Terence Tao
and some mathematical numerical pattern might show up in the behavior of of of insects um in a swarm or in a stock market or whatever. Um and then um sometimes once you understand it you you can actually convert it into some useful technology or you can have a company that actually makes some money out of out of somehow some service related to to that uh exploiting that phenomenon. We often don't uh we don't see that. I mean that that's that that's much further down the pipeline. What you do need is that you do need the people doing the basic sciences to talk to the people doing applied sciences and they have to talk to people who are doing engineering and they have to talk to people in industry. If you didn't have one of these communities, then you wouldn't have this pipeline of getting from curiosity driven questions to actual, you know, commercial results, you know, you know, like um like the ability to communicate across the across the planet with almost zero cost. This is part of what Eugene Wigner calls the unreasonable effectiveness of
某个数学上的数值模式可能出现在昆虫群体的行为中，或股票市场中，诸如此类。然后，有时一旦你理解了它，你实际上可以把它转化成某种有用的技术，或者可以有一家公司从利用那个现象的某种服务中赚钱。我们常常看不到那一步。我的意思是，那在管道中要靠后得多。你所需要的是，做基础科学的人必须与做应用科学的人交谈，他们必须与做工程的人交谈，他们必须与产业中的人交谈。如果你缺少这些共同体中的一个，你就不会有这条从好奇心驱动的问题通向实际商业成果的管道，比如以几乎为零的成本在全球通信的能力。这是 Eugene Wigner 所谓数学在物理科学中。
Terence Tao
mathematics in the physical sciences. He observed that mathematicians often discover concepts such as complex numbers or curved space or whatever just because it it seems to be a natural extension of um the mathematical objects they're already studying. Uh and then 10 20 50 years later some scientists discover that that these concepts that introduced for fun or for play were in fact exactly what was or almost exactly what was needed to understand some new new type of science.
不合理的有效性的一部分。他观察到，数学家常常发现像复数或弯曲空间之类的概念，只是因为它似乎是他们已经在研究的数学对象的自然延伸。然后过了 10 年、20 年、50 年，一些科学家发现，这些为了好玩或游戏而引入的概念，实际上正好、或几乎正好是理解某种新科学所需要的。
Terence Tao
So that's a really amazing phenomenon and we still don't have a good explanation really for for why uh that actually works. So one historical example of how curiosity driven mathematics led to a really deep uh scientific advance uh was the story of the parallel postulate Euclid in like the 3rd century BC introduced the notion of proof of being able to explain complicated results in geometry in this case from simpler axioms. for example that the sum of angles of a triangle always added up to 180° He was able to explain that in terms of simpler axioms. He laid out five axioms of geometry that he thought he reduced all the other facts he knew about points and angles and lines to these five statements and four of them were very straightforward like if I give you two points there's always a line you can draw between them right things like that. These are very straightforward non-controversial axioms. But there was this one axiom which is called the
这是一个真正惊人的现象，我们仍然没有真正好的解释，说明为什么这实际上有效。好奇心驱动的数学如何导向真正深刻的科学进展，一个历史例子是平行公设的故事。Euclid 在大约公元前 3 世纪引入了证明的概念：能够用更简单的公理来解释几何中的复杂结果。例如，三角形的内角和总是等于 180 度。他能够用更简单的公理来解释这一点。他列出了几何的五条公理，他认为他把他所知道的关于点、角和线的所有其他事实都还原成了这五个陈述，其中四条非常直截了当：比如如果我给你两点，你总可以在它们之间画一条线，诸如此类。这些是非常直截了当、没有争议的公理。但有一条公理叫做
Terence Tao
parallel postulate which gave him a lot of grief and in fact his original version was very very complicated. Um it got simplified but even even the simplified version always was controversial. The simplified version is that if you have a line and you have a point and the point is not on the line then there's exactly one line you can draw through that point which is parallel to the first line. By parallel I mean that it never crosses this first line. So that was this axiom that you can always draw a parallel line through any other point and there's only one. You cannot draw two parallel lines. Once you have that you can do all kinds of things. You can derive what I just said the angles of a triangle at 180° and all the other classic results of Euclidean geometry. But it was a very ugly axiom compared to the other four which were which are really elegant. Eventually they realized that that what they had done was that they there actually were multiple geometries beyond Euclidean geometry. There's something called spherical geometry where there's no parallel lines at all. Um so in spherical geometry instead of lines you have great
平行公设，给他带来了很多苦恼。事实上，他原来的版本非常非常复杂。它被简化了，但即使简化版也总是有争议。简化版是：如果你有一条线，你有一个点，而这个点不在这条线上，那么恰好有一条线你可以过那个点画出，与第一条线平行。所谓平行，我是指它永远不会与第一条线相交。所以这条公理是：你总可以通过任何其他点画出一条平行线，而且只有一条。你不能画出两条平行线。一旦你有了那个，你就可以做各种各样的事情。你可以推导出我刚才说的三角形内角和 180 度，以及欧几里得几何的所有其他经典结果。但它与其他四条相比是一条非常丑陋的公理，那四条非常优美。最终他们意识到，他们所做的是：实际上存在着欧几里得几何之外的多种几何。有一种叫做球面几何，其中根本没有平行线。在球面几何中，不是线，而是大圆，
Terence Tao
circles like the equator or a longitude and these great circles on the sphere they always intersect. You can never make two parallel great circles. So there are no parallel lines and then there's this weirder geometry which is harder to visualize called hyperbolic geometry where lines actually diverge from each other and have lines that start off looking parallel but they they move further and further apart and and in fact now there are actually multiple parallel lines you can draw from one point to a given line. And those two geometries are entirely self-consistent.
比如赤道或经线，球面上的这些大圆总是相交。你永远无法做出两条平行的大圆。所以没有平行线。然后还有这种更奇怪、更难可视化的几何，叫做双曲几何，其中线实际上彼此发散，有些线开始看起来平行，但它们越离越远，事实上现在你可以从一个点向一条给定的线画出多条平行线。而那两种几何是完全自洽的。
Terence Tao
And eventually it was just accepted that there was there were more geometries out there than just Euclidean geometry. So these were the first two non-Euclidean geometries to be discovered. Spherical geometry and hyperbolic geometry. But once we had sort of freed of our notion that there's only one geometry, this opened all the floodgates and people studied all kinds of other geometries. So all kinds of curved spaces, you know, spaces that were shaped like donuts or had twists in them. There are geometries where um you know if you're right-handed you can go off explore the universe and come back and if you start off right-handed you will come back left-handed. Um that there there are geometries where where you can change your orientation just by by by travel. So which is very unintuitive or you can come back smaller or larger than than than what you started with. People developed all these geometries and they developed um a very nice language for for describing all um all these geometries. uh it's called Riemannian geometry after Bernard Riemann and but it was a curiosity I mean there were these abstract curved
最终人们接受了：在欧几里得几何之外还有更多几何。这是被发现的前两种非欧几何：球面几何和双曲几何。但一旦我们从“只有一种几何”的观念中解放出来，这就打开了所有闸门，人们研究了各种各样的其他几何。各种各样的弯曲空间，形状像甜甜圈或有扭转的空间。有些几何中，如果你是右撇子，你可以出去探索宇宙再回来，如果你出发时是右撇子，回来时会变成左撇子。有些几何中，你仅仅通过旅行就可以改变你的定向。这非常反直觉，或者你回来时会比出发时更小或更大。人们发展了所有这些几何，并发展了一种非常好的语言来描述所有这些几何。它叫做黎曼几何，以 Bernard Riemann 命名，但它是一种好奇心。我的意思是，有这些抽象的弯曲
Terence Tao
spaces but you know the the universe we lived in seemed completely flat but then Einstein when he was trying to understand gravity he eventually came to conclusion that what gravity was doing was it was bending space and time in a certain way and he needed a language to to describe how space and time could bend in such a way that light rays would would become not straight and sometimes times you know they would hit each other or or diverge. He asked his mathematician friend if there was any existing mathematics that would describe this and he he said oh yeah there's this bright chap Bernard Riemann who developed this theory but it turned out to be almost exactly the right language to describe the Einstein equations there's a notion of curvature that some space can have positive curvature negative curvature in Riemannian geometry and the Einstein equations turned out extremely simple in to state in this language that basically mass and energy create curvature that the curvature of space and time is proportional to how how much mass and energy
空间，但我们所生活的宇宙似乎完全是平的。然后 Einstein 在试图理解引力时，最终得出结论：引力所做的是以某种方式弯曲空间和时间，他需要一种语言来描述空间和时间如何能够弯曲，使得光线不再是直的，有时它们会相遇或发散。他问他的数学家朋友。是否有任何现成的数学可以描述这一点，朋友说，哦，是的，有一位出色的人 Bernard Riemann 发展了。这个理论，结果它几乎正好是描述 Einstein 方程的正确语言。在黎曼几何中有曲率的概念：有些空间可以有正曲率、负曲率，而 Einstein 方程用这种语言来陈述变得极其简单：基本上，质量和能量产生曲率，时空的曲率与你系统中有多少质量和能量成正比。
Terence Tao
you have in your system. And that's basically the Einstein equations. Now, solving them is a different matter. They're extremely hard to model. Even the question of for how to model two colliding black holes, we can barely do it with modern supercomputers. But but stating the the equations is actually extremely natural once we had this language. Another example of um of how mathematical curiosity led to like really practical developments centuries later is the story of sphere packing. There was uh some uh British sailor um who was just curious about the question of you know there's a certain number of cannonballs they had to stack in the hold of of their ship and these are these are round cannonballs they're not they're not square so when you when you stack them there's a certain amount of wasted space and he was curious what is the most efficient way to pack cannonballs so you can get the most cannonballs into a certain amount
那基本上就是 Einstein 方程。现在，求解它们是另一回事。它们极其难以建模。即使是如何为两个相撞的黑洞建模这个问题，我们用现代超级计算机也几乎勉强能做。但一旦我们有了这种语言，陈述这些方程实际上是极其自然的。数学好奇心如何在几个世纪后导向真正实用的发展，另一个例子是球体堆积的故事。有一位英国水手，只是对这个问题感到好奇：他们必须在船舱里堆放一定数量的炮弹，这些是圆炮弹，不是方的，所以当你堆放它们时，会有一定量的浪费空间，他好奇最有效的堆积炮弹的方式是什么，从而能在一定空间里放入最多的炮弹。
Terence Tao
of space and so he asked a physician friend who happened to be Johannes Kepler eventually proposed that the most efficient packing should be the same packing that that you see in nowadays in supermarkets when you pack oranges. It's what's called a hexagonal close packing. You pack layer by layer. Each layer is sort of the triangular grid of cannonballs or oranges and then you stack another triangular grid on top of it just shifted by a little bit and then you stack it back and there's a a regular pattern which is the most natural pattern and it's about 76% efficient. And Kepler thought this was the um the best you could do that. there was no clever way to to to squeeze in any any any more space but he couldn't actually prove it. So this became known as the Kepler conjecture. Um and it was one of the most famous unsolved problems in geometry for for centuries in
于是他问了一位医生朋友，碰巧是 Johannes Kepler。Kepler 最终提出，最有效的堆积应当与你如今在超市里堆橙子时看到的堆积相同。这叫做六角密堆积。你一层一层地堆。每一层是炮弹或橙子的某种三角形网格，然后你再在上面堆另一层三角形网格，只是稍微错开一点，然后再堆回去，有一种规则模式，是最自然的模式，效率大约是 76%。Kepler 认为这是你能做到的最好结果：没有巧妙的办法再挤进任何更多空间，但他实际上无法证明它。于是这被称为 Kepler 猜想。几个世纪以来，它是几何中最著名的未解问题之一。
Terence Tao
two dimensions. Uh I think it was solved by about 1900 or something like you're packing discs in a plane. That's a simpler problem. That one there's a similar lattice of triangular lattice and and that was relatively easy to prove that this was the optimal one. three dimensions are just too many possibilities. There was no way there. Um, in fact, we still do not have a nice simple proof of the Kepler conjecture that that that humans can completely understand by themselves. The conjecture was eventually solved. It eventually got published I think in 1998, but it required uh computers. It was one of the first computer assisted proofs. There was a team of referees who I think uh said that they could not verify all of the computations but they at least believe that the the strategy is correct but there were still lingering doubts. It is only much more recently 2014 I think and finally um the proof was converted to what is called a a proof assistant
在二维中，我想大约在 1900 年左右被解决了，那是在平面上堆积圆盘。那是一个更简单的问题。那里有类似的三角形点阵，相对容易证明这是最优的。三维则可能性太多了。没有办法。事实上，我们仍然没有一个人类可以完全靠自己理解的、关于 Kepler 猜想的漂亮简单证明。这个猜想最终被解决了。我想它最终在 1998 年发表，但它需要计算机。这是最早的计算机辅助证明之一。有一组审稿人，我想他们说他们无法核验所有计算，但他们至少相信策略是正确的，但仍有挥之不去的疑虑。直到更近的时候，我想是 2014 年，证明终于被转换成所谓的证明助手
Terence Tao
language computer language that is specifically designed to check proofs with 100% certainty. So the Kepler conjecture is now formally verified. we are now 100% certain it is true but mathematicians were not content with just that the three dimensional problem so they also asked what happens if you're in four dimensions or five dimensions or six dimensions so here of course there is no practical you know I mean there are no four dimensional oranges or or cannonballs that you would like to like to pack but people still ask this question people also ask what happened if instead of a continuous space um you have a discrete space in particular computer scientists once computer computer science became developed We realized that in addition to the geometry of regular space where XYZ where coordinates are given by real numbers, we're interested in studying the geometry of of strings of bits. So this is now very um divorced sounding from the original sphere packing problem both because now you have many many dimensions like like thousands
语言，一种专门设计用来以 100% 确定性检查证明的计算机语言。所以 Kepler 猜想现在已被形式化验证。我们现在 100% 确定它是真的。但数学家并不满足于仅仅那个三维问题，他们还问：如果你在四维、五维或六维中会怎样。当然，这里没有实际用途：没有四维橙子或炮弹是你想要堆积的，但人们仍然问这个问题。人们还问：如果不是连续空间，而是离散空间会怎样。特别是，一旦计算机科学发展起来，计算机科学家意识到，除了普通空间的几何——XYZ 坐标由实数给出——我们还对研究比特串的几何感兴趣。这现在听起来与原来的球体堆积问题非常脱节，既。因为现在有非常多的维数，比如成千上万。
Terence Tao
and thousands of dimensions and space is now discreet rather than continuous but still it is geometry and and many of the techniques to understand sphere packing still work. uh when you when you're in a setting. And then it turned out that this setting of this problem of of packing spheres as efficiently as possible into um on this big huge cube of of bit strings it turns out to be extremely practical. When cell phones became digital, they um every signal that you that you send, you know, like an image or a text or whatever is encoded as some bitstring which is then sent over um over um some wireless network. But there are other people also sending you the signals and you don't want your signal to be corrupted by interference and be mistaken for for someone else's signal. So you want to keep each different signal that that comes out. You want to keep them as as separated from each other in this space of of bitstrings as possible. And it turns
维，而且空间现在是离散的而不是连续的，但它仍然是几何，许多理解球体堆积的技术在这种设定中仍然有效。然后结果是，把球体尽可能有效地堆积进这个由比特串组成的巨大立方体中的这个问题，结果极。其实用。当手机变成数字的，你发送的每一个信号，。比如一张图像或一条文本等等，都被编码成某个比特串，然后通过某个无线网络发送。但也有其他人在向你发送信号，你不希望你的信号被干扰破坏，并被误认为是别人的信号。所以你希望让每一个发出的不同信号，在这个比特串空间中尽可能彼此分开。结果
Terence Tao
out mathematically this problem of separating all these signals so that there's no way that one can be confused for another is almost exactly the sphere packing problem except that it's in high dimensions and it's discreet. All the mathematics, well, not all of it, but a lot of the mathematics that was used to understand sphere packings could be used to design uh really efficient sphere packing codes and not just to design codes, but also to to to tell engineers what was the what is the theoretical limit of communication like what what is the maximum number of of bits per second you could possibly hope to um to send in a certain wireless spectrum. So that gave really good um benchmarks to measure how efficient your your protocol was. They you could price how many billions of dollars you should pay for a certain spectrum wireless spectrum because now you know exactly how much uh data you can push through that. And so like the entire wireless telecommunication industry um is is based on being able to to to pack oranges in really high
从数学上讲，把所有这些信号分开，使得一个不可能被误认为另一个，这个问题几乎正好就是球体堆积问题，只。不过它是在高维中，而且是离散的。所有数学——好吧？，不是全部，但很多用来理解球体堆积的数学——都可以用来设计真正高效的球体堆积码，不仅是设计码，还可以告诉工程师通信的理论极限是什么：在某个无线频谱中，你最多可能希望每秒发送多少比特。于是这给出了非常好的基准，来衡量你的协议有多高效。他们可以定价：你应当为一块特定的无线频谱支付多少亿美元，因为现在你确切知道你可以推送多少数据通过它。所以整个无线电信产业，都基于能够在非常高的维数中堆积橙子。
Terence Tao
dimensions. One of the applications of mathematics that I was involved in that I'm most proud of is the story of compressed sensing. So I was once at an interdisciplinary program at a math institute here in Los Angeles um and I met with a friend of mine who is a statistician and he was working with an electrical engineer and trying to improve um imaging medical imaging um and specifically MRI scans at the time uh MRI scans were were quite slow. uh you had to sit in this scanning machine for like 3 minutes so that you could collect enough data from all different angles that the the scan could reconstruct a good image of your body and be able to pick up tumors or cysts or anything else which is sort of medically important uh to uh to um to resolve. But if you only sat in the machine for a short period of time like say half a minute you will not get enough data. Um so if you tried the standard reconstruction algorithm from all the data to recover the image uh using
我参与过的、我最自豪的数学应用之一，是压缩感知的故事。我曾经在洛杉矶这里的一个数学研究所参加一个跨学科项目，遇到了我的一位朋友，他是统计学家，他当时在与一位电气工程师合作，试图改进医学成像，特别是当时的 MRI 扫描。当时 MRI 扫描相当慢。你必须在这台扫描机器里坐大约 3 分钟，才能从各个不同角度收集足够的数据，扫描才能重建出你身体的良好图像，并能够发现肿瘤或囊肿或任何其他在医学上重要、需要分辨的东西。但如果你只在机器里坐很短一段时间，比如半分钟，你就得不到足够的数据。所以如果你尝试当时的标准重建算法，从所有数据恢复图像，使用。
Terence Tao
what's called least squares approximation which is the standard technique at the time you would get an image that was so blurry and so res so so low resolution that you could not tell anything useful for diagnosis purposes. So we had to sit in these machines for for minutes and minutes and if you were a kid sometimes they you had to be sedated because the kid would wriggle around and not follow instructions after like the second minute. So they were trying a new technique not least squares uh something called total variation minimization. They had a hunch that that this other method might perform a little bit better but so they they tried it on some test data. They were expecting a slightly sharper image than the least squares approximation but they got perfect resolution that they got back almost exactly the correct image even though they only took a few measurements. It would be like giving someone a crossword puzzle where you had only filled in, you know, 10% of of of of the letters and suddenly they could fill in all the other letters without having to look up the clues. They couldn't explain this and and they showed this to
所谓的最小二乘近似，这是当时的标准技术，你会得到一张非常模糊、分辨率非常低的图像，对诊断目的没有任何有用信息。所以我们必须在这些机器里坐上许多分钟。如果你是个孩子，有时必须被镇静，因为孩子会扭来扭去，大约到第二分钟之后就不听指令了。于是他们在尝试一种新技术，不是最小二乘，而是所谓的全变差最小化。他们有一种直觉，这种方法可能表现得稍好一些，于是他们在一些测试数据上试了它。他们预期会得到比最小二乘近似稍微更清晰一点的图像，但他们得到了完美的分辨率：尽管他们只取了少量测量，却几乎正好得到了正确的图像。这就像给某人一个填字游戏，你只填了 10% 的字母，突然他们就能填完所有其他字母，而不必去看线索。他们无法解释这一点，他们把这个给我看，。
Terence Tao
me and my first instinct was you made a mistake. You could not possibly have done what you said you did. And in fact, I'm going to prove to you that that there was not enough information in the data that you you took to make to make this uh measurement possible. So I I I went home that night. I tried to write down a proof that that there was no way to just correctly guess the the right image from the small amount of data that they were measuring. And while writing it down,I found one of my steps does not work. And in fact, in fact, it showed the opposite that if a certain measurement matrix had a certain property, then actually what they were doing was actually going to work. And then I checked that what that their measurement matrix actually did seem to obey this property. So I actually understood uh how um how the method worked and so I went back to them the next day and explained this and they got very excited and we wrote a couple papers and that got
我的第一反应是：你弄错了。你不可能做到你所说的那样。事实上，我要向你证明，你所取的数据中没有足够的信息使这种测量成为可能。于是那天晚上我回家，试图写下证明：没有办法仅从他们测量的那少量数据正确猜出正确的图像。而在写下来的过程中，我发现我的某一步行不通。事实上，它显示了相反的结论：如果某个测量矩阵具有某种性质，那么他们所做的实际上将会有效。然后我检查了他们的测量矩阵，它实际上似乎确实服从这个性质。于是我实际上理解了这个方法如何工作，第二天我回去向他们解释，他们非常兴奋，我们写了几篇论文，那让
Terence Tao
everyone else excited. This method that they had stumbled upon was not completely new. There were seismologists who had discovered a similar method. So they had a se a different problem where they were trying to understand the fault lines to locate the fault lines of a crust based on on a small amount of of seismic data. And astronomers had a similar problem that they were trying to to measure the location of stars or something using a very small amount of of observed data. And there are a couple other disciplines where a similar problem of trying to extract a high quality image from a very small amount of signal had had occurred. each case they had found some ad hoc fix that could kind of squeeze more data out of um a more uh better resolution out of the data they had but they could not explain mathematically why it worked. The seismologist thought, "Oh, here's a trick, but it only works for seismographs." And the astronomers had a trick, but it only worked for astronomy. But once we found the mathematical explanation, we found
所有其他人都兴奋起来。他们偶然发现的这种方法并非全新。有地震学家发现过类似的方法。他们有一个不同的问题：他们试图根据少量地震数据理解断层线，定位地壳的断层线。天文学家有类似的问题：他们试图用非常少量的观测数据测量恒星的位置之类。还有其他几个学科也出现过类似的问题：试图从非常少量的信号中提取高质量图像。在每种情形中，他们都找到了某种临时办法，能够从已有数据中挤出更多数据、更好的分辨率，但他们无法从数学上解释为什么它有效。地震学家想：哦，这是一个技巧，但只对地震仪有效。天文学家有一个技巧，但只对天文学有效。但一旦我们找到了数学解释，我们发现
Terence Tao
that this was a general technique that we we now call compressed sensing. And it's useful for MRI, but it's also useful for wireless broadband. It's useful for for certain types of sensor networks. Once we figured out the underlying mathematics, we could see all the other applications that it's useful for. And so now compressed sensing is taught in textbooks right next to least squares. So sometimes at least squares it's the right thing to do and sometimes compressed sensing is the right thing to do. Sometimes neither. It's now a a very very well developed theory.
这是一种一般技术，我们现在称之为压缩感知。它对 MRI 有用，对无线宽带也有用，对某些类型的传感器网络也有用。一旦我们弄清了底层数学，我们就能看到它有用的所有其他应用。所以现在压缩感知被写进教科书，就在最小二乘旁边。有时最小二乘是该做的正确事情，有时压缩感知是该做的正确事情，有时两者都不是。它现在是一个发展得非常非常好的理论。
Terence Tao
I was quite pleased to be involved at the very beginning of that. It's a fascinating interplay between mathematics and science. So we have this unreasonable effectiveness of of mathematics where mathematical discoveries are often end up being the the best way to explain physical phenomena. Philosophers and historians have have debated why this is the case. Um, one of my theories is that whenever we learn anything uh whether it's math or science or any other subject, the first theories or explanations that we um we make are often um not the best. We don't understand what is the cleanest way to to to express something. And before you understand something completely, you might have an overly elaborate explanation for why something is true. But often the the the true
我很高兴能在一开始就参与其中。这是数学与科学之间迷人的相互作用。我们有数学这种不合理的有效性：数学发现常常最终成为解释物理现象的最佳方式。哲学家和历史学家一直在争论为什么会是这样。我的理论之一是：每当我们学习任何东西，无论是数学还是科学或任何其他学科，我们做出的第一批理论或解释常常并不是最好的。我们不理解表达某件事最干净的方式是什么。在你完全理解某件事之前，你可能对为什么某件事为真有一个过于繁复的解释。但常常，真正的
Terence Tao
explanation is is is more elegant and shorter and simpler than our initial attempts to describe the um the phenomenon. But finding the the short explanation takes time because you have to sort of unlearn certain um assumptions that you might have that turn out to be to be incorrect. For example, with with Einstein's theory of relativity, one of the key assumptions that people had before Einstein was was that was that time was universal. everyone had the same notion of time.
解释比我们最初试图描述这个现象的尝试更优美、更短、更简单。但找到那个简短的解释需要时间，因为你必须忘掉某些结果证明是不正确的假设。例如，对于 Einstein 的相对论，Einstein 之前人们持有的一个关键假设是：时间是普遍的。每个人都有相同的时间概念。
Terence Tao
You know that that an hour for me is the same as an hour for you. That mindset uh really blocks you from from from finding the right way to explain gravity in particular. But once you accept that everyone has their own relative notion of time um then you can find the right language to explain things properly. Mathematicians also try to take phenomena that they first understand using very inefficient language and try to condense it. try to find the the the most concise elegant explanation of a mathematical phenomenon. And I think just because there's only so many ways you can you can say things concisely. It just happens that um that uh that often the concise way to describe some mathematical phenomenon is also a concise way to describe a physical phenomenon. So that is that is my theory. Unfortunately, we only have one timeline of science. If you only have one history of scientific development and we have maybe a 100 turning points in science and that's a little bit of data and you can make some theories. I would love in the future in the
对我来说的一小时与对你来说的一小时是一样的。那种心态真的会阻碍你找到解释引力的正确方式。但一旦你接受每个人都有自己相对的时间概念，你就可以找到正确的语言来恰当地解释事情。数学家也试图把他们最初用非常低效的语言理解的现象加以浓缩，试图找到对一个数学现象最简洁、最优美的解释。我认为，正因为用简洁的方式说话只有那么多种方式，结果常常是：描述某个数学现象的简洁方式，也是描述某个物理现象的简洁方式。这就是我的理论。不幸的是，我们只有一条科学时间线。如果你只有一部科学发展史，我们也许有大约 100 个科学转折点，那只是一点点数据，你可以提出一些理论。我希望在未来，在
Terence Tao
far future maybe we will meet other civilizations and we will see their history of science and we will see whether they also had ke their version of Kepler and Einstein and Newton and whether it um what they they followed a similar track or a completely different track I don't know the funny thing about doing mathematics is that so you know there's this stereotype that you know we're all geniuses and you know we're stuck at a problem and then you get this eureka insight, a light bulb goes on and um and like you get this genius idea out of nowhere. I would love for that to happen to me actually. This this does not happen um so often to me. What what does happen when I do work on a problem is that I try something and it doesn't work. Okay, I try something else and it kind of works but it gets stuck at a certain point. Um but now at least I know kind of there's at least one obstacle and I need to find some tool that is will will help me deal with that obstacle. And so maybe
遥远的未来，也许我们会遇见其他文明，我们会看到他们的科学史，会看到他们是否也有他们版本的 Kepler、Einstein 和 Newton，以及他们是走了类似的轨道，还是完全不同的轨道，我不知道。做数学这件事有趣的地方在于：有一种刻板印象，说我们都是天才，卡在一个问题上，然后你得到这种尤里卡洞见，灯泡。亮了，你从无处得到这个天才想法。我其实很希望这发生在我身上。这对我来说并不经常发生。当我真正在一个问题上工作时，实际发生的是：我试某件事，它行不通。好，我试别的，它有点行得通，但在某一点卡住了。但现在至少我知道至少有一个障碍，我需要找到某种工具来帮助我处理那个障碍。于是也许
Terence Tao
um I will now identify a sub problem which has the same type of difficulty but is simpler and I try to solve that one first. Uh and if I can do that one then I can try to scale up back to the original problem and I go back and forth. Often a lot of what you're doing is that you are you are exploring the negative space of the problem like all the the techniques that don't work and um eventually if you have enough of the negative space the the path forward becomes clear almost by elimination that there's only sort of one thing to do that could work. Um, some sometimes there's nothing that can work and then you give up on the problem. But there's this repeated what you might call failure, but it really is kind of just just really understanding the limitations of what different approaches can do. And then after weeks or months, you know, of of this, the answer becomes clear. But by that point, it's so internalized what the difficulties are that it doesn't feel amazing to you anymore. It feels natural. Like, of course, you had to do
我会现在识别出一个子问题，它有同样类型的困难，但更简单，我先试图解决那个。如果我能做到那个，我就可以试图放大，回到原来的问题，我来来回回。你所做的很多事情，常常是在探索问题的负空间：所有那些行不通的技术。最终，如果你有足够多的负空间，前进的道路几乎通过排除法就变得清晰：只有那么一件可能行得通的事可做。有时没有任何东西能行得通，然后你放弃这个问题。但这是反复的、你也许会称为失败的东西，但它其实只是真正理解不同方法能做什么的局限。然后经过数周或数月这样的工作，答案变得清晰。但到那时候，困难已经如此内化，对你来说不再感觉惊人。它感觉很自然：当然，你必须这样做，
Terence Tao
this because there's this difficulty. you must go around this this pothole. You must do this step first because you know in five lines you're going to need this hypothesis to to solve this problem. You you just become so um attuned to the problem that everything becomes natural. Um yeah or sometimes you never solve the problem because you never attune and and you and you give up. The the feeling I get is never so much eureka but it's always oh how come I missed this early?
因为有这个困难。你必须绕过这个坑洼。你必须先做这一步，因为你知道再过五行你就会需要这个假设来解决这个问题。你变得对这个问题如此合拍，一切都变得自然。是的，或者有时你永远解决不了这个问题，因为你从未合拍，于是你放弃。我得到的感觉从来不是那么多尤里卡，而总是：我怎么这么早就错过了这个？
Terence Tao
I was so stupid. Um but often it it's it's the constant experimentation and and and failure that really primes you to find uh to accept the right solution. There's this dramatic contrast between the standards we assign to outcomes and the standards we assign to process. So for outcomes, mathematics very famously has a very high standard of correctness. You know, for a given problem, there's a correct answer and lots of incorrect answers. And and when we grade our um students homework in mathematics, you know, if you didn't if you get your sign wrong or whatever and you got the wrong answer, you got all these all these negative marks, you get criticized if you make math mistakes in your final answer. One consequence of that is that many students who go through, you know, say a high school level of mathematics become very averse to making any mistakes whatsoever in um when they try to approach a math problem. Paradoxically, um the process of arriving at the answer is is almost the complete opposite where you almost
我真蠢。但常常正是持续的实验和失败，真正让你准备好去发现、去接受正确的解。我们对结果设定的标准与对过程设定的标准之间，有一种戏剧性的反差。对于结果，数学非常著名地有非常高的正确性标准。对一个给定问题，有一个正确答案和许多不正确的答案。当我们批改学生的数学作业时，如果你把符号弄错了之类，得到了错误答案，你就会得到所有这些负分，如果你在最终答案中犯数学错误，你就会被批评。其后果之一是，许多经过比如说高中水平数学的学生，在试图处理数学问题时，变得非常厌恶犯任何错误。吊诡的是，到达答案的过程几乎完全相反：你几乎。
Terence Tao
have to make mistakes over and over again and you have to try the stupid things first to appreciate why the clever things work. There's a quote by Niels Bohr who's a physicist who said that an expert is someone who has made all the mistakes that can be made in a very narrow field. You don't publish these mistakes. you execute these mistakes in your process in order to locate the correct answer but only after exploring a lot of incorrect answers first. It's it's very important I think to to normalize uh failure in the process to disclose that behind every successful solution to a problem there are dozens of of of incorrect attempts and it's not because the people trying these problems were were stupid but this is often just part of of the learning process. Chapter 3. How AI is changing math and science forever.
必须一遍又一遍地犯错，你必须先试那些愚蠢的事情，才能体会为什么巧妙的事情有效。物理学家 Niels Bohr 有一句名言：专家是在一个非常狭窄的领域里犯过所有可能犯的错误的人。你并不发表这些错误。你在过程中执行这些错误，以便定位正确答案，但只有在先探索大量不正确的答案之后。我认为非常重要的是把过程中的失败正常化，公开这一点：在每一个成功的问题解答背后，都有几十次不正确的尝试，而这不是因为尝试这些问题的人愚蠢，而这常常只是学习过程的一部分。第三章。人工智能如何永远改变数学与科学。
Terence Tao
Science and mathematics has uh changed a lot over the centuries. Traditionally in science uh the two major paradigms were theory and experiment. um like you would you would create a theory like you know Kepler might create a theory of how planets move or or Newton might create a theory of gravity and then there's this experimental um data that you would run an experiment and and see what happens and then you try to see if the theory and the experiment fit. Math was a little different in that it was almost entirely theory. Um there's there are very very few experiments that you would do purely in mathematics. There were a few for example Gauss famously computed the first 100,000 prime numbers and that was a data set that he used to make predictions. he predicted what we now call the prime number theorem. But science and experiment were the two major forms of science. Then later on simulation came along um that you didn't have to run a a big expensive experiment. Sometimes you could just simulate uh let's say um a hurricane in
科学和数学几个世纪以来已经改变了很多。传统上，在科学中，两大范式是理论与实验。比如你会创造一个理论，Kepler 也许会创造一个行星如何运动的理论，或者 Newton 也许会创造一个引力理论，然后有实验数据：你会做实验，看会发生什么，然后试图看理论与实验是否吻合。数学有点不同，因为它几乎完全是理论。在纯数学中你会做的实验非常非常少。有一些，例如 Gauss 著名地计算了前 10 万个素数，那是他用来做预测的数据集，他预测了我们现在所谓的素数定理。但理论和实验是科学的两种主要形式。后来模拟出现了：你不必做一项又大又贵的实验。有时你只需在超级计算机里模拟一场飓风，而不是在现实生活中。
Terence Tao
in a supercomputer instead of in real life. And then later on big data came along that um rather than than just do a small number of experiments to try to confirm or deny um a specific theory, you could take megabytes or petabytes of data and try to discern patterns, try to extract out laws from just massive data sets. Um and that's a more emerging type of science. But now all these modes of science are being transformed because we now also have AI to to to help us. So so in the past every one of these ways of doing science had to be done by human scientists. You know you had to have someone to perform the experiments or someone to do the theoretical calculations or run the simulations um or go through the data and you could use computers for some of that but you have to but even then you have to program the um the data analysis tool or whatever and you still need a lot of expertise. You know, we have automated labs that can that can perform experiments
再后来大数据出现了：与其只做少量实验来试图确认或否定某个特定理论，你可以取兆字节或拍字节的数据，试图辨别模式，试图仅从海量数据集中提取出定律。那是一种更新兴的科学类型。但现在所有这些科学模式都在被转变，因为我们现在还有人工智能来帮助我们。所以在过去，每一种做科学的方式都必须由人类科学家来做。你必须有人来做实验，或有人来做理论计算，或运行模拟，或梳理数据，你可以用计算机做其中一些，但即使那样，你也必须编程数据分析工具之类，你仍然需要大量专业知识。我们有自动化实验室，可以自动做实验。
Terence Tao
automatically. Um, you can you can get a coding agent to run a simulation for you and you can try to to also run automated data analysis. Um, and increasingly you can also do automated theory. You can take some mathematical problem and ask what are the consequences of these hypotheses and these axioms. What what conclusion should you get? These tools can now be done at scale um much faster. He could potentially run many more theoretical analyses than than any one human scientist could.
你可以让一个编码智能体为你运行模拟，你也可以尝试运行自动化数据分析。而且越来越多地，你也可以做自动化理论。你可以拿某个数学问题，问这些假设和这些公理的后果是什么，你应当得到什么结论。这些工具现在可以大规模、快得多地完成。原则上可以运行比任何一位人类科学家所能做的多得多的理论分析。
Terence Tao
On the other hand, this is not the only thing that we want. There's value in doing things a slow way, you know. So, a scientist who spending hours and hours working things out on pen and paper uh doing the experiments in the field with with their bare hands and actually debugging the simulations that show up, they often learn a lot of of extra insight beyond just getting the answer that they're trying to seek. They can find they can discover new phenomena. They can see connections. They can see similarities to some some previous thing that's been studied elsewhere in the literature and they can communicate what what they uh what they are finding to other people. So there is this paradox that on the one hand AI are becoming more powerful and more capable um and and making fewer mistakes and they are ostensibly achieving a lot of the goals that we want we think scientists are trying to to do. They're they're running experiments. They're analyzing data. They're writing papers. But it may be that it comes at the cost of the AI becomes
另一方面，这并不是我们想要的唯一事情。以缓慢的方式做事是有价值的。一位科学家花许多小时用纸笔把事情推演出来，在现场亲手做实验，实际调试出现的模拟，他们常常学到大量额外洞见，超出他们试图寻求的那个答案。他们可以发现新现象，可以看到联系，可以看到与文献中别处已经研究过的某件先前之事的相似之处，他们可以把他们正在发现的东西传达给其他人。所以有这样一种悖论：一方面，人工智能正变得更强大、更有能力，犯更少的错误，它们表面上正在实现我们认为科学家试图做的许多目标。它们在做实验，在分析数据，在写论文。但代价可能是：人工智能变得
Terence Tao
picks up some skill, but but no human scientist gets any better at doing the science. No human can communicate exactly what just happened and and and why this this scientific discovery is interesting, why this proof is new and and and what features it has and how it connects. We may have to sort of redesign our um conception of what science is and what we actually want out of science. What exactly is science for? And and what are we trying to do? And is there a danger that we are optimizing the wrong thing when we are pointing our AI tools at science? One analogy I've given in the past is that science is a little bit like going on a hike. You've heard there's some interesting waterfall, some beautiful waterfall out there. So you decide to to to to go hike with some friends to to uh to find it, but you need to make a map. you got lost. You get lost a little bit, but maybe while getting lost, you discover something else which is interesting and you make a note of it. Um, on the way to to this waterfall, you find an even more spectacular waterfall in the distance. You
掌握了某种技能，但没有人类科学家在做科学上变得更好。没有人能确切传达刚才发生了什么，以及为什么这个科学发现有趣，为什么这个证明是新的，它有什么特征，它如何关联。我们可能必须重新设计我们对科学是什么、我们实际上想从科学中得到什么的构想。科学究竟是为了什么？我们试图做什么？当我们把人工智能工具指向科学时，是否有危险我们在优化错误的东西？我过去给出过一个类比：科学有点像去徒步。你听说外面有某个有趣的瀑布，某个美丽的瀑布。于是你决定和一些朋友去徒步找到它，但你需要做一张地图。你迷路了。你有点迷路，但也许在迷路的过程中，你发现了别的有趣的东西，你把它记下来。在去这个瀑布的路上，你发现远处有一个更壮观的瀑布。你
Terence Tao
can't get there yet, but maybe some future hiker will figure out a way to get there, too. And so, there's there's a whole process to get to your goal, which um is also very valuable. But these tools, these AI tools, they can be like helicopters that will just fly you directly to this waterfall and you can see it and then you fly back, but you learn nothing about how to get there. You you you may not see any other interesting phenomena than the specific thing that you asked for. And so even though technically you achieve your goal much more efficiently,there may be something that that is lost. Modern AIs are powered by a a type of algorithm known as machine learning, which is trying to predict patterns in data. So a very simple example of machine learning is regression. So if you have some inputs and you some outputs like let's say you observe that if you feed um some animal more food they get they get bigger right? So you can plot how much food you give various animals and and you plot their weight or something
还到不了那里，但也许未来的徒步者会找出办法也到达那里。所以到达你目标的整个过程本身也非常有价值。但这些工具，这些人工智能工具，它们可以像直升机一样，直接把你飞到这个瀑布，你可以看见它，然后飞回去，但你对如何到达那里一无所知。你可能看不到任何其他有趣的现象，除了你所要求的那件特定的事。所以即使从技术上讲你更高效地实现了目标，也可能有某种东西失去了。现代人工智能由一种叫做机器学习的算法驱动，它试图预测数据中的模式。机器学习一个非常简单的例子是回归。如果你有一些输入和一些输出，比如说你观察到，如果你给某只动物更多食物，它们会变得更大。所以你可以画出你给各种动物多少食物，画出它们的体重之类，
Terence Tao
and you get some dots on on a graph and if you're lucky they will they will fit some line and then um and that line becomes your prediction. So then if you give this dogs this sort this much food you they would gain this much weight. Now in the real world um you don't always get these nice linear relationships. Often there are many many inputs and there's many many outputs and the relationship can be can be really complicated. But sometimes the data has a shape and and um there we now have all kinds of clever ways to kind of detect this shape and try to fit curves to to these input output pairs. What uh large language models which power chat bots and things like that they're just playing the game of naming the next word in a sentence. Roughly speaking, if I say roses are red, violets are blank. What is the next word to fill the sentence? You can probably guess the answer is blue. That's an output. You can imagine this giant plot where the inputs are all these
你在图上得到一些点，如果你幸运，它们会拟合某条线，然后那条线就成为你的预测。于是如果你给这些狗这么多食物，它们会增加这么多体重。现在在现实世界中，你并不总是得到这些漂亮的线性关系。常常有许多许多输入，有许多许多输出，关系可以非常复杂。但有时数据有一种形状，我们现在有各种各样巧妙的方法来检测这种形状，并试图为这些输入输出对拟合曲线。为聊天机器人之类提供动力的大语言模型，它们只是在玩给句子里下一个词命名的游戏。粗略地说，如果我说玫瑰是红的，紫罗兰是空白。填这个句子的下一个词是什么？你大概能猜到答案是蓝的。那是一个输出。你可以想象这个巨大的图，输入是所有这些
Terence Tao
incomplete sentences and the outputs are the words you want to complete and you got all these dots in this high dimensional space and you want to fit some curve to it that will um try to explain what is the most likely word to come out. Sometimes there's more than one answer. Hello, my name is you know there could be many many names you could put after the end of the sentence. So you don't always get a single answer but you could try to get the most plausible answer. People have tried this and and you know the the autocomplete feature on your phone does this you know like you you text something and it will suggest the next word and sometimes it's kind of right sometimes it's silly. Once you have any kind of operation like this it creates some dynamics and you can just keep you know many people have just played on their phone just press autocomplete over and over again and but you get these these gibberish sentences. Okay. Um you get monkeys typing on typewriters. But the uh the magic of LLMs is that if you um train these LLMs on enough data, okay, so trillions and
不完整的句子，输出是你想补全的词，你在这个高维空间里有所有这些点，你想拟合某条曲线，试图解释最可能出现的词是什么。有时答案不止一个。“你好，我的名字是”——句子末尾你可以放许多许多名字。所以你并不总是得到单一答案，但你可以试图得到最合理的答案。人们试过这个，你手机上的自动补全功能就是这样做的：你发短信，它会建议下一个词，有时有点对，有时很可笑。一旦你有了任何这类运算，它就会产生某种动力学，你可以一直做下去：许多人只是在手机上玩，一遍又一遍地按自动补全，但你会得到这些胡言乱语的句子。好。你得到在打字机上打字的猴子。但大语言模型的魔力在于：如果你在足够多的数据上训练这些大语言模型，好，数万亿
Terence Tao
trillions of data points and you you um you really try to fit as good a curve as possible and this takes like millions and millions of dollars of of computing power and months and months of of time. Then suddenly uh even when you iterate, it stays coherent. it begins to sound not like monkeys, but it actually sounds like a human speaking. And somehow we don't fully understand why that's the case. But what seems to be true is that language like like English or other natural languages contains a lot of hidden patterns that we're not consciously aware of. I mean, we know some of the laws of English, you know, there's laws of grammar and things, but there are there are sort of unspoken unwritten rules of of language that humans pick up. you know, a a human child, even though they're not taught, you know, what a noun is, what a verb is or whatever, they can they can pick up what order uh English words go in, just by continual exposure to the language, it
万亿个数据点，你真正试图拟合尽可能好的曲线，而这需要数百万美元的算力和数月的时间，然后突然，即使你迭代，它仍然保持连贯。它开始听起来不像猴子，而实际上听起来像人在说话。不知怎么，我们并不完全理解为什么会是这样。但似乎为真的是：语言，比如英语或其他自然语言，包含许多我们并未有意识觉察到的隐藏模式。我的意思是，我们知道一些英语的定律，有语法定律之类，但还有某种未说出口、未写下来的语言规则，是人类掌握的。一个人类孩子，即使没有人教他们什么是名词、什么是动词之类，他们也可以仅通过持续接触语言，掌握英语词按什么顺序排列。
Terence Tao
seems like you can teach these models to also pick up patterns in language to the point where you can give them math questions. The answer to 2 plus 3 is, and they will they will say five. They have been trained to to get the the correct answer to to at least simple math questions. Once you have a little bit of ability to to speak English, you can kind of go in loops and and sort of check your work and make fewer mistakes and you can prompt these models to to to proceed step by step and and and not say something unless it's been double checked and so forth. And so they become a little bit smarter, quote unquote, to the point where they can solve many, many complicated tasks, but they're still just guessing the next word to say. It's not really grounded in any deep understanding of the real world. It's just that they they just have seen the patterns in in the English language or other language that they've absorbed so well that they can mimic people who are speaking uh in
似乎你也可以教这些模型掌握语言中的模式，达到你可以给他们数学问题的程度。“2 加 3 的答案是”，他们会说五。他们已经被训练成至少对简单数学问题给出正确答案。一旦你有了一点说英语的能力，你就可以有点循环，检查你的工作，少犯错误，你可以提示这些模型一步一步进行，除非经过双重检查，否则不说某件事，等等。于是它们变得稍微更聪明一点，加引号的聪明，达到它们可以解决许多许多复杂任务的程度，但它们仍然只是在猜测下一个要说的词。它并没有真正扎根于对现实世界的任何深刻理解。只是它们把英语或其他它们所吸收的语言中的模式看得太好了，以至于它们可以模仿那些以智能方式说话的人，
Terence Tao
intelligent fashion and they can present as being intelligent long enough that they can fool us but long enough they can actually do useful things. So you know we can now solve certain math problems by asking the LLM to provide a proof and sometimes the proof is complete rubbish but if you loop it enough and you have enough checks um you can actually uh start having a positive success rate. Um so it's it's a very strange way of solving problems like it is it is completely orthogonal to the way we normally think of intelligence as being very grounded methodical thinking first principles. You know, it's like having someone who knows a lot and is but is slightly drunk and is sort of throwing out ideas, but with enough guidance, uh, you you can actually extract useful output. It's not the most advanced mathematics out there actually. Um, but you give it a lot of data and a lot of time and a lot of other band-aids and things and it actually works pretty well. I
它们可以足够长时间地表现为智能，以至于可以愚弄我们，但也足够长时间地实际做有用的事情。所以我们现在可以通过让大语言模型提供一个证明来解决某些数学问题，有时证明完全是垃圾，但如果你足够循环，有足够的检查，你实际上可以开始有一个正的成功率。所以这是一种非常奇怪的解决问题的方式：它与我们通常所想的智能完全正交，。智能是非常扎根的、有条理的思考、第一原理。这就像有一个知道很多、但有点醉、在抛出想法的人，但有足够的引导，你实际上可以提取有用的输出。它实际上并不是那里最先进的数学。但你给它大量数据、大量时间，以及大量其他权宜之计之类，它实际上工作得相当好。我
Terence Tao
find that some of the debate on AI's uh role in in science and other disciplines is we often default to a one dimensional view of thinking like this. there's there's easy tasks and and and hard tasks and very hard tasks and and and humans are can can do tasks up to a certain level and AIs can do task to a certain level and so which one is which one is better right that's kind of a one dimensional way of thinking but what I found uh when when sort of working with AIs and and comparing their way of solving problems to humans way of solving problems is that they are really quite complementary human experts um uh will will focus on depth you know so like a human mathematician which will solve thousands and thousands of problems they can work on but they will pick one or two problems that they think are are difficult but not so difficult that they're impossible but they're difficult enough that the the exercise of trying to make a bit of progress towards them will reveal all kinds of insights that they can share and maybe their students or some other collaborators or or
发现，关于人工智能在科学和其他学科中角色的一些辩论，我们常常默认成一种一维的思考方式：有简单任务、困难任务和非常困难的任务，人类可以做到某个水平的任务，人工智能可以做到某个水平的任务，那么哪一个更好？那是一种一维的思考方式。但我发现，当与人工智能一起工作、比较它们解决问题的方式与人类解决问题的方式时，。它们实际上相当互补。人类专家会聚焦于深度。比如一位人类数学家会解决成千上万个他们能处理的问题，但他们会挑选一两个他们认为困难、但还没有难到不可能的问题，但又足够困难，以至于试图朝它们推进一点的练习会揭示各种各样的洞见，他们可以分享，也许他们的学生或其他合作者或其他人可以建立在他们所做的之上。当我们把人工智能指向真正困难的问题，那里没有任何标准技术适用，。
Terence Tao
other people can build upon what they do when we point the AIs at really difficult problems where none of the standard techniques apply they are still very very bad at I mean they're just randomly guessing but they excel at breadth. So if you if you point them at a thousand problems um of various difficulties now some may be just too hard but there will be some which actually are within reach of existing methods and there's some method out there in the literature which will solve your problem or maybe you have to combine together two separate methods and it's just that there there's just not enough human experts to look at all these problems and the human experts that do look at these problems they may not realize that there was this obscure paper from a journal um in 1970 that actually has the key idea that will solve this problem. They don't have the patience or the time to sort of go through all the different combinations of how which technique might work on which problem. But the AIs, you know, they will somewhat randomly
它们仍然非常非常差，我的意思是它们只是在随机猜测，但它们擅长广度。所以如果你把它们指向一千个难度各异的问题，现在有些可能只是太难了，但会有一些实际上在现有方法的触及范围内，文献中有某种方法可以解决你的问题，或者也许你必须把两种分开的方法结合起来，只是没有足够的人类专家去看所有这些问题，而那些确实去看这些问题的人类专家，可能意识不到有一篇 1970 年某期刊上的冷僻论文实际上有解决这个问题的关键想法。他们没有耐心或时间去过所有不同组合：哪种技术可能对哪个问题有效。但人工智能会有点随机地，会对什么技术可能对一个问题有效做出有根据的猜测，其中一些会是愚蠢的，但其中一些可能有效，通过所有这些组合，我们发现有时它们可以抓住所有其他人类都错过的一个解。偶尔，专家们的共识、常规智慧是错的。你知道，我们都认为一个问题有一个肯定的答案，但实际上有一个否定的答案，我们只是没有太多看否定情形，因为我们以为每个人。
Terence Tao
they will take sort of educated guesses as to what techniques might work for a problem and some of them will be stupid and but some of them might work and through all these combinations we're finding that sometimes they can catch they can catch a solution that that the rest of all the humans have missed. Occasionally the consensus the conventional wisdom on of of the experts is wrong. You know that we all think that that a problem has a positive answer but actually the has a negative answer and we just didn't look at the negative case too much because we thought everyone thought that the the answer was true but an AI may not have that preconception. So uh sometimes the AI just serves as an independent pair of eyes. And so some problems that we thought were very difficult had a surprisingly simple solution which in retrospect we should have as maybe we should have gotten ourselves too. They're beginning to become successful at when you point them at a very broad range of problems and they solve some percentage of them. Like maybe you point them at a thousand problems and they solve 5% of those problems. That's still 50 problems solved. you can
都认为答案为真，但人工智能可能没有那种先入之见。所以有时人工智能只是充当一双独立的眼睛。于是有些我们以为非常困难的问题，有一个出人意料的简单解，事后看来我们也许自己也应当得到。它们开始在你把它们指向非常广的问题范围、它们解决其中某个百分比时变得成功。比如也许你把它们指向一千个问题，它们解决其中 5%。那仍然是 50 个问题解决了。你已经可以有工具在某种意义上以解决的问题原始数量超过人类数学家。现在，被解决的那 50 个问题可能不是你最想解决的 50 个问题。它们可能是 50 个随机问题。但这仍然非常令人印象深刻。我认为作为一门职业，我们将必须找到办法，把这种新能力纳入进来，在广的尺度上解决一些
Terence Tao
already have tools that in some sense outperform humans mathematicians by raw number of problems solved. Um now the 50 problems that get solved may not be the 50 problems that you most want solved. They could be 50 random problems. Um but still it it is it is very impressive. What I think we will have to do as a profession is find ways to um to incorporate this new capability to to solve some problems at broad scales um and somehow figure out how to to to make that mesh with our existing capability to solve a few deep problems very slowly. Kepler's story of how he found his famous laws of motion is a is a fascinating one. It shows how important the process is. Kepler learned of Copernicus' theory of the motion of the planets. And Copernicus had roughly worked out how far the Earth was from the Sun, how far Mars was and so forth. And Kepler noticed that the ratios of these
问题，并设法弄清楚如何让它与我们现有的、非常缓慢地解决少数深刻问题的能力啮合。Kepler 发现他著名运动定律的故事是一个迷人的故事。它表明过程有多么重要。Kepler 得知了 Copernicus 的行星运动理论。Copernicus 大致算出了地球离太阳有多远，火星有多远等等。Kepler 注意到这些轨道的比看起来有点像几何中向他显示的某些比。于是最终他提出：实际上，如果你取球体，每个行星一个球体，当时他知道六颗行星，他可以在这六个球体之间内接五个正多面体，比如十二面体、立方体、四面体等等，他认为会得到完美拟合，这就用五个正多面体解释了太阳系的形状。
Terence Tao
um orbits looked a little bit like the like certain ratios that showed him in geometry. And so eventually he proposed that actually if you take spheres one sphere for every planet and he had six planets known at the time that he could inscribe five platonic solids you like a dodecahedron and a cube and a tetrahedron and so forth between these six spheres and he thought it would get a perfect fit and this explained the shape of the solar system in terms of the five platonic solids.
这是他优美的几何想法。直到他设法拿到 Tycho Brahe 一些真正高质量的观测数据——他实际上必须争取，甚至可能还得偷——他试图把他的理论拟合到这些数据上，他发现它实际上并不完全拟合：以 Tycho 数据所提供的精度，他无法让这些球体完全贴合，事实上他从那个过程中发现，火星。
Terence Tao
This was his beautiful geometric idea. It was only after he managed to get his hands on some really high quality um observational data of Tycho Brahe which he had to fight for actually and possibly even steal and he tried to fit it his theory to this this data and he found that it didn't actually quite fit that with the precision that Tycho's data um offered he could not quite get these spheres to fit and in fact he discovered from that process that the the orbit of Mars and Earth could not be circles at all that there had to be some other shape. He spent many years um figuring out what to do. I think I I don't know how long he held on to this this theory of of the platonic solids and you can see in his writings he tried many other things. He tried to to make the circles off center and at some point he landed on the ellipse and then suddenly everything fit. It does show that there is an interplay between theory and experiment. You know that that you
和地球的轨道根本不可能是圆，必须是某种其他形状。他花了许多年弄清楚该做什么。我想我不知道他抓住这个正多面体理论有多久，你可以在他的著作中看到他试了许多其他事情。他试图让圆偏离中心，到某个时候他落到了椭圆上，然后突然一切都拟合了。这确实表明理论与实验之间有一种相互作用。你知道，你可以提出一个理论，如果它不拟合数据，它可能就不是一个好理论。但这也比那更复杂。在 Kepler 之前，对 Copernicus 理论的批评之一是：Copernicus 自己已经承认，他的测量比当时能得到的最佳预测更差。所以当时最好的模型是地心模型，由希腊人发展，然后由阿拉伯人和印度人发展。有许多许多调整和微调，
Terence Tao
can pose a theory if it doesn't fit the data it's it it may not be a good theory. But um it's it's more complicated than that too. Before Kepler, one of the criticisms of Copernicus's theory was that already Copernicus acknowledged that that his measurements were worse than the best predictions available at the time. So the best models were the geocentric models which had been developed by the Greeks and then by the Arabs and Indians. There were many many adjustments and fine-tuning and they had a very very precise model that could predict in a very complicated way where all the planets would be. uh and Kepler Copernicus's model was worse. Just knowing agreement of data is not necessarily um the the only metric. It was only after Kepler found his his revised model where the orbits were not circles but ellipses that the heliocentric model became more accurate than the geocentric model. What this tells you is that is that science is um you can't always get instant
他们有一个非常非常精确的模型，能以非常复杂的方式预测所有行星会在哪里。而 Kepler、Copernicus 的模型更差。仅仅知道与数据吻合，并不一定是唯一的度量。直到 Kepler 找到他的修正模型，轨道不是圆而是椭圆，日心模型才变得比地心模型更准确。这说明的是：科学是，你并不能总是立刻得到关于你是否已经解决一个科学问题的反馈。如果 Kepler 和 Copernicus 有人工智能，他们让人工智能预测一个宇宙模型，有可能生成正确日心模型的那些人工智能会被丢掉，因为起初它们的预测不如地心模型那么好。真正消化所有这些理论，看它们如何与我们关于行星、运动、引力以及。
Terence Tao
feedback as to whether you've solved a scientific problem or not. If Kepler and and Copernicus had AIs and they asked them to predict a model for for for the universe, it could be that the AIs that generated the correct heliocentric model would be discarded because initially their predictions were not as good as as as the geocentric ones. It takes time to really digest all these theories and see how they fit with everything else that we know about planets and motion and gravity and everything. One concern actually is that AI are too fast. there's a danger that these AIs will do what's called overfitting and and create a very complicated model which is not which has nothing to do with what's actually going on but just fits your data extremely extremely well um but it doesn't extrapolate beyond that that that data set how we incorporate AI into the scientific discovery process will be a challenge it can certainly accelerate individual steps of of the process you know you can make experimentation faster um you can you can write code faster you
一切其他已知之事相拟合，是需要时间的。一个实际的担忧是：人工智能太快了。有一种危险是，这些人工智能会做所谓的过拟合，创造出一个非常复杂的模型，它与实际发生的事情毫无关系，只是把你的数据拟合得极其极其好，但它并不能外推到那个数据集之外。我们如何把人工智能纳入科学发现过程，将是一个挑战。它当然可以加速过程中的个别步骤：你可以让实验更快，你可以写代码更快，你可以写论文更快，但科学作为整体并不一定仅仅因为每一个组成部分都变快了就会加速。有一种危险是，当我们把人工智能指向科学时，我们会优化错误的东西，我们会在纸面上得到所有这些惊人的成功，却发现科学并没有以它过去的方式实际推进。但我们会发现，有这些工具仍然比没有它们更好。我们仍在学习
Terence Tao
can write your papers faster but science as a whole may not necessarily accelerate just because every single component gets faster. There's a danger that that we will optimize uh the wrong thing when we when we point AI at science and we will on paper get all these amazing successes and but find out that science is not actually advancing as it in the way that it used to. But we will find out it's still better to have these tools than not have them. But we're still learning how to use them most efficiently. Part of what we do is is we solve problems and and we we try to find solutions to problems and and and prove things. Uh and proofs go through a certain life cycle. First of all, you have to to to generate a proof for solution. And this used to be quite hard, but but some of the proofs that you generate are incorrect. Um so then you have to to verify them, check which one check that is correct. Uh and that also used to be quite tedious. But both of these of these tasks are becoming more and more automated. So we we beginning to see more and more
如何最有效地使用它们。我们所做的一部分是解决问题，我们试图找到问题的解，并证明事情。证明会经历某种生命周期。首先，你必须生成一个解的证明。这曾经相当困难，但你生成的一些证明是不正确的。于是你必须核验它们，检查哪一个、检查它是正确的。那也曾相当繁琐。但这两项任务都正变得越来越自动化。所以我们开始看到越来越多对各种问题的提议解，而且其中许多实际上是正确的，但证明也在变长。当它们由人工智能写出时，它们常常并不好读。一份人工智能生成的证明可能花大量时间谈论非常琐碎的东西，而花很少时间谈论论文中最有趣的部分。我认为这是因为人工智能无法区分什么是难的、什么是困难的，因为靠蛮力，对它们来说一切都花同样多的时间。一个人类自然地必须在论文最困难的那一步上挣扎，
Terence Tao
proposed solutions to various problems and many of them are actually correct but proofs are also getting longer. Uh and when when when they're written by AIs they are often not very pleasant to read. An AI generated proof might might spend a lot of time talking about something very trivial and spend very little time talking about the most interesting portion of of of the paper. I think because the AI can't distinguish sort of what is hard what is difficult because by brute force everything takes the same amount of time for them. A human who has sort of naturally had to struggle at the most difficult step of a paper would naturally spend a lot of time on that step. And so you need to write out the paper in a proof in a way that it reads well and it can be explained to other people. And then other people have to get excited by it. they they they they have to accept it as oh this is really interesting that this this will help me solve my own problems or it really um clarifies why this phenomenon was true that it has to be accepted and this is where we we
就会自然地在那一步上花大量时间。所以你需要以一种读起来顺畅、并能向其他人解释的方式写出证明中的论文。然后其他人必须因此而兴奋，他们必须接受它：哦，这真的很有趣，这会帮助我解决我自己的问题，或者它真正澄清了为什么这个现象为真。它必须被接受，而这就是我们传统上有同行评审过程的地方：我们把论文送给审稿人，如果审稿人对这个结果感到兴奋，论文就会被接受。但你知道，可能有些论文在技术上是正确的，也可读、也没问题，但它们回答的是一个没有人关心的问题。然后最后，它需要被完全打磨，放进教科书，教给学生。而且常常，一个证明的第一版
Terence Tao
traditionally have the peer review process where we we send papers to referees and if the referees are are excited by this result then um the paper gets accepted but you know there could be papers that are technically they are correct and and they are readable and fine but but they could answering a question that no one cares about. And then finally, it needs to be sort of completely polished and put into textbooks and and taught to students. And often the first version of a proof is not suitable for writing on textbooks. It often is is done in a very inefficient way and is not the the ordering of steps is not quite logical. There's a certain digestion process where someone has to spend a lot of time thinking very hard to what is completely the the right way to organize to edit the paper to sort of flow in the same way. a little bit like how you would you would edit a documentary or a movie. And so what we're finding is that AI tools are accelerating the early stages of this process, but not the late stages. We are now generating many proofs.
并不适合写进教科书。它常常以非常低效的方式完成，步骤的排序也不完全合乎逻辑。有某种消化过程：必须有人花大量时间非常努力地思考，什么才是完全正确的组织方式，如何编辑这篇论文，让它以同样的方式流动，有点像你会如何剪辑一部纪录片或一部电影。所以我们发现，人工智能工具正在加速这个过程的早期阶段，而不是后期阶段。我们现在正在生成许多证明。我们正在核验其中一批，但理解它们并把它们放进最终教科书形式的节奏，仍然由人类完成。事实上，我们现在正经历你也许会称为证明消化不良的情况：突然有大量待处理的问题解答，它们应当被理解，应当进入教科书，但我们只是被太多东西淹没了。我们必须挑选，我们必须分诊。而这是某种
Terence Tao
We are verifying a bunch of them, but the pace of understanding them and putting them into the final textbook form is still done by humans. Um, and in fact, we're now experiencing what you might call proof indigestion where suddenly there's lots and lots of pending uh solutions to problems that should be understood and should be go into textbooks, but they just we're just flooded now with with too many of them. Uh, and we have to pick and we have to triage. And this is something that has never had to happen before. Um it used to be that solutions came out so rarely that um if a solution to a major problem got got solved, all the experts would sort of drop everything and read it and and try to digest it as quickly as possible because it was so rare and so valuable that it was worth doing. And now we're just getting flooded with with all these possible solutions. I myself, you know, I've had to stop trying to stay current with all the latest developments in in in in my
以前从未不得不发生的事情。过去，解答出现得如此稀少，以至于如果一个重大问题得到了解决，所有专家都会放下手头的一切去读它，并尽快消化它，因为它如此稀少、如此宝贵，值得这样做。而现在我们只是被所有这些可能的解答淹没。我自己，你知道，我已经不得不停止试图跟上我领域里所有最新进展。有时发生的事情太多了。我现在无法承诺去读出现的每一个进展。我的意思是，这在人工智能之前就已经开始成为一个问题，但人工智能真正加速了正在生成的内容的绝对体量。所以我们将需要好得多的策展和过滤。这是一个好问题。我的意思是，有吃不完的食物，总比没有足够的食物吃要好。
Terence Tao
field. Sometimes there's just so much going on. I I can't promise now to read every single development that that that shows up. I mean, this is already beginning to be a problem before AI, but AI has really accelerated the sheer volume of of content being generated. And so, we're going to need much better curation and uh and filtering. It's a good problem to have. I mean, it's like it's better to have a to have too much food, more food than you can eat than than not enough food to eat. But it is still a problem. AI have become increasingly capable in mathematics. For people like me who have been following the developments for the last three years, um there's been a kind of steady progression, you know, so four years ago they could solve middle school math problems and then they could solve high school math problems and then uh high school Olympiad level problems and then um some problems you like graduate student um level qualifying exam problems and then they're starting to solve a few of the the minor
但它仍然是一个问题。人工智能在数学中已经变得越来越有能力。对像我这样过去三年一直关注这些发展的人来说，一直有一种稳定的进展：四年前它们能解决中学数学问题，然后它们能解决高中数学问题，然后是高中奥林匹克水平的问题，然后是一些像研究生资格考试水平的问题，然后它们开始解决一些较小的。未解问题，也许是像 Paul Erdős 会提出但没有人真正看过的那种。所以有很多低垂的果实。然后就在最近，有一两次场合，它们设法解决了一些人们实际上真的非常努力去解决的问题。不知怎么，人类集体都走错了弯，而有不同偏见集合的人工智能设法拼凑出一个相当
Terence Tao
um unsolved problems that maybe someone like Paul Erdős would have proposed but no one really looked at. So it's a lot of low hanging fruit. And then just recently there's been one or two occasions where they they managed to solve some problems that people actually really did try very hard to solve. Somehow collectively the humans all had were taking the wrong turn and the AIs which had a different set of biases had managed to cobble together um a solution which was quite clever and has been quite um has already had some impact. there's been some nearby problems to uh the unit distance problem for instance which have also been solved by humans who have adapted the uh the AI's technique. I found that quite exciting. Uh so I think for some my colleagues it was very concerning especially if they hadn't been following the previous developments and and not realizing that this was where they were at. If a colleague had only seen say what ChatGPT could do in 2023 and if you asked it a difficult math question then it would give you complete
巧妙的解，并且已经产生了相当大的影响。例如单位距离问题附近的一些问题，也已经被采用了人工智能技术的人类解决了。我发现这相当令人兴奋。所以我认为对我的一些同事来说，这非常令人担忧，尤其是如果他们没有关注先前的发展，没有意识到它们已经到了这个程度。如果一位同事只见过比如说 2023 年 ChatGPT 能做什么，如果你问它一个困难的数学问题，它会给你完全垃圾，而它们现在已经相当不同了。它在根本上仍然是同一项技术，但它们已经找到办法降低错误率，并变得真正有用。现在，这有多可复制仍然不清楚。这些成就中的许多是由私营公司完成的。它们没有披露花费了多少资源来使用。我的意思是，是 10 万美元的算力？还是 100 万美元？我们并不真正知道。而且我们
Terence Tao
rubbish and they they are quite different now. And it's still fundamentally the same technology, but but they have found ways to reduce the error rate and and become genuinely useful. Now, it's still unclear how replicable this is. Many of these achievements um they're done by private companies. They they not disclosing how much resources they spent to to use. I mean, is it was it $100,000 of commute compute? Was it a million dollars? We don't really know. And we don't know their success rate. um was this problem that they solved the only problem that they looked at or did they look at 10 problems? They look at 100 problems. While the results are impressive, um we don't have enough data to to really gauge whether this will become a completely regular occurrence going forward or whether it's only if you spend $100,000 over several months with a team of 10 people that you can get results like this. And maybe it's only 1% of all problems that we
不知道它们的成功率。它们解决的这个问题，是它们看过的唯一问题，还是它们看了 10 个问题？它们看了 100 个问题？尽管结果令人印象深刻，我们没有足够的数据来真正衡量，这今后会不会变成完全常规的事情，还是只有当你花 10 万美元、几个月、一个 10 人团队，才能得到这样的结果。而且也许只有我们关心的所有问题中的 1%适合这种方法。是的，我们不知道，但有人在努力以科学的方式更恰当地做基准测试。这些挑战中最近的一个叫做 FrontierMath 挑战。他们用一组 10 个研究级问题测试最新模型，最好的模型能做这些中等难度数学问题中的大约 5 个或 6 个，这些问题已经有解，但解被保密了。我认为有
Terence Tao
care about are amenable to this method. uh yeah we don't know but there are efforts to more properly benchmark in a scientific way. The most recent of these challenges is called the FrontierMath challenge. They tested the latest models against a test set of 10 research level questions and the best models could do like five or six out of 10 of these sort of medium level difficulty math problems which already had a solution but the solution was kept secret that I think there's a lot of routine tasks tasks that we we do every day as a as a as in our research that some percentage of those can now be done by by AIs. it could be expensive. Uh many of of these tools they require say a couple hundred dollars to run to before they can get a solution and sometimes they fail. They they spend all this compute and and they end up with with nothing useful. We are seeing now in programming that many expert programmers are reporting that their ability to write code
很多常规任务，我们每天在研究中做的那些任务，其中某个百分比现在可以由人工智能来做。它可能很贵。这些工具中的许多需要比如说几百美元才能运行到得到一个解，而且有时它们会失败。它们花掉所有这些算力，最终得不到任何有用的东西。我们现在在编程中看到，许多专家程序员报告说，他们写代码的能力用这些工具提高了 5 倍、10 倍或 100 倍。但他们也感觉自己正在失去亲手编码的能力，有时他们无法审查这些智能体产出的代码。这是一种权衡：速度并不是一切。我非常喜欢数学的合作方面。直到职业生涯相对较晚，我才意识到这有多么重要：我学到的很多数学，是我
Terence Tao
has increased by a factor of five or 10 or 100 with these with these tools. But they are also they can they can feel themselves learn losing the ability to code by hand and sometimes they cannot review the code that that that comes out of these agents. There's a trade-off you know speed and is not everything. I very much like the collaborative aspect of mathematics. I didn't realize was so important until relatively late in my career that uh like a lot of the mathematics I've learned I I learned after grad school by by working with uh with mathematicians and and and scientists in different fields. I teach them what I know, they teach me what what what they know and I become much broader as as as a as a as a result. When you're working with a collaborator that you've been working with a long time, um there's a point where you become almost mentally attuned, perhaps you've been familiar, like if it's a really close friend or family member that you talk to for a long time, sometimes you can complete each other's sentences. You know what the other person's going
在研究生院之后，通过与不同领域的数学家和科学家合作学到的。我教他们我所知道的，他们教我他们所知道的，结果我变得宽广得多。当你与一位已经合作很久的合作者一起工作时，会有一个点，你们几乎在心智上合拍，也许你们已经熟悉，就像如果是一位真正亲近的朋友或家人，你谈了很长时间，有时你们可以补全彼此的句子。你知道对方要说什么。合作时有时也能得到那种感觉。你可以抛出一个想法，还没说完这句话，对方就明白了，并能接着往下跑。人们试图这样使用人工智能，而人工智能，你无法与它们交谈。当然它们会犯错，有时它们会谄媚，只告诉你你想听的，而且现在与这些工具的互动也有点私人化，比如我
Terence Tao
to say? And you can sometimes get that when you collaborate. You can you can throw out an idea and before you even finish the sentence, the other person gets it and and can run with it. People have tried to use AIs like this and AIs they are you can't converse with them. Of course they make mistakes sometimes they they sometimes will be sycophantic and and only tell you what what you want to hear but also the interactions right now with these tools are kind of personal like I've tried to collaborate with people um in person and also have an AI present but breaks up the rhythm. these these tools they don't they're not really conversational like not as fluid conversation as as as they are with human collaborators yet. uh maybe they will get they will get there until recently they don't learn from your conversations you know so with a collaborator when when you you know you can resume the next day you can pick up very quickly and sometimes even for a call
试过与人当面合作，同时也让一个人工智能在场，但它会打断节奏。这些工具并不真正是对话式的，还没有与人类合作者那样流畅的对话。也许它们会到达那里。直到最近，它们并不会从你的对话中学习。所以与一位合作者，你知道，你可以第二天接着来，很快就能接上，有时甚至一次通话你们多年未见，你也可以接上一些非常老的线索。人工智能有一定量的上下文，它们能记住一些事情，它们可以记录，可以做笔记，并某种程度上模拟这种记忆，但你无法以对待真正紧密合作的同样方式去与人工智能合拍，更谈不上更好。我实际上并不怎么把这些工具用于真正的问题解决过程。我迄今发现，这些工具在次要任务上好得多，比如做
Terence Tao
that you haven't met for years you can pick up some very old threads AIs have a certain amount of context and they can remember some things and and they can record they can make notes and kind of simulate this memory but you can't attune uh to an AI the same way that you can to a really close collaboration better yet. I actually don't use these tools so much for the actual um problem solving process. I found to date that these tools are much better at secondary tasks like like doing literature searches or or checking a proof or writing some code, proofreading uh something that I wrote to see if there's any opportunity to make things a little bit tighter. I find, yeah, the rhythm of of working with an AI is not quite the the rhythm I I prefer working with a human collaborator, but that could just be the the current state of current technology. Maybe future AIs will be much more conversational and and much more human to interact with.
文献检索，或检查一个证明，或写一些代码，校对我写的东西，看有没有机会把事情收得更紧一点。我发现，是的，与人工智能一起工作的节奏，并不完全是我更喜欢与人类合作者一起工作的节奏，但那可能只是当前技术的当前状态。也许未来的人工智能会更对话式，与之互动会更像人。所以我们现在处于科学事业整体资助结构上一个有些危险的点，因为一方面这些工具让我们能以快得多的速度创造科学的产出，或看似科学的产出，但它可能以牺牲培育下一代科学家的种子为代价。例如，有一种真正的担忧：我们给我们研究生去做的训练问题，。
Terence Tao
So we're at a somewhat risky point in in sort of the uh the structure of of funding the scientific enterprise in general because on the one hand these tools are allowing us to create the outputs of science or what seems to be the outputs of science at at a much accelerated rate but it could come at the cost of of nurturing our seed corn for the next generation of scientists. For example, there there is a real concern that um the training problems that we give our graduate students to work on to as as their as their first projects to to get a little bit of of of recognition and and career training and experience. These are the types of problems which now AIs can um they can replicate many of the papers. But if you replace the grad students by by these AIs, you know, the AIs will generate these grad student level papers, but then we won't get the next generation of of of students. But if we if we don't continue this process of digesting,
作为他们的第一批项目，让他们获得一点认可、职业训练和经验。这些正是现在人工智能可以复制许多论文的那类问题。但如果你用这些人工智能替换研究生，你知道，人工智能会生成这些研究生水平的论文，然后我们就不会得到下一代学生。但如果我们不继续这个消化过程，把所有这些人工智能产出消化掉，为下一代人类和人工智能建立起下一层知识基础，我们作为科学社会可能最终会停滞。也就是说，我们将能够优化我们用当前技术能做的一切，但我们可能不再真正发展真正原创的新想法。我们将需要真正更开放地讨论基础科学是什么，它对什么有用，。
Terence Tao
you know, all this AI output to build the next base of knowledge for the next generation of humans and AIs to to build upon that, we may end up stagnating as as as a a scientific society. You know, that we we'll be able to optimize everything that we can we can do with our current technology, but we may not actually develop really original new ideas anymore. we will need to to really um have a much more open discussion about what basic science is and what is useful for and why it's important to still have curiosity driven research. why we still need a community of of of humans to explore things sometimes slowly, sometimes you know in in in ways that are not as efficient as the latest model AIs and also the um the insights that that we that we we we gain. I think um we should share them more and we should do more outreach to to the general public. I think the general public today, you know, they can see the visible outputs of science. you know,
以及为什么仍然需要好奇心驱动的研究。为什么我们仍然需要一个人类共同体，有时缓慢地、有时以不如最新模型人工智能那么高效的方式去探索事物，以及我们所获得的那些洞见。我认为我们应当更多地分享它们，我们应当对公众做更多外展。我认为今天的公众可以看到科学的可见产出。你知道，他们有手机，他们有互联网，他们有 GPS 之类。许多人看不到整个过程，以及对数与科学的基本理解如何实际上让他们周围的世界少了很多可怕，也清晰了很多。我认为现在很多人只是生活在一种焦虑状态中。世界如此复杂。我们没有像强调硬的
Terence Tao
they have a cell phone, they have the internet, they they they have GPS or whatever. Many people, they they don't see the um the whole process and and how a basic understanding of math and science actually makes the world around them a lot less scary and just a a lot a lot clearer. I think a lot of people now are just living in in a state of anxiety. The world is so complicated. We haven't emphasized these s of softer values of science as much as sort of the hard you know like um technological outputs and things but science does add a certain amount of clarity to to to one's thinking and you know these are valuable things and they need to be supported.
技术产出之类那样，同样强调科学这些更软的价值，但科学确实会给一个人的思考增加一定量的清晰。你知道，这些是有价值的东西，它们需要得到支持。技术产出之类那样，同样强调科学这些更软的价值，但科学确实会给一个人的思考增加一定量的清晰。你知道，这些是有价值的东西，它们需要得到支持。