Two Researchers Are Rebuilding Mathematics From the Ground Up
Kristina Armitage/ Quanta Magazine
克里斯蒂娜·阿米蒂奇/广达杂志
L et’s start with what’s probably the most tired, overused joke in math: A topologist is someone who can’t tell a coffee cup from a doughnut.
让我们从数学中最令人厌倦、最过度使用的笑话开始：拓扑学家是一个无法区分咖啡杯和甜甜圈的人。
Both, you see, have a hole in them.
你看，两者都有一个洞。
Topology is usually described as a sort of “rubber sheet” geometry in which two shapes are considered the same if one can be stretched or compressed into the other without tearing it.
拓扑通常被描述为一种“橡胶板”几何形状，其中如果一个形状可以拉伸或压缩到另一个形状而不撕裂它，则认为两个形状是相同的。
But this summary leaves out something essential: How do topologists, and the many other mathematicians who rely on their methods, rigorously account for all this stretching?
但这个总结遗漏了一些重要的东西：拓扑学家和许多其他依赖他们的方法的数学家如何严格解释所有这些拉伸？
They don’t look at a doughnut and a coffee cup, squint, and say to themselves, “Sure, I can intuitively see how to squeeze one into the other, so they must be the same.” Rather, they describe a shape in a way that can “forget” about distance while respecting the underlying structure in a flexible way, allowing it to bend and stretch.
他们不会看着甜甜圈和咖啡杯，而是眯着眼睛对自己说：“当然，我可以直观地看到如何将其中一个挤入另一个，所以它们一定是相同的。”相反，他们以一种可以“忘记”距离的方式描述形状，同时以灵活的方式尊重底层结构，允许其弯曲和拉伸。
When these “topological spaces” were developed over 100 years ago, they played a major part in the revolutions in logic and set theory that marked the boundary between 19th-century and modern mathematics.
当这些“拓扑空间”在 100 多年前被开发出来时，它们在标志着 19 世纪和现代数学界限的逻辑和集合论革命中发挥了重要作用。
Their birth was a crucial waypoint on math’s inexorable march from the numbers and shapes that people encounter in everyday life into ever more abstract caverns of thought.
它们的诞生是数学不可阻挡地从人们在日常生活中遇到的数字和形状迈向更加抽象的思想洞穴的关键路点。
Topological spaces have since become the foundation for huge chunks of mathematics.
拓扑空间从此成为大量数学的基础。
If you think of math as a skyscraper, topological spaces are concrete pilings, driven deep into the bedrock of common sense that all of math ultimately rests on.
如果你把数学想象成一座摩天大楼，那么拓扑空间就是混凝土桩，深入到所有数学最终依赖的常识的基石中。
But disconcertingly, topological spaces turn out to be extremely poorly suited for a big chunk of modern math: They are an awkward setting in which to do algebra, which is something mathematicians quite like doing.
但令人不安的是，拓扑空间非常不适合现代数学的一大块：在它们中进行代数是一个尴尬的环境，而代数是数学家非常喜欢做的事情。
For years, mathematicians figured they just had to live with the limitations of topological spaces.
多年来，数学家们认为他们只能忍受拓扑空间的局限性。
If you’re working on the 87th story of a skyscraper, fixing the foundations in the subbasement is a scary proposition.
如果您正在建造摩天大楼的第 87 层，那么修复地下室的地基是一个可怕的提议。
Peter Scholze prefers coming up with new definitions rather than coming up with new proofs.
彼得·舒尔茨更喜欢提出新的定义，而不是提出新的证明。
As he put it, he’s “trying to give names to what is there.”
正如他所说，他“试图为那里的事物命名”。
Barbara Frommann/Hausdorff Center for Mathematics, University of Bonn
波恩大学芭芭拉·弗洛曼/豪斯多夫数学中心
But over the past decade, Peter Scholze of the Max Planck Institute for Mathematics in Bonn and Dustin Clausen of the Institute of Advanced Scientific Studies in France have sought to replace topological spaces.
但在过去的十年里，波恩马克斯·普朗克数学研究所的彼得·舒尔茨和法国高级科学研究所的达斯汀·克劳森一直在寻求取代拓扑空间。
They have defined a new category of mathematical objects called condensed sets, which resemble a sort of infinitely fine dust and retain all the nicest properties of topological spaces without the drawbacks.
他们定义了一种新的数学对象类别，称为凝聚集，它类似于一种无限细小的灰尘，保留了拓扑空间的所有最好的属性，而没有缺点。
Dust, it turns out, is a better foundational material than the pebbly, well-understood soil of topological spaces.
事实证明，与拓扑空间中众所周知的卵石土壤相比，灰尘是更好的基础材料。
“They are solving a problem we didn’t know we had,” said Ravi Vakil , a mathematician at Stanford University and president of the American Mathematical Society, “because we already had what we thought were reasonable solutions.” As a result, “a whole slate of mathematics has become much simpler.”
斯坦福大学数学家、美国数学会主席拉维·瓦基尔 (Ravi Vakil) 表示：“他们正在解决一个我们不知道的问题，因为我们已经有了我们认为合理的解决方案。”结果，“整个数学变得更加简单。”
It’s an ambitious project.
这是一个雄心勃勃的项目。
The new definitions and concepts that Scholze and Clausen have introduced are powerful but also complicated and hard to learn.
舒尔茨和克劳森引入的新定义和概念非常强大，但也很复杂且难以学习。
Scholze, for his part, is not sure how widely used they will become.
就舒尔茨而言，他不确定它们的使用范围有多大。
On the other hand, he sees them as just the first step in a far bigger program to understand why numbers behave the way they do.
另一方面，他认为它们只是一个更大的计划的第一步，旨在理解为什么数字会这样表现。
Doing math can be a little like rock climbing: Just as the route you take up a sheer face can incorporate creativity and even elegance in the way it sequences technical maneuvers, so too can a proof.
做数学有点像攀岩：正如你所走的路线可以在排列技术动作的方式中融入创造力甚至优雅一样，证明也可以。
Both traverse existing terrain.
两者都穿越现有地形。
Most research — even some of the best research — takes the form of finding new routes to known peaks.
大多数研究——甚至一些最好的研究——都采取寻找通往已知山峰的新路线的形式。
But in mathematics, there is a weird relationship between the equipment and the landscape, as though developing a new type of ice axe causes hitherto unknown mountain ranges to emerge.
但在数学上，设备和景观之间存在着一种奇怪的关系，就好像开发一种新型冰镐会导致迄今为止未知的山脉出现一样。
As those new ranges appear on the horizon, older mountains that had seemed forbiddingly steep begin to resemble gentle hills.
当这些新山脉出现在地平线上时，那些看似陡峭得令人生畏的古老山脉开始变得像平缓的山丘。
Dustin Clausen, along with Scholze, has spent the past decade developing a new mathematical framework.
达斯汀·克劳森（Dustin Clausen）和舒尔茨（Scholze）在过去十年中一直在开发一种新的数学框架。
Their “condensed mathematics” is already helping to connect topology, category theory, algebra, and other areas.
他们的“浓缩数学”已经在帮助连接拓扑、范畴论、代数和其他领域。
Developing these new tools takes a certain revolutionary confidence — especially when it requires setting aside implements that have been used in the community for so long that they seem to be part of the mountains themselves.
开发这些新工具需要一定的革命性信心——尤其是当它需要搁置社区中已经使用了很长时间的工具时，这些工具似乎本身就是山脉的一部分。
It’s possible to discover powerful mathematical truths without having a good language to work in.
即使没有良好的工作语言，也有可能发现强大的数学真理。
Which is to say that topology predates topological spaces.
也就是说，拓扑早于拓扑空间。
As far back as 1735, Leonhard Euler proved that it was impossible to traverse the city of Königsberg by crossing each of its seven bridges only once.
早在 1735 年，莱昂哈德·欧拉 (Leonhard Euler) 就证明，只穿过柯尼斯堡市的七座桥梁一次是不可能的。
This is a recognizably topological result — the size of each of the city’s landmasses doesn’t matter, nor does the length of the bridges between them.
这是一个明显的拓扑结果——城市每块土地的大小并不重要，它们之间桥梁的长度也不重要。
Only the pattern of how they connect to each other does.
只有它们如何相互连接的模式才起作用。
For nearly 200 years, research in topology proceeded in fits and starts.
近200年来，拓扑学的研究断断续续地进行着。
In the mid-19th century, August Ferdinand Möbius analyzed the strip that bears his name: a ribbon twisted on itself one time before its ends are joined.
19 世纪中叶，奥古斯特·费迪南德·莫比乌斯 (August Ferdinand Möbius) 分析了以他的名字命名的条带：一条丝带在其两端连接之前先自我扭曲了一圈。
It is arguably the strangest topological object to have practical utility in the real world — for instance, in one-sided conveyor belts that wear evenly as they drive machines.
它可以说是现实世界中具有实际用途的最奇怪的拓扑对象——例如，在驱动机器时均匀磨损的单面传送带中。
Around the same time, Möbius began introducing some of the field’s key ideas, such as how to classify shapes with varying numbers of holes by looking at how loops can be drawn on them.
大约在同一时间，莫比乌斯开始介绍该领域的一些关键思想，例如如何通过研究如何在其上绘制循环来对具有不同数量孔的形状进行分类。
Shortly afterward, Bernhard Riemann, Henri Poincaré, and others made further advances.
不久之后，伯恩哈德·黎曼（Bernhard Riemann）、亨利·庞加莱（Henri Poincaré）等人取得了进一步的进展。
But they struggled for lack of the right language.
但他们因缺乏正确的语言而苦苦挣扎。
As the Australian mathematician John Stillwell wrote in 2009 of Poincaré’s groundbreaking work in topology: “Along with great breakthroughs, there is also confusion.” Poincaré had ideas that he lacked the vocabulary to properly express.
正如澳大利亚数学家约翰·斯蒂尔威尔 (John Stillwell) 在 2009 年谈到庞加莱在拓扑学方面的开创性工作时所写的那样：“在取得巨大突破的同时，也存在着困惑。”庞加莱的想法缺乏适当的词汇来表达。
To an outsider, it seems as if the branch of math closest to topology ought to be geometry.
对于外行人来说，似乎最接近拓扑的数学分支应该是几何。
It seems as if topology is geometry, just with flexible objects instead of rigid ones.
看起来拓扑学就是几何学，只不过是柔性物体而不是刚性物体。
But the resolution to Poincaré’s confusion would come not from geometry, but from a nascent branch of logic called set theory.
但庞加莱困惑的解决方案不是来自几何学，而是来自一个名为集合论的新生逻辑分支。
The German mathematician Felix Hausdorff, who laid the foundations for modern topology, was also a poet, philosopher, and playwright.
德国数学家费利克斯·豪斯多夫为现代拓扑学奠定了基础，同时也是一位诗人、哲学家和剧作家。
At the turn of the 20th century, researchers were trying to wrestle mathematics onto firmer footing.
20 世纪初，研究人员试图为数学奠定更坚实的基础。
They had only recently realized that their everyday intuition about numbers was completely wrong ; now they were fervently debating which axioms , or obvious truths, they should build their theories on.
他们直到最近才意识到，他们对数字的日常直觉是完全错误的。现在他们正在激烈地争论他们的理论应该建立在哪些公理或明显的真理之上。
Seemingly small differences in how they stated their most straightforward assumptions had major consequences for what would be possible or impossible to prove.
他们在表达最直接的假设时看似微小的差异，却对可能或不可能证明的内容产生了重大影响。
They used set theory to hash out these debates about the foundations of mathematics.
他们使用集合论来解决这些关于数学基础的争论。
In 1912, Felix Hausdorff, who had recently started teaching at the University of Bonn — where Scholze would end up generations later — set out to write the first comprehensive treatment of set theory.
1912 年，刚开始在波恩大学任教的菲利克斯·豪斯多夫（Felix Hausdorff）（数代之后舒尔茨最终在该大学任教）开始撰写第一个关于集合论的综合论述。
At the time, Hausdorff, then in his mid-40s, was already an accomplished writer: Under the pseudonym Paul Mongré, he had published a collection of poetry, two books of philosophy that tried to reconcile Nietzsche and Kant, and a play that was produced in 40 cities.
当时，40 多岁的豪斯多夫已经是一位颇有成就的作家：他以保罗·蒙格雷 (Paul Mongré) 的笔名出版了一本诗集、两本试图调和尼采和康德的哲学书，以及一部在 40 个城市上演的戏剧。
As a mathematician, he was successful but not yet a superstar.
作为一名数学家，他很成功，但还不是超级明星。
That changed after the 1914 publication of his book Fundamentals of Set Theory .
1914 年他的书《集合论基础》出版后，情况发生了变化。
In it, he gave the first description of topological spaces.
在其中，他首次描述了拓扑空间。
A topological space is simply a collection of items that are grouped together into what Hausdorff called neighborhoods — today known as open sets.
拓扑空间只是一组项目的集合，这些项目被组合在一起形成豪斯多夫所谓的邻域（今天称为开集）。
The open sets give structure to the space.
开放式的布景赋予了空间结构。
Open sets must satisfy just two conditions.
开集必须仅满足两个条件。
First, any combination of open sets must also be an open set.
首先，开集的任何组合也必须是开集。
(If Brooklyn forms a neighborhood and Queens forms a neighborhood, Brooklyn and Queens together must count as a single, bigger neighborhood.) And second, any finite overlap between open sets must also be an open set.
（如果布鲁克林形成一个邻域，而皇后区形成一个邻域，则布鲁克林和皇后区一起必须算作一个更大的邻域。）其次，开集之间的任何有限重叠也必须是开集。
Topological spaces can be finite or infinite.
拓扑空间可以是有限的，也可以是无限的。
They can encode intricate structure, or no structure at all.
它们可以编码复杂的结构，或者根本没有结构。
As Scholze put it, topological spaces are “just everywhere.
正如 Scholze 所说，拓扑空间“无处不在”。
If you have this intuitive idea of points being close to each other, you have topology.”
如果你有这种点彼此靠近的直观想法，那么你就拥有了拓扑。”
Consider a more familiar object: the number line.
考虑一个更熟悉的对象：数轴。
When we think about the way numbers relate to one another, we’re thinking about just one topological space: the so-called standard topology, in which every possible interval (not including its end points) forms a neighborhood, or open set.
当我们考虑数字彼此关联的方式时，我们考虑的只是一个拓扑空间：所谓的标准拓扑，其中每个可能的区间（不包括其端点）形成一个邻域或开集。
All these intervals are open sets:
所有这些区间都是开集：
This topology gives the real numbers the structure we’re used to.
这种拓扑结构为实数提供了我们所习惯的结构。
But you can instead take the numbers you are used to dealing with, forget all the everyday intuition you’ve built up about them, and define entirely different topological spaces on them.
但你可以使用你习惯处理的数字，忘记你对它们建立的所有日常直觉，并在它们上定义完全不同的拓扑空间。
If you think of each number as a book in a library, this would be like taking all the books off their usual shelves and organizing them in a completely different way.
如果您将每个数字视为图书馆中的一本书，这就像将所有书籍从平常的书架上取下并以完全不同的方式组织它们一样。
For instance, you might crumple the number line into a ball and squeeze that ball down to a point, making every number arbitrarily close to every other number, like so:
例如，您可以将数轴揉成一个球并将该球挤压到一个点，使每个数字任意接近其他每个数字，如下所示：
This is akin to throwing all your books into a disorderly heap.
这类似于把你所有的书都乱七八糟地堆起来。
The relationships between the books — historical fiction all being on the same shelf, and so on — are lost, because all the books are now neighbors.
这些书之间的关系——历史小说都在同一个书架上，等等——都消失了，因为所有的书现在都是邻居了。
This is called the indiscrete topology, and we create it by declaring that there are only two open sets: the empty set and the entire number line.
这称为不离散拓扑，我们通过声明只有两个开集来创建它：空集和整个数轴。
At the other extreme is the “discrete” topology, in which every point forms its own neighborhood:
另一个极端是“离散”拓扑，其中每个点都形成自己的邻域：
In this topology, instead of every point being in contact with every other point, no point touches any other one.
在这种拓扑中，不是每个点都与其他点接触，而是没有点接触任何其他点。
It’s like putting every book in your library on its own private island.
这就像把图书馆里的每一本书都放在自己的私人岛屿上。
The relationships are again lost, because each book is isolated.
由于每本书都是孤立的，因此关系再次消失。
In a way, the old joke about the doughnut and coffee cup misunderstands the power of topological thinking.
在某种程度上，关于甜甜圈和咖啡杯的老笑话误解了拓扑思维的力量。
It’s not so much that it is possible to change distances by stretching or compressing things.
并不是说可以通过拉伸或压缩物体来改变距离。
It’s that it’s possible to meaningfully think about structure in spaces where distance simply does not exist.
而是可以在根本不存在距离的空间中有意义地思考结构。
In this way, topological spaces made it possible to explore novel parts of the mathematical landscape.
通过这种方式，拓扑空间使得探索数学领域的新颖部分成为可能。
For instance, they provided a new, distance-free way of understanding concepts like continuity and connectedness, which is a profoundly powerful and counterintuitive capability to have.
例如，它们提供了一种新的、无距离的方式来理解连续性和连通性等概念，这是一种非常强大且违反直觉的能力。
This has allowed mathematicians to generalize those ideas to a much broader range of settings — and to prove important statements in a wide variety of fields.
这使得数学家能够将这些想法推广到更广泛的环境中，并在各个领域证明重要的陈述。
For example, results like the fundamental theorem of algebra, whose proof tripped up even mathematical giants like Carl Friedrich Gauss, end up with very simple proofs once topological arguments come into play.
例如，像代数基本定理这样的结果，其证明甚至连像卡尔·弗里德里希·高斯这样的数学巨人都绊倒了，一旦拓扑论证发挥作用，最终就会得到非常简单的证明。
The introduction of topological spaces also led mathematicians to ask new questions they might not otherwise have thought to ask.
拓扑空间的引入也导致数学家提出了他们原本可能不会想到的新问题。
As is true of any good mathematical definition, topological spaces both opened new vistas and made it much easier to traverse known ones.
正如任何好的数学定义一样，拓扑空间既开辟了新的前景，也使穿越已知的前景变得更加容易。
Hausdorff’s book was arguably the beginning of modern topology.
豪斯多夫的书可以说是现代拓扑学的开端。
As the mathematical collective known as Bourbaki would later write, his well-chosen definitions imbued “his theory with both the full precision and full generality desired.
正如布尔巴基 (Bourbaki) 数学团体后来所写的那样，他精心挑选的定义使“他的理论既具有所需的完全精确性，又具有完全的普遍性。
The chapter in which he develops the consequences of these axioms remains a model of axiomatic theory — abstract, yet anticipatory.”
他阐述这些公理推论的章节仍然是公理理论的模型——抽象，但具有预见性。”
There are many subdisciplines within modern mathematics.
现代数学有许多分支学科。
Each has its own vocabulary, grammar, and intellectual flavor.
每个都有自己的词汇、语法和知识风格。
But they are never wholly separate — they interact in strange ways.
但它们从来都不是完全分离的——它们以奇怪的方式相互作用。
In part, this is because many mathematical objects, like the real numbers, exist as objects of study in multiple disciplines: They have an algebraic structure, an analytic structure, a combinatorial structure, and a topological structure (among others!).
在某种程度上，这是因为许多数学对象（如实数）作为多个学科的研究对象而存在：它们具有代数结构、解析结构、组合结构和拓扑结构（等等！）。
Often, areas of overlap end up becoming distinct areas of study.
通常，重叠的领域最终会成为不同的研究领域。
We are just chewing at the corners of this vast territory.
我们只是在咀嚼这片广阔领土的角落。
Clark Barwick, University of Edinburgh
克拉克·巴威克，爱丁堡大学
In 1945, two American mathematicians, Samuel Eilenberg and Saunders MacLane, published an audacious paper that created an entirely new discipline, now known as category theory.
1945 年，两位美国数学家塞缪尔·艾伦伯格 (Samuel Eilenberg) 和桑德斯·麦克莱恩 (Saunders MacLane) 发表了一篇大胆的论文，创建了一门全新的学科，即现在的范畴论。
In one fell swoop, category theory created a set of expressways between other, existing areas of math.
范畴论一举在其他现有数学领域之间创建了一系列高速公路。
Eilenberg and MacLane defined “categories” as collections of objects and the relationships (called morphisms) between them.
艾伦伯格和麦克莱恩将“类别”定义为对象及其之间的关系（称为态射）的集合。
For instance, a category might consist of sets and functions that relate these sets to each other, or of vector spaces and the linear maps that transform one vector space into another.
例如，一个类别可能由集合和将这些集合相互关联的函数组成，或者由向量空间和将一个向量空间转换为另一个向量空间的线性映射组成。
But the real power of the pair’s theory rested on the next level of abstraction they introduced: What if, they asked, you mapped one whole category onto another, with something they called a functor?
但这对理论的真正力量在于他们引入的下一个抽象层次：他们问，如果你用他们称之为函子的东西将一个整个类别映射到另一个类别上怎么办？
A functor takes objects to objects and morphisms to morphisms, in an orderly way.
函子以有序的方式将对象带到对象，将态射带到态射。
In other words, it doesn’t just give you a way to translate from one group of objects to another; it also preserves the relationships between them — allowing you to connect different areas of math to each other.
换句话说，它不仅为您提供了一种从一组对象转换为另一组对象的方法，而且还为您提供了一种方法。它还保留了它们之间的关系——允许您将不同的数学领域相互联系起来。
Clark Barwick, along with his graduate student Peter Haine, independently defined “pyknotic sets,” objects that are very similar to condensed sets.
克拉克·巴威克（Clark Barwick）和他的研究生彼得·海恩（Peter Haine）独立定义了“固缩集”，即与凝聚集非常相似的对象。
Among other aims, Eilenberg and MacLane wanted to connect topology with the rest of math.
除其他目标外，艾伦伯格和麦克莱恩还希望将拓扑与数学的其他部分联系起来。
Topology was already known not to gel particularly well with one of math’s most important areas: algebra.
众所周知，拓扑学与数学最重要的领域之一：代数的结合并不特别好。
Although “topology has been this enormous gift to algebra,” said Clark Barwick of the University of Edinburgh, it has also “obstructed progress because topology and algebra don’t play all that nicely with each other, because of the particular way topology was built up.”
爱丁堡大学的克拉克·巴威克（Clark Barwick）说，虽然“拓扑学是代数的巨大礼物”，但它也“阻碍了进步，因为拓扑学和代数由于拓扑的特殊构建方式而不能很好地相互配合。”
Category theory quickly goes from stating things that sound obvious to drawing powerful mathematical conclusions on the basis of what its practitioners typically call, with affection, “abstract nonsense.” Some categories have particular properties that make them more useful to mathematicians than others.
范畴论很快就从陈述听起来显而易见的事情，转变为在其实践者通常亲切地称之为“抽象废话”的基础上得出强有力的数学结论。有些类别具有特定的属性，这使得它们比其他类别对数学家更有用。
In those categories, abstract nonsense becomes powerful — in other categories, it ends up just being nonsense.
在这些类别中，抽象的废话变得强大——而在其他类别中，它最终只是废话。
In topology, you can define a category consisting of topological spaces and the continuous maps between them.
在拓扑中，您可以定义一个由拓扑空间及其之间的连续映射组成的类别。
In algebra, meanwhile, an important category is that of abelian groups — groups that have useful symmetries.
与此同时，在代数中，一个重要的类别是阿贝尔群——具有有用对称性的群。
If you want to produce a grand synthesis of the ways in which algebraic things behave topologically, or, conversely, how algebraic considerations govern topological constructions, the natural thing to do is to form a category out of objects that have both topological and algebraic structure, called topological abelian groups.
如果你想对代数事物的拓扑行为方式进行大综合，或者相反，代数考虑如何控制拓扑构造，自然要做的就是从同时具有拓扑和代数结构的对象中形成一个类别，称为拓扑阿贝尔群。
But topological abelian groups lack the particular properties that category theorists desire.
但拓扑阿贝尔群缺乏范畴论学家所期望的特殊性质。
If category theory unveiled a hitherto hidden network of highways between different mountain ranges of math, topologists who wanted to travel on that highway were stuck driving a janky car that kept breaking down and needing repairs.
如果范畴论揭示了不同数学山脉之间迄今为止隐藏的高速公路网络，那么想要在这条高速公路上旅行的拓扑学家就只能驾驶一辆不断抛锚且需要修理的破旧汽车。
That’s the problem that Scholze and Clausen wound up trying to solve.
这就是舒尔茨和克劳森最终试图解决的问题。
As Scholze told me, “I think topologists don’t actually like topological spaces, because it’s not a convenient category.” What if they could define new objects to replace topological spaces — ones that would retain their power, but also create a better kind of category that would finally allow mathematicians to connect topology to algebra and other fields?
正如 Scholze 告诉我的那样，“我认为拓扑学家实际上并不喜欢拓扑空间，因为它不是一个方便的类别。”如果他们能够定义新的对象来取代拓扑空间——这些对象将保留其力量，但也创建一种更好的类别，最终允许数学家将拓扑连接到代数和其他领域？
Sometimes new ideas seem ready to burst into the world.
有时，新的想法似乎即将爆发。
In 2013, Scholze was 25 years old and already making waves as a deep mathematical thinker , hailed as the youngest full professor in Germany .
2013年，25岁的舒尔茨已经作为一位深刻的数学思想家引起轰动，被誉为德国最年轻的正教授。
He and Bhargav Bhatt , then a researcher at the Institute for Advanced Study in Princeton, New Jersey, coauthored a paper in which they came up with a new definition for a particular type of category.
他和时任新泽西州普林斯顿高级研究所研究员的巴尔加夫·巴特 (Bhargav Bhatt) 共同撰写了一篇论文，其中他们为特定类型的类别提出了新的定义。
In passing, the pair defined a somewhat abstruse mathematical set, a “sheaf on the pro-étale site of a point.” Scholze didn’t think much of it at the time.
顺便说一句，两人定义了一个有点深奥的数学集，“一个点的原始站点上的一捆”。舒尔茨当时并没有想太多。
Those sets, which they didn’t name, “felt like a weird aspect of the story that I didn’t fully comprehend,” Scholze told me.
他们没有透露这些场景的名称，“感觉像是故事中一个我没有完全理解的奇怪的方面，”舒尔茨告诉我。
Bhargav Bhatt, along with Scholze, defined the objects that would later become known as condensed sets years before anyone realized their enormous theoretical significance.
巴尔加夫·巴特 (Bhargav Bhatt) 与舒尔茨 (Scholze) 一起定义了后来被称为凝聚集的物体，早在人们意识到它们巨大的理论意义之前的几年。
At the time, Clausen was completing his doctorate at the Massachusetts Institute of Technology.
当时，克劳森正在麻省理工学院完成博士学位。
After spending five years as a postdoctoral fellow at the University of Copenhagen, he moved to Bonn in 2018 to work with Scholze, who had just been awarded a Fields Medal , math’s highest honor.
在哥本哈根大学做了五年博士后后，他于 2018 年搬到波恩，与刚刚获得数学最高荣誉菲尔兹奖的 Scholze 一起工作。
Clausen had independently come up with the same kind of set for different reasons, and he persuaded Scholze that they should study it more closely.
克劳森出于不同的原因独立地提出了同样的一套，他说服舒尔茨他们应该更仔细地研究它。
By this time, Scholze recalled, Clausen had glimpsed a potential opportunity to replace topological sets.
舒尔茨回忆道，此时克劳森已经看到了取代拓扑集的潜在机会。
Around the same time, Barwick and his then-student, Peter Haine , independently came up with a slightly different definition in order to answer a particular question in category theory that interested them.
大约在同一时间，巴威克和他当时的学生彼得·海恩独立提出了一个略有不同的定义，以回答他们感兴趣的范畴论中的一个特定问题。
“We wanted to solve one problem,” Haine said.
“我们想解决一个问题，”海恩说。
“We kind of understood that this theory should be useful for quite a lot of stuff, but there is one thing we really wanted to do: prove this kind of specific result which generalized what we had previously done.”
“我们知道这个理论应该对很多东西有用，但我们真正想做的一件事是：证明这种具体的结果概括了我们之前所做的事情。”
On the other hand, he said of Clausen and Scholze, “I think they had a lot more that they wanted to do.”
另一方面，他谈到克劳森和舒尔茨时说：“我认为他们还有很多想做的事情。”
Indeed they did.
他们确实做到了。
They gave their sets a name — “condensed sets” — and got to work.
他们给自己的集合起了个名字——“压缩集合”——然后开始工作。
They didn’t publish their incremental progress.
他们没有公布他们的增量进展。
Instead, in April 2019, Scholze began giving lectures on “condensed mathematics” at the University of Bonn; in May, he posted a 77-page set of notes that culminated in a new, elegant proof of an important theorem called coherent duality.
相反，2019 年 4 月，舒尔茨开始在波恩大学讲授“浓缩数学”； 5 月，他发布了一套 77 页的笔记，最终为一个名为相干对偶性的重要定理提供了一个新的、优雅的证明。
As Johan Commelin , who would go on to collaborate with Scholze, remembers, “coherent duality had an extremely roundabout and technical proof before.” Scholze and Clausen’s proof was clean and elegant.
正如后来与舒尔茨合作的约翰·科梅林（Johan Commelin）回忆的那样，“相干二元性之前有过极其迂回和技术性的证明。”舒尔茨和克劳森的证明干净而优雅。
That, Commelin said, “gave people the motivation to say, ‘Let’s organize reading groups and study seminars all over the world to go through these lecture notes and find out what’s happening.’” Commelin organized one such group at the University of Freiburg, but it was challenging.
科梅林说，这“让人们有动力说，‘让我们在世界各地组织阅读小组和研究研讨会，浏览这些讲义并了解正在发生的事情。’”科梅林在弗莱堡大学组织了一个这样的小组，但它具有挑战性。
“I don’t think anybody in our group understood all the details,” he said.
“我认为我们团队中没有人了解所有细节，”他说。
Although condensed sets had started as a tool for proving useful results, Scholze recalled, they quickly became something more.
舒尔茨回忆道，尽管凝聚集最初是作为证明有用结果的工具，但它们很快就变得更加重要。
“For me personally, condensed sets changed something very basic about how I think about mathematics,” he said.
“对我个人来说，凝聚集改变了我对数学的看法的一些非常基本的东西，”他说。
“Replacing the topological perspective by the condensed one,” he added, “creeps into everything I’m doing.”
“用简洁的观点取代拓扑观点，”他补充道，“渗透到我所做的每一件事中。”
Jeffrey Bergfalk , a set theorist at the University of Barcelona, remembers first meeting Scholze and Clausen in 2019 after he and his fellow set theorist Chris Lambie-Hanson of the Czech Academy of Sciences posted a technical paper online.
巴塞罗那大学的集合论学家 Jeffrey Bergfalk 记得第一次见到 Scholze 和 Clausen 是在 2019 年，当时他和捷克科学院的集合论同事 Chris Lambie-Hanson 在网上发布了一篇技术论文。
Set theorists form a particularly small and tight-knit community, somewhat removed from the mathematical mainstream.
集合理论家形成了一个特别小而紧密的社区，在某种程度上远离了数学主流。
“We were only expecting to hear from people that we knew,” Bergfalk said.
“我们只希望听到我们认识的人的消息，”伯格法尔克说。
But they got an email from Clausen, who had noticed the paper.
但他们收到了克劳森发来的一封电子邮件，克劳森注意到了这篇论文。
He and Scholze, the email said, were thinking about similar things in the context of condensed mathematics — a nascent subject that neither Bergfalk nor Lambie-Hanson had heard of.
电子邮件称，他和舒尔茨正在浓缩数学的背景下思考类似的事情——这是伯格法克和兰比-汉森都没有听说过的新兴学科。
It took a moment for Bergfalk to realize the scale of Scholze and Clausen’s ambitions for condensed math.
伯格法克过了一会儿才意识到舒尔茨和克劳森对浓缩数学的雄心壮志。
“Dustin and Peter were just wonderfully responsive, really cool and communicative,” he said.
“达斯汀和彼得的反应非常好，非常酷并且善于沟通，”他说。
“They don’t have to be as nice as they are.” And even though Scholze and Clausen were working well outside their areas of expertise, Bergfalk noted, they asked about the sorts of things you wouldn’t expect anybody but a set theorist to care about.
“他们不必像现在这样友善。”伯格法克指出，尽管舒尔茨和克劳森在他们的专业领域之外工作得很好，但他们询问的问题是除了集合论学家之外任何人都不会关心的事情。
“They were asking us to think in the direction we would want to think anyway,” he said.
“他们要求我们按照我们想要的方向思考，”他说。
“Which meant we had to learn condensed math.”
“这意味着我们必须学习浓缩数学。”
For two people who are reinventing a big chunk of 20th-century mathematics, Clausen and Scholze are unassuming.
对于正在重塑 20 世纪数学一大块内容的两个人来说，克劳森和舒尔茨是谦逊的。
“To a large extent, what I am doing is rephrasing what others have done in my own words,” Scholze told the mathematician Maria Yakerson in a 2021 interview .
“在很大程度上，我所做的就是用我自己的话重新表述别人所做的事情，”舒尔茨在 2021 年的一次采访中告诉数学家玛丽亚·雅克森 (Maria Yakerson)。
“I’m not that much interested in theorems or proofs.” What he wanted to do, he said, was to come up with new definitions: “They must make it easy to state interesting theorems, and they must make it easy to prove them.” Scholze doesn’t see himself as creative.
“我对定理或证明不太感兴趣。”他说，他想做的是提出新的定义：“它们必须使有趣的定理变得容易陈述，并且必须使证明它们变得容易。”舒尔茨并不认为自己有创造力。
He is, he said, just “trying to give names to what is there.”
他说，他只是“试图给那里的东西命名”。
Clausen, for his part — as he told Yakerson in a separate interview around the same time — avoids publishing papers, because he believes that the scientific publishing industry is fundamentally flawed.
就克劳森而言，正如他在同一时间的另一次采访中告诉亚克森的那样，他避免发表论文，因为他认为科学出版业存在根本缺陷。
He also largely avoids even informally writing up results, leaving that to collaborators.
他还很大程度上避免非正式地写出结果，将其留给合作者。
He just wants to focus on the math; like Scholze, he’s constantly looking for the right names, the right language.
他只想专注于数学；和 Scholze 一样，他一直在寻找合适的名字、合适的语言。
(At one point, in fact, he considered pursuing a career in literary translation.)
（事实上，他曾一度考虑从事文学翻译事业。）
“I was never completely convinced by topological spaces,” Clausen said.
“我从来没有完全相信拓扑空间，”克劳森说。
They couldn’t give him an understanding of “this world that’s there, this rich world that we’re trying to get at but we don’t have the proper language to talk about.”
他们无法让他理解“这个世界就在那里，这个我们试图到达的丰富世界，但我们没有合适的语言可以谈论。”
But that only motivated him further.
但这只会进一步激励他。
“I’m extremely happy not understanding,” he said, “because I’m even happier when I finally do understand.”
“我非常高兴不理解，”他说，“因为当我最终理解时，我会更高兴。”
That’s where condensed sets come in.
这就是凝聚集的用武之地。
Condensed sets can be seen as a sort of recipe for building continuous objects, such as the real numbers, out of “totally disconnected” spaces — like making a cake out of disparate grains of flour and sugar, in Scholze’s telling.
凝聚集可以被视为一种在“完全不相连”的空间中构建连续对象（例如实数）的方法——用舒尔茨的话说，就像用不同的面粉和糖粒制作蛋糕一样。
Take the so-called Cantor set.
以所谓的康托集为例。
Start with the line segment containing all the real numbers between 0 and 1, and remove the middle third.
从包含 0 到 1 之间所有实数的线段开始，删除中间三分之一。
Then remove the middle third from the remaining line segments.
然后从剩余的线段中删除中间三分之一。
Repeat this process infinitely many times, and you’ll end up with a “dust” of points.
无限次地重复这个过程，你最终会得到点的“灰尘”。
No point is right next to any other.
没有一个点是与其他任何点相邻的。
The space of points is totally disconnected.
点的空间是完全不连通的。
The Cantor set is the simplest kind of condensed set, as well as a building block for making other condensed sets.
康托集是最简单的凝聚集，也是构建其他凝聚集的基础。
You can make more complicated condensed sets, Scholze said, by smashing together clouds of points like the Cantor set in a weird way.
舒尔茨说，你可以通过以一种奇怪的方式将像康托集这样的点云粉碎在一起来制作更复杂的压缩集。
Such dust might seem foreign, but Scholze points out that we use it all the time.
这些灰尘可能看起来很陌生，但舒尔茨指出我们一直在使用它。
When you represent numbers as decimal expansions, for instance, you’re essentially thinking of the numbers as a similar kind of dust.
例如，当您将数字表示为十进制扩展时，您本质上是将数字视为类似的灰尘。
It’s like taking each number in the expansion and cutting out its section of the number line, like so:
这就像取出展开式中的每个数字并剪掉数轴上的相应部分，如下所示：
In this way, producing a given number actually involves infinitely “dissecting” the number line.
这样，产生一个给定的数字实际上涉及到无限地“解剖”数轴。
As Scholze put it, the “decimal expansion describes a totally disconnected space because with every new digit, you are chopping up your line more and more.” Every number is totally disconnected from every other one.
正如 Scholze 所说，“十进制扩展描述了一个完全不相连的空间，因为每出现一个新数字，你就会越来越多地分割你的行。”每个数字都与其他数字完全无关。
How, then, can you use such a disconnected set to get a continuous object like the real number line we’re so used to?
那么，如何使用这样一个不连通集来获得一个连续的对象，就像我们习惯的实数轴一样呢？
You have to glue all the disconnected segments back together by equating, say, 0.49999999999999999…
你必须将所有断开的部分粘合在一起，例如，0.49999999999999999...
with 0.5 (and 0.50999999999999999…
0.5（和 0.50999999999999999…
with 0.51, and so on).
0.51，依此类推）。
Scholze and Clausen’s condensed sets work similarly: They’re disconnected, but they can be used to build and study continuous objects, like those you want to understand in topology.
Scholze 和 Clausen 的凝聚集的工作原理类似：它们是断开的，但可以用来构建和研究连续对象，就像你想在拓扑中理解的那些对象一样。
And if you start with them instead of topological spaces, you get an additional benefit: It turns out, Scholze explained, that “these totally disconnected pieces are extremely simple to describe algebraically.”
如果你从它们而不是拓扑空间开始，你会得到一个额外的好处：舒尔茨解释说，事实证明，“这些完全不相连的部分用代数描述起来非常简单。”
Condensed sets form a special type of category that, according to Juan Esteban Rodríguez Camargo , a collaborator of Scholze’s at the Max Plank Institute for Mathematics, finally makes it possible to mix topology, algebra, and other fields “in a very practical and precise way.”
马克斯·普朗克数学研究所舒尔茨的合作者胡安·埃斯特班·罗德里格斯·卡马戈表示，凝聚集形成了一种特殊类型的范畴，最终使得“以一种非常实用和精确的方式”混合拓扑、代数和其他领域成为可能。
Scholze and Clausen started by using their condensed sets to re-prove old results that previously relied on topological spaces — like the fundamental theorem of algebra.
舒尔茨和克劳森首先使用他们的凝聚集来重新证明以前依赖于拓扑空间的旧结果——就像代数的基本定理一样。
These new proofs have given mathematicians a novel understanding “suitably oiled and greased up and made supple,” Vakil said.
瓦基尔说，这些新的证明给了数学家一种新颖的理解，“适当地涂油、润滑，变得柔软”。
“The more you can fit into your intuition, the better you understand.”
“你越能适应自己的直觉，你就越能理解。”
Then the pair decided to push even further.
然后两人决定更进一步。
Since 2019, Scholze and Clausen have kept building new types of structures out of their condensed sets — and sharing new sets of lecture notes.
自 2019 年以来，Scholze 和 Clausen 不断从他们的压缩集中构建新型结构，并分享新的讲义集。
“The ideas were evolving in Bonn way faster than the rest of the world could consume them,” Commelin said.
“这些想法在波恩的发展速度远远快于世界其他地方的消费速度，”科梅林说。
There were “light” condensed sets, and also solid, then liquid, then gaseous spaces — an entire condensed mathematics.
有“光”凝聚集合，也有固体、液体、气体空间——一整套凝聚的数学。
Neither Clausen nor Scholze thinks of himself as a topologist.
克劳森和舒尔茨都不认为自己是拓扑学家。
If the two were less friendly, or their ideas less effective, there might be some resentment over their attempt to rebuild the foundations of a field they don’t tend to work in.
如果两人不太友好，或者他们的想法不太有效，那么他们试图重建一个他们不喜欢从事的领域的基础可能会引起一些不满。
“I wouldn’t want to impose anything,” Scholze said when asked what he thinks the impact of condensed math will be.
当被问及他认为浓缩数学会产生什么影响时，舒尔茨说：“我不想强加任何东西。”
He and Clausen sound as though they are having fun, trying to come up with ideas that they themselves find useful.
他和克劳森听起来好像他们很开心，试图想出他们自己认为有用的想法。
Alexander Grothendieck rewrote the field of algebraic geometry before abandoning mathematics to become a recluse in the French Pyrenees.
亚历山大·格洛腾迪克重写了代数几何领域，然后放弃了数学，隐居法国比利牛斯山脉。
Paul R.
保罗·R。
Halmos photograph collection, e_ph_01181_pub, The Dolph Briscoe Center for American History, The University of Texas at Austin
Halmos 照片集，e_ph_01181_pub，德克萨斯大学奥斯汀分校多尔夫·布里斯科美国历史中心
But some mathematicians are likening their work to a similar mathematical revolution that took place in the 1950s and ’60s — when Alexander Grothendieck reimagined the field of algebraic geometry in accordance with category theory, vastly extending its reach and power.
但一些数学家将他们的工作比作 20 世纪 50 年代和 60 年代发生的一场类似的数学革命，当时亚历山大·格洛腾迪克 (Alexander Grothendieck) 根据范畴论重新构想了代数几何领域，极大地扩展了其范围和力量。
Grothendieck’s impact on modern mathematics was profound.
格洛腾迪克对现代数学的影响是深远的。
And now, according to Dagur Asgeirsson , a postdoctoral researcher at the University of Alberta and Clausen’s former graduate student, “I think it’s fair to compare Peter to Grothendieck in this sense.
现在，阿尔伯塔大学博士后研究员、克劳森的前研究生达古尔·阿斯盖尔松 (Dagur Asgeirsson) 表示，“我认为从这个意义上将彼得与格洛腾迪克进行比较是公平的。
He is reinventing everything somehow.”
他正在以某种方式重塑一切。”
“The real excitement about condensed stuff for me is the possibility of defining new objects of study,” Barwick said.
“对我来说，浓缩的东西真正令人兴奋的是定义新研究对象的可能性，”巴威克说。
It’s “showing you there is this natural class of objects that you just never looked at before, like an unclimbed mountain.
它“向你展示了一些你以前从未见过的自然物体，就像一座未曾攀登过的山峰。
We are just chewing at the corners of this vast territory.”
我们只是在咀嚼这片广阔领土的角落。”
In one set of lecture notes , Clausen and Scholze quoted a well-known saying by the prominent mathematician David Mumford.
在一套讲义中，克劳森和舒尔茨引用了著名数学家大卫·芒福德的一句名言。
Mumford’s field of algebraic geometry, he said, “seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics.” Clausen and Scholze went on to note that their plan was to use condensed math — also esoteric, exclusive, and abstract — to continue where this effort had left off.
他说，芒福德的代数几何领域“似乎已经获得了深奥、排他性和非常抽象的声誉，其追随者正在秘密密谋接管所有其他数学领域。”克劳森和舒尔茨接着指出，他们的计划是使用浓缩数学——同样是深奥的、排他性的和抽象的——来继续这项工作停止的地方。
They were not entirely joking.
他们并不完全是在开玩笑。
Regardless, their not-so-secret “takeover of mathematics” is continuing.
不管怎样，他们不那么秘密的“接管数学”仍在继续。
In the last few years, Clausen and Scholze have defined other novel mathematical objects, such as “analytic stacks” and “gestalten.” Some mathematicians consider these to be even more significant than condensed sets.
在过去的几年中，克劳森和舒尔茨定义了其他新颖的数学对象，例如“分析堆栈”和“格式塔”。一些数学家认为这些比凝聚集更重要。
Scholze suspects that condensed mathematics might prove useful even in areas distant from his and Clausen’s core interest in number theory.
舒尔茨怀疑，即使在远离他和克劳森对数论核心兴趣的领域，浓缩数学也可能被证明是有用的。
Quantum field theory — a central aspect of modern physics, which has long struggled with its foundations — makes use of very sophisticated algebra and category theory, Scholze noted.
舒尔茨指出，量子场论——现代物理学的一个核心方面，长期以来一直在其基础上苦苦挣扎——利用了非常复杂的代数和范畴论。
“At the same time,” he added, “quantum field theories are by their nature very analytic and topological.
“与此同时，”他补充道，“量子场论本质上是非常分析性和拓扑性的。
Mixing these two worlds is a nontrivial matter, but condensed mathematics gives a possible framework to do so.”
混合这两个世界是一件不平凡的事情，但浓缩数学提供了一个可能的框架来做到这一点。”
Scholze and Clausen’s body of work shows just how much choosing the right language matters — how reframing concepts makes it possible to traverse known terrain more easily and to explore new mathematical vistas.
Scholze 和 Clausen 的研究成果表明，选择正确的语言有多么重要——重构概念如何使人们能够更轻松地穿越已知领域并探索新的数学前景。
“Trying to get to the bottom of these phenomena is trying to find a language in which to express them,” Scholze said.
“试图弄清楚这些现象的真相就是试图找到一种语言来表达它们，”舒尔茨说。
In a memoir Grothendieck published late in life, he described mathematicians as builders, even though he argued that they are most certainly not inventing anything, only finding structures that are already there, waiting to be discovered.
在格洛腾迪克晚年出版的回忆录中，他将数学家描述为建设者，尽管他认为他们肯定没有发明任何东西，只是寻找已经存在、等待被发现的结构。
He wrote: “The most beautiful house, that in which the love of the builder is most evident, is not that which is larger or higher than the others.
他写道：“最美丽的房子，最能体现建造者的爱的房子，并不是比其他房子更大或更高的房子。
Rather, a house is beautiful if it faithfully reflects the structure and beauty hidden in things.”
相反，如果一座房子忠实地反映了隐藏在事物中的结构和美丽，那么它就是美丽的。”
Correction: May 20, 2026 Peter Scholze is now at the Max Planck Institute for Mathematics in Bonn, not the University of Bonn.
更正：2026 年 5 月 20 日 Peter Scholze 现在在波恩的马克斯·普朗克数学研究所，而不是波恩大学。
Correction: May 26, 2026 The story has been updated to remove a parenthetical that incorrectly stated that Leonhard Euler lived in Königsberg.
更正：2026 年 5 月 26 日 该报道已更新，删除了错误地表述莱昂哈德·欧拉住在柯尼斯堡的括号。
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