I would even say that the future is for quantum mathematicians.

我甚至想說，未來是量子數學家的天下。

Mathematics is also what mathematicians do, and we do this with our brain, and so how is our brain created?

數學也是數學家所做的，我們用我們的大腦來做這件事，那麼我們的大腦是如何產生的呢？

Well, many many million years of evolution. And so I think our concepts

嗯，很多很多年的進化。所以我認為我們的概念

are very much influenced by the things to see around us.

在很大程度上受到我們周圍事物的影響。

So we talk about space because we kind of move in space,

所以我們談論空間，因為我們在空間中移動。

We move in time. We count because we see objects. We throw stones,

我們在時間中移動。我們計數是因為我們看到了物體。我們扔石頭。

we want to describe the path of a stone and its velocity.

我們想描述一個石頭的路徑和它的速度。

And so calculus, for instance, is based on the idea of speed or slope.

是以，舉例來說，微積分是基於速度或斜率的概念。

So I think actually our mind is very much kind of set by everyday experience.

是以，我認為實際上我們的思想在很大程度上是由日常經驗設定的。

So you might think that mathematics kind of stops being effective if we move into a new territory.

所以你可能認為，如果我們進入一個新的領域，數學就不再有效了。

For instance, the very large, the universe, or the very small, the world of elementary particles.

例如，非常大的宇宙，或非常小的基本粒子的世界。

But a remarkable thing: this hasn't happened. So mathematics has been

但有一件了不起的事情：這並沒有發生。所以數學一直是

extremely successful in conquering, certainly I think, the physical experience.

在征服方面極為成功，當然，我認為，物理經驗。

General relativity,

廣義相對論。

Einstein's theory is a good example.

愛因斯坦的理論就是一個很好的例子。

He needed a concept of curved space. And it turns out that Riemann and others,

他需要一個彎曲空間的概念。而事實證明，黎曼和其他人。

in the 19th century, had already thought about it. They thought, you have two dimensions, three dimensions,

在19世紀，已經想到了這個問題。他們認為，你有兩個維度，三個維度。

why not have an arbitrary number of dimensions?

為什麼不能有一個任意數量的維度？

Abstract thinking of mathematicians was kind of ahead. In quantum theory,

數學家的抽象思維有點超前。在量子理論方面。

you know, even if you sit and meditate for a million years you wouldn't come up with the laws of quantum

你知道，即使你坐下來冥想一百萬年，你也不會想出量子定律的。

mechanics because it's so bizarre. So in that sense nature had to force that upon us,

機械學，因為它是如此離奇。所以從這個意義上說，大自然不得不把它強加給我們。

and it was like a very painful process. We had to let go of many very cherished concepts.

而這就像是一個非常痛苦的過程。我們不得不放棄許多非常珍惜的概念。

For instance, the fact that the world is a predictable system. Quantum mechanics is just the way nature works.

例如，世界是一個可預測的系統這一事實。量子力學只是自然界的運作方式。

You see it very clearly at the level of atoms and elementary particles,

你在原子和基本粒子的層面上非常清楚地看到它。

and if you take many many atoms together, then in some sense many of the features of quantum mechanics

而如果你把許多許多原子放在一起，那麼在某種意義上，量子力學的許多特徵

are kind of washed away. We don't observe it because we are made out of like 10 to the 24 molecules typically.

被洗掉了。我們沒有觀察到它，因為我們通常是由10到24個分子組成的。

There are several things that you have to give up in quantum mechanics. The first is that you cannot know the answer to all questions.

在量子力學中，有幾件事是你必須放棄的。首先是你不能知道所有問題的答案。

So, for instance, typically, I would say, well, this is pen, I know where it is, and if I drop it,

是以，例如，通常情況下，我會說，嗯，這是筆，我知道它在哪裡，如果我掉了。

I know what the velocity is; that it's moving. Quantum mechanics says that's all fine,

我知道速度是什麼；它在移動。量子力學說這一切都很好。

you can know either the position or the velocity, so roughly half the questions are unanswerable.

你可以知道位置或速度，所以大約有一半的問題是無法回答的。

That's really bizarre, because how would I then describe a pen?

這真的很奇怪，因為這樣我怎麼會描述一支筆呢？

But there's a second thing, which is that even if you want to describe

但還有第二件事，那就是即使你想描述

how things move, they never have a specific answer. So, like if you are a

事物如何運動，他們從來沒有一個具體的答案。是以，像如果你是一個

classical object like a pen, now you would say "How do I go from A to B?"

像筆這樣的經典物體，現在你會說 "我如何從A到B？"

Well there will be a shortest path. If I throw a stone it will go like this from A to B.

那麼會有一條最短的路徑。如果我扔一塊石頭，它會像這樣從A到B。

And I will know exactly how it went,

而且我將清楚地知道它是如何進行的。

and this is like the optimal path, and I can calculate this, and mathematics is about this. Now, if you're a

而這就像是最優路徑，我可以計算出這個，數學就是關於這個。現在，如果你是一個

quantum mechanical particle, you are basically free to go on any possible

量子力學粒子，你基本上可以自由地進行任何可能的

path, so you know, and you could even have really very bizarre paths that goes all the way like that.

路徑，所以你知道，你甚至可以有真正非常離奇的路徑，像這樣一路走下去。

And the thing that physics will do, it will give a certain probability for all these

而物理學將做的事情，它將為所有這些提供一定的概率

possible scenarios. If I for instance want to go from my work to home

可能出現的情況。例如，如果我想從我的工作地點到家裡去

I know pretty good what the path is I'm taking,

我很清楚自己要走的路是什麼。

but if I'm an electron that's moving, say, from one atom to another atom,

但如果我是一個電子，正在移動，比如說，從一個原子到另一個原子。

it's more like these, these curves, so everything is in terms of chance.

它更像這些，這些曲線，所以一切都在機會方面。

The quantum probability is something, it's like an, it's a complex number, and if you take the absolute value squared,

量子概率是一種東西，它就像一個，它是一個複數，如果你取絕對值的平方。

it becomes actual probabilities.

它成為實際的概率。

So for instance, one thing that, because of these kind of funny things, what can happen is that

是以，例如，有一件事，由於這些種類的有趣的事情，可能發生的情況是，

instead, you know, even if you think about where a particle is,

相反，你知道，即使你考慮到一個粒子在哪裡。

now it could be either here, or it could be here, and you might say, well there's a certain probability

現在它可能在這裡，也可能在這裡，你可能會說，有一定的概率

it's an A, and a probability it's in B.

它是一個A，和一個概率它是在B。

But a quantum mechanical particle can be for half be here and half be there, at the same time.

但一個量子力學粒子可以在同一時間一半在這裡，一半在那裡。

So this is very strange.

所以這非常奇怪。

It's like you take a point, and you divide it in two fractions of a point, put one here and put one there.

這就像你拿一個點，把它抽成兩個分數，這裡放一個，那裡放一個。

And so the rules of quantum mechanics are partly captured probability,

於是，量子力學的規則部分地捕捉到了概率。

but they have some kind of very strange phenomena built in them, and it's ... you have to kind of dig deep and basically things

但它們有一些非常奇怪的現象，而且......你必須深入挖掘，基本上是這樣的。

happening, you know, all at the same time

同時發生，你知道，所有的事情都在發生

makes quantum mechanics very difficult to absorb.

使得量子力學非常難以吸收。

It's been said, you know, if you say you understand quantum mechanics that you're just lying, because this is

有人說，你知道，如果你說你瞭解量子力學，你就是在撒謊，因為這是

something that in our intuition we simply cannot understand.

在我們的直覺中，有些東西我們根本無法理解。

Physicists were thinking about, well, we need some objects, for instance, to describe the transitions in the atom,

物理學家們在想，好吧，我們需要一些對象，比如說，來描述原子中的轉變。

how we go from one state to another state,

我們如何從一個州到另一個州。

so well, we need something, and then they discovered it's called a matrix, and it's actually known, of course, in the mathematical literature.

所以，我們需要一些東西，然後他們發現它被稱為矩陣，實際上它是已知的，當然，在數學文獻中。

But to understand the

但要了解

concepts that came in, like Hilbert spaces and wave functions, that was mathematics that partly was being developed at that time,

來的概念，如希爾伯特空間和波函數，那是當時部分正在發展的數學。

but I would say then the physicists really ran off with it,

但我想說的是，當時物理學家們真的帶著它跑掉了。

and at this point, I would say mathematicians are

而在這一點上，我想說數學家是

trying to catch up and

試圖追趕和

understand, you know,

瞭解，你知道。

not only the language, but also the power of the concept. Because there's something incredibly

不僅是語言，而且是概念的力量。因為有一些令人難以置信的

natural in quantum theory which is perfect for the mathematical mind.

量子理論中的自然，這對數學思維來說是完美的。

Because mathematicians, they're not studying one object.

因為數學家們，他們不是在研究一個物體。

They actually really always think about the whole family; the category, all possible objects. So, all numbers at the same time, or all possible

他們實際上真的總是在考慮整個家族；類別，所有可能的對象。是以，所有的數字在同一時間，或所有可能的

geometrical shapes. And what quantum mechanics says is, that's like perfect, because I'm not considering one path,

幾何形狀。而量子力學說的是，這就像是完美的，因為我不是在考慮一條路徑。

I'm considering the space of all paths. And in fact, I'm giving you something else. I'm giving you a way,

我正在考慮所有路徑的空間。而事實上，我在給你別的東西。我在給你一條路。

indeed, a probability of each possible path,

事實上，每個可能路徑的概率。

so in some way, you can sum all over these different paths. We call this the sum over histories

所以在某種程度上，你可以對這些不同的路徑進行求和。我們把這稱為歷史的總和

which is a ...

這是一個...

again, very funny concept, because we think of history as something, well,

再次，非常有趣的概念，因為我們認為歷史是一種東西，嗯。

certain things, certain facts that happens. For the quantum mechanical mathematician,

某些事情，某些發生的事實。對於量子力學數學家來說。

history is all possible scenarios at the same time. And so there's a certain way to

歷史是所有可能的情況在同一時間。是以，有一種特定的方式來

consider many different things at the same time. And so one

在同一時間考慮許多不同的事情。是以，一個

wonderful application on all of this was in, in knot theory. So if you have a knot, like this,

這一切的美妙應用是在結理論中。是以，如果你有一個結，像這樣。

connected like this, so it's a ... it's a... I'm just

像這樣的連接，所以它是一個......它是一個......。我只是

drawing here the projection, but there's a certain trajectory in three dimensions, and, you know,

這裡畫的是投影，但在三維空間有一定的軌跡，而且，你知道。

mathematicians want to distinguish. This one is knotted, this one is not. And so, if you're a knot

數學家想要區分。這個是打結的，這個不是。是以，如果你是一個結

theorist, you want to study knots in generality; the space of all possible knots. Actually

理論家，你想在一般情況下研究結；所有可能的結的空間。實際上

quantum theory comes to the rescue.

量子理論來拯救。

It's a very efficient way to describe these knots. And young physicists are not that surprised anymore as

這是一種描述這些結的非常有效的方法。而年輕的物理學家們也不再那麼驚訝了，因為

for instance Niels Bohr or

例如，尼爾斯-玻爾或

Albert Einstein was, who were really fighting the concept. I often think, you know, at some point there will be a generation of mathematicians

阿爾伯特-愛因斯坦是，他們真的在與這個概念作鬥爭。我經常想，你知道，在某些時候，會有一代數學家

that kind of grow, grow up with quantum theory, and they will probably apply these kinds of rules in a very natural way.

那種成長，與量子理論一起成長，他們可能會以一種非常自然的方式應用這類規則。

Brady: "They are starting to steal your stuff, arent they?"

布萊迪："他們開始偷你的東西了，是嗎？"

Exactly. So, the remarkable thing is that there are some very deep

正是如此。是以，引人注目的是，有一些非常深刻的

mathematical problems; abstract mathematical problems, like this classification of all knots,

數學問題；抽象的數學問題，如這個所有結的分類。

or thinking of four- or six-dimensional spaces, really purely interested from mathematical arguments,

或想到四維或六維空間，真的純粹是對數學論證感興趣。

and now there is a bunch of these problems that actually have been solved using quantum theory. I think of it like a good mathematician

而現在有一堆這樣的問題，實際上已經用量子理論解決了。我認為這就像一個好的數學家一樣

has a toolbox that's filled with everything: geometry,

有一個工具箱，裡面裝滿了一切：幾何。

calculus, number theory,

微積分，數論。

and, you know, the rules of modern physics should be there.

而且，你知道，現代物理學的規則應該在那裡。

It's just a very efficient way to deal with certain problems, and I am actually ... I think many mathematicians were surprised that the

這只是處理某些問題的一種非常有效的方法，而我實際上 ...我想很多數學家都對

information was flowing in the other direction. Physicists are surprised, because they see the quantum

資訊是向另一個方向流動的。物理學家們很驚訝，因為他們看到量子

mathematicians, so to say, take these ideas and

數學家們，可以說，把這些想法和

generalise them even further.

甚至進一步歸納它們。

It's like you have a very brilliant student and you're teaching the student everything you know,

這就像你有一個非常出色的學生，你把你知道的一切都教給了這個學生。

and then the student takes that much further, because again,

然後學生就會更進一步，因為再次。

and that's the beauty of math. It is, in some sense, rooted in reality, but then it's not. It's hard to meet a

而這正是數學的魅力所在。從某種意義上說，它是植根於現實的，但又不是。很難遇到一個

mathematician that's not ...

不是的數學家...

doesn't think

不認為

platonic; thinks that, you know, the prime numbers are just out there, and they have nothing to do with the fact that there are

柏拉圖式的；認為，你知道，素數只是在那裡，它們與存在的事實沒有關係

three pens here. That the number three is there.

這裡有三支筆。那數字三就在那裡。

And so, there's this kind of

是以，有這樣一種

interesting kind of an

有趣的一種

oscillation between math

數學之間的震盪

absorbing

吸收性

facts from reality,

事實來自於現實。

and then kind of take a flight of the imagination, and go in directions that the physicist says, wait a moment that was not

然後採取一種飛行的想象力，並在物理學家說的方向上走，等一下這不是

actually our intent to go and do, to do this kind of crazy stuff.

實際上我們的意圖是去做，去做這種瘋狂的事情。

Brady: "Has this emergence of mathematics coming out of physics been a difficult birth? Has it been

布拉迪："從物理學中出現的數學是否是一個艱難的誕生？它是否是

"something that has caused problems?"

"造成問題的東西？"

Yes. In two directions, I think.

是的，在兩個方向，我想。

You know, the physicists would say math is very attractive,

你知道，物理學家會說數學很有吸引力。

but it's like a black hole in which you can disappear, because you will be divorced from reality.

但它就像一個黑洞，你可以在其中消失，因為你將脫離現實。

Odysseus, you have to tie yourself to the to the mast of the ship not to be seduced by the beauty of mathematics

奧德修斯，你必須把自己綁在船的桅杆上，不要被數學的美麗所誘惑。

because

因為

many of the great insights in physics were looked very painful and inelegant and ugly

物理學中的許多偉大見解看起來都非常痛苦，不優雅和醜陋

to begin with. It took some time to appreciate

來開始。花了一些時間來欣賞

quantum theory as something beautiful. So that's the rivalry from the physics part. From the math part,

量子理論是美麗的東西。所以這就是來自物理學部分的競爭。從數學的部分。

I think it is that, you know, physicists are sometimes seen as kind of plumbers, they, you know, they make their hands dirty,

我認為這是，你知道，物理學家有時被看作是一種水管工，他們，你知道，他們使他們的手髒。

they deal with this messy thing called reality.

他們處理這個叫做現實的混亂的東西。

It's beautiful, as a mathematician, you can go this platonic world where there's a certain perfectness. So there's also

這是美麗的，作為一個數學家，你可以去這個柏拉圖式的世界，那裡有某種完美性。所以也有

I think quite a few mathematicians that weren't originally

我想有不少數學家原本並不是

impressed with the physical ideas. They might say, well,

對物理概念印象深刻。他們可能會說，好吧。

you know, you might kind of be ... this might ... one or two areas where this is relevant, but not to my field,

你知道，你可能有點......這可能......有一兩個領域與此有關，但與我的領域無關。

which is pristine. And then certainly,

這是最原始的。然後當然。

out of left field,

從左手邊出來。

there were some physical ideas that proved to be very very very powerful,

有一些物理想法被證明是非常非常強大的。

and I feel it's like a wave, you know, you're, you think it won't hurt you until the wave hits you, and then suddenly you are underwater.

我覺得它就像一個波浪，你知道，你，你認為它不會傷害你，直到波浪擊中你，然後突然你就在水下了。

In physics we have really a crisis, and the crisis is that we have two

在物理學中，我們確實有一個危機，而危機在於我們有兩個

wonderful theories that describe the world.

描述世界的精彩理論。

One is quantum

一個是量子

theory: It's perfect for elementary particles. The other one is relativity, and it's used to describe large structures in the universe.

理論：它對基本粒子來說是完美的。另一個是相對論，它被用來描述宇宙中的大型結構。

But the point is these two things clash, and they clash because these physical laws of the large. They include

但問題是這兩件事發生了衝突，而它們發生衝突是因為這些大的物理規律。它們包括

events like black holes, and perhaps more important, the big bang, where the rules of physics break down.

像黑洞這樣的事件，也許更重要的是大爆炸，物理學的規則在這裡被打破。

So this is quite remarkable that the theory that you have to describe reality has its own failure built into it; and

是以，這是相當了不起的，你所擁有的描述現實的理論有其自身的失敗；而且

we think that, in some sense, the only solution is

我們認為，在某種意義上，唯一的解決辦法是

to bring quantum mechanics into it. So we somehow have to marry these two pictures. It's not enough to do quantum mechanics.

將量子力學引入其中。是以，我們必須以某種方式將這兩張圖片結合起來。光做量子力學是不夠的。

It's not enough to do geometry. And we know this is happening, because the universe is working,

光是做幾何學是不夠的。而我們知道這正在發生，因為宇宙正在發揮作用。

you know, right now while we speaking, the laws of nature are perfectly working,

你知道，現在當我們說話時，自然法則正在完美地發揮作用。

and there's no breakdown anywhere. So that means that we have to come up with something

而且沒有任何地方出現故障。是以，這意味著我們必須想出一些辦法來

deeper than geometry.

比幾何學更深刻。

Now what is that?

現在那是什麼？

That's, that's, I think, the big mystery. And we have some ideas.

那是，那是，我認為，大的謎團。而我們有一些想法。

So, the idea could be that geometry is something like

所以，這個想法可能是，幾何學是這樣的

thermodynamics, which is the theory that describes, basically, materials. So if I think of this table, it feels pretty solid to me,

熱力學，這是描述，基本上，材料的理論。是以，如果我想到這張表，我就覺得很踏實。

but I know it's a large collection of molecules.

但我知道這是一個大分子的集合。

I feel the air around me, and I feel a certain temperature.

我感覺到我周圍的空氣，我感覺到某種溫度。

But there's no such thing as temperature, these are just molecules bouncing up and down.

但是沒有溫度這回事，這些只是分子在上下跳動而已。

So could it be that just as this physical sheet of paper, if I zoom in, zoom in, zoom in,

所以會不會就像這張有形的紙一樣，如果我把它放大，放大，放大。

I will certainly see molecules and atoms and elementary particles, and there's no paper anymore,

我一定會看到分子、原子和基本粒子，而且沒有紙了。

could it be that if I zoom into space itself, at some point I see the pixels?

會不會是如果我放大空間本身，在某個點上我看到了像素？

I see something else? You know, we were thinking perhaps it's pure informations, like zeroes and ones. So, I find that absolutely

我看到了別的東西？你知道，我們在想，也許這是純粹的資訊，像零和一。所以，我發現這絕對是

breathtaking. Because if you are able to come up with a piece of mathematics

歎為觀止。因為如果你能想出一個數學的方法

that is more fundamental than space, than geometry, I think it would be

比空間、比幾何更基本的東西，我想這將是

revolutionising all of math.

徹底改變了所有的數學。

Just imagine

試想一下

trying to imagine these two huge objects what they're doing to the matter, to the space-time around it as

試圖想象這兩個巨大的物體對物質、對周圍的時空做了什麼，因為

they, and the space-time must be going "oh my god what's happening here", and it's flipping up and down up and down.

他們，而時空一定會 "哦，我的上帝，這裡發生了什麼"，它在上下翻動著，上下翻動著。

And they're just generating, these waves are beginning to propagate out. 350 kilometers is about the distance,

而且他們剛剛產生，這些波開始向外傳播。350公里是大約的距離。

what, to London? Then you've got two thirty solar mass black holes sort of orbiting one another in this region

什麼，到倫敦？然後你有兩個30個太陽品質的黑洞，在這個區域內互相繞行。

And so they're going...

是以，他們要...