line by line that every single line of your proof still works.

几周后，有人说：“哦，我们实际上可以通过重新设计其中一些步骤来将 12 改进为 11。”当用笔和纸发生这种情况时，每次更改参数时，您都必须逐行检查证明的每一行是否仍然有效。

And there can be subtle things that you didn’t quite realize, some properties, not number 12, that you didn’t even realize that you were taking advantage of.

可能有一些微妙的事情你没有完全意识到，有些属性，不是 12 号，你甚至没有意识到你正在利用它们。

So a proof can break down at a subtle place.

所以证明可能会在一个微妙的地方崩溃。

1:26:29 So we had formalized the proof with this constant 12, and then when this new paper came out, we said, “Oh,” so that took three weeks to formalize and 20 people to formalize this original proof.

因此，我们用这个常数 12 形式化了证明，然后当这篇新论文发表时，我们说，“哦，”所以花了三周的时间来形式化，并花了 20 个人来形式化这个原始证明。

I said, “Now let’s update the proof to 11.” And what you can do with Lean is in your headline theorem, you change your 12 to 11, you run the compiler and off the thousands of lines of code, you have 90% of them still work and there’s a couple that are lined in red.

我说：“现在让我们将证明更新为 11。”你可以用精益做的就是在你的标题定理中，你将 12 更改为 11，运行编译器并关闭数千行代码，其中 90% 仍然可以工作，并且有几行是红色的。

Now, I can’t justify these steps, but immediately isolates which steps you need to change, but you can skip over everything, which works just fine.

现在，我无法证明这些步骤的合理性，但会立即隔离出您需要更改的步骤，但您可以跳过所有内容，这样效果就很好。

1:27:04 And if you program things correctly with good programming practices, most of your lines will not be red.

如果您使用良好的编程实践正确地进行编程，那么大多数线条都不会是红色的。

And there’ll just be a few places where you, if you don’t hard code your constants, but you use smart tactics and so forth, you can localize the things you need to change to a very small period of time.

在某些地方，如果您不对常量进行硬编码，而是使用聪明的策略等等，您可以将需要更改的内容本地化到很短的时间内。

So within a day or two, we had updated our proof because it’s this very quick process, you make a change.

因此，在一两天之内，我们就更新了我们的证明，因为这是一个非常快的过程，你可以做出改变。

There are 10 things now that don’t work.

现在有 10 件事不起作用。

For each one, you make a change and now there’s five more things that don’t work, but the process converges much more smoothly then with pen and paper.

对于每一项，你都做了改变，现在还有五件事不起作用，但这个过程比用笔和纸更顺利地融合。

So that’s for writing?

那么这是为了写作吗？

1:27:40 Are you able to read it?

你能读懂吗？

If somebody else has a proof, are you able to, what’s the versus paper and?

如果别人有证明，你能做吗？纸质证明和纸质证明是什么？

Yeah, so the proofs are longer, but each individual piece is easier to read.

是的，所以校样更长，但每一篇都更容易阅读。

1:27:48 So if you take a math paper and you jump to page 27 and you look at paragraph six and you have a line of text of math, I often can’t read it immediately because it assumes various definitions, which I have to go back and maybe on 10 pages earlier this was defined and the proof is scattered all over the place, and you basically are forced to read fairly sequentially.

因此，如果你拿一篇数学论文，跳到第 27 页，然后你看第六段，你会看到一行数学文本，我通常无法立即阅读它，因为它假设了各种定义，我必须回去，也许在 10 页之前就已经定义了这个定义，并且证明分散在各处，你基本上被迫按顺序阅读。

It’s not like say a novel where in a theory you could open up a novel halfway through and start reading.

这不像小说，理论上你可以打开一半开始阅读。

There’s a lot of context.

有很多背景。

But when [inaudible 01:28:23] Lean, if you put your cursor on a line code, every single object there, you can hover over it and it would say what it is, where it came from, where stuff is justified.

但是，当 [听不清 01:28:23] 精益时，如果你将光标放在一行代码上，那里的每个对象上，你可以将鼠标悬停在它上面，它会说它是什么，它来自哪里，东西是合理的。

You can trace things back much easier than flipping through a math paper.

你可以比翻阅数学试卷更容易地追溯事情。

1:28:34 So one thing that Lean really enables is actually collaborating on proofs at a really atomic scale that you really couldn’t do in the past.

因此，精益真正能够实现的一件事实际上是在原子尺度上进行合作证明，而这是过去无法做到的。

So traditionally with pen and paper, when you want to collaborate with another mathematician, either you do it at a blackboard where you can really interact, but if you’re doing it sort of by email or something, basically, yeah, you have to segment it.

所以传统上用笔和纸，当你想与另一位数学家合作时，要么你在黑板上进行，这样你就可以真正互动，但如果你通过电子邮件或其他方式进行，基本上，是的，你必须对其进行分段。

Say, “I’m going to finish section three, you do section four,” but you can’t really work on the same thing, collaborate at the same time.

说，“我要完成第三部分，你完成第四部分”，但你们不能真正做同一件事，同时进行协作。

1:29:03 But with Lean, you can be trying to formalize some portion of proof and say, “I got stuck at line 67 here.

但对于精益，你可以尝试将证明的某些部分形式化，然后说：“我被困在第 67 行了。

I need to prove this thing, but it doesn’t quite work.

我需要证明这一点，但这不太有效。

Here’s the three lines of code I’m having trouble with.” But because all the context is there, someone else can say, “Oh, okay, I recognize what you need to do.

这是我遇到问题的三行代码。”但因为所有的上下文都在那里，其他人可以说：“哦，好吧，我知道你需要做什么。

You need to apply this trick or this tool,” and you can do extremely atomic-level conversations.

你需要应用这个技巧或这个工具”，你就可以进行极其原子级的对话。

So because of Lean, I can collaborate with dozens of people across the world, most of who I have never met in person, and I may not know actually even whether they’re how reliable they are in the proof-taking field, but Lean gives me a certificate of trust so I can do trustless mathematics.

因此，由于 Lean，我可以与世界各地的数十个人合作，其中大多数人我从未见过面，而且我实际上可能不知道他们在校对领域是否可靠，但 Lean 给了我一份信任证书，这样我就可以进行无需信任的数学计算。

So there’s so many interesting questions there.

所以那里有很多有趣的问题。

1:29:43 So one, you’re known for being a great collaborator.

因此，您以出色的合作者而闻名。

So what is the right way to approach solving a difficult problem in mathematics when you’re collaborating?

那么，当你们合作时，解决数学难题的正确方法是什么？

Are you doing a divide and conquer type of thing?

你在做分而治之的事情吗？

Or are you focused in on a particular part and you’re brainstorming?

或者您是否专注于某个特定部分并正在集思广益？

There’s always a brainstorming process first.

首先总是有一个集思广益的过程。

1:30:05 Yeah, so math research projects, by their nature, when you start, you don’t really know how to do the problem.

是的，所以数学研究项目，就其本质而言，当你开始时，你并不真正知道如何解决这个问题。

It’s not like an engineering project where somehow the theory has been established for decades and it’s implementation is the main difficulty.

它不像一个工程项目，理论已经建立了几十年，而其实施是主要困难。

You have to figure out even what is the right path.

你甚至必须弄清楚什么是正确的道路。

So this is what I said about cheating first.

所以这就是我首先说的关于作弊的事情。

It’s like to go back to the bridge building analogy.

这就像回到建造桥梁的类比。

So first assume you have infinite budget and unlimited amounts of workforce and so forth.

因此，首先假设您有无限的预算和无限数量的劳动力等等。

Now can you build this bridge?

现在你能建造这座桥吗？

Okay, now have infinite budget, but only finite workforce, right?

好吧，现在预算无限，但劳动力有限，对吧？

Now can you do that?

现在你能做到吗？

And so forth.

等等。

So, of course no engineer can actually do this.

所以，当然没有工程师能够真正做到这一点。

Like I say, they have fixed requirements.

就像我说的，他们有固定的要求。

Yes, there’s this sort of jam sessions always at the beginning where you try all kinds of crazy things and you make all these assumptions that are unrealistic, but you plan to fix later.

是的，一开始总会有这种即兴演奏会，你会尝试各种疯狂的事情，做出所有这些不切实际的假设，但你计划稍后修复。

1:30:57 And you try to see if there’s even some skeleton of an approach that might work.

然后你尝试看看是否有一些可行的方法的框架。

And then hopefully that breaks up the problem into smaller sub problems, which you don’t know how to do.

然后希望将问题分解成更小的子问题，而你不知道该怎么做。

But then you focus on the sub ones.

但随后你会专注于次要的。

And sometimes different collaborators are better at working on certain things.

有时不同的合作者更擅长处理某些事情。

So one of my themes I’m known for is a theme of Ben Green, which now called the Green-Tao theorem.

因此，我著名的主题之一是本·格林的主题，现在称为格林-陶定理。

It’s a statement that the primes contain mathematic progressions of an event.

它声明素数包含一个事件的数学级数。

So it was a modification of his [inaudible 01:31:26] already.

所以这已经是他的[听不清 01:31:26] 的修改。

And the way we collaborated was that Ben had already proven a similar result for progressions of length three.

我们合作的方式是 Ben 已经证明了长度为 3 的级数的类似结果。

He showed that such like the primes contain loss and loss of progressions of length three, even subsets of the primes, certain subsets do, but his techniques only worked for the three progressions.

他表明，像素数一样包含长度为三的级数的丢失和丢失，甚至素数的子集，某些子集也如此，但他的技术只适用于三个级数。

They didn’t work for longer.

他们没有工作更长的时间。

1:31:46 But I had these techniques coming from a [inaudible 01:31:48] theory, which is something that I had been playing with and I knew better than Ben at the time.

但我拥有的这些技术来自 [听不清 01:31:48] 理论，这是我一直在研究的东西，而且当时我比本更了解。

And so if I could justify certain randomness properties of some set relating for primes, there’s a certain technical condition, which if I could have it, if Ben could supply me to this fact, I could conclude the theorem.

因此，如果我能够证明与素数相关的某些集合的某些随机性属性是合理的，那么就存在一定的技术条件，如果我可以拥有它，如果本可以向我提供这个事实，我就可以得出定理。

But what I asked was a really difficult question in number theory, which he said, “There’s no way we can prove this.” So he said, “Can you prove your part of the theorem using a weak hypothesis that I have a chance to prove it?” And he proposed something which he could prove, but it was too weak for me.

但我问的是数论中的一个非常困难的问题，他说，“我们无法证明这一点。”所以他说：“你能用一个弱假设来证明你的定理部分吗？我有机会证明它？”他提出了一些他可以证明的东西，但对我来说太弱了。

I can’t use this.

我不能用这个。

So there was this conversation going back and forth, a hacker-

于是就有了这样的对话，一个黑客——

Different cheats to-

不同的作弊方式-

Yeah, yeah, I want to cheat more.

是啊是啊，我还想再骗一点。

1:32:31 He wants to cheat less, but eventually we found a property which A, he could prove, and B, I could use, and then we could prove our theorem.

他想少作弊，但最终我们找到了一个属性，A，他可以证明，B，我可以使用，然后我们可以证明我们的定理。

So there are all kinds of dynamics.

所以就有了各种各样的动力。

Every collaboration has some story.

每一次合作都有一些故事。

No two are the same.

没有两个是相同的。

And then on the flip side of that, like you mentioned with Lean programming, now that’s almost like a different story because you can create, I think you’ve mentioned a blueprint for a problem, and then you can really do a divide and conquer with Lean where you’re working on separate parts and they’re using the computer system proof checker essentially to make sure that everything is correct along the way.

另一方面，就像你提到的精益编程一样，现在这几乎就像一个不同的故事，因为你可以创建，我认为你已经提到了问题的蓝图，然后你可以真正用精益进行分而治之，你在不同的部分上工作，他们本质上使用计算机系统证明检查器来确保一路上一切都是正确的。

So it makes everything compatible and trustable.

因此它使一切兼容且值得信赖。

1:33:17 Yeah, so currently only a few mathematical projects can be cut up in this way.

是的，所以目前只有少数数学项目可以用这种方式进行分割。

At the current state of the art, most of the Lean activity is on formalizing proofs that have already been proven by humans.

在目前的技术水平下，大多数精益活动都是在形式化已经被人类证明的证明上。

A math paper basically is a blueprint in a sense.

从某种意义上说，数学论文基本上是一个蓝图。

It is taking a difficult statement like big theorem and breaking up into me a hundred little lemmas, but often not all written with enough detail that each one can be sort of directly formalized.

它需要像大定理这样的困难陈述，并将其分解为一百个小引理，但通常不会全部写得足够详细，以至于每个引理都可以直接形式化。

1:33:46 A blueprint is a really pedantically written version of a paper where every step is explained as much detail as possible and just trying to make each step kind of self-contained or depending on only a very specific number of previous statements that been proven so that each node of this blueprint graph that gets generated can be tackled independently of the others.

蓝图是论文的一个真正迂腐的书面版本，其中每个步骤都被尽可能详细地解释，并且只是试图使每个步骤都是独立的，或者仅依赖于先前已被证明的特定数量的语句，以便生成的蓝图图的每个节点都可以独立于其他节点进行处理。

And you don’t even need to know how the whole thing works.

你甚至不需要知道整个事情是如何运作的。

So it’s like a modern supply chain.

所以它就像一个现代供应链。

If you want to create an iPhone or some other complicated object, no one person can build up a single object, but you can have specialists who just, if they’re given some widgets from some other company, they can combine them together to form a slightly bigger widget.

如果你想创建一部 iPhone 或其他一些复杂的物体，没有人可以构建一个单一的物体，但你可以让专家，如果他们从其他公司获得一些小部件，他们可以将它们组合在一起形成一个稍大的小部件。

I think that’s a really exciting possibility because you can have, if you can find problems that could be broken down in this way, then you could have thousands of contributors, right?

我认为这是一个非常令人兴奋的可能性，因为如果你能找到可以通过这种方式分解的问题，那么你可以拥有数千名贡献者，对吗？

1:34:27 To be completely distributed.

完全分布式。

So I told you before about this split between theoretical and experimental mathematics.

所以我之前告诉过你们理论数学和实验数学之间的分歧。

1:34:39 And right now most mathematics is theoretical, only a tiny bit it’s experimental.

目前大多数数学都是理论性的，只有一小部分是实验性的。

I think the platform that Lean and other software tools, so GitHub and things like that will allow experimental mathematics to scale up to a much greater degree than we can do now.

我认为精益和其他软件工具的平台，比如 GitHub 和类似的东西，将使实验数学的规模扩大到比我们现在能做的更大的程度。

So right now, if you want to do any mathematical exploration of some mathematical pattern or something, you need some code to write out the pattern.

所以现在，如果您想对某些数学模式或其他东西进行任何数学探索，您需要一些代码来写出该模式。

And I mean, sometimes there are some computer algebra packages that could help, but often it’s just one mathematician coding lots and lots of Python or whatever.

我的意思是，有时有些计算机代数包可以提供帮助，但通常只是一位数学家编写大量 Python 或其他代码。

And because coding is such an error-prone activity, it’s not practical to allow other people to collaborate with you on writing modules for your code because if one of the modules has a bug in it, the whole thing is unreliable.

而且由于编码是一项容易出错的活动，因此允许其他人与您合作为您的代码编写模块是不切实际的，因为如果其中一个模块存在错误，则整个事情都是不可靠的。

So you get these bespoke spaghetti code written by non-professional programmers, mathematicians, and they’re clunky and slow.

所以你会得到这些由非专业程序员、数学家编写的定制意大利面条代码，它们笨重且缓慢。

And so because of that, it’s hard to really mass-produce experimental results.

因此，很难真正批量生产实验结果。

1:35:45 But I think with Lean, I’m already starting some projects where we are not just experimenting with data, but experimenting with proofs.

但我认为通过精益，我已经开始了一些项目，我们不仅在试验数据，而且在试验证明。

So I have this project called the Equational Theories Project.

所以我有一个名为“方程理论项目”的项目。

Basically we generated about 22 million little problems in abstract algebra.

基本上我们生成了大约 2200 万个抽象代数小问题。

Maybe I should back up and tell you what the project is.

也许我应该备份并告诉你这个项目是什么。

Okay, so abstract algebra studies operations like multiplication, addition and the abstract properties.

好的，抽象代数研究乘法、加法和抽象属性等运算。

So multiplication for example, is commutative.

例如，乘法是可交换的。

X times Y is always Y times X, at least for numbers.

X 乘以 Y 始终等于 Y 乘以 X，至少对于数字来说是这样。

And it’s also associative.

而且它也是关联的。

X times Y times Z is the same as X times Y times Z.

X 乘以 Y 乘以 Z 与 X 乘以 Y 乘以 Z 相同。

So these operations obey some laws that don’t obey others.

因此，这些操作遵守一些不遵守其他法律的法律。

For example, X times X is not always equal to X.

例如，X 乘以 X 并不总是等于 X。

So that law is not always true.

所以这条法律并不总是正确的。

So given any operation, it obeys some laws and not others.

因此，对于任何操作，它都会遵守一些法律，而不是其他法律。

And so we generated about 4,000 of these possible laws of algebra that certain operations can satisfy.

因此，我们生成了大约 4,000 个可能的代数定律，某些运算可以满足这些定律。

1:36:38 And our question is which laws imply which other ones, so for example, does commutativity imply associativity?

我们的问题是哪些定律意味着其他哪些定律，例如，交换性是否意味着结合性？

And the answer is no, because it turns out you can describe an operation which obeys the commutative law but doesn’t obey the associative law.

答案是否定的，因为事实证明你可以描述一个遵守交换律但不遵守结合律的运算。

So by producing an example, you can show that commutativity does not imply associativity.

因此，通过生成一个示例，您可以证明交换性并不意味着结合性。

But some other laws do imply other laws by substitution and so forth, and you can write down some algebraic proof.

但是其他一些定律确实通过替换等暗示了其他定律，并且您可以写下一些代数证明。

So we look at all the pairs between these 4,000 laws and this up 22 million of these pairs.

因此，我们查看这 4,000 条定律之间的所有对，以及其中的 2200 万对。

And for each pair we ask, does this law imply this law?

对于我们询问的每一对，这条定律是否意味着这条定律？

If so, give a proof.

如果有，请给出证明。

If not, give a counterexample.

如果不是，请举出反例。

So 22 million problems, each one of which you could give to an undergraduate algebra student, and they had a decent chance of solving the problem, although there are a few, at least 22 million, like a hundred or so that are really quite hard, but a lot are easy.

2200 万个问题，每一个问题你都可以交给一个代数本科生，他们有很大的机会解决这个问题，尽管有一些，至少 2200 万个问题，比如一百个左右，确实相当困难，但很多都很简单。

And the project was just to work out to determine the entire graph which ones imply which other ones.

该项目只是为了确定整个图表中哪些图表暗示了哪些图表。

That’s an incredible project, by the way.

顺便说一句，这是一个令人难以置信的项目。

1:37:31 Such a good idea, such a good test that the very thing we’ve been talking about at a scale that’s remarkable.

这是一个多么好的想法，一个多么好的测试，我们一直在谈论的事情规模是惊人的。

So it would not have been feasible.

所以这是不可行的。

1:37:38 The state of the art in the literature was like 15 equations and sort of how they applied, that’s at the limit of what a human with pen and paper can do.

文献中最先进的技术大概有 15 个方程以及它们的应用方式，这已经是人类用笔和纸所能做到的极限了。

So you need to scale that up.

所以你需要扩大规模。

So you need to crowdsource, but you also need to trust all the, no one person can check 22 million of these proofs.

所以你需要众包，但你也需要信任所有这些，没有人可以检查 2200 万份这样的证据。

You need it to be computerized.

您需要将其计算机化。

And so it only became possible with Lean.

因此，只有通过精益，这一切才成为可能。

We were hoping to use a lot of AI as well.

我们也希望使用大量人工智能。

So the project is almost complete.

所以该项目已基本完成。

So at these 22 million, all but two had been settled.

因此，在这 2200 万人中，除了两人之外，其他人都已经解决了。

Wow.

哇。

Well, actually, and of those two, we have a pen and paper proof of the two, and we’re formalizing it.

嗯，实际上，对于这两者，我们有两者的笔和纸证明，并且我们正在将其形式化。

1:38:12 In fact, this morning I was working on finishing it, so we’re almost done on this.

事实上，今天早上我正在努力完成它，所以我们快完成了。

It’s incredible.

太不可思议了。

Yeah.

是的。

1:38:22 Fantastic.

极好的。

How many people were you able to get?

你能找到多少人？

About 50, which in mathematics is considered a huge number.

大约有50个，这在数学上被认为是一个巨大的数字。

It’s a huge number.

这是一个巨大的数字。

1:38:30 That’s crazy.

这太疯狂了。

Yeah.

是的。

1:38:32 So we’re going to have a paper of 50 authors and a big appendix of who contributed what.

因此，我们将有一篇由 50 位作者组成的论文，以及一个很大的附录，说明谁贡献了什么。

Here’s an question, not to maybe speak even more generally about it.

这是一个问题，也许不是为了更笼统地谈论它。

1:38:38 When you have this pool of people, is there a way to organize the contributions by level of expertise of the people, of the contributors?

当您拥有这样的人员库时，是否有办法根据人员和贡献者的专业水平来组织贡献？

Now, okay, I’m asking a lot of pothead questions here, but I’m imagining a bunch of humans, and maybe in the future, some AIs, can there be an ELO rating type of situation?

现在，好吧，我在这里问了很多傻瓜问题，但我想象着一群人类，也许在未来，一些人工智能，会不会出现 ELO 评级类型的情况？

1:39:00 Can there be an Elo-rating type of situation where a gamification of this.

是否存在一种 Elo-Rating 类型的情况，可以将其游戏化。

The beauty of these lean projects is that automatically you get all this data, so everything’s being uploaded for this GitHub.

这些精益项目的美妙之处在于，您可以自动获取所有这些数据，因此所有内容都会上传到此 GitHub。

1:39:07 GitHub tracks who contributed what.

GitHub 跟踪谁贡献了什么。

So you could generate statistics at any later point in time.

因此您可以在以后的任何时间点生成统计数据。

You can say, “Oh, this person contributed this many lines of code” or whatever.

你可以说，“哦，这个人贡献了这么多行代码”或者其他什么。

These are very crude metrics.

这些都是非常粗略的指标。

I would definitely not want this to become part of your tenure review or something.

我绝对不希望这成为你的任期审查或其他内容的一部分。

But I mean, I think already in enterprise computing, people do use some of these metrics as part of the assessment of performance of an employee.

但我的意思是，我认为在企业计算中，人们确实使用其中一些指标作为员工绩效评估的一部分。

Again, this is the direction which is a bit scary for academics to go down.

再说一次，这对于学术界来说是一个有点可怕的下降方向。

We don’t like metrics so much.

我们不太喜欢指标。

And yet academics use metrics.

然而学术界却使用指标。

1:39:49 They just use old ones, number of papers.

他们只使用旧的、一些纸张。

Yeah, it’s true that…

是的，确实如此……

It feels like this is a metric, while flawed, is going in more in the right direction.

感觉这是一个指标，虽然有缺陷，但正在朝着正确的方向发展。

1:39:59 Right.

正确的。

Yeah.

是的。

It’s interesting.

这很有趣。

1:40:06 At least it’s a very interesting metric.

至少这是一个非常有趣的指标。

Yeah, I think it’s interesting to study.

是的，我觉得学习很有趣。

1:40:08 I think you can do studies of whether these are better predictors.

我认为你可以研究这些是否是更好的预测因素。

There’s this problem called Goodhart’s Law.

有一个问题叫做古德哈特定律。

If a statistic is actually used to incentivize performance, it becomes gamed, and then it’s no longer a useful measure.

如果一项统计数据实际上被用来激励绩效，那么它就会被欺骗，然后它就不再是一个有用的衡量标准。

Oh, humans.

哦，人类。

1:40:22 Always gaming the…

总是玩游戏……

It’s rational.

这是理性的。

1:40:25 So what we’ve done for this project is self-report.

所以我们为这个项目所做的就是自我报告。

So there are actually standard categories from the sciences of what types of contributions people give.

因此，科学上实际上有标准的类别来描述人们做出的贡献类型。

So there’s the concept and validation and resources and coding and so forth.

所以有概念、验证、资源和编码等等。

So there’s a standard list of twelve or so categories, and we just ask each contributor to…

因此，有一个包含十二个左右类别的标准列表，我们只要求每个贡献者……

There’s a big matrix of all the authors and all the categories just to tick the boxes where they think that they contributed, and just give a rough idea.

有一个包含所有作者和所有类别的大矩阵，只是为了在他们认为自己做出贡献的方框中打勾，并给出一个粗略的想法。

Also, you did some coding and you provided some compute, but you didn’t do any of the pen- and-paper verification or whatever.

另外，你做了一些编码并提供了一些计算，但你没有进行任何纸笔验证或其他任何操作。

1:41:02 And I think that that works out.

我认为这是可行的。

Traditionally, mathematicians just order alphabetically by surname.

传统上，数学家只是按姓氏的字母顺序排列。

So we don’t have this tradition as in the sciences of “lead author” and “second author” and so forth, which we’re proud of.

因此，我们没有像科学领域那样的“第一作者”和“第二作者”等传统，我们对此感到自豪。

We make all the authors equal status, but it doesn’t quite scale to this size.

我们让所有作者享有平等的地位，但它并没有完全扩展到这个规模。

So a decade ago I was involved in these things called polymath projects.

所以十年前我参与了这些被称为博学项目的事情。

It was the crowdsourcing mathematics but without the lean component.

这是众包数学，但没有精益成分。

So it was limited by, you needed a human moderator to actually check that all the contributions coming in were actually valid.

因此，它受到限制，您需要一名人工审核员来实际检查所有收到的贡献是否实际上有效。

And this was a huge bottleneck, actually, but still we had projects that were 10 authors or so.

实际上，这是一个巨大的瓶颈，但我们的项目仍然有 10 名左右的作者。

But we had decided, at the time, not to try to decide who did what, but to have a single pseudonym.

但当时我们决定不试图决定谁做了什么，而是使用一个化名。

So we created this fictional character called DHJ Polymath in the spirit of [inaudible 01:41:51].

因此，我们本着 [听不清 01:41:51] 的精神创造了这个名为 DHJ Polymath 的虚构人物。

This is the pseudonym for a famous group of mathematicians in the 20th century.

这是20世纪一群著名数学家的笔名。

1:41:56 And so the paper was authored on the pseudonym, so none of us got the author credit.

所以这篇论文是用笔名写的，所以我们都没有得到作者的认可。

This actually turned out to be not so great for a couple of reasons.

由于几个原因，事实证明这实际上并不是那么好。

So one is that if you actually wanted to be considered for tenure or whatever, you could not use this paper in your…

所以一个是，如果你真的想被考虑获得终身教职或其他什么，你不能在你的……中使用这篇论文。

As you submitted as one your publications, because it didn’t have the formal author credit.

当您作为出版物提交时，因为它没有正式的作者信用。

But the other thing that we’ve recognized much later is that when people referred to these projects, they naturally referred to the most famous person who was involved in the project.

但我们后来认识到的另一件事是，当人们提到这些项目时，他们自然会提到参与该项目的最著名的人。

So “This was Tim Gower’s playoff project.” “This was Terence Tao’s playoff project,” and not mention the other 19 or whatever people that were involved.

所以“这是蒂姆·高尔的季后赛项目。” “这是陶哲轩的季后赛项目”，更不用说其他 19 人或任何参与其中的人了。

Oh, yeah.

哦，是的。

So we’re trying something different this time around where we have, everyone’s an author, but we will have an appendix with this matrix, and we’ll see how that works.

所以这次我们尝试一些不同的东西，每个人都是作者，但我们将有一个包含这个矩阵的附录，我们将看看它是如何工作的。

So both projects are incredible, just the fact that you’re involved in such huge collaborations.

所以这两个项目都令人难以置信，事实上你们参与了如此大规模的合作。

1:42:45 But I think I saw a talk from Kevin Buzzard about the Lean Programming Language just a few years ago, and you’re saying that this might be the future of mathematics.

但我想几年前我看到 Kevin Buzzard 关于精益编程语言的演讲，你说这可能是数学的未来。

And so it’s also exciting that you’re embracing one of the greatest mathematicians in the world embracing this, what seems like the paving of the future of mathematics.

因此，令人兴奋的是，您正在拥抱世界上最伟大的数学家之一，拥抱这一点，这似乎为数学的未来铺平了道路。

1:43:12 So I have to ask you here about the integration of AI into this whole process.

所以我必须在这里问你关于人工智能融入整个过程的问题。

So DeepMind’s alpha proof was trained using reinforcement learning on both failed and successful formal lean proofs of IMO problems.

因此，DeepMind 的 alpha 证明是使用强化学习对 IMO 问题的失败和成功的正式精益证明进行训练的。

So this is sort of high-level high school?

那么这是一所高级高中吗？

Oh, very high-level, yes.

噢，非常高水平，是的。

Very high-level, high-school level mathematics problems.

非常高水平的高中水平的数学问题。

1:43:33 What do you think about the system and maybe what is the gap between this system that is able to prove the high-school level problems versus gradual-level problems?

您对这个系统有何看法？这个能够证明高中水平问题与渐进水平问题的系统之间的差距是什么？

Yeah, the difficulty increases exponentially with the number of steps involved in the proof.

是的，难度随着证明中涉及的步骤数量呈指数级增加。

1:43:47 It’s a commentarial explosion.

这是评论爆炸。

So the thing of large language models is that they make mistakes and so if a proof has got 20 steps and your [inaudible 01:44:01] has a 10% failure rate at each step of going the wrong direction, it’s extremely unlikely to actually reach the end.

所以大型语言模型的问题在于它们会犯错误，所以如果一个证明有 20 个步骤，而你的 [听不清 01:44:01] 在走错方向的每一步都有 10% 的失败率，那么它实际上不太可能到达终点。

Actually, just to take a small tangent here, how hard is the problem of mapping from natural language to the formal program?

实际上，在这里稍微切一下，从自然语言到正式程序的映射问题有多难？

Oh yeah.

哦，是的。

1:44:19 It’s extremely hard, actually.

实际上，这非常困难。

Natural language, it’s very fault-tolerant.

自然语言，它是非常容错的。

You can make a few minor grammatical errors and speak in the second language, can get some idea of what you’re saying.

你可以犯一些小语法错误并用第二语言说话，可以了解你在说什么。

But formal language, if you get one little thing wrong, then the whole thing is nonsense.

但在正式语言中，如果你犯了一点小错误，那么整件事都是无稽之谈。

Even formal to formal is very hard.

即使是正式到正式也是非常困难的。

There are different incompatible prefaces and languages.

有不同的不兼容的前言和语言。

There’s Lean, but also Cox and Isabelle and so forth.

有 Lean，还有 Cox 和 Isabelle 等等。

And even converting from a formal action to formal language is an unsolved problem.

甚至从正式动作转换为正式语言也是一个未解决的问题。

That is fascinating.

这很有趣。

1:44:52 Okay.

好的。

But once you have an informal language, they’re using their RL trained model, something akin to AlphaZero that they used to go to then try to come up with tools, also have a model.

但是一旦你有了一种非正式的语言，他们就会使用他们的强化学习训练模型，类似于他们过去常常使用的 AlphaZero，然后尝试拿出工具，也有一个模型。

I believe it’s a separate model for geometric problems.

我相信这是一个单独的几何问题模型。

Yes.

是的。

So what impresses you about the system, and what do you think is the gap?

那么这个系统给您留下的印象是什么？您认为差距是什么？

Yeah, we talked earlier about things that are amazing over time become kind of normalized.

是的，我们之前谈到过随着时间的推移，一些令人惊奇的事情会变得常态化。

1:45:18 So now somehow, it’s of course geometry is a silver bullet problem.

所以现在不知何故，几何当然是一个银弹问题。

Right.

正确的。

1:45:27 That’s true, that’s true.

确实如此，确实如此。

I mean, it’s still beautiful to…

我的意思是，它仍然很美丽......

Yeah, these are great work that shows what’s possible.

是的，这些都是伟大的作品，展示了一切的可能性。

1:45:31 The approach doesn’t scale currently.

该方法目前无法扩展。

Three days of Google’s server time can solve one high school math format there.

谷歌服务器三天的时间可以解决一种高中数学问题。

This is not a scalable prospect, especially with the exponential increase as the complexity increases.

这不是一个可扩展的前景，尤其是随着复杂性的增加而呈指数级增长。

We should mention that they got a silver medal performance.

值得一提的是，他们的表演获得了银牌。

1:45:49 The equivalent of the silver medal performance.

相当于银牌的表现。

So first of all, they took way more time than was allotted, and they had this assistance where the humans started helped by formalizing, but also they’re giving themselves full marks for the solution, which I guess is formally verified.

首先，他们花费的时间比分配的时间多得多，他们得到了人类通过形式化开始提供帮助的帮助，但他们也给自己的解决方案打了满分，我想这已经得到了形式化的验证。

1:45:55 So I guess that’s fair.

所以我想这是公平的。

There are efforts, there will be a proposal at some point to actually have an AI math Olympiad where at the same time as the human contestants get the actual Olympiad problems, AI’s will also be given the same problems, the same time period and the outputs will have to be graded by the same judges, which means that it’ll have be written in natural language rather than formal language.

我们已经做出了努力，在某个时候将会有人提议真正举办一次人工智能数学奥林匹克竞赛，在人类参赛者得到实际的奥林匹克题目的同时，人工智能也会遇到同样的问题，同样的时间段，输出结果必须由同样的评委进行评分，这意味着它将用自然语言而不是正式语言编写。

Oh, I hope that happens.

哦，我希望这会发生。

1:46:37 I hope that this IMO happens.

我希望这会发生。

I hope next one.

我希望下一篇。

It won’t happen this IMO.

在我看来，这不会发生。

1:46:41 The performance is not good enough in the time period.

时间段内表现不够好。

But there are smaller competitions, there are competitions where the answer is a number rather than a long form proof.

但也有一些较小的比赛，有些比赛的答案是一个数字而不是一个长形式的证明。

And AI is actually a lot better at problems where there’s a specific numerical answer, because it’s easy to do reinforcement learning on it.” Yeah, you’ve got the right answer, you’ve got the wrong answer.” It’s a very clear signal, but a long form proof either has to be formal, and then the Lean can give it thumbs up or down, or it’s informal, but then you need a human to create it to tell.

事实上，人工智能更擅长解决有具体数字答案的问题，因为很容易对其进行强化学习。”是的，你有正确的答案，也有错误的答案。”这是一个非常明确的信号，但长篇证明要么必须是正式的，然后精益可以给予它赞成或反对的意见，要么它是非正式的，但你需要一个人来创建它来告诉你。

And if you’re trying to do billions of reinforcement learning runs, you can’t hire enough humans to grade those.

如果你试图进行数十亿次强化学习运行，你就无法雇用足够的人员来对这些进行评分。

It’s already hard enough for the last language to do reinforcement learning on just the regular text that people get.

对于最后一种语言来说，仅对人们获得的常规文本进行强化学习已经足够困难了。

But now we actually hire people, not just give thumbs up, thumbs down, but actually check the output mathematically, yeah, that’s too expensive.

但现在我们实际上雇佣了人，不仅仅是竖起大拇指，竖起大拇指，而是实际上用数学方法检查输出，是的，这太昂贵了。

So if we just explore this possible future, what is the thing that humans do that’s most special in mathematics, that you could see AI not cracking for a while?

因此，如果我们只是探索这个可能的未来，人类所做的在数学中最特别的事情是什么，你可能会看到人工智能暂时无法破解？

1:47:45 So inventing new theories?

那么发明新理论？

Coming up with new conjectures versus proving the conjectures?

提出新猜想还是证明猜想？

Building new abstractions?

构建新的抽象？

New representations?

新的表述？

Maybe an AI turnstile with seeing new connections between disparate fields?

也许人工智能旋转门可以看到不同领域之间的新联系？

That’s a good question.

这是一个好问题。

1:48:17 I think the nature of what mathematicians do over time has changed a lot.

我认为随着时间的推移，数学家所做的事情的本质已经发生了很大的变化。

So a thousand years ago, mathematicians had to compute the date of Easter, and they really complicated calculations, but it is all automated, the order of centuries, we don’t need that anymore.

所以一千年前，数学家必须计算复活节的日期，他们的计算确实很复杂，但它都是自动化的，几个世纪的顺序，我们不再需要它了。

They used to navigate to do spherical navigation, circle trigonometry to navigate how to get from the Old World to the New or something, like very complicated calculation.

他们以前导航是做球面导航，圆三角学来导航如何从旧世界到新世界什么的，比如非常复杂的计算。

Again, have been automated.

再次，已经自动化。

Even a lot of undergraduate mathematics even before AI, like Wolfram Alpha for example.

甚至很多本科生数学甚至在人工智能之前就已经存在，例如 Wolfram Alpha。

It’s not a language model, but it can solve a lot of undergraduate-level math tasks.

它不是一个语言模型，但它可以解决很多本科水平的数学任务。

So on the computational side, verifying routine things, like having a problem and saying, ” Here’s a problem in partial differential equations, could you solve it using any of the 20 standard techniques?” And say, “Yes, I’ve tried all 20 and here are the 100 different permutations and my results.”

因此，在计算方面，验证常规事物，例如遇到问题并说：“这是一个偏微分方程问题，你能使用 20 种标准技术中的任何一种来解决它吗？”并说：“是的，我已经尝试了所有 20 种排列，这是 100 种不同的排列和我的结果。”

1:49:12 And that type of thing, I think it will work very well, type of scaling to once you solve one problem to make the AI attack a hundred adjacent problems.

我认为这种事情会非常有效，一旦你解决了一个问题，人工智能就会攻击一百个相邻的问题。

The things that humans do still…

人类所做的事情仍然......

So where the AI really struggles right now is knowing when it’s made a wrong turn.

因此，人工智能目前真正遇到的困难是知道自己何时走错了方向。

It can say, “Oh, I’m going to solve this problem.

它可以说：“哦，我要解决这个问题。

I’m going to split up this one into these two cases.

我将把这个问题分为这两种情况。

I’m going to try this technique.” And sometimes, if you’re lucky and it’s a simple problem, it’s the right technique and you solve the problem and sometimes it will have a problem, it would propose an approach which is just complete nonsense, but it looks like a proof.

我要尝试一下这个技术。”有时，如果你很幸运，这是一个简单的问题，这是正确的技术，你解决了问题，有时它会出现问题，它会提出一种完全是无稽之谈的方法，但它看起来像是一个证明。

1:49:53 So this is one annoying thing about LLM-generated mathematics.

所以这是法学硕士生成的数学的一件恼人的事情。

So yeah, we’ve had human generated mathematics as a very low quality, like submissions who don’t have the formal training and so forth, but if a human proof is bad, you can tell it’s bad pretty quickly.

所以，是的，我们的人类生成的数学质量非常低，比如没有经过正式培训的提交等，但如果人类的证明很糟糕，你很快就能看出它很糟糕。

It makes really basic mistakes.

它犯了非常基本的错误。

But the AI-generated proofs, they can look superficially flawless.

但人工智能生成的证据，表面上看起来完美无缺。

And it’s partly because what the reinforcement learning has actually trained them to do, to make things to produce tech that looks like what is correct, which for many applications is good enough.

部分原因是强化学习实际上训练了他们去做，让事情产生看起来正确的技术，这对于许多应用程序来说已经足够好了。

So the air is often really subtle and then when you spot them, they’re really stupid.

所以空气往往非常微妙，当你发现它们时，它们真的很愚蠢。

Like no human would’ve actually made that mistake.

就像没有人会真正犯这样的错误一样。

Yeah, it’s actually really frustrating in the programming context, because I program a lot, and yeah, when a human makes low quality code, there’s something called “code smell”, right?

是的，在编程环境中这实际上真的很令人沮丧，因为我经常编程，是的，当一个人编写出低质量的代码时，就会有一种叫做“代码味道”的东西，对吧？

1:50:36 You can tell immediately there’s signs, but with the AI generated code…

你可以立即看出有迹象，但使用人工智能生成的代码......

[inaudible 01:50:53].

[听不清 01:50:53]。

And you’re right, eventually you find an obvious dumb thing that just looks like good code.

你是对的，最终你会发现一个明显愚蠢的东西，只是看起来像是好的代码。

Yeah.

是的。

It’s very tricky, too, and frustrating, for some reason, to have to work.

由于某种原因，不得不工作也很棘手，而且令人沮丧。

So the sense of smell, this is one thing that humans have, and there’s a metaphorical mathematical smell that it’s not clear how to get the AI to duplicate that eventually.

所以嗅觉，这是人类所拥有的一件事，并且有一种隐喻的数学气味，目前尚不清楚如何让人工智能最终复制它。

1:51:05 So the way AlphaZero and so forth make progress on Go and chess and so forth, is in some sense they have developed a sense of smell for Go and chess positions, that this position is good for white, it’s good for black.

所以AlphaZero等在围棋和国际象棋等方面取得进展的方式，从某种意义上说，他们已经开发了对围棋和国际象棋位置的嗅觉，这个位置对白有利，对黑有利。

They can’t initiate why, but just having that sense of smell lets them strategize.

他们无法找出原因，但只要有嗅觉就可以让他们制定策略。

So if AIs gain that ability to a sense of viability of certain proof strategies, because I’m going to try to break up this problem into two small subtasks and they can say, “Oh, this looks good.

因此，如果人工智能获得了某种证明策略的可行性的能力，因为我将尝试将这个问题分解为两个小子任务，他们可以说，“哦，这看起来不错。

The two tasks look like they’re simpler tasks than your main task and they’ve still got a good chance of being true.

这两个任务看起来比你的主要任务更简单，而且它们仍然有很大的机会成为现实。

So this is good to try.” Or “No, you’ve made the problem worse, because each of the two subproblems is actually harder than your original problem,” which is actually what normally happens if you try a random thing to try normally it’s very easy to transform a problem into an even harder problem.

所以这是很好的尝试。”或者“不，你让问题变得更糟了，因为这两个子问题实际上都比你原来的问题更难”，这实际上是通常发生的情况，如果你尝试随机的事情来正常尝试，很容易将问题变成更难的问题。

Very rarely do you transform into a simpler problem.

你很少会转化为更简单的问题。

So if they can pick up a sense of smell, then they could maybe start competing with a human level of mathematicians.

因此，如果他们能够获得嗅觉，那么他们也许可以开始与人类水平的数学家竞争。

So this is a hard question, but not competing but collaborating.

所以这是一个很难的问题，但不是竞争而是合作。

1:52:24 Okay, hypothetical.

好吧，假设的。

If I gave you an Oracle that was able to do some aspect of what you do and you could just collaborate with it, what would you like that Oracle to be able to do?

如果我给你一个 Oracle，它能够完成你所做的某些事情，并且你可以与它协作，你希望 Oracle 能够做什么？

Would you like it to maybe be a verifier, like check, do the codes?

你希望它成为一个验证器，比如检查代码吗？

Like “Yes, Professor Tao, correct, this is a promising fruitful direction”?

就像“是的，陶教授，正确的，这是一个有前途、富有成果的方向”？

Or would you like it to generate possible proofs and then you see which one is the right one?

或者您希望它生成可能的证明，然后您看看哪一个是正确的？

Or would you like it to maybe generate different representation, different totally different ways of seeing this problem?

或者你希望它能够生成不同的表示形式，不同的完全不同的方式来看待这个问题？

Yeah, I think all of the above.

是的，我想以上都是。

1:53:10 A lot of it is we don’t know how to use these tools, because it’s a paradigm that…

很多是我们不知道如何使用这些工具，因为它是一个范例......

We have not had in the past.

我们过去没有过。

Systems that are competent enough to understand complex instructions that can work at massive scale, but are also unreliable.

系统有足够的能力理解可以大规模运行的复杂指令，但也不可靠。

It’s an interesting…

这是一个有趣的……

A bit unreliable in subtle ways, whereas was providing sufficiently good output.

虽然提供了足够好的输出，但在微妙的方面有点不可靠。

It’s an interesting combination.

这是一个有趣的组合。

I mean, you have graduate students that you work with who kind of like this, but not at scale.

我的意思是，你有和你一起工作的研究生，他们有点像这样，但规模不大。

And we had previous software tools that can work at scale, but very narrow, so we have to figure out how to use, so Tim Gowers is actually, you mentioned he actually foresaw in 2000.

我们以前有可以大规模工作的软件工具，但非常狭窄，所以我们必须弄清楚如何使用，所以 Tim Gowers 实际上，你提到他实际上在 2000 年就预见到了。

He was envisioning what mathematics would look like in actually two and a half decades.

他想象着两年半后数学会是什么样子。

That’s funny.

那很好笑。

Yeah, he wrote his article, a hypothetical conversation between a mathematical assistant of the future and himself.

是的，他写了他的文章，这是未来的数学助手和他自己之间的假设对话。

1:54:09 He’s trying to solve a problem and they would have a conversation.

他正在尝试解决问题，他们会进行对话。

Sometimes the human would propose an idea and the AI would evaluate it, and sometimes the AI would propose an idea and sometimes a competition was required and AI would just go and say, “Okay, I’ve checked the 100 cases needed here,” or “The first you set this through for all N, I’ve checked N up to 100 and it looks good so far,” or “Hang on, there’s a problem at N equals 46.” So just a freeform conversation where you don’t know in advance where things are going to go, but just based on, “I think ideas are going to propose on both sides.” Calculations could propose on both sides.

有时人类会提出一个想法，人工智能会评估它，有时人工智能会提出一个想法，有时需要进行竞赛，人工智能会说，“好吧，我已经检查了这里需要的 100 个案例”，或者“你第一次为所有 N 设置这个，我已经检查了 N 到 100，到目前为止看起来不错”，或者“等等，N 等于 46 时有问题。”因此，这只是一场自由形式的对话，你事先不知道事情会走向何方，但只是基于“我认为双方都会提出想法”。双方都可以提出计算建议。

1:54:53 I’ve had conversations with AI where I say, “Okay, we’re going to collaborate to solve this math problem,” and it’s a problem that I already know the solution to, so I try to prompt it.

我与人工智能进行过对话，我说，“好吧，我们将合作解决这个数学问题”，这是一个我已经知道解决方案的问题，所以我尝试提示它。

“Okay, so here’s the problem.” I suggest using this tool, and it’ll find this.” Okay, it might start using this, and then it’ll go back to the tool that it wanted to do before.

“好吧，那么问题来了。”我建议使用这个工具，它会找到这个。”好吧，它可能会开始使用这个，然后它会回到它之前想做的工具。

You have to keep railroading it onto the path you want, and I could eventually force it to give the proof I wanted, but it was like herding cats.

你必须不断地把它带到你想要的道路上，我最终可以强迫它给出我想要的证据，但这就像放牧猫一样。

And the amount of personal effort I had to take to not just prompt it but also check its output because a lot of what it looked like is going to work, I know there’s a problem on line 17, and basically arguing with it.

我必须付出大量的个人努力，不仅要提示它，还要检查它的输出，因为它看起来很多东西都会起作用，我知道第 17 行有问题，并且基本上与它争论。

It was more exhausting than doing it unassisted, but that’s the current state of the art.

这比独自完成更累人，但这就是当前的技术水平。

I wonder if there’s a phase shift that happens to where it’s no longer feels like herding cats.

我想知道是否会发生一个相变，让人们不再感觉像在放牧猫。

1:55:44 And maybe you’ll surprise us how quickly that comes.

也许你会让我们惊讶这一切来得如此之快。

I believe so.

我相信是这样。

1:55:54 In formalization, I mentioned before that it takes 10 times longer to formalize a proof than to write it by hand.

在形式化方面，我之前提到过，形式化证明的时间比手写要长 10 倍。

With these modern AI tools and also just better tooling, the Lean developers are doing a great job adding more and more features and making it user-friendly going from nine to eight to seven…

借助这些现代人工智能工具和更好的工具，精益开发人员正在出色地添加越来越多的功能，并使其用户友好性从朝九晚八到七……

Okay, no big deal, but one day it’ll drop a little one.

好吧，没什么大不了的，但有一天它会掉一点点。

And that’s a phase shift, because suddenly it makes sense when you write a paper to write it in Lean first, or through a conversation with AI, which is generally on the fly with you, and it becomes natural for journals to accept.

这是一个阶段性的转变，因为突然间，当你写一篇论文时，先用精益来写，或者通过与人工智能的对话，这通常是与你一起进行的，期刊自然会接受。

Maybe they’ll offer expedite refereeing.

也许他们会提供快速裁判。

If a paper has already been formalized in Lean, they’ll just ask the referee to comment on the significance of the results and how it connects to literature and not worry so much about the correctness, because that’s been certified.

如果一篇论文已经在精益中正式化，他们只会要求审稿人评论结果的重要性以及它如何与文献联系起来，而不必太担心正确性，因为这已经得到了认证。

Papers are getting longer and longer in mathematics, and it’s harder and harder to get good refereeing for the really long ones unless they’re really important.

数学领域的论文越来越长，除非它们真的很重要，否则很难为很长的论文获得良好的审稿。

It is actually an issue, and the formalization is coming in at just the right time for this to be.

这实际上是一个问题，而正式化的出现恰逢其时。

And the easier and easier to guess because of the tooling and all the other factors, then you’re going to see much more like math lib will grow potentially exponentially, as it’s a virtuous cycle.

由于工具和所有其他因素，越来越容易猜测，那么您将看到更多类似的数学库将呈指数级增长，因为这是一个良性循环。

I mean, one phase shift of this type that happened in the past was the adoption of LaTeX.

我的意思是，过去发生的这种类型的一个阶段性转变是 LaTeX 的采用。

1:57:16 So LaTeX is this typesetting language that all mathematicians use now.

所以LaTeX就是现在所有数学家都使用的这种排版语言。

So in the past people used all kinds of word processors and typewriters and whatever, but at some point LaTeX became easier to use than all other competitors, and people would switch within a few years.

因此，过去人们使用各种文字处理程序和打字机等等，但在某些时候，LaTeX 变得比所有其他竞争对手更容易使用，人们会在几年内更换。

It was just a dramatic base shift.

这只是一个戏剧性的基础转变。

It’s a wild, out-there question, but what year, how far away are we from a AI system being a collaborator on a proof that wins the Fields Medal?

这是一个疯狂而遥远的问题，但我们距离人工智能系统成为赢得菲尔兹奖证明的合作者还有多远？

1:57:37 So that level.

那么那个水平。

Okay, well it depends on the level of collaboration, right?

好吧，这取决于合作的水平，对吗？

No, it deserves to be get the Fields Medal.

不，它应该获得菲尔兹奖。

1:57:58 So half-and-half

所以一半一半

Already.

已经。

1:58:03 I can imagine if it was a medal-winning paper, having some AI assistance in writing it just like the order complete alone is already, I use it speeds up my own writing.

我可以想象，如果这是一篇获奖论文，有一些人工智能辅助来写作，就像单独完成订单一样，我用它来加快我自己的写作速度。

You can have a theorem and you have a proof, and the proof has three cases, and I write down the proof of first case and the autocomplete just suggests that.

你可以有一个定理，也有一个证明，这个证明有三种情况，我写下第一种情况的证明，自动完成只是建议这一点。

Now here’s how the proof of second case could work.

现在介绍第二种情况的证明是如何工作的。

And it was exactly correct.

这是完全正确的。

That was great.

那太好了。

Saved me like five, ten minutes of typing.

节省了我大约五到十分钟的打字时间。

But in that case, the AI system doesn’t get the Fields Medal.

但那样的话，人工智能系统就不会获得菲尔兹奖。

1:58:30 Are we talking 20 years, 50 years, a hundred years?

我们是在谈论20年、50年还是100年？

What do you think?

你觉得怎么样？

Okay, so I gave a prediction in print by 2026, which is now next year, there will be math collaborations with the AI, so not Fields-Medal winning, but actual research-level papers.

好吧，我在印刷版上做出了预测，到 2026 年，也就是明年，将会有与人工智能的数学合作，所以不是菲尔兹奖获得者，而是实际的研究水平论文。

Published ideas that are in part generated by AI.

已发表的想法部分由人工智能产生。

Maybe not the ideas, but at least some of the computations, the verifications.

也许不是想法，但至少是一些计算和验证。

Has that already happened?

这已经发生了吗？

That already happened.

那已经发生了。

1:59:04 There are problems that were solved by a complicated process conversing with AI to propose things and the human goes and tries it and the contract doesn’t work, but it might pose a different idea.

有些问题是通过与人工智能对话来提出建议的复杂过程来解决的，人类去尝试它，但合同不起作用，但它可能会提出不同的想法。

It’s hard to disentangle exactly.

很难准确地解开。

There are certainly math results which could only have been accomplished because there was a human authentication and an AI involved, but it’s hard to disentangle credit.

当然，有些数学结果只能通过人工验证和人工智能的参与才能实现，但很难区分信用。

I mean, these tools, they do not replicate all the skills needed to do mathematics, but they can replicate some non-trivial percentage of them, 30, 40%, so they can fill in gaps.

我的意思是，这些工具并不能复制做数学所需的所有技能，但它们可以复制其中的一些重要比例，例如 30%、40%，因此它们可以填补空白。

So coding is a good example.

所以编码就是一个很好的例子。

So it’s annoying for me to code in Python.

所以用 Python 编写代码对我来说很烦人。

I’m not a native, I’m a professional programmer, but with AI, the friction cost of doing it is much reduced.

我不是本地人，我是一名专业程序员，但有了人工智能，这样做的摩擦成本就大大降低了。

So it fills in that gap for me.

所以它填补了我的空白。

AI is getting quite good at literature review.

人工智能在文献综述方面越来越擅长。

2:00:15 I mean, it’s still a problem with hallucinating references that don’t exist, but this, I think, is a solvable problem.

我的意思是，这仍然是一个不存在的幻觉参考的问题，但我认为这是一个可以解决的问题。

If you train in the right way and so forth and verify using the internet, you should, in a few years, get to the point where you have a lemma that you need and say, “Has anyone proven this lemma before?” And it will do basically a fancy web search and say, yeah, there are these six papers where something similar has happened.

如果你以正确的方式进行训练并使用互联网进行验证，那么几年后，你应该达到你需要的引理并说：“以前有人证明过这个引理吗？”它基本上会进行一次奇特的网络搜索，然后说，是的，有这六篇论文发生了类似的事情。

I mean, you can ask it right now and it’ll give you six papers of which maybe one is legitimate and relevant, one exists but is not relevant, and four are hallucinated.

我的意思是，你现在就可以问它，它会给你六篇论文，其中一篇可能是合法且相关的，一篇存在但不相关，四篇是幻觉的。

It has a non-zero success rate right now, but there’s so much garbage, so much the signal-to-noise ratio is so poor, that it’s most helpful when you already somewhat know the relationship, and you just need to be prompted to be reminded of a paper that was already subconsciously in your memory.

现在它的成功率不为零，但是垃圾太多，信噪比太差，当你已经对其中的关系有所了解时，它是最有帮助的，只需要提示你想起一篇已经在你潜意识里记忆中的论文。

Versus helping you discover new you were not even aware of, but is the correct citation.

与帮助您发现您甚至没有意识到的新事物相比，这是正确的引用。

Yeah, that it can sometimes do, but when it does, it’s buried in a list of options for which the other-

是的，它有时可以做到，但当它发生时，它会被埋在一个选项列表中，而其他选项-

That are bad.

那很糟糕。

2:01:26 I mean, being able to automatically generate a related work section that is correct.

我的意思是，能够自动生成正确的相关工作部分。

That’s actually a beautiful thing.

这其实是一件很美好的事情。

That might be another phase shift because it assigns credit correctly.

这可能是另一个阶段性转变，因为它正确地分配了功劳。

It breaks you out of the silos of thought.

它让你摆脱思想的束缚。

Yeah, no, there’s a big hump to overcome right now.

是的，不，现在有一个很大的困难需要克服。

2:01:42 I mean it’s like self-driving cars.

我的意思是这就像自动驾驶汽车。

The safety margin has to be really high for it to be feasible.

安全裕度必须非常高才可行。

So yeah, there’s a [inaudible 02:01:54]-Morrow problem with a lot of AI applications that they can develop tools that work 20%, 80% of the time, but it’s still not good enough.

所以，是的，有一个 [听不清 02:01:54]-Morrow 问题，许多人工智能应用程序可以开发出在 20%、80% 的情况下工作的工具，但仍然不够好。

And in fact, even worse than good, in some ways.

事实上，在某些方面，甚至比好的还要糟糕。

I mean, another way of asking the Fields Medal question is what year do you think you’ll wake up and be like real surprised?

我的意思是，问菲尔兹奖问题的另一种方式是，你认为你会在哪一年醒来并感到非常惊讶？

2:02:08 You read the headline, the news or something happened that AI did, real breakthrough.

你读到了人工智能所做的标题、新闻或发生的事情，真正的突破。

Something.

某事。

Like Fields Medal, even a hypothesis.

就像菲尔兹奖一样，甚至是一个假设。

It could be really just this AlphaZero Go moment would go that kind of thing.

可能真的只是 AlphaZero Go 时刻就会发生这样的事情。

Yeah, this decade I can see it making a conjecture between two things that people would thought was unrelated.

是的，这十年我可以看到它在人们认为无关的两件事之间做出了推测。

Oh, interesting.

哦，有趣。

2:02:42 Generating a conjecture.

产生一个猜想。

That’s a beautiful conjecture.

这是一个美丽的猜想。

Yeah.

是的。

2:02:45 And actually has a real chance of being correct and meaningful.

并且实际上有真正的机会是正确且有意义的。

Because that’s actually kind of doable, I suppose, but the word of the data is…

因为我认为这实际上是可行的，但数据上的说法是……

Yeah.

是的。

No, that would be truly amazing.

不，那真是太神奇了。

The current models struggle a lot.

当前的模型有很多困难。

2:02:59 I mean, so a version of this…

我的意思是，所以这个版本......

The physicists have a dream of getting the AI to discover new laws of physics.

物理学家的梦想是让人工智能发现新的物理定律。

The dream is you just feed it all this data, and this is here is a new patent that we didn’t see before, but it actually, even the current state of the art even struggles to discover old laws of physics from the data.

梦想是你只需向它提供所有这些数据，这是一项我们以前没有见过的新专利，但实际上，即使是当前的最先进技术，甚至很难从数据中发现旧的物理定律。

Or if it does, there’s a big concern of contamination, that it did it only because it’s somewhere in this training, somehow new, Boyle’s Law or whatever you’re trying to reconstruct.

或者，如果确实如此，人们会非常担心污染，因为它这样做只是因为它在训练中的某个地方，某种新的东西，波义耳定律或任何你试图重建的东西。

2:03:35 Part of it is we don’t have the right type of training data for this.

部分原因是我们没有合适类型的训练数据。

So for laws of physics, we don’t have a million different universes with a million different laws of nature.

因此，对于物理定律来说，我们并没有一百万个不同的宇宙和一百万个不同的自然定律。

And a lot of what we are missing in math is actually the negative space of…

我们在数学中缺少的很多东西实际上是负空间……

So we have published things of things that people have been able to prove, and conjectures that end up being verified or we counter examples produced, but we don’t have data on things that were proposed and they’re kind of a good thing to try, but then people quickly realized that it was the wrong conjecture and then they said, “Oh, but we should actually change our claim to modify it in this way to actually make it more plausible.”

因此，我们发布了人们已经能够证明的事情，以及最终被验证的猜想，或者我们制作了反例，但是我们没有关于所提出的事情的数据，它们是值得尝试的一件好事，但人们很快意识到这是错误的猜想，然后他们说，“哦，但我们实际上应该改变我们的主张，以这种方式修改它，使其实际上变得更加可信。”

2:04:16 There’s a trial and error process, which is a real integral part of human mathematical discovery, which we don’t record because it’s embarrassing.

有一个反复试验的过程，这是人类数学发现真正不可或缺的一部分，但我们没有记录下来，因为这很尴尬。

We make mistakes, and we only like to publish our wins.

我们会犯错误，但我们只喜欢公布我们的胜利。

And the AI has no access to this data to train on.

人工智能无法访问这些数据进行训练。

I sometimes joke that basically AI has to go through grad school and actually go to grad courses, do the assignments, go to office hours, make mistakes, get advice on how to correct the mistakes and learn from that.

我有时开玩笑说，基本上人工智能必须通过研究生院，实际上去读研究生课程，做作业，去办公时间，犯错误，获得如何纠正错误的建议并从中学习。

Let me ask you if I may, about Grigori Perelman, you mentioned that you try to be careful in your work and not let a problem completely consume you just you’ve really fall in love with the problem and it really cannot rest until you solve it.

让我问你是否可以，关于格里戈里·佩雷尔曼（Grigori Perelman），你提到你在工作中要小心谨慎，不要让问题完全吞噬你，因为你真的爱上了这个问题，并且在你解决它之前它真的不能休息。

2:04:47 But you also hastened to add that sometimes this approach actually can be very successful, and an example you gave is Grigori Perelman who proved the Poincare Conjecture and did so by working alone for seven years, with basically little contact with the outside world.

但您也赶紧补充说，有时这种做法实际上可以非常成功，您举的一个例子是格里戈里·佩雷尔曼（Grigori Perelman），他证明了庞加莱猜想，他独自工作了七年，基本上与外界接触很少。

Can you explain this one Millennial Prize problem that’s been solved, Poincare Conjecture, and maybe speak to the journey that Grigori Perelman has been on?

您能否解释一下这个已解决的千禧年奖问题：庞加莱猜想，或者谈谈格里戈里·佩雷尔曼 (Grigori Perelman) 所经历的旅程？

All right, so it’s a question about curved spaces.

好吧，这是一个关于弯曲空间的问题。

2:05:31 Earth is a good example.

地球就是一个很好的例子。

So think of Earth as a 2-D surface.

因此，可以将地球视为二维表面。

Injecting around you could maybe be a torus with a hole in it or can have many holes and there are many different topologies, a priori, that a surface could have, even if you assume that it’s bounded and smooth and so forth.

在你周围注入可能是一个有孔的环面，或者可以有很多孔，并且表面可能具有许多不同的先验拓扑，即使你假设它是有界的和光滑的等等。

So we have figured out how to classify surfaces as a first approximation.

因此，我们已经弄清楚如何将表面分类为第一近似值。

Everything is determined by some called the genus, how many holes it has.

一切都是由一些叫做属的东西决定的，它有多少个洞。

So a sphere has genus zero, or a donut has genus one, and so forth.

因此，球体的属数为零，或者甜甜圈的属数为一，依此类推。

And one way you can tell the surfaces apart, probably the sphere has, which is called simply connected.

区分曲面的一种方法可能是球体，这称为简单连通。

If you take any closed loop on the sphere, like a big closed loop of rope, you can contract it to a point while staying on the surface.

如果你在球体上取任何闭环，比如一个大的闭环绳子，你可以将它收缩到一个点，同时保持在表面上。

And the sphere has this property, but a torus doesn’t.

球体具有此属性，但环面没有。

If you’re on a torus and you take a rope that goes around say the outer diameter of torus, there’s no way…

如果你在一个圆环上，并且拿着一根绕圆环外径的绳子，那就没有办法……

It can’t get through the hole.

它无法通过孔。

There’s no way to contract it to a point.

没有办法将其收缩到某个点。

2:06:25 So it turns out that the sphere is the only surface with this property of contractibility, up to continuous deformations of the sphere.

因此，事实证明，球体是唯一具有这种可收缩性的表面，直到球体发生连续变形。

So things that are what called topologically equivalent of the sphere.

因此，所谓的球体的拓扑等价物。

So Poincare asks the same question, higher dimensions, so it becomes hard to visualize because surface you can think of as embedded in three dimensions, but a curved three-space, we don’t have good intuition of four-dimensional space to live it.

所以庞加莱提出了同样的问题，更高的维度，所以它变得很难可视化，因为你可以认为表面嵌入在三维空间中，但是弯曲的三空间，我们没有对四维空间的良好直觉来生活它。

And there are also three-dimensional spaces that can’t even fit into four dimensions.

还有一些三维空间甚至无法容纳四个维度。

You need five or six or higher.

你需要五六个或更多。

But anyway, mathematically you can still pose this question, that if you have a bounded three- dimensional space now, which also has this simply connected property that every loop can be contracted, can you turn it into a three-dimensional version of the sphere?

但无论如何，从数学上讲，你仍然可以提出这个问题，如果你现在有一个有界的三维空间，它也具有每个环都可以收缩的单连通属性，你能把它变成球体的三维版本吗？

And so this is the Poincare conjecture.

这就是庞加莱猜想。

2:07:09 Weirdly, in higher dimensions, four and five was actually easier.

奇怪的是，在更高的维度中，四和五实际上更容易。

So it was solved first in higher dimensions, there’s somehow more room to do the deformation.

所以它首先在更高的维度中解决，有更多的空间来进行变形。

It is easier to move things around to your sphere.

将事物移动到您的范围内会更容易。

But three was really hard.

但三级真的很难。

So people tried many approaches.

于是人们尝试了很多方法。

There’s sort of commentary approaches where you chop up the surface into little triangles or tetrahedra and you just try to argue based on how the faces interact each other.

有一种评论方法，你把表面切成小三角形或四面体，然后你只是试图根据面孔如何相互作用来争论。

There were algebraic approaches, there’s various algebraic objects like things called the fundamental group that you can attach to these homologies and co-homology and all these very fancy tools.

有代数方法，有各种代数对象，例如称为基本群的东西，您可以将其附加到这些同调和上同调以及所有这些非常奇特的工具上。

They also didn’t quite work, but Richard Hamilton’s proposed a partial differential equations approach.

它们也不太有效，但理查德·汉密尔顿提出了一种偏微分方程方法。

2:07:52 So the problem is that…

所以问题是……

So you have this object, which is secret is a sphere, but it’s given to you in a weird way.

所以你有这个物体，它是一个秘密的球体，但它是以一种奇怪的方式给你的。

So I think of a ball that’s being crumpled up and twisted, and it’s not obvious that it’s the ball, but if you have some sort of surface, which is a deformed sphere, you could for example, think that as a surface of a balloon, you could try to inflate it, you blow it up and naturally as you fill it with air, the wrinkles will sort of smooth out and it will turn into a nice round sphere, unless of course it was a torus or something, which case it would get stuck at some point.

所以我想到一个被弄皱和扭曲的球，它并不明显是球，但是如果你有某种表面，这是一个变形的球体，你可以例如认为作为气球的表面，你可以尝试给它充气，你把它吹起来，当你充满空气时，皱纹会自然地平滑，它会变成一个漂亮的圆形球体，当然，除非它是一个环面或其他东西，这种情况会在某个时刻陷入困境。

2:08:32 If you inflate a torus, there be a point in the middle when the inner ring shrinks to zero, you get a singularity and you can’t blow up any further and you can’t flow further.

如果你膨胀一个环面，当内环收缩到零时，中间会有一个点，你会得到一个奇点，你不能进一步膨胀，也不能进一步流动。

So he created this flow, which is now called Ricci Flow, which is a way of taking an arbitrary surface or space and smoothing it out to make it rounder and rounder to make it look like a sphere.

所以他创造了这种流，现在被称为“里奇流”，这是一种获取任意表面或空间并将其平滑化以使其变得越来越圆以使其看起来像球体的方法。

And he wanted to show that either this process would give you a sphere, or it would create a singularity, actually very much like how PDEs either they have global regularity or finite and blow up.

他想表明，这个过程要么会给你一个球体，要么会创建一个奇点，实际上非常像偏微分方程，它们要么具有全局正则性，要么有限并爆炸。

Basically, it’s almost exactly the same thing.

基本上，这几乎是完全相同的事情。

It’s all connected.

一切都是相连的。

And he showed that for two dimensions, two-dimensional surfaces, if you start to simply connect it, no singularities ever formed, you never ran into trouble and you could flow and it will give you a sphere.

他表明，对于二维、二维表面，如果你开始简单地连接它，就不会形成奇点，你永远不会遇到麻烦，你可以流动，它会给你一个球体。

So he got a new proof of the two-dimensional result.

于是他得到了二维结果的新证明。

But by the way, that’s a beautiful explanation of Ricci flow in its application in this context.

但顺便说一句，这是对利玛窦流在这种情况下的应用的一个很好的解释。

2:09:20 How difficult is the mathematics here for the 2D case?

对于 2D 情况，这里的数学有多困难？

Is it?

是吗？

Yeah, these are quite sophisticated equations on par with the Einstein equations.

是的，这些都是与爱因斯坦方程相当复杂的方程。

2:09:27 Slightly simpler, but they were considered hard nonlinear equations to solve, and there’s lots of special tricks in 2D that helped.

稍微简单一些，但它们被认为是难以求解的非线性方程，并且二维中有很多特殊技巧可以提供帮助。

But in 3D, the problem was that this equation was actually super critical.

但在 3D 中，问题是这个方程实际上非常关键。

The same problem as [inaudible 02:09:48].

与 [听不清 02:09:48] 相同的问题。

As you blow up, maybe the curvature could get concentrated in smaller and smaller regions, and it looked more and more nonlinear and things just looked worse and worse.

当你爆炸时，也许曲率会集中在越来越小的区域，它看起来越来越非线性，事情看起来越来越糟糕。

And there could be all kinds of singularities that showed up.

可能会出现各种各样的奇点。

Some singularities, these things called neck pinches where the surface behaves like a barbell and it pinches at a point.

一些奇点，这些被称为“颈部挤压”的东西，其表面表现得像杠铃，并且在一个点上挤压。

Some singularities are simple enough that you can sort of see what to do next.

有些奇点非常简单，您可以知道下一步该做什么。

You just make a snip and then you can turn one surface into two and e-bolt them separately.

您只需进行剪断，然后即可将一个表面变成两个表面，并分别用电子螺栓固定它们。

But there was the prospect that there’s some really nasty knotted singularities showed up that you couldn’t see how to resolve in any way, that you couldn’t do any surgery to.

但有可能会出现一些非常令人讨厌的打结奇点，你无法看到如何以任何方式解决，你无法对其进行任何手术。

So you need to classify all the singularities, like what are all the possible ways that things can go wrong?

所以你需要对所有的奇点进行分类，比如事情可能出错的所有可能方式是什么？

So what Perelman did was, first of all, he made the problem, he turned the problem from a super critical problem to a critical problem.

所以佩雷尔曼所做的就是，首先他提出了问题，他把问题从超级关键问题变成了关键问题。

I said before about how the invention of energy, the Hamiltonian, really clarified Newtonian mechanics.

我之前说过能量的发明，哈密顿量，如何真正澄清了牛顿力学。

So he introduced something which is now called Perelman’s reduced volume and Perelman’s entropy.

因此，他引入了现在称为佩雷尔曼约化体积和佩雷尔曼熵的东西。

He introduced new quantities, kind of like energy, that looked the same at every single scale, and turned the problem into a critical one where the non-linearities actually suddenly looked a lot less scary than they did before.

他引入了新的量，有点像能量，在每个尺度上看起来都是一样的，并将问题变成了一个关键的问题，其中非线性实际上突然看起来比以前不那么可怕了。

And then he had to solve…

然后他必须解决……

He still had to analyze the singularities of this critical problem.

他仍然需要分析这个关键问题的奇点。

And that itself was a problem similar to this wave map thing I worked on actually.

这本身就是一个与我实际研究的波图类似的问题。

So on the level of difficulty of that.

那么就其难度而言。

2:11:18 So he managed to classify all the singularities of this problem, and show how to apply surgery to each of these.

因此，他设法对这个问题的所有奇点进行分类，并展示如何对每个奇点进行手术。

And through that was able to resolve the Poincare Conjecture.

从而解决了庞加莱猜想。

So quite a lot of really ambitious steps, and nothing that a large language model today, for example, could…

因此，有很多真正雄心勃勃的步骤，而当今的大型语言模型无法做到这一点……

At best, I could imagine a model proposing this idea as one of hundreds of different things to try, but the other 99 would be complete dead ends.

充其量，我可以想象一个模型将这个想法作为数百种不同的尝试之一，但其他 99 种将是完全死胡同。

But you’d only find out after months of work, he must have had some sense that this was the right track to pursue.

但经过几个月的工作你才会发现，他一定已经意识到这是一条正确的道路。

It takes years to get from A to B.

从 A 到 B 需要数年时间。

So you’ve done, like you said, actually, you see even strictly mathematically, but more broadly in terms of the process, you’ve done similar-

所以你已经做到了，就像你说的，实际上，你甚至可以从严格的数学角度来看，但从更广泛的过程来看，你已经做了类似的事情——

2:12:01 In terms of the process, you’ve done similarly difficult things.

从过程上来说，你也做过类似困难的事情。

What can you infer from the process he was going through because he was doing it alone?

你能从他独自一人经历的过程中推断出什么？

What are some low points in a process like that when you start to, you’ve mentioned hardship, AI doesn’t know when it’s failing.

当你开始的时候，你提到过困难，人工智能不知道什么时候会失败。

What happens to you, you’re sitting in your office when you realize the thing you did for the last few days, maybe weeks is a failure?

当你坐在办公室里，当你意识到你过去几天（也许几周）所做的事情是失败的时候，你会发生什么？

Well, for me, I switch to a different problem.

好吧，对我来说，我会转向另一个问题。

2:12:27 So I’m a fox, I’m not a hedgehog.

所以我是狐狸，不是刺猬。

But you’re generally, that is a break that you can take, is to step away and look at a different problem?

但一般来说，这是你可以采取的休息方式，是离开并看看不同的问题吗？

Yeah, yeah.

是啊是啊。

2:12:37 You can modify the problem too.

你也可以修改问题。

I mean, you can ask some cheater if there’s a specific thing that’s blocking you that some bad case keeps showing up, that for which your tool doesn’t work.

我的意思是，你可以问一些作弊者，是否有特定的事情阻碍了你，导致一些糟糕的情况不断出现，而你的工具对此不起作用。

You can just assume by fiat this bad case doesn’t occur.

你可以通过命令假设这种糟糕的情况不会发生。

So you do some magical thinking, but strategically okay for the point to see if the rest of the argument goes through.

所以你做了一些神奇的思考，但从策略上来说是可以的，看看论证的其余部分是否成立。

If there’s multiple problems with your approach, then maybe you just give up.

如果您的方法存在多个问题，那么您可能会放弃。

But if this is the only problem but everything else checks out, then it’s still worth fighting.

但如果这是唯一的问题，而其他一切都已解决，那么它仍然值得一战。

So yeah, you have to do some forward reconnaissance sometimes too.

所以，是的，有时你也必须进行一些前方侦察。

And that is sometimes productive to assume like, “Okay, we’ll figure it out eventually”?

有时假设“好吧，我们最终会解决这个问题”是有成效的吗？

Oh, yeah, yeah.

哦，是啊，是啊。

2:13:21 Sometimes actually it’s even productive to make mistakes.

事实上，有时犯错误甚至是富有成效的。

So one of, there was a project which actually we won some prizes for with four other people.

其中之一是，有一个项目实际上我们和另外四个人一起赢得了一些奖项。

We worked on this PDE problem.

我们研究了这个偏微分方程问题。

Again, actually this blow-off regularity type problem, and it was considered very hard.

再说一次，实际上这种吹出规律性类型的问题，被认为是非常困难的。

Jean Bourgiugnon was another Fields mathematist who worked on a special case of this, but he could not solve the general case.

让·布尔久农（Jean Bourgiugnon）是另一位菲尔兹数学家，他研究了这个问题的特殊情况，但他无法解决一般情况。

And we worked on this problem for two months and we thought we solved it.

我们研究这个问题两个月了，我们认为我们已经解决了。

We had this cute argument that if everything fit, and we were excited, we were planning celebration, to all get together and have champagne or something, and we started writing it up.

我们有一个可爱的争论，如果一切都合适，我们很兴奋，我们正在计划庆祝活动，大家聚在一起喝香槟什么的，然后我们开始把它写下来。

And one of us, not me actually, but another co-author said, “Oh, in this lemma here, we have to estimate these 13 terms that show up in this expansion.

我们中的一个人，实际上不是我，而是另一位合著者说，“哦，在这个引理中，我们必须估计这个扩展中出现的这 13 个项。

2:14:13 And we estimate 12 of them, but in our notes, I can’t find the estimation of the 13th.

我们估计了其中的 12 个，但在我们的笔记中，我找不到第 13 个的估计。

Can someone supply that?” And I said, “Sure, I’ll look at this.” Yeah, we didn’t cover it, we completely omitted this term and this term turned out to to be worse than the other 12 terms put together.

有人可以提供吗？”我说：“当然，我会看看这个。”是的，我们没有涵盖它，我们完全省略了这个术语，结果这个术语比其他 12 个术语加起来更糟糕。

In fact, we could not estimate this term.

事实上，我们无法估计这个术语。

And we tried for a few more months and all different permutations, and there was always this one term that we could not control.

我们又尝试了几个月和所有不同的排列，但总有一个术语是我们无法控制的。

And so this was very frustrating.

所以这非常令人沮丧。

But because we had already invested months and months of effort in this already, we stuck at this, which we tried increasingly desperate things and crazy things.

但因为我们已经在这方面投入了数月的努力，所以我们坚持了下来，我们尝试了越来越绝望和疯狂的事情。

And after two years we found an approach that was somewhat different, but quite a bit from our initial strategy, which actually didn’t generate these problematic terms and actually solve the problem.

两年后，我们发现了一种有点不同的方法，但与我们最初的策略有很大不同，它实际上没有生成这些有问题的术语，也没有真正解决问题。

2:14:58 So we solve the problem after two years, but if we hadn’t had that initial false dawn of nearly solving a problem, we would’ve given up by month two or something and worked on an easier problem.

因此，我们在两年后解决了这个问题，但如果我们没有最初的那种几乎解决问题的虚假曙光，我们就会在两个月左右放弃并致力于解决一个更简单的问题。

If we had known it would take two years, not sure we would’ve started the project.

如果我们知道需要两年的时间，我们不确定是否会开始这个项目。

Sometimes actually having the incorrect, it’s like Columbus struggling in the new world, they had an incorrect measurement of the size of the Earth.

有时实际上是不正确的，就像哥伦布在新世界中挣扎一样，他们对地球大小的测量是错误的。

He thought he was going to find a new trade route to India, or at least that was how he sold it in his prospectus.

他认为他将找到一条通往印度的新贸易路线，或者至少他在招股说明书中是这样出售的。

I mean, it could be that he actually secretly knew, but.

我的意思是，他可能实际上偷偷知道，但是。

Just from a psychological element, do you have emotional or self-doubt that just overwhelms you in moments like that?

仅从心理因素来看，在这样的时刻，你是否有情绪或自我怀疑让你不知所措？

2:15:31 Because this stuff, it feels like math is so engrossing that it can break you when you invest so much of yourself in the problem and then it turns out wrong.

因为这些东西，感觉数学是如此引人入胜，以至于当你在这个问题上投入如此多的精力而结果却是错误的时，它可能会击垮你。

You could start to…

你可以开始……

A similar way chess has broken some people.

国际象棋也以类似的方式让一些人崩溃了。

Yeah, I think different mathematicians have different levels of emotional investment in what they do.

是的，我认为不同的数学家对他们所做的事情有不同程度的情感投入。

2:15:59 I mean, I think for some people it’s as a job, you have a problem and if it doesn’t work out, you go on the next one.

我的意思是，我认为对某些人来说，这是一项工作，你遇到问题，如果解决不了，你就会继续下一个。

So the fact that you can always move on to another problem, it reduces the emotional connection.

因此，事实上，你总是可以继续解决另一个问题，这会减少情感联系。

I mean, there are cases, so there are certain problems that are what are called mathematical diseases where just latch onto that one problem and they spend years and years thinking about nothing but that one problem.

我的意思是，在某些情况下，有些问题就是所谓的数学疾病，他们只是抓住那个问题，然后花了很多年的时间除了那个问题之外什么也不思考。

And maybe their career suffers and so forth, but they say, “Okay, I’ve got this big win.

也许他们的职业生涯会受到影响等等，但他们说，“好吧，我已经取得了巨大的胜利。

Once I finish this problem, it will make up for all the years of lost opportunity.” I mean, occasionally it works, but I really don’t recommend it for people without the right fortitude.

一旦我解决了这个问题，就可以弥补多年来失去的机会。”我的意思是，偶尔它会起作用，但我真的不推荐给没有适当毅力的人。

2:16:54 So I’ve never been super invested in any one problem.

所以我从来没有对任何一个问题投入过多的精力。

One thing that helps is that we don’t need to call our problems in advance.

有帮助的一件事是我们不需要提前提出我们的问题。

Well, when we do grant proposals, we say we will study this set of problems, but even though we don’t promise, definitely by five years I will supply a proof of all these things.

好吧，当我们授予提案时，我们说我们将研究这一系列问题，但即使我们不承诺，我肯定会在五年内提供所有这些事情的证明。

You promise to make some progress or discover some interesting phenomena.

你承诺会取得一些进展或发现一些有趣的现象。

And maybe you don’t solve the problem, but you find some related problem that you can say something new about and that’s a much more feasible task.

也许你没有解决问题，但你发现了一些相关的问题，你可以说一些新的东西，这是一个更可行的任务。

But I’m sure for you, there’s problems like this.

但我确信对你来说，存在这样的问题。

2:17:27 You have made so much progress towards the hardest problems in the history of mathematics.

您在数学史上最困难的问题上取得了如此大的进展。

So is there a problem that just haunts you?

那么有一个问题一直困扰着您吗？

It sits there in the dark corners, twin prime conjecture, Riemann hypothesis, Goldbach’s conjecture?

它坐在黑暗的角落里，孪生素数猜想、黎曼猜想、哥德巴赫猜想？

Twin prime, that sounds…

双素数，听起来……

2:17:48 Look, again, I mean, the problems like the Riemann hypothesis, those are so far out of reach.

再说一次，我的意思是，像黎曼猜想这样的问题，这些都是遥不可及的。

You think so?

你这么认为吗？

Yeah, there’s no even viable stretch.

是的，甚至没有可行的延伸。

2:17:55 Even if I activate all the cheats that I know of in this book, there’s just still no way to get from A to B.

即使我激活了本书中我所知道的所有秘籍，仍然无法从 A 到达 B。

I think it needs a breakthrough in another area of mathematics to happen first and for someone to recognize that it would a useful thing to transport into this problem.

我认为首先需要在数学的另一个领域取得突破，并且有人认识到将其转移到这个问题将是有用的事情。

So we should maybe step back for a little bit and just talk about prime numbers.

所以我们也许应该退后一步，只讨论素数。

Okay.

好的。

So they’re often referred to as the atoms of mathematics.

因此它们通常被称为数学原子。

2:18:23 Can you just speak to the structure that these atoms provide?

您能谈谈这些原子提供的结构吗？

So the natural numbers have two basic operations, addition, and multiplication.

所以自然数有两个基本运算，加法和乘法。

2:18:31 So if you want to generate the natural numbers, you can do one of two things.

因此，如果你想生成自然数，你可以做以下两件事之一。

You can just start with one and add one to itself over and over again.

您可以从一个开始，然后一遍又一遍地添加一个。

And that generates you the natural numbers.

这就生成了自然数。

So additively, they’re very easy to generate one, two, three, four, five.

所以加起来，它们很容易生成一、二、三、四、五。

Or you can take the prime number if you want to generate multiplicatively, you can take all the prime numbers, two, three, five, seven and multiply them all together.

或者，如果你想乘法生成，你可以取素数，你可以取所有素数，两个，三个，五个，七个，然后将它们全部相乘。

Together that gives you all the natural numbers except maybe for one.

加在一起就可以得到除一个之外的所有自然数。

So there are these two separate ways of thinking about the natural numbers from an additive point of view and a multiplicative point of view.

因此，从加法的角度和乘法的角度来思考自然数有两种不同的方式。

And separately, they’re not so bad.

单独来看，它们并没有那么糟糕。

So any question about that natural was it only was addition, it’s relatively easy to solve.

所以任何关于自然数的问题是否只是加法，都相对容易解决。

2:19:11 And any question that only was multiplication is relatively easy to solve.

任何仅涉及乘法的问题都相对容易解决。

But what has been frustrating is that you combine the two together and suddenly you get the extremely rich…

但令人沮丧的是，你将两者结合在一起，突然间你就变得非常富有......

I mean, we know that there are statements in number theory that are actually as undecidable.

我的意思是，我们知道数论中的某些陈述实际上是不可判定的。

There are certain polynomials in some number of variables.

某些数量的变量存在某些多项式。

Is there a solution in the natural numbers?

自然数有解吗？

And the answer depends on an undecidable statement whether the axioms of mathematics are consistent or not.

答案取决于数学公理是否一致的不可判定的陈述。

But even the simplest problems that combine something more applicative such as the primes with something additives such as shifting by two, separately we understand both of them well, but if you ask when you shift the prime by two, can you get up?

但是，即使是最简单的问题，将更适用的东西（例如素数）与附加的东西（例如移位二）结合起来，我们单独地对它们都很好理解，但是如果你问当你将素数移位二时，你能站起来吗？

How often can you get another prime?

您多久可以获得一次素数？

It’s been amazingly hard to relate the two.

将两者联系起来非常困难。

And we should say that the twin prime conjecture is just that, it pauses that there are infinitely many pairs of prime numbers that differ by two.

我们应该说，孪生素数猜想就是这样的，它暂停了存在无限多对相差二的素数。

2:19:59 Now the interesting thing is that you have been very successful at pushing forward the field in answering these complicated questions of this variety.

有趣的是，您在回答这些复杂问题方面非常成功地推动了这一领域的发展。

Like you mentioned the Green-Tao Theorem.

就像你提到的格林陶定理。

It proves that prime numbers contain arithmetic progressions of any length.

它证明素数包含任意长度的算术级数。

Right.

正确的。

It’s just mind-boggling that you could prove something like that.

你能证明这样的事情真是令人难以置信。

Right.

正确的。

2:20:27 Yeah.

是的。

So what we’ve realized because of this type of research is that different patterns have different levels of indestructibility.

因此，通过此类研究，我们意识到不同的模式具有不同程度的不可破坏性。

What makes the twin prime problem hard is that if you take all the primes in the world, three, five, seven, 11, and so forth, there are some twins in there, 11 and 13 is a twin prime, pair of twin primes and so forth.

孪生素数问题之所以困难，是因为如果你取世界上所有的素数，3、5、7、11等等，其中会有一些孪生，11和13是一个孪生素数，一对孪生素数等等。

But you could easily, if you wanted to redact the primes to get rid of these twins.

但如果你想编辑素数来摆脱这些双胞胎，你可以很容易地做到这一点。

The twins, they’d show up and they’re infinitely many of them, but they’re actually reasonably sparse.

双胞胎，他们会出现，而且数量无限多，但实际上相当稀疏。

There’s not, I mean, initially there’s quite a few, but once you got to the millions, the trillions, they become rarer and rarer.

我的意思是，最初并没有很多，但一旦达到数百万、数万亿，它们就会变得越来越稀有。

And you could actually just, if someone was given access to the database of primes, you just edit out a few primes here and there.

实际上，如果有人有权访问素数数据库，您只需在这里或那里编辑一些素数即可。

2:21:15 They could make the twin prime conjecture false by just removing 0.01% of the primes or something, just well-chosen to do this.

他们只需删除 0.01% 的素数或其他东西就可以使孪生素数猜想为假，这只是精心选择的做法。

And so you could present a censored database of the primes, which passes all of these statistical tests of the primes.

因此，您可以提供一个经过审查的素数数据库，该数据库通过了所有这些素数统计测试。

It obeys things like the polynomial theorem and other effects of the primes, but doesn’t contain any twin primes anymore.

它遵循多项式定理和素数的其他效应，但不再包含任何孪生素数。

And this is a real obstacle to the twin-prime conjecture.

这是孪生素数猜想的真正障碍。

It means that any proof strategy to actually find twin primes in the actual primes must fail when applied to these slightly edited primes.

这意味着任何在实际素数中实际找到孪生素数的证明策略在应用于这些稍微编辑过的素数时都必须失败。

And so it must be some very subtle, delicate feature of the primes that you can’t just get from aggregate statistical analysis.

因此，它一定是素数的一些非常微妙、微妙的特征，你不能从总体统计分析中得到这些特征。

Okay, so that’s out.

好吧，就这样了。

Yeah.

是的。

2:22:02 On the other hand, progressions has turned out to be much more robust.

另一方面，事实证明进步要强劲得多。

You can take the primes and you can eliminate 99% of the primes actually, and you can take any 90% you want.

你可以取素数，实际上你可以消除 99% 的素数，你可以取任何你想要的 90%。

And it turns out, and another thing we proved is that you still get arithmetic progressions.

事实证明，我们证明的另一件事是你仍然可以获得算术级数。

Arithmetic progressions are much, they’re like cockroaches.

算术级数很多，就像蟑螂一样。

Of arbitrary length though.

但任意长度。

Yes.

是的。

2:22:23 Yes.

是的。

That’s crazy.

这太疯狂了。

Yeah.

是的。

So for people who don’t know, arithmetic progressions is a sequence of numbers that differ by some fixed amount.

因此，对于不知道的人来说，算术级数是相差某个固定数量的数字序列。

Yeah.

是的。

2:22:32 But it’s again like, it’s an infinite monkey type phenomenon.

但这又像是，这是一种无限的猴子类型的现象。

For any fixed length of your set, you don’t get arbitrary length progressions.

对于任何固定长度的集合，您都不会获得任意的长度级数。

You only get quite short progressions.

你只能得到很短的进展。

But you’re saying twin-prime is not an infinite monkey phenomena.

但你是说孪生素数并不是无限的猴子现象。

2:22:40 I mean, it’s a very subtle monkey.

我的意思是，这是一只非常狡猾的猴子。

It’s still an infinite monkey phenomena.

这仍然是一个无限的猴子现象。

Right.

正确的。

2:22:48 Yeah.

是的。

If the primes were really genuinely random, if the primes were generated by monkeys, then yes, in fact the infinite monkey theorem would-

如果素数真的是随机的，如果素数是由猴子生成的，那么是的，事实上无限猴子定理会-

Oh, but you’re saying that twin prime, you can’t use the same tools.

哦，但你是说双胞胎素数，你不能使用相同的工具。

2:22:56 It doesn’t appear random almost.

它看起来几乎不是随机的。

Well, we don’t know.

好吧，我们不知道。

2:23:05 We believe the primes behave like a random set.

我们相信素数的行为就像一个随机集。

And so the reason why we care about the twin prime conjecture is a test case for whether we can genuinely confidently say with 0% chance of error that the primes behave like a random set.

因此，我们关心孪生素数猜想的原因是一个测试用例，看看我们是否可以真正自信地说，素数的行为就像一个随机集，错误率为零。

Random versions of the primes we know contain twins at least with 100% probably, or probably tending to 100% as you go out further and further.

我们所知道的素数的随机版本至少有 100% 可能包含双胞胎，或者随着你走得越来越远，可能趋向于 100%。

So the primes, we believe that they’re random.

所以质数，我们相信它们是随机的。

The reason why arithmetic progressions are indestructible is that regardless of whether it looks random or looks structured like periodic, in both cases the arithmetic progressions appear, but for different reasons.

等差数列之所以坚不可摧，是因为无论它看起来是随机的还是看起来像周期性的结构，在这两种情况下，等差数列都会出现，但原因不同。

And this is basically all the ways in which the thing was…

这基本上就是事情发生的所有方式……

There are many proofs of this sort of arithmetic progression-type theorems.

这种算术级数型定理有很多证明。

2:23:54 And they’re all proven by some sort of dichotomy where your set is either structured or random and in both cases you can say something and then you put the two together.

它们都是通过某种二分法来证明的，你的集合要么是结构化的，要么是随机的，在这两种情况下你都可以说些什么，然后将两者放在一起。

But in twin primes, if the primes are random, then you are happy, you win.

但在孪生素数中，如果素数是随机的，那么你很高兴，你赢了。

If the primes are structured, they could be structured in a specific way that eliminates the twins.

如果素数是结构化的，那么它们可以以消除孪生子的特定方式进行结构化。

And we can’t rule out that one conspiracy.

我们不能排除这一阴谋。

And yet you were able to make, as I understand, progress on the K-tuple version

然而，据我所知，您能够在 K 元组版本上取得进展

Right.

对。

2:24:21 Yeah.

是的。

So the one funny thing about conspiracies is that any one conspiracy theory is really hard to disprove.

因此，关于阴谋论的一件有趣的事情是，任何一种阴谋论都很难被反驳。

That if you believe the word is one by lizards is that here’s some evidence that it’s not [inaudible 02:24:32] work, that it was just talked about lizards.

如果你相信这个词是蜥蜴一词，那么这里有一些证据表明它不是[听不清02:24:32]作品，它只是谈论蜥蜴。

You might have encountered this kind of phenomena.

您可能遇到过这种现象。

Yes.

是的。

There’s almost no way to definitively rule out a conspiracy.

几乎没有办法明确排除阴谋。

2:24:41 And the same is true in mathematics.

数学也是如此。

A conspiracy that is solely devoted to eliminating twin primes, you would have to also infiltrate other areas of mathematics, but it could be made consistent at least as far as we know.

这是一个专门致力于消除孪生素数的阴谋，你还必须渗透到数学的其他领域，但至少据我们所知，它可以保持一致。

But there’s a weird phenomenon that you can make one conspiracy rule out other conspiracies.

但有一个奇怪的现象，你可以让一个阴谋排除其他阴谋。

So if the world is run by lizards, it can’t also be one by aliens, right?

因此，如果世界是由蜥蜴统治的，那么它也不可能是由外星人统治的，对吗？

Right.

正确的。

So one unreasonable thing is hard to disprove, but more than one, there are tools.

所以一件不合理的事情很难反驳，但不止一件，还有工具。

2:25:09 So yeah, so for example, we know there’s infinitely many primes that no two, which…

是的，例如，我们知道有无限多个素数，没有两个，这......

So there are infinite pairs of primes which differ by at most, 246 actually is the code.

所以存在无限对素数，它们最多相差246，实际上就是代码。

Oh, so there’s like a bound on the-

哦，所以有一个界限-

Right.

对。

2:25:28 So there’s twin primes, there’s a thing called cousin primes that differ by four.

所以有孪生素数，有一种叫做表弟素数的东西，它们相差四。

There’s a thing called sexy primes that differ by six.

有一种东西叫性感素数，相差六倍。

What are sexy primes?

什么是性感素数？

Primes that differ by six.

相差六的素数。

2:25:38 The name is much less…

名字就少了很多……

It causes much less exciting than the name suggests.

它的结果远没有名字所暗示的那么令人兴奋。

Got it.

知道了。

So you can make a conspiracy rule out one of these, but once you have 50 of them, it turns out that you can’t rule out all of them at once.

所以你可以制定一个阴谋来排除其中一个，但一旦你有了 50 个，你就无法一次排除所有这些。

2:25:45 It requires too much energy somehow in this conspiracy space.

在这个阴谋空间里，不知怎的，它需要太多的能量。

How do you do the bound part?

装订部分怎么做？

2:25:55 How do you develop a bound for the differented team deposit-

如何为不同的团队存款建立界限-

Okay.

好的。

…

……

2:26:01 that there’s an infinite number of?

有无限多个吗？

So it’s ultimately based on what’s called the pigeonhole principle.

所以它最终是基于所谓的鸽巢原理。

2:26:03 So the pigeonhole principle is a statement that if you have a number of pigeons, and they all have to go into pigeonholes and you have more pigeons than pigeonholes, then one of the pigeonholes has to have at least two pigeons in it.

因此，鸽舍原则是这样的一种说法：如果你有很多鸽子，并且它们都必须进入鸽舍，并且你的鸽子多于鸽舍，那么其中一个鸽舍必须至少有两只鸽子。

So there has to be two pigeons that are close together.

所以必须有两只靠得很近的鸽子。

So for instance, if you have 100 numbers and they all range from one to 1,000, two of them have to be at most 10 apart because you can divide up the numbers from one to 100 into 100 pigeonholes.

举例来说，如果您有 100 个数字，并且它们的范围都在 1 到 1,000 之间，那么其中两个最多相距 10，因为您可以将从 1 到 100 的数字分为 100 个鸽格。

Let’s say if you have 101 numbers.

假设您有 101 个号码。

101 numbers, then two of them have to be a distance less than 10 apart because two of them have to belong to the same pigeonhole.

101 个数字，那么其中两个的距离必须小于 10，因为它们必须属于同一个鸽舍。

So it’s a basic feature of a basic principle in mathematics.

所以这是数学基本原理的基本特征。

2:26:45 So it doesn’t quite work with the primes already because if the primes get sparser and sparser as you go out, that there are fewer and fewer numbers are prime.

所以它已经不太适合素数了，因为如果当你出去时素数变得越来越稀疏，那么素数就会越来越少。

But it turns out that there’s a way to assign weights to numbers.

但事实证明，有一种方法可以为数字分配权重。

So there are numbers that are kind of almost prime, but they don’t have no factors at all other than themselves and one.

所以有些数字几乎是素数，但除了它们本身和一之外，它们根本没有任何因数。

But they have very few factors.

但他们的因素很少。

And it turns out that we understand almost primes a lot better than primes.

事实证明，我们对素数的理解比对素数的理解要好得多。

And so for example, it was known for a long time that there were twin almost primes.

例如，人们很早就知道存在孪生几乎素数。

This has been worked out.

这已经解决了。

So almost primes are something we can understand.

所以几乎素数都是我们可以理解的。

So you can actually restrict the attention to a suitable set of almost primes.

因此，您实际上可以将注意力限制在一组合适的几乎素数上。

And whereas the primes are very sparse overall relative to the almost primes actually are much less sparse.

尽管相对于几乎素数而言，素数总体上非常稀疏，但实际上素数的稀疏程度要低得多。

2:27:33 You can set up a set of almost primes where the primes have density like say 1%.

您可以设置一组几乎素数，其中素数的密度例如为 1%。

And that gives you a shot at proving by applying some sort of pigeonhole principle that there’s pairs of primes that are just only 100 apart.

这让你有机会应用某种鸽巢原理来证明存在相距仅 100 的素数对。

But in order to prove the twin prime conjecture, you need to get the density of primes, this having also up to a threshold of 50%.

但为了证明孪生素数猜想，你需要得到素数的密度，这个阈值也达到了50%。

Once you get up to 50%, you will get twin primes.

一旦达到50%，你将获得孪生素数。

But unfortunately, there are barriers.

但不幸的是，存在障碍。

We know that no matter what kind of good set of almost primes you pick, the density of primes can never get above 50%.

我们知道，无论你选择什么样的好素数集，素数的密度永远不会超过 50%。

It’s what the parity barrier and I would love to fight.

这就是平价障碍，我愿意与之抗争。

So one of my long-term dreams is to find a way to breach that barrier because it would open up not only the twin prime conjecture but the Goldbach conjecture.

因此，我的长期梦想之一就是找到一种突破这一障碍的方法，因为它不仅会开启孪生素数猜想，还会开启哥德巴赫猜想。

2:28:12 And many other problems in number theory are currently blocked because our current techniques would require going beyond this theoretical parity barrier.

数论中的许多其他问题目前都受到阻碍，因为我们当前的技术需要超越这个理论奇偶校验障碍。

It’s like going fast as the speed of light.

就像光速一样快。

Yeah.

是的。

2:28:24 So we should say a twin prime conjecture, one of the biggest problems in the history of mathematics.

所以我们应该说孪生素数猜想，数学史上最大的问题之一。

Goldbach conjecture also.

哥德巴赫猜想也。

They feel like next-door neighbors.

他们感觉就像隔壁邻居。

Has there been days when you felt you saw the path?

是否有过几天你感觉自己看到了那条路？

Oh, yeah.

哦，是的。

2:28:37 Yeah.

是的。

Sometimes you try something and it works super well.

有时你尝试一些东西，效果会非常好。

You again, get the sense of mathematical smell we talked about earlier.

你再次体会一下我们之前讨论过的数学嗅觉。

You learn from experience when things are going too well because there are certain difficulties that you sort of have to encounter.

当事情进展顺利时，你会从经验中学习，因为你必须遇到某些困难。

I think the way of colleague might put it is that if you are on the streets of New York and you’re put in a blindfold and you’re put in a car and after some hours the blindfold is off and then you’re in Beijing.

我认为同事的说法可能是，如果你在纽约的街道上，你被蒙上眼睛，然后被放进车里，几个小时后，眼罩被摘掉，那么你就到了北京。

I mean that was too easy somehow.

我的意思是这太容易了。

There was no ocean being crossed.

没有跨越海洋。

Even if you don’t know exactly what was done, you’re suspecting that something wasn’t right.

即使你不知道到底做了什么，你也会怀疑有些事情不对劲。

But is that still in the back of your head?

但这仍然在你的脑海中吗？

2:29:21 Do you return to the prime numbers every once in a while to see?

你会时不时地回过头来看看素数吗？

Yeah, when I have nothing better to do, which is less and less.

是的，当我无事可做的时候，事情就越来越少。

2:29:29 I get busy with so many things these days.

这些天我要忙很多事情。

But when I have free time and I’m not, I’m too frustrated to work on my real research projects, and I also don’t want to do my administrative stuff or I don’t want to do some errands for my family.

但是，当我有空闲时间而没有空闲时间时，我会因为太沮丧而无法从事真正的研究项目，而且我也不想做我的行政工作，或者我不想为我的家人做一些差事。

I can play with these things for fun.

我可以玩这些东西来取乐。

And usually you get nowhere.

通常你一事无成。

You have to just say, “Okay, fine.

你必须说：“好吧，好吧。

Once again, nothing happened.

再次，什么也没发生。

I will move on.” Very occasionally one of these problems or actually solved.

我会继续前进。”偶尔这些问题之一或实际上已经解决。

Well, sometimes as you say, you think you solved it and then you forward for maybe 15 minutes and then you think, “I should check this.

好吧，有时正如你所说，你认为你解决了这个问题，然后你转发了大约 15 分钟，然后你想，“我应该检查一下这个。

This is too easy, too good to be true.” And it usually is.

这太简单了，好得令人难以置信。”通常是这样。

What’s your gut say about when these problems would be solved, the twin prime and Goldbach?

对于孪生素数和哥德巴赫这些问题何时得到解决，您的直觉是什么？

The twin prime, I’ll think we’ll-

孪生素数，我想我们会-

10 years?

10年？

…

……

2:30:19 keep getting more partial results.

不断获得更多的部分结果。

It does need at least one…

它确实需要至少一个……

This parity barrier is the biggest remaining obstacle.

这个奇偶性障碍是剩下的最大障碍。

There are simpler versions of the conjecture where we are getting really close.

这个猜想有一些更简单的版本，我们已经非常接近了。

So I think in 10 years we will have many more much closer results, we may not have the whole thing.

所以我认为十年后我们将会有更多更接近的结果，但我们可能不会拥有全部。

So twin primes is somewhat close.

所以孪生素数有些接近。

The Riemann hypothesis I have no clue.

黎曼假设我不知道。

It has happened by accident I think.

我认为这是偶然发生的。

So the Riemann hypothesis is a kind of more general conjecture about the distribution of prime numbers, right?

那么黎曼假设是一种关于素数分布的更一般的猜想，对吗？

Right.

正确的。

2:30:53 Yeah.

是的。

It’s states that sort of viewed multiplicatively, for questions only involving multiplication, no addition.

这是一种乘法看待的状态，对于只涉及乘法而不涉及加法的问题。

The primes really do behave as randomly as you could hope.

素数的行为确实如您所希望的那样随机。

So there’s a phenomenon in probability called square root cancellation that if you want to poll, say America on some issue, and you ask one or two voters and you may have sampled a bad sample, and then you get a really imprecise measurement of the full average.

因此，概率中有一种称为平方根抵消的现象，如果你想就某个问题进行民意调查，比如说美国，你询问一两个选民，你可能会抽取到一个不好的样本，然后你会得到一个非常不精确的总体平均值测量。

But if you sample more and more people, the accuracy gets better and better.

但如果你对越来越多的人进行采样，准确性就会越来越好。

And it accuracy improves the square root of the number of people you sample.

它的准确性提高了抽样人数的平方根。

So if you sample 1, 000 people, you can get a 2 or 3% margin of error.

因此，如果您对 1, 000 人进行抽样，则可以获得 2% 或 3% 的误差范围。

So in the same sense, if you measure the primes in a certain multiplicative sense, there’s a certain type of statistic you can measure and it’s called the Riemann’s data function, and it fluctuates up and down.

所以在同样的意义上，如果你在某种乘法意义上测量素数，你可以测量某种类型的统计量，它被称为黎曼数据函数，它会上下波动。

2:31:42 But in some sense, as you keep averaging more and more, if you sample more and more, the fluctuation should go down as if they were random.

但从某种意义上说，随着你不断平均，如果你采样越来越多，波动应该会下降，就好像它们是随机的一样。

And there’s a very precise way to quantify that.

有一种非常精确的方法可以量化这一点。

And the Riemann hypothesis is a very elegant way that captures this.

黎曼假设是捕捉这一点的一种非常优雅的方式。

But as with many other ways in mathematics, we have very few tools to show that something really genuinely behaves really random.

但与数学中的许多其他方法一样，我们几乎没有工具可以证明某些东西的行为确实是随机的。

And this is actually not just a little bit random, but it’s asking that it behaves as random as it actually random set, this square root cancellation.

这实际上不仅仅是一点点随机，而是要求它的行为与实际随机设置一样随机，即平方根抵消。

And we know because of things related to the parity problem actually, that most of us’ usual techniques cannot hope to settle this question.

我们知道，由于实际上与奇偶校验问题相关的事情，我们大多数人常用的技术都无法解决这个问题。

The proof has to come out of left field.

证明必须来自左领域。

But what that is, no one has any serious proposal.

但那是什么，没有人提出任何认真的建议。

And there’s various ways to solve.

并且有多种解决方法。

As I said, you can modify the primes a little bit and you can destroy the Riemann hypothesis.

正如我所说，你可以稍微修改素数，你就可以摧毁黎曼假设。

2:32:37 So it has to be very delicate.

所以它必须非常精致。

You can’t apply something that has huge margins of error.

你不能应用有巨大误差范围的东西。

It has to just barely work.

它必须勉强工作。

And there’s all these pitfalls that you dodge very adeptly.

你可以非常熟练地避开所有这些陷阱。

The prime numbers is just fascinating.

质数真是令人着迷。

Yeah.

是的。

2:32:52 Yeah, yeah.

是啊是啊。

What to you is most mysterious about the prime numbers?

对你来说，素数最神秘的是什么？

That’s a good question.

这是一个好问题。

2:33:00 Conjecturally, we have a good model of them.

据推测，我们有一个很好的模型。

I mean, as I said, I mean they have certain patterns, like the primes are usually odd, for instance.

我的意思是，正如我所说，我的意思是它们有一定的模式，例如素数通常是奇数。

But apart from there’s some obvious patterns, they behave very randomly and just assuming that they behave.

但除了一些明显的模式外，它们的行为非常随机，并且只是假设它们的行为。

So there’s something called the Kramer random model of the primes that after a certain point, primes just behave like a random set.

因此，有一种称为素数的克莱默随机模型的东西，在某个点之后，素数的行为就像一个随机集。

And there’s various slight modifications to this model.

这个模型有各种细微的修改。

But this has been a very good model.

但这是一个非常好的模式。

It matches the numerics.

它与数字匹配。

It tells us what to predict.

它告诉我们要预测什么。

I can tell you with complete certainty the twin prime conjecture is true.

我可以完全肯定地告诉你，孪生素数猜想是正确的。

The random model gives overwhelming odds it is true, I just can’t prove it.

随机模型给出了压倒性的可能性，它是真的，我只是无法证明它。

Most of our mathematics is optimized for solving things with patterns in them.

我们的大多数数学都是为了解决具有模式的问题而优化的。

2:33:39 And the primes have this anti-pattern, as do almost everything really, but we can’t prove that.

素数有这种反模式，几乎所有事物都有这种反模式，但我们无法证明这一点。

I guess it’s not mysterious that the primes be random because there’s no reason for them to have any kind of secret pattern.

我想素数是随机的并不神秘，因为它们没有理由有任何秘密模式。

But what is mysterious is what is the mechanism that really forces the randomness to happen?

但神秘的是真正迫使随机性发生的机制是什么？

This is just absent.

这只是不存在的。

Another incredibly surprisingly difficult problem is the Collatz conjecture.

另一个令人难以置信的困难问题是科拉茨猜想。

Oh yes.

哦是的。

Simple to state, beautiful to visualize in it simplicity and yet extremely difficult to solve.

说起来很简单，想象起来很简单，但解决起来却极其困难。

2:34:10 And yet you have been able to make progress.

然而你已经取得了进步。

Paul Erdos said about the Collatz conjecture that mathematics may not be ready for such problems.

保罗·埃尔多斯（Paul Erdos）谈到科拉茨猜想时表示，数学可能还没有准备好解决此类问题。

Others have stated that it is an extraordinarily difficult problem, completely out of reach, this is in 2010, out of reach of present-day mathematics, and yet you have made some progress.

还有人说这是一个非常困难的问题，完全遥不可及，这是2010年，现在的数学遥不可及的，而你却取得了一些进展。

Why is it so difficult to make?

为何制作如此困难？

Can you actually even explain what it is, is the key to-

你真的能解释一下它是什么吗？关键是——

Oh, yeah.

哦，是的。

2:34:41 So it’s a problem that you can explain.

所以这是一个你可以解释的问题。

It helps with some visual aids.

它有助于一些视觉辅助工具。

But yeah, so you take any natural number, like say 13, and you apply the following procedure to it.

但是，是的，所以你可以取任何自然数，比如 13，然后对其应用以下过程。

So if it’s even, you divide it by two, and if it’s odd, you multiply it by three and add one.

因此，如果是偶数，则将其除以二；如果是奇数，则将其乘以三并加一。

So even numbers get smaller, odd numbers get bigger.

所以偶数会变小，奇数会变大。

So 13 would become 40 because 13 times 3 is 39, add one you get 40.

所以 13 会变成 40，因为 13 乘以 3 等于 39，加一就得到 40。

So it’s a simple process.

所以这是一个简单的过程。

For odd numbers and even numbers, they’re both very easy operations.

对于奇数和偶数来说，它们都是非常简单的运算。

And then you put it together, it’s still reasonably simple.

然后你把它们放在一起，它仍然相当简单。

But then you ask what happens when you iterate it?

但是你会问当你迭代它时会发生什么？

You take the output that you just got and feed it back in.

您获取刚刚获得的输出并将其反馈回来。

So 13 becomes 40, 40 is now even divide by two is 20.

所以 13 变成 40，40 现在甚至除以二就是 20。

20 is still even divide by 2, 10, five, and then five times three plus one is 16, and then eight, four, two, one.

20 仍然是除以 2、10、5 的偶数，然后五乘以三加一是 16，然后是八、四、二、一。

And then from one it goes one, four, two, one, four, two, one.

然后从一开始，变成一、四、二、一、四、二、一。

It cycles forever.

它永远循环下去。

So this sequence I just described, 13, 40, 20, 10, so both, these are what is known hailstorm sequences, because there’s an oversimplified model of hailstorm formation which is not actually quite correct but it’s still somehow taught to high school students as a first approximation, is that a little nugget of ice gets an ice crystal forms and clouded.

所以我刚才描述的这个序列，13、40、20、10，所以这两个都是所谓的冰雹序列，因为冰雹形成有一个过于简单的模型，实际上并不完全正确，但它仍然以某种方式教授给高中生，作为第一个近似值，即一小块冰块形成冰晶形式并被云化。

It goes up and down because of the wind.

由于风的作用，它会上下移动。

And sometimes when it’s cold it acquires a bit more mass and maybe it melts a little bit.

有时，当天气寒冷时，它会获得更多的质量，并且可能会融化一点。

And this process of going up and down creates this partially melted ice which eventually causes hailstorm, and eventually it falls down to the earth.

这个上下移动的过程产生了部分融化的冰，最终导致冰雹，最终落到地球上。

So the conjecture is that no matter how high you start up, you take a number which is in the millions or billions, this process that goes up, if you are odd and down, it eventually comes down to Earth all the time.

所以推测是，无论你从多高开始，你都会采取数百万或数十亿的数字，这个上升的过程，如果你是奇数和下降，它最终会一直下降到地球。

No matter where you start with very simple algorithm, you end up at one.

无论您从哪里开始使用非常简单的算法，最终都会得到一个。

2:36:23 And you might climb for a while-

你可能会爬一段时间-

Right.

对。

…

……

2:36:29 you come down.

你下来吧。

Yeah.

是的。

So yeah, if you plotted these sequences, they look like Brownian motion.

所以，是的，如果你绘制这些序列，它们看起来就像布朗运动。

They look like the stock market.

它们看起来就像股票市场。

They just go up and down in a seemingly random pattern.

它们只是以看似随机的模式上下移动。

And in fact, usually that’s what happens, that if you plug in a random number, you can actually prove, at least initially, that it would look like a random walk.

事实上，通常会发生这样的情况，如果你插入一个随机数，你实际上可以证明，至少在最初，它看起来像随机游走。

And that’s actually a random walk with a downward drift.

这实际上是一个向下漂移的随机游走。

It’s like if you are always gambling on a roulette at the casino with odds slightly weighted against you.

这就像您总是在赌场玩轮盘赌，而赔率对您稍稍不利。

So sometimes you win, sometimes you lose.

所以有时你会赢，有时你会输。

But over in the long run, you lose a bit more than you win.

但从长远来看，你输的比赢的多一些。

And so normally your wallet will go to zero if you just keep playing over and over again.

因此，如果你一遍又一遍地玩游戏，通常你的钱包就会变为零。

So statistically it makes sense that we go here?

从统计上看，我们去这里有意义吗？

Yes.

是的。

2:37:11 So the result that I proved roughly speaking such that statistically like 99% of all inputs would drift down to maybe not all the way to one, but to be much, much smaller than what you started.

因此，我粗略地证明了结果，从统计上看，所有输入的 99% 可能不会一直下​​降到 1，而是比你开始时要小得多。

So it’s like if I told you that if you go to a casino, most of the time you end up, if you keep playing it for long enough, you end up with a smaller amount in your wallet then when you started.

所以就像我告诉你，如果你去赌场，大多数时候，如果你玩得足够长，你最终钱包里的钱会比你开始时少。

That’s kind of like the result that I proved.

这有点像我证明的结果。

So why is that result…

那么为什么会是这样的结果呢……

2:37:35 Can you continue down that thread to prove the full conjecture?

你能继续沿着这条线索来证明完整的猜想吗？

Well, the problem is that I used arguments from probability theory, and there’s always this exceptional event.

好吧，问题是我使用了概率论的论据，而且总是有这种特殊事件。

2:37:42 So in probability, we have this law of large numbers, which tells you things like if you play a casino with a game at a casino with a losing expectation over time you are guaranteed, almost surely with probability as close to 100% as you wish, you’re guaranteed to lose money.

所以在概率上，我们有这个大数定律，它告诉你一些事情，比如如果你在赌场玩游戏，随着时间的推移，你的预期会输钱，那么你肯定会输钱，几乎可以肯定，概率接近 100%，如你所愿，你肯定会输钱。

But there’s always this exceptional outlier.

但总有这种特殊的异常值。

It is mathematically possible that even in the game is the odds are not in favor, you could just keep winning slightly more often than you lose.

从数学上来说，即使在赔率不高的游戏中，你也可能会继续获胜的次数略多于失败的次数。

Very much like how in Navier-Stokes it could be, most of the time your waves can disperse, there could be just one outlier choice of initial conditions that would lead you to blow up.

与纳维-斯托克斯的情况非常相似，大多数时候你的波可以分散，可能只有一个初始条件的异常选择会导致你爆炸。

And there could be one outlier choice of a special number they stick in that shoots off infinity while all other numbers crash to Earth, crash to one.

他们所坚持的一个特殊数字可能会出现一个异常选择，该数字会射出无穷大，而所有其他数字都会坠落到地球，坠落到一。

2:38:40 In fact, there’s some mathematicians who, Alex Kontorovich for instance, who’ve proposed that actually these collapse iterations are like these similar Automator.

事实上，有一些数学家，例如 Alex Kontorovich，提出实际上这些崩溃迭代就像这些类似的 Automator。

Actually, if you look at what they happen in binary, they do actually look a little bit like these game of life type patterns.

实际上，如果你看看它们在二进制中发生的情况，它们实际上看起来有点像这些生命类型模式的游戏。

And in analogy to how the game of life can create these massive self-replicating objects and so forth, possibly you could create some sort of heavier-than-air flying machine.

与生命游戏如何创造这些巨大的自我复制物体等类似，你可能可以创造某种比空气重的飞行器。

A number which is actually encoding this machine, which is just whose job it’s to encode, is to create a version of something which is larger.

实际上对这台机器进行编码的数字，这正是编码的工作，是创建更大的版本。

Heavier-than-air machine encoded in a number-

比空气重的机器以数字编码

Yeah.

是的。

…

……

2:39:20 that flies forever.

永远飞翔。

So Conway in fact, worked on this problem as well.

事实上，康威也致力于解决这个问题。

Oh wow.

哦哇。

Conway, so similar, in fact, that was more on inspirations for the Navier-Stokes project.

康威，如此相似，事实上，这更多的是纳维-斯托克斯项目的灵感。

2:39:26 Conway studied generalizations of the collapse problem where instead of multiplying by three and adding one or dividing by two, you have more complicated branching list.

康威研究了崩溃问题的概括，其中不是乘以三加一或除以二，而是更复杂的分支列表。

But instead of having two cases, maybe you have 17 cases and then you go up and down.

但是，您可能有 17 个案例，而不是两个案例，然后上下波动。

And he showed that once your iteration gets complicated enough, you can actually encode Turing machines and you can actually make these problems undecidable and do things like this.

他表明，一旦你的迭代变得足够复杂，你实际上可以对图灵机进行编码，并且实际上可以使这些问题变得不可判定，并做这样的事情。

In fact, he invented a programming language for these kind of fractional linear transformations.

事实上，他发明了一种用于此类分数线性变换的编程语言。

He called it frac-trat as a play on full-trat.

他将其称为 frac-trat，作为 full-trat 的一种发挥。

And he showed that you can program, it was Turing-complete, you could make a program that if your number you insert in was encoded as a prime, it would sink to zero.

他证明你可以编程，它是图灵完备的，你可以编写一个程序，如果你插入的数字被编码为素数，它就会沉为零。

2:40:13 It would go down, otherwise it would go up and things like that.

它会下跌，否则就会上涨，诸如此类。

So the general class of problems is really as complicated as all the mathematics.

所以一般的问题实际上和所有数学一样复杂。

Some of the mystery of the cellular automata that we talked about, having a mathematical framework to say anything about cellular automata, maybe this same kind of framework is required.

我们谈到的元胞自动机的一些奥秘，有一个数学框架来谈论元胞自动机的任何事情，也许需要同样类型的框架。

2:40:23 Yeah, Goldbach’s conjecture.

是的，哥德巴赫猜想。

Yeah.

是的。

2:40:35 If you want to do it, not statistically, but you really want 100% of all inputs to for the earth.

如果你想这样做，不是统计上的，但你确实希望 100% 的所有投入都用于地球。

Yeah.

是的。

So what might be feasible is, yeah, statistically 99% go to one, but everything, that looks hard.

所以可能可行的是，是的，统计上 99% 的人都会选择其中之一，但一切看起来都很难。

What would you say is out of these within reach famous problems is the hardest problem we have today?

您认为在这些触手可及的著名问题中，什么是我们今天遇到的最难的问题？

2:40:50 Is it the Riemann hypothesis?

是黎曼猜想吗？

Well, it’s up there.

嗯，它就在那里。

2:40:59 P equals NP is a good one because that’s a meta problem.

P 等于 NP 是一个很好的问题，因为这是一个元问题。

If you solve that in the positive sense that you can find a P equals NP algorithm, potentially, this solves a lot of other problems as well.

如果你从积极的意义上解决这个问题，你可以找到一个 P 等于 NP 的算法，那么这也可能解决很多其他问题。

And we should mention some of the conjectures we’ve been talking about.

我们应该提一下我们一直在谈论的一些猜想。

2:41:14 A lot of stuff is built on top of them now.

现在很多东西都是建立在它们之上的。

There’s ripple effects.

有连锁反应。

P equals NP has more ripple effects than basically any other-

P 等于 NP 的连锁反应比基本上任何其他因素都多-

Right.

对。

2:41:24 If the Riemann hypothesis is disproven, that’d be a big mental shock to the number theorists.

如果黎曼假设被证伪，那对数论学家来说将是一个巨大的精神冲击。

But it would have follow-on effects for cryptography, because a lot of cryptography uses number theory, uses number-theory constructions involving primes and so forth.

但它会对密码学产生后续影响，因为许多密码学使用数论，使用涉及素数的数论结构等等。

And it relies very much on the intuition that number-theories are built over many, many years of what operations involving primes behave randomly and what ones don’t?

它在很大程度上依赖于这样的直觉：数论是经过很多年建立起来的，涉及素数的哪些运算是随机的，哪些不是？

And in particular, encryption methods are designed to turn text-written information on it into text, which is indistinguishable from random noise.

特别是，加密方法旨在将其上的文本写入信息转换为文本，这与随机噪声无法区分。

And hence, we believe to be almost impossible to crack, at least mathematically.

因此，我们认为几乎不可能破解，至少在数学上是这样。

But if something has caught our beliefs as the Riemann hypothesis is wrong, it means that there are actual patterns of the primes that we’re not aware of.

但如果我们相信黎曼假设是错误的，那就意味着素数存在一些我们不知道的实际模式。

2:42:21 And if there’s one, there’s probably going to be more.

如果有的话，可能还会有更多。

And suddenly a lot of our crypto systems are in doubt.

突然之间，我们的许多加密系统都受到了质疑。

Yeah.

是的。

2:42:27 But then how do you then say stuff about the primes-

但是那么你如何谈论素数呢？

Yeah.

是的。

…

……

2:42:34 that you’re going towards because of the Collatz conjecture again?

你又因为科拉茨猜想而走向这个方向吗？

Because, do you want it to be random, right?

因为，你希望它是随机的，对吧？

Yes.

是的。

You want it to be random?

你希望它是随机的吗？

Yeah.

是的。

2:42:43 So more broadly, I’m just looking for more tools, more ways to show that things are random.

更广泛地说，我只是在寻找更多的工具、更多的方法来证明事物是随机的。

How do you prove a conspiracy doesn’t happen?

如何证明阴谋没有发生？

Right.

正确的。

2:42:49 Is there any chance to you that P equals NP?

你有可能认为 P 等于 NP 吗？

Can you imagine a possible universe?

你能想象一个可能的宇宙吗？

It is possible.

这是可能的。

2:42:57 I mean, there’s various scenarios.

我的意思是，有各种各样的场景。

I mean, there’s one where it is technically possible, but in fact it’s never actually implementable.

我的意思是，有一种技术在技术上是可行的，但事实上它永远无法真正实现。

The evidence is sort of slightly pushing in favor of no, that probably P is not a good NP.

证据有点倾向于否定，即 P 可能不是一个好的 NP。

I mean, it seems like it’s one of those cases similar to Riemann hypothesis.

我的意思是，这似乎是类似于黎曼假设的情况之一。

2:43:11 I think the evidence is leaning pretty heavily on the no.

我认为证据非常倾向于否定。

Certainly more on the no than on the yes.

当然，更多的是“不”而不是“是”。

2:43:20 The funny thing about P equals NP is that we have also a lot more obstructions than we do for almost any other problem.

P 等于 NP 的有趣之处在于，我们遇到的障碍比几乎任何其他问题都多得多。

So while there’s evidence, we also have a lot of results ruling out many, many types of approaches to the problem.

因此，虽然有证据，但我们也有很多结果，排除了解决该问题的多种方法。

This is the one thing that the computer science has actually been very good at.

这是计算机科学实际上非常擅长的一件事。

It’s actually saying that certain approaches cannot work.

这实际上是在说某些方法行不通。

No-go theorems.

不可行定理。

It could be undecidable, yeah, we don’t know.

这可能是无法确定的，是的，我们不知道。

There’s a funny story I read that when you won the Fields Medal, somebody from the internet wrote you and asked, what are you going to do now that you’ve won this prestigious award?

我读过一个有趣的故事，当你赢得菲尔兹奖时，互联网上有人写信给你，问你，既然你赢得了这个享有盛誉的奖项，你打算做什么？

2:43:43 And then you just quickly, very humbly said that a shiny metal is not going to solve any of the problem I’m currently working on, so I’m going to keep working on them.

然后你很快地、非常谦虚地说，闪亮的金属无法解决我目前正在研究的任何问题，所以我将继续研究它们。

First of all, it’s funny to me that you would answer an email in that context, and second of all, it just shows your humility.

首先，对我来说很有趣的是，你会在这种情况下回复电子邮件，其次，这只是显示了你的谦虚。

But anyway, maybe you could speak to the Fields Medal, but it’s another way for me to ask about Gregorio Perlman.

但无论如何，也许你可以谈论菲尔兹奖，但这是我询问格雷戈里奥·帕尔曼的另一种方式。

What do you think about him famously declining the Fields Medal and the Millennial Prize, which came with a $1 million of prize money.

您如何看待他拒绝获得菲尔兹奖和奖金为 100 万美元的千禧年奖的著名事件？

He stated that, “I’m not interested in money or fame.

他说：“我对金钱或名誉不感兴趣。

The prize is completely irrelevant for me.

奖品与我完全无关。

If the proof is correct, then no other recognition is needed.”

如果证明是正确的，那么就不需要其他的认可。”

Yeah, no, he’s somewhat of an outlier, even among mathematicians who tend to have somewhat idealistic views.

是的，不，他在某种程度上是一个异类，即使在那些倾向于抱有理想主义观点的数学家中也是如此。

2:44:40 I’ve never met him.

我从未见过他。

I think I’d be interested to meet him one day, but I’ve never had the chance.

我想有一天我会有兴趣见到他，但我一直没有机会。

I know people who met him.

我认识见过他的人。

He’s always had strong views about certain things.

他对某些事情总是有强烈的看法。

I mean, it’s not like he was completely isolated from the math community.

我的意思是，他并不是完全与数学界隔绝。

I mean, he would give talks and write papers and so forth, but at some point he just decided not.

我的意思是，他会发表演讲并撰写论文等等，但在某些时候他决定不这样做。

2:45:00 …

……

He talks and write papers and so forth, but at some point he just decided not to engage with the rest of the community.

他会说话、写论文等等，但在某个时候，他决定不与社区的其他成员接触。

He was disillusioned or something, I don’t know.

我不知道他是幻灭了还是怎么的。

And he decided to peace out and collect mushrooms in St. Petersburg or something.

他决定平静下来，去圣彼得堡或其他地方采集蘑菇。

And that’s fine, you can do that.

没关系，你可以做到。

That’s another sort of flip side.

这是另一种反面。

A lot of our problems that we solve, some of them do have practical application and that’s great.

我们解决了很多问题，其中一些确实有实际应用，这很棒。

But if you stop thinking about a problem, so he hasn’t published since in this field, but that’s fine.

但如果你停止思考一个问题，那么他从那以后就没有在这个领域发表过文章，但这也没关系。

There’s many, many other people who’ve done so as well.

还有很多很多其他人也这样做过。

2:45:39 Yeah.

是的。

So I guess one thing I didn’t realize initially with the Fields Medal is that it sort of makes you part of the establishment.

所以我想我最初没有意识到菲尔兹奖的一件事是它让你成为了机构的一部分。

So most mathematicians, just career mathematicians, you just focus on publishing the next paper, maybe promote it one rank, and starting a few projects, may have taken some students or something.

所以大多数数学家，只是职业数学家，你只专注于发表下一篇论文，也许将其提升一个级别，并开始一些项目，可能会吸引一些学生或其他什么。

But then suddenly people want your opinion on things and you have to think a little bit about things that you might just foolishly say, because you know no one’s going to listen to you, it’s more important now.

但突然人们想要你对事情的看法，你必须考虑一下你可能愚蠢地说的事情，因为你知道没有人会听你的，现在更重要。

Is it constraining to you?

它对你有限制吗？

2:46:11 Are you able to still have fun and be a rebel and try crazy stuff and play with ideas?

你还能享受乐趣、成为叛逆者、尝试疯狂的东西、发挥创意吗？

I have a lot less free time than I had previously, mostly by choice.

我的空闲时间比以前少了很多，这主要是我的选择。

2:46:19 I always say I have the option to sort of decline, so I decline a lot of things.

我总是说我可以选择拒绝，所以我拒绝了很多事情。

I could decline even more or I could acquire a reputation of being so unreliable that people don’t even ask anymore.

我可能会拒绝更多，或者我可能会获得不可靠的名声，以至于人们甚至不再询问。

I love the different algorithms here.

我喜欢这里不同的算法。

2:46:38 This is great.

这太棒了。

It’s always an option, but there are things that I don’t spend as much time as I do as a postdoc, just working on one problem at a time or fooling around.

这总是一种选择，但有些事情我不会像博士后那样花那么多时间，一次只解决一个问题或闲逛。

2:46:41 I still do that a little bit.

我现在仍然这样做一点。

But yeah, as you advance in your career, the more soft skills, so math somehow front-loads all the technical skills to the early stages of your career.

但是，是的，随着你职业生涯的进步，软技能也会越来越多，所以数学在某种程度上将所有技术技能提前加载到你职业生涯的早期阶段。

So as a postdoc, you publish or perish.

因此，作为一名博士后，你要么发表论文，要么灭亡。

You’re incentivized to basically focus on proving very technical theorems, so prove yourself as well as prove the algorithms.

你会被激励基本上专注于证明非常技术性的定理，所以在证明算法的同时也要证明你自己。

But then as you get more senior, you have to start mentoring and giving interviews and trying to shape direction of field both research-wise and sometimes you have to do various administrative things.

但当你变得更高级时，你必须开始指导和接受采访，并试图在研究方面塑造领域方向，有时你必须做各种行政工作。

And it’s kind the right social contract because you need to work in the trenches to see what can help mathematicians.

这是一种正确的社会契约，因为你需要在第一线工作才能看到什么可以帮助数学家。

The other side of the establishment, the really positive thing is that you get to be a light that’s an inspiration to a lot of young mathematicians or young people that are just interested in mathematics.

另一方面，真正积极的事情是，你可以成为一盏明灯，激励许多年轻数学家或只对数学感兴趣的年轻人。

2:47:40 It’s like-

就像-

Yeah, yeah.

是啊，是啊。

…

……

2:47:52 just how the human mind works.

人类思维是如何运作的。

This is where I would probably say that I like the Fields Medal, that it does inspire a lot of young people somehow.

这就是我可能会说我喜欢菲尔兹奖的地方，它确实以某种方式激励了很多年轻人。

This is just how human brains work.

这就是人类大脑的工作原理。

At the same time, I also want to give sort of respect to somebody like Grigori Perlman, who is critical of awards.

与此同时，我也想向像格里戈里·帕尔曼这样对奖项持批评态度的人表示敬意。

In his mind, those are his principles and any human that’s able for their principles to do the thing that most humans would not be able to do, it’s beautiful to see.

在他看来，这些就是他的原则，任何能够按照自己的原则去做大多数人无法做到的事情的人，都是很美好的。

Some recognition is necessary and important, but yeah, it’s also important to not let these things take over your life and only be concerned about getting the next big award or whatever.

一些认可是必要且重要的，但是，是的，同样重要的是不要让这些事情占据你的生活，而只关心获得下一个大奖或其他什么。

2:48:25 So again, you see these people try to only solve really big math problems and not work on things that are less sexy, if you wish, but actually still interesting and instructive.

再说一遍，你会看到这些人只尝试解决真正大的数学问题，而不是研究那些不那么性感的事情，如果你愿意的话，但实际上仍然很有趣和有启发性。

As you say, the way the human mind works, we understand things better when they’re attached to humans, and also if they’re attached to a small number of humans.

正如你所说，人类思维的运作方式，当事物依附于人类时，或者如果它们依附于少数人时，我们会更好地理解事物。

The way our human mind is wired, we can comprehend the relationships between 10 or 20 people.

根据我们人类思维的连接方式，我们可以理解 10 到 20 个人之间的关系。

But once you get beyond like 100 people, there’s a limit, I think there’s a name for it, beyond which it just becomes the other.

但一旦超过 100 人，就会有一个限制，我认为它有一个名称，超过这个名称就变成了另一个。

2:49:18 And so you have to simplify the [inaudible 02:49:21] 99.9% of humanity becomes the other.

所以你必须简化 [听不清 02:49:21] 99.9% 的人类变成了另一个人。

Often these models are incorrect, and this causes all kinds of problems.

这些模型通常是不正确的，这会导致各种问题。

So yeah, to humanize a subject, if you identify a small number of people and say these are representative people of a subject, role models, for example, that has some role, but it can also be too much of it can be harmful because I’ll be the first to say that my own career path is not that of a typical mathematician.

所以，是的，为了使一个学科人性化，如果你确定一小部分人，并说这些人是某个学科的代表人物，例如，榜样，他们有一定的作用，但过多也可能是有害的，因为我将是第一个说我自己的职业道路不是典型数学家的职业道路的人。

The very accelerated education, I skipped a lot of classes.

非常加速的教育，我逃了很多课。

I think I always had very fortunate mentoring opportunities, and I think I was at the right place at the right time.

我认为我一直有非常幸运的指导机会，而且我认为我在正确的时间出现在正确的地点。

Just because someone doesn’t have my trajectory, it doesn’t mean that they can’t be good mathematicians.

仅仅因为某人没有我的轨迹，并不意味着他们不能成为优秀的数学家。

They would be, but in a very different style, and we need people of a different style.

他们会的，但是风格非常不同，我们需要不同风格的人。

2:50:16 And sometimes too much focus is given on the person who does the last step to complete a project in mathematics or elsewhere that’s really taken centuries or decades with lots and lots of, building on lots of previous work.

有时，人们过多地关注完成数学或其他领域的项目的最后一步的人，这些项目确实需要几个世纪或几十年的时间，需要花费很多很多的时间，建立在以前的大量工作的基础上。

But that’s a story that’s difficult to tell if you’re not an expert.

但如果你不是专家，这是一个很难讲述的故事。

It’s easier to just say one person did this one thing.

只说一个人做了这件事更容易。

It makes for a much simpler history.

它使历史变得更加简单。

I think on the whole, it is a hugely positive thing.

我认为总的来说，这是一件非常积极的事情。

2:50:40 To talk about Steve Jobs as a representative of Apple, when I personally know and of course everybody knows the incredible design, the incredible engineering teams, just the individual humans on those teams.

谈论史蒂夫·乔布斯作为苹果的代表，我个人知道，当然每个人都知道令人难以置信的设计，令人难以置信的工程团队，以及这些团队中的个人。

They’re not a team.

他们不是一个团队。

They’re individual humans on a team, and there’s a lot of brilliance there, but it’s just a nice shorthand, like π, Steve Jobs, π.

他们是团队中的个体，有很多才华，但这只是一个很好的简写，就像 π、史蒂夫·乔布斯、π。

Yeah, as a starting point, as a first approximation that’s how you-

是的，作为一个起点，作为第一个近似值，你就是这样-

And then read some biographies and then look into much deeper first approximation.

然后阅读一些传记，然后研究更深入的第一近似。

Yeah.

是的。

That’s right.

这是正确的。

2:51:17 So you mentioned you were at Princeton too.

所以你提到你也在普林斯顿大学。

Andrew Wiles at that time-

安德鲁·怀尔斯当时——

Oh yeah.

哦，是的。

…

……

2:51:22 he was a professor there.

他是那里的一名教授。

It’s a funny moment how history is just all interconnected, and at that time, he announced that he proved Fermat’s Last Theorem.

这是一个有趣的时刻，历史是如何相互关联的，就在那时，他宣布他证明了费马大定理。

What did you think, maybe looking back now with more context about that moment in math history?

你觉得怎么样，也许现在回顾一下数学历史上那一刻的更多背景？

Yeah, so I was a graduate student at the time.

是的，所以当时我还是一名研究生。

2:51:37 I vaguely remember there was press attention and we all had the same, we had pigeonholes in the same mail room, so we all got mail and suddenly Andrew Wiles’ mailbox exploded to be overflowing.

我依稀记得有媒体的关注，我们都有同样的情况，我们在同一个邮件室里有文件柜，所以我们都收到了邮件，突然安德鲁·怀尔斯的邮箱爆炸了，溢出来了。

That’s a good metric.

这是一个很好的指标。

Yeah.

是的。

2:51:54 We all talked about it at tea and so forth.

我们在喝茶等时都谈论过这件事。

We didn’t understand.

我们不明白。

Most of us sort of didn’t understand the proof.

我们大多数人都不太理解这个证明。

We understand high level details.

我们了解高层次的细节。

In fact, there’s an ongoing project to formalize it in Lean.

事实上，有一个正在进行的项目将其正式化为精益。

Kevin Buzzard is actually-

凯文·巴扎德实际上是——

Yeah.

是的。

2:52:09 Can we take that small tangent?

我们可以采用那个小切线吗？

How difficult is that ’cause as I understand the proof for Fermat’s Last Theorem has super complicated objects?

这有多困难，因为据我了解，费马大定理的证明具有超级复杂的对象？

Yeah.

是的。

It’s really difficult to formalize now.

现在真的很难正式化。

Yeah, I guess.

是的，我想。

2:52:22 Yeah, you’re right.

是的，你说得对。

The objects that they use, you can define them.

他们使用的对象，您可以定义它们。

So they’ve been defined in Lean, so just defining what they are can be done.

所以它们已经在精益中被定义了，所以只要定义它们是什么就可以完成。

That’s really not trivial, but it’s been done.

这确实不是一件小事，但它已经完成了。

But there’s a lot of really basic facts about these objects that have taken decades to prove in all these different math papers.

但是，关于这些物体的许多非常基本的事实，在所有这些不同的数学论文中都花了几十年的时间来证明。

And so lots of these have to formalized as well.

因此，其中许多也必须正式化。

Kevin Buzzard’s goal, actually he has a five-year grant to formalize Fermat’s Last Theorem, and his aim is that he doesn’t think he will be able to get all the way down to the basic axioms, but he wants to formalize it to the point where the only things that he needs to rely on is black boxes, are things that were known by 1980 to a number of theorists at the time, and then some other person or some other work would have to be done to get from there.

Kevin Buzzard 的目标，实际上他有五年的资助来形式化费马大定理，他的目标是，他不认为自己能够一路深入到基本公理，但他想将其形式化到他唯一需要依赖的东西是黑匣子，是 1980 年当时许多理论家已经知道的东西，然后必须做一些其他人或其他一些工作来实现从那里得到。

2:53:13 So it’s a different area of mathematics than the type of mathematics I’m used to.

所以这是一个与我习惯的数学类型不同的数学领域。

In analysis, which is my area, the objects we study are kind of much closer to the ground.

在我的领域的分析中，我们研究的对象更接近地面。

I study things like prime numbers and functions and things that are within scope of a high school math education to at least define.

我研究素数和函数之类的东西，以及高中数学教育范围内的东西，至少要定义这些东西。

But then, there’s this very advanced algebraic side of number theory where people have been building structures upon structures for quite a while, and it’s a very sturdy structure.

但是，数论中有一个非常先进的代数方面，人们长期以来一直在结构上构建结构，而且这是一个非常坚固的结构。

It’s been very…

已经很…

At the base, at least it’s extremely well-developed with textbooks and so forth.

在基础上，至少它的教科书等非常发达。

But it does get to the point where if you haven’t taken these years of study and you want to ask about what is going on at level six of this tower, you have to spend quite a bit of time before they can even get to the point where you can see something that you recognize.

但确实到了这样的地步，如果你没有经过这些年的学习，想要询问这座塔的第六层到底发生了什么，你就得花相当长的时间，才能到达你能看到你认识的东西的地步。

What inspires you about his journey that was similar, as we talked about, seven years mostly working in secret?

正如我们所说，他的七年大部分时间都在秘密工作，这段相似的旅程对您有何启发？

Yeah, so it kind of fits with the romantic image I think people have of mathematicians to the extent that they think of them all as these kind of eccentric wizards or something.

是的，所以这有点符合我认为人们对数学家的浪漫形象，因为他们认为他们都是古怪的巫师之类的。

2:54:15 So that’s certainly kind of accentuated that perspective.

所以这肯定强调了这种观点。

It is a great achievement.

这是一项伟大的成就。

His style of solving problems is so different from my own, which is great.

他解决问题的风格与我的非常不同，这很棒。

We need people like that.

我们需要这样的人。

Can you speak to it, like in terms of you like the collaborative?

你能谈谈它吗，比如你喜欢合作吗？

I like moving on from a problem if it’s giving too much difficulty.

如果问题太困难，我喜欢继续前进。

Got it.

知道了。

But you need the people who have the tenacity and the fearlessness.

但你需要有坚韧和无畏精神的人。

2:54:55 I’ve collaborated with people like that where I want to give up ’cause the first approach that we tried didn’t work and the second one didn’t work.

我曾与这样的人合作过，但我想放弃，因为我们尝试的第一种方法不起作用，第二种方法也不起作用。

But they’re convinced and they have third, fourth, and the fifth, which works.

但他们对此深信不疑，而且他们有第三个、第四个和第五个，这很有效。

And I’d have to eat my words, “Okay.

我不得不收回我的话，“好吧。

I didn’t think this was going to work, but yes, you were right all along.”

我不认为这会起作用，但是是的，你一直都是对的。”

And we should say for people who don’t know, not only are you known for the brilliance of your work, but the incredible productivity, just the number of papers, which are all very high quality.

我们应该对那些不知道的人说，你不仅因为工作的辉煌而闻名，而且因为令人难以置信的生产力、论文的数量而闻名，这些都是非常高质量的。

2:55:16 So there’s something to be said about being able to jump from topic to topic.

因此，关于能够从一个主题跳到另一个主题，有一些话要说。

Yeah, it works for me.

是的，这对我有用。

2:55:31 But there are also people who are very productive and they focus very deeply.

但也有一些人非常富有成效，而且他们的注意力非常集中。

I think everyone has to find their own workflow.

我认为每个人都必须找到适合自己的工作流程。

One thing which is a shame in mathematics is that mathematics has a sort a one-size-fits-all approach to teaching mathematics, and so we have a certain curriculum and so forth.

数学中令人遗憾的一件事是，数学有一种一刀切的数学教学方法，所以我们有一定的课程等等。

Maybe if you do math competitions or something, you get a slightly different experience.

也许如果你参加数学竞赛什么的，你会得到稍微不同的体验。

But I think many people, they don’t find their native math language until very late or usually too late.

但我认为很多人直到很晚或通常太晚才找到自己的母语数学语言。

So they stop doing mathematics and they have a bad experience with a teacher who’s trying to teach them one way to do mathematics that they don’t like it.

因此，他们不再做数学，而且他们对一位试图教他们一种他们不喜欢的数学方法的老师有一段糟糕的经历。

2:56:12 My theory is that humans don’t come, evolution has not given us a math center of a brain directly.

我的理论是，人类没有出现，进化并没有直接给我们大脑的数学中心。

We have a vision center and a language center and some other centers, which evolution has honed, but we don’t have an innate sense of mathematics.

我们有视觉中心、语言中心和其他一些中心，这些中心是进化磨练出来的，但我们没有天生的数学感。

But our other centers are sophisticated enough that we can repurpose other areas of our brain to do mathematics.

但我们的其他中心足够复杂，我们可以重新利用大脑的其他区域来进行数学计算。

So some people have figured out how to use the visual center to do mathematics, and so they think things very visually when they do mathematics.

所以有些人已经弄清楚如何使用视觉中枢来做数学，所以他们在做数学时非常直观地思考问题。

Some people have repurposed their language center and they think very symbolically.

有些人重新调整了他们的语言中心，他们的思考非常具有象征意义。

Some people, if they are very competitive and they’re gaming, there’s a part of your brain that’s very good at solving puzzles and games, and that can be repurposed.

有些人，如果他们非常有竞争力并且喜欢玩游戏，那么你大脑的一部分非常擅长解决谜题和游戏，并且可以重新利用。

2:57:02 But when I talk about the mathematicians, they don’t quite think that, I can tell that they’re using some other different styles of thinking, not disjoint, but they may prefer visual.

但当我谈论数学家时，他们不太这么认为，我可以看出他们正在使用其他一些不同的思维方式，不是脱节的，但他们可能更喜欢视觉。

I don’t actually prefer visual so much.

我实际上不太喜欢视觉效果。

I need lots of visual aids myself.

我自己需要很多视觉辅助工具。

Mathematics provides a common language, so we can still talk to each other even if we are thinking in different ways.

数学提供了一种共同语言，所以即使我们的思维方式不同，我们仍然可以互相交谈。

But you could tell there’s a different set of subsystems being used in the thinking process?

但你能看出思维过程中使用了一组不同的子系统吗？

Yeah, they take different paths.

是的，他们走的是不同的路。

2:57:32 They’re very quick at things that I struggle with and vice versa, and yet they still get to the same goal.

他们对我遇到的困难很快，反之亦然，但他们仍然达到相同的目标。

That’s beautiful.

那好美丽。

But the way we educate, unless you have a personalized tutor or something, education, sort of just financial skill has to be mass-produced, you have to teach the 30 kids.

但我们的教育方式，除非你有一个个性化的导师什么的，教育，某种财务技能必须大规模生产，你必须教 30 个孩子。

2:57:41 If they have 30 different styles, you can’t teach 30 different ways.

如果他们有 30 种不同的风格，你就无法教授 30 种不同的方式。

On that topic, what advice would you give to students, young students who are struggling with math, but are interested in it and would like to get better?

在这个话题上，您会给那些在数学上遇到困难但对数学感兴趣并希望取得更好成绩的学生和年轻学生什么建议？

2:57:55 Is there something in this complicated educational context?

在如此复杂的教育背景下，有什么东西吗？

What would you advise?

你有什么建议？

Yeah, it’s a tricky problem.

是的，这是一个棘手的问题。

2:58:10 One nice thing is that there are now lots of sources for mathematical enrichment outside the classroom.

一件好事是，现在有很多课堂之外的数学丰富资源。

So in my days, there were math competitions and there are also popular math books in the library.

所以在我那个时代，有数学竞赛，图书馆里也有流行的数学书籍。

But now you have YouTube.

但现在你有了 YouTube。

There are forums just devoted to solving math puzzles.

有些论坛专门致力于解决数学难题。

And math shows up in other places.

数学也出现在其他地方。

For example, there are hobbyists who play poker for fun and they, for very specific reasons, are interested in very specific probability questions.

例如，有些业余爱好者玩扑克是为了好玩，他们出于非常具体的原因对非常具体的概率问题感兴趣。

And actually, there’s a community of amateur probabilists in poker, in chess, in baseball.

事实上，在扑克、国际象棋和棒球领域有一个业余概率学家社区。

There’s math all over the place, and I’m hoping actually with these new tools for Lean and so forth, that actually we can incorporate the broader public into math research projects.

数学无处不在，我希望通过这些精益等新工具，我们实际上可以将更广泛的公众纳入数学研究项目。

This almost doesn’t happen at all currently.

目前这种情况几乎根本不会发生。

2:59:13 So in the sciences, there’s some scope for citizen science, like astronomers.

因此，在科学领域，公民科学有一定的发挥空间，比如天文学家。

There are amateurs who would discover comets, and there’s biologists that people who could identify butterflies and so forth.

有些业余爱好者会发现彗星，有些生物学家可以识别蝴蝶等等。

And in math, there are a small number of activities where amateur mathematicians can discover new primes and so forth.

在数学中，有一些活动可供业余数学家发现新的素数等等。

But previously, because we had to verify every single contribution, most mathematical research projects, it would not help to have input from the general public.

但以前，由于我们必须验证每一项贡献，大多数数学研究项目，因此获得公众的意见并无帮助。

In fact, it’ll just be time-consuming because just error checking and everything.

事实上，这会很耗时，因为只是错误检查等等。

But one thing about these formalisation projects is that they are bringing in more people.

但这些正规化项目的一件事是它们引进了更多的人。

So I’m sure there are high school students who’ve already contributed to some of these formalizing projects, who’ve contributed to mathlib.

所以我确信有些高中生已经为其中一些正式化项目做出了贡献，他们也为 mathlib 做出了贡献。

You don’t need to be a PhD holder to just work on one atomic thing.

你不需要成为一名博士学位持有者就可以只研究一件原子的事情。

There’s something about the formalisation here that also, as a very first step, opens it up to the programing community too.

作为第一步，这里的形式化也向编程社区开放。

3:00:03 The people who are already comfortable with program.

已经对程序感到满意的人。

It seems like programing is somehow maybe just the feeling, but it feels more accessible to folks than math.

似乎编程在某种程度上可能只是一种感觉，但对人们来说，它比数学更容易理解。

Math is seen as this extreme, especially modern mathematics is seen as this extremely difficult-to-enter area, and programing is not.

数学被视为这种极端，尤其是现代数学被视为极其难以进入的领域，而编程则不然。

So that could be just an entry point.

所以这可能只是一个切入点。

You can execute code and you can get results.

您可以执行代码并获得结果。

3:00:31 You can print out the world pretty quickly.

你可以很快地打印出世界。

If programing was taught as an almost entirely theoretical subject where you’re just taught the computer science, the theory of functions and routines and so forth, and outside of some very specialized homework assignments, you’re not actually programing, like on the weekend for fun, they would be as considered as hard as math.

如果编程几乎完全是作为一门理论学科来教授的，你只教授计算机科学、函数和例程理论等，除了一些非常专业的家庭作业之外，你实际上并没有编程，就像周末为了好玩，它们会被认为和数学一样难。

So as I said, there are communities of non-mathematicians where they’re deploying math for some very specific purpose, like optimizing their poker game, and for them, then math becomes fun for them.

正如我所说，有一些非数学家社区，他们为了某些非常具体的目的而部署数学，比如优化他们的扑克游戏，对他们来说，数学对他们来说变得很有趣。

What advice would you give in general to young people how to pick a career, how to find themselves, what they could be good at?

您对年轻人如何选择职业、如何找到自己以及他们可以擅长什么有什么建议？

That’s a tough, tough, tough question.

这是一个很难、很难、很难的问题。

3:01:25 Yeah, so there’s a lot of uncertainty now in the world.

是的，所以现在世界上有很多不确定性。

There was this period after the war where, at least in the West, if you came from a good demographic, there was a very stable path to it, to a good career.

战后的一段时期，至少在西方，如果你来自良好的人口结构，就有一条非常稳定的道路通向良好的职业生涯。

You go to college, you get an education, you pick one profession and you stick to it.

你上大学，接受教育，选择一个职业并坚持下去。

It’s becoming much more a thing of the past.

这已经成为过去式了。

So I think you just have to be adaptable and flexible.

所以我认为你必须具有适应能力和灵活性。

I think people will have to get skills that are transferable, like learning one specific programing language or one specific subject of mathematics or something.

我认为人们必须获得可转移的技能，例如学习一种特定的编程语言或一门特定的数学学科或其他东西。

That itself is not a super transferable skill, but sort of knowing how to reason with abstract concepts or how to problem solve when things go wrong.

这本身并不是一种超级可转移的技能，而是知道如何用抽象概念进行推理或在出现问题时如何解决问题。

Anyway, these are things which I think we will still need even as our tools get better, and you’ll be working with AI supports and so forth.

无论如何，我认为即使我们的工具变得更好，我们仍然需要这些东西，并且您将使用人工智能支持等等。

But actually you’re an interesting case study.

但实际上你是一个有趣的案例研究。

3:02:13 You’re one of the great living mathematicians, and then you had a way of doing things, and then all of a sudden you start learning.

你是当今最伟大的数学家之一，然后你有了一种做事的方式，然后突然你开始学习。

First of all, you kept learning new fields, but you learned Lean.

首先，你不断地学习新的领域，但你学会了精益。

That’s not a non-trivial thing to learn.

这不是一件值得学习的事情。

For a lot of people, that’s an extremely uncomfortable leap to take, right?

对于很多人来说，这是一个非常不舒服的飞跃，对吧？

Yeah.

是的。

A lot of mathematicians.

很多数学家。

First of all, I’ve always been interested in new ways to do mathematics.

首先，我一直对数学的新方法感兴趣。

3:02:42 I feel like a lot of the ways we do things right now are inefficient.

我觉得我们现在做事的很多方式都是低效的。

Many of my colleagues, who spend a lot of time doing very routine computations or doing things that other mathematicians would instantly know how to do and we don’t know how to do them, like how we search and get a quick response and so forth.

我的许多同事花费大量时间进行非常常规的计算或做其他数学家立即知道如何做而我们不知道如何做的事情，例如我们如何搜索并获得快速响应等等。

So that’s why I’ve always been interested in exploring new workflows.

这就是为什么我一直对探索新的工作流程感兴趣。

3:03:09 About four or five years ago, I was on a committee where we had to ask for ideas for interesting workshops to run at a math institute.

大约四五年前，我是一个委员会的成员，我们必须征求在数学研究所举办有趣的研讨会的想法。

And at the time, Peter Scholze had just formalized one of his new theorems, and there were some other developments in computer-assisted proof that look quite interesting.

当时，Peter Scholze 刚刚形式化了他的一个新定理，并且计算机辅助证明方面还有其他一些看起来非常有趣的进展。

And I said, “Oh, we should run a workshop on this.

我说：“哦，我们应该就此举办一个研讨会。

This would be a good idea.” And then I was a bit too enthusiastic about this idea, and so I got volun-told to actually run it.

这将是一个好主意。”然后我对这个想法有点太热情了，所以我自愿去实际运行它。

So I did with a bunch of other people, Kevin Buzzard and Jordan Ellenberg and a bunch of other people, and it wasn’t a nice success.

所以我和其他一些人一起做了，凯文·巴扎德（Kevin Buzzard）、乔丹·埃伦伯格（Jordan Ellenberg）和其他一些人，但这并不是一个很好的成功。

We pulled together a bunch of mathematicians and computer scientists and other people, and we got up to speed on state of the yard, and it was really interesting developments that most mathematicians didn’t know was going on, lots of nice proofs of concept, just hints of what was going to happen.

我们召集了一群数学家、计算机科学家和其他人，我们快速了解了院子里的状况，这是大多数数学家不知道正在发生的非常有趣的发展，有很多很好的概念证明，只是对即将发生的事情的暗示。

This was just before ChatGPT, but even then there was one talk about language models and the potential capability of those in the future.

那是在 ChatGPT 之前，但即便如此，也有人谈论过语言模型以及未来的潜在能力。

3:04:11 So that got me excited about the subject.

这让我对这个话题感到兴奋。

So I started giving talks about this is something more of us should start looking at, now that I had arranged, run this conference.

所以我开始就这个问题进行演讲，这是我们更多人应该开始考虑的事情，既然我已经安排好了，举办这次会议。

And then ChatGPT came out and suddenly AI was everywhere.

然后 ChatGPT 出现，人工智能突然无处不在。

And so I got interviewed a lot about this topic and in particular, the interaction between AI and [inaudible 03:04:33].

因此，我接受了很多关于这个话题的采访，特别是人工智能和 [听不清 03:04:33] 之间的互动。

I said, “Yeah, they should be combined.

我说：“是的，他们应该结合起来。

This is perfect synergy to happen here.” And at some point I realized that I have to actually not just talk the talk, but walk the walk.

这是在这里发生的完美协同效应。”在某些时候，我意识到我不仅要说说而已，还要付诸行动。

I don’t work in machine learning and I don’t work in proof formalisation, and there’s a limit to how much I can just rely on authority and say, “I’m a mathematician.

我不从事机器学习工作，也不从事证明形式化工作，我只能在多大程度上依赖权威并说：“我是一名数学家。

Just trust me when I say that this is going to change mathematics,” and I don’t do any of it myself.

当我说这将改变数学时，请相信我。”而我自己并没有这样做。

So I felt like I had to actually justify it.

所以我觉得我必须真正证明它的合理性。

3:05:03 A lot of what I get into, actually, I don’t quite see in advance as how much time I’m going to spend on it, and it’s only after I’m sort of waist deep in a project that I realize, but at that point, I’m committed.

事实上，我所涉及的很多事情，我并没有提前预见到我要花多少时间在上面，只有在我全身心投入一个项目之后我才意识到，但在那一点上，我已经承诺了。

Well, that’s deeply admirable that you’re willing to go into the fray, be in some small way a beginner, or have some of the challenges that a beginner would, right?

嗯，你愿意加入战斗，在某种程度上成为一个初学者，或者接受一些初学者会遇到的挑战，这真是令人钦佩，对吧？

Yeah.

是的。

New concepts, new ways of thinking, also sucking at a thing that others…

新概念，新思维方式，还吸别人的东西……

3:05:27 I think in that talk, you could be a Fields Medal-winning mathematician and an undergrad knows something better than you.

我认为在那次演讲中，你可能是一位获得菲尔兹奖的数学家，而一个本科生比你懂得更多。

Yeah, I think mathematics inherently, mathematics is so huge these days that nobody knows all of modern mathematics.

是的，我认为数学本质上是数学，如今数学是如此庞大，以至于没有人了解所有的现代数学。

3:05:42 And inevitably, we make mistakes and you can’t cover up your mistakes with just bravado because people will ask for your proofs, and if you don’t have the proofs, you don’t have the proofs.

我们不可避免地会犯错误，你不能仅仅用虚张声势来掩盖你的错误，因为人们会要求你提供证据，如果你没有证据，你就没有证据。

I love math.

我喜欢数学。

Yeah, so it does keep us honest.

是的，所以它确实让我们保持诚实。

3:06:04 It’s not a perfect panacea, but I think we do have more of a culture of admitting error because we’re forced to all the time.

这不是完美的灵丹妙药，但我认为我们确实有更多承认错误的文化，因为我们一直被迫这样做。

Big ridiculous question.

大可笑的问题。

3:06:17 I’m sorry for it once again.

我再次为此感到抱歉。

Who is the greatest mathematician of all time, maybe one who’s no longer with us?

谁是有史以来最伟大的数学家，也许他已经不在我们身边了？

Who are the candidates?

谁是候选人？

Euler, Gauss, Newton, Ramanujan, Hilbert?

欧拉、高斯、牛顿、拉马努金、希尔伯特？

So first of all, as mentioned before, there’s some time dependence.

首先，如前所述，存在一定的时间依赖性。

On the day.

当天。

Yeah.

是的。

3:06:38 Like if you plot cumulatively over time, for example, Euclid is one of the leading contenders, and then maybe some unnamed anonymous mathematicians before that, whoever came up with the concept of numbers.

例如，如果你随着时间的推移累积绘图，欧几里得是主要的竞争者之一，然后可能是在此之前的一些无名的匿名数学家，无论是谁提出了数字的概念。

Do mathematicians today still feel the impact of Hilbert, just-

今天的数学家是否仍然感受到希尔伯特的影响，只是——

Oh, yeah.

哦，是的。

Directly of what?

直接什么？

3:06:58 Everything that’s happened in the 20th century?

20世纪发生的一切？

Yeah, Hilbert spaces, we have lots of things that are named after him of course.

是的，希尔伯特空间，我们当然有很多东西以他的名字命名。

3:07:00 Just the arrangement of mathematics and just the introduction of certain concepts, 23 problems have been extremely influential.

仅仅数学的安排和某些概念的引入，23个问题就已经产生了极大的影响。

There’s some strange power to the declaring which problems are hard to solve, the statement of the open problems.

宣布哪些问题难以解决、公开问题的陈述有一种奇怪的力量。

Yeah, this is bystander effect everywhere.

是的，这到处都是旁观者效应。

3:07:19 If no one says you should do X, everyone just mills around waiting for somebody else to do something, and nothing gets done.

如果没有人说你应该做某事，那么每个人都会四处转悠，等待其他人做某事，但什么也做不了。

And the one thing that actually you have to teach undergraduates in mathematics is that you should always try something.

实际上你必须教本科生数学的一件事是你应该总是尝试一些东西。

So you see a lot of paralysis in an undergraduate trying a math problem.

所以你会看到本科生在解决数学问题时表现得很瘫痪。

If they recognize that there’s a certain technique that can be applied, they will try it.

如果他们认识到有某种技术可以应用，他们就会尝试。

But there are problems which they see and none of their standard techniques obviously applies and the common reaction is than just paralysis, I don’t know what to do.

但他们看到了一些问题，而且他们的标准技术显然都不适用，而且常见的反应不仅仅是瘫痪，我不知道该怎么办。

I think there’s a quote from the Simpsons, “I’ve tried nothing and I’m all out of ideas.” So the next step then is to try anything no matter how stupid and in fact almost the stupider, the better, which technically is almost guaranteed to fail, but the way it fails is going to be instructive.

我记得《辛普森一家》中有一句话，“我什么也没尝试过，我已经没有想法了。”因此，下一步就是尝试任何事情，无论多么愚蠢，事实上，越愚蠢越好，从技术上讲，这几乎肯定会失败，但失败的方式将具有启发性。

It fails ’cause you are not at all taking into account this hypothesis.

它失败了，因为你根本没有考虑到这个假设。

Oh, this hypothesis must be useful.

哦，这个假设一定有用。

That’s a clue.

这是一个线索。

I think you also suggested somewhere this fascinating approach, which really stuck with me as they’re using it, and it really works, I think you said it’s called structured procrastination.

我想你也在某个地方提出了这种令人着迷的方法，当他们使用它时，它真的让我印象深刻，而且它确实有效，我想你说过这就是所谓的结构化拖延。

No, yes.

不，是的。

It’s when you really don’t want to do a thing that you imagine a thing you don’t want to do more that’s worse than that and then in that way, you procrastinate by not doing the thing that’s worse.

当你真的不想做一件事时，你就会想象一件你不想做的事情比这更糟糕，然后这样，你就会拖延，不做更糟糕的事情。

3:08:37 It’s a nice hack, it actually works.

这是一个很好的技巧，它确实有效。

Yeah, yeah.

是啊是啊。

3:08:51 With anything, psychology is really important.

对于任何事情，心理学都非常重要。

You talk to athletes like marathon runners and so forth and they talk about what’s the most important thing, is it the training regimen or the diet and so forth?

你与马拉松运动员等运动员交谈，他们会谈论最重要的事情，是训练方案还是饮食等等？

So much of it is psychology, just tricking yourself to think that the problem is feasible so that you’re motivated to do it.

其中大部分是心理学，只是欺骗自己认为这个问题是可行的，这样你就有动力去做它。

Is there something our human mind will never be able to comprehend?

有什么东西是我们人类的头脑永远无法理解的吗？

Well, as a mathematician, [inaudible 03:09:23].

好吧，作为一名数学家，[听不清 03:09:23]。

3:09:21 There must be some large number that you can’t understand.

一定有很多你无法理解的数字。

That was the first thing that came to mind.

这是我首先想到的事情。

So that, but even broadly, is there something about our mind that we’re going to be limited even with the help of mathematics?

那么，更广泛地说，我们的思维是否存在即使在数学的帮助下也会受到限制的东西？

Well, okay, how much augmentation are you willing.

好吧，你愿意增加多少。

3:09:41 Like for example, if I didn’t even have a pen and paper, if I had no technology whatsoever, so I’ve not allowed blackboard, pen and paper-

比如说，如果我连笔和纸都没有，如果我没有任何技术，那么我不允许黑板、笔和纸——

You’re already much more limited than you would be.

你已经比你想象的要受限制得多了。

…

……

3:09:55 Incredibly limited.

极其有限。

Even language, the English language is a technology.

即使是语言，英语也是一门技术。

It’s one that’s been very internalized.

这是一个非常内化的东西。

So you’re right, the formulation of the problem is incorrect ’cause there really is no longer just a solo human already augmented in extremely complicated intricate ways, right?

所以你是对的，问题的表述是不正确的，因为真的不再只是一个已经以极其复杂的方式增强的单独人类，对吧？

Yeah.

是的。

3:10:17 Yeah.

是的。

So like a collective intelligence?

那么像集体智慧吗？

Yes.

是的。

3:10:20 Yeah, I guess, so humanity plural has much more intelligence in principle on its good days than the individual humans put together.

是的，我猜，所以原则上，人类在美好的日子里比个体人类的智慧加起来要多得多。

It can have less, but yeah, so the mathematical community plural is incredibly super intelligent entity that no single human mathematician can come closer to replicating.

它可以有更少，但是，是的，所以数学社区复数是令人难以置信的超级智能实体，没有任何一个人类数学家可以接近复制。

You see it a little bit on these question analysis sites.

你可以在这些问题分析网站上看到一点。

So this math overflow, which is the math version of stackable flow, sometimes you get this very quick response to very difficult questions from the community, and it’s a pleasure to watch actually, as an expert.

所以这个数学溢出，这是可堆叠流程的数学版本，有时你会得到对社区中非常困难的问题的快速答复，作为专家，实际上很高兴观看。

I’m a fan spectator of that site, just seeing the brilliance of the different people, the depth and knowledge that people have.

我是该网站的粉丝观众，只是看到不同人的才华、人们所拥有的深度和知识。

3:11:01 And the willingness to engage in the rigor and the nuance of the particular question, it’s pretty cool to watch.

并且愿意参与特定问题的严格性和细微差别，这看起来很酷。

It’s almost like just fun to watch.

观看起来几乎很有趣。

What gives you hope about this whole thing we have going on with human civilization?

是什么让你对人类文明正在发生的这一切充满希望？

I think the younger generation is always really creative and enthusiastic and inventive.

我认为年轻一代总是非常有创造力、热情和创造力。

3:11:25 It’s a pleasure working with young students.

与年轻学生一起工作很愉快。

The progress of science tells us that the problems that used to be really difficult can become trivial to solve.

科学的进步告诉我们，以前很难解决的问题现在变得微不足道。

Like navigation, just knowing where you work on the planet was this horrendous problem.

就像导航一样，仅仅知道你在地球上的哪个地方工作就是一个可怕的问题。

People died or lost fortunes because they couldn’t navigate.

人们因为无法导航而死亡或失去财富。

And we have devices in our pockets that do this automatically for us, like it is a completely solved problem.

我们口袋里有设备可以自动为我们完成这些工作，就像这是一个完全解决的问题一样。

So things that seem unfeasible for us now, could be maybe just homework exercises.

所以现在对我们来说似乎不可行的事情可能只是家庭作业。

Yeah.

是的。

3:12:13 One of the things I find really sad about the finiteness of life is that I won’t get to see all the cool things we create as a civilization because in the next 100 years, 200 years, just imagine showing up in 200 years.

对于生命的有限性，我感到真正悲伤的一件事是，我无法看到我们作为一个文明所创造的所有很酷的东西，因为在接下来的 100 年、200 年里，想象一下 200 年后出现的情况。

Yeah, well, already plenty has happened.

是的，好吧，已经发生了很多事情。

3:12:27 If you could go back in time and talk to your teenage self or something, the internet and now AI, again, they’re getting to internalize and yeah, of course, AI can understand our voice and give reasonable slightly incorrect answers to any question.

如果你能回到过去，与青少年时期的自己或其他什么东西交谈，互联网和现在的人工智能，他们就会内化，是的，当然，人工智能可以理解我们的声音，并对任何问题给出合理的、稍微不正确的答案。

But yeah, this was mind-blowing even two years ago.

但是，是的，即使在两年前，这也是令人兴奋的。

And in the moment, it’s hilarious to watch on the internet and so on, the drama, people take everything for granted very quickly, and then we humans seem to entertain ourselves with drama.

此刻，在互联网上观看戏剧等很搞笑，人们很快就认为一切都是理所当然的，然后我们人类似乎用戏剧来娱乐自己。

3:12:50 Out of anything that’s created, somebody needs to take one opinion, another person needs to take an opposite opinion, argue with each other about it.

对于任何被创造出来的东西，需要有人采取一种意见，另一个人需要采取相反的意见，互相争论。

But when you look at the arc of things, just even in the progress of robotics, just to take a step back and be like, “Wow, this is beautiful, that we humans are able to create this.”

但当你看到事物的发展轨迹时，即使是在机器人技术的进步中，只要退后一步，就会说：“哇，这太美了，我们人类能够创造这个。”

When the infrastructure and the culture is healthy, the community of humans can be so much more intelligent and mature and rational than the individuals within it.

当基础设施和文化健康时，人类社会就会比其中的个人更加聪明、成熟和理性。

Well, one place I can always count on rationality is the Comment section of your blog, which I’m a big fan of.

好吧，我总是可以依靠理性的一个地方是你博客的评论部分，我是你的忠实粉丝。

3:13:31 There’s a lot of really smart people there.

那里有很多非常聪明的人。

And thank you, of course, for putting those ideas out on the blog.

当然，感谢您将这些想法发布在博客上。

And I can’t tell you how honored I am that you would spend your time with me today.

我无法告诉你我是多么荣幸你今天能和我一起度过。

I was looking forward to this for a long time.

我对此期待已久。

Terry, I’m a huge fan.

特里，我是它的超级粉丝。

You inspire me, you inspire millions of people.

你激励了我，也激励了数百万人。

Thank you so much for time.

非常感谢您抽出时间。

Thank you.

谢谢。

3:13:58 It was a pleasure.

这是一种乐趣。

Thanks for listening to this conversation with Terrence Tao.

感谢您收听陶哲轩的对话。

3:14:00 To support this podcast, please check out our sponsors in the description or at lexfridman.com/sponsors.

为了支持此播客，请在说明中或访问 lexfridman.com/sponsors 查看我们的赞助商。

And now, let me leave you with some words from Galileo Galilei, “Mathematics is a language with which God has written the universe.”

现在，让我向你们传达伽利略·伽利雷的话：“数学是上帝书写宇宙的语言。”

3:14:21 Thank you for listening and hope to see you next time.

感谢您的聆听，并希望下次再见。