The Man Who Stole Infinity
Kristina Armitage, Michael Kanyongolo/ Quanta Magazine
克里斯蒂娜·阿米蒂奇、迈克尔·坎永格洛/Quanta 杂志
W hen Demian Goos followed Karin Richter into her office on March 12 of last year, the first thing he noticed was the bust.
去年3月12日，当德米安·古斯跟随卡琳·里克特走进她的办公室时，他首先注意到的是这座半身像。
It sat atop a tall pedestal in the corner of the room, depicting a bald, elderly gentleman with a stoic countenance.
它坐落在房间角落的一个高高的基座上，描绘了一位秃顶的老绅士，面容坚忍。
Goos saw no trace of the anxious, lonely man who had obsessed him for over a year.
古斯没有看到那个困扰他一年多的焦虑、孤独的男人的踪迹。
Instead, this was Georg Cantor as history saw him.
相反，这就是历史所见的格奥尔格·康托 (Georg Cantor)。
An intellectual giant: steadfast, strong-willed, determined to bring about a mathematical revolution over the clamorous objections of his peers.
一位知识巨人：坚定、意志坚强，决心不顾同行的强烈反对，掀起一场数学革命。
It was here, at the University of Halle in Germany, that Cantor launched his revolution 150 years ago.
150 年前，正是在德国哈勒大学，康托尔发起了他的革命。
Here, in 1874, he published one of the most important papers in math’s 4,000-year history.
1874 年，他在这里发表了数学 4000 年历史上最重要的论文之一。
That paper crystallized a concept that had long been viewed as a mathematical malignancy to be shunned at all costs: infinity.
这篇论文具体化了一个长期以来被视为数学毒瘤、必须不惜一切代价回避的概念：无穷大。
It forced mathematicians to question some of their longest-held assumptions, rocking mathematics to its very foundations.
它迫使数学家质疑他们长期以来坚持的一些假设，从根本上动摇了数学的基础。
And it gave rise to a new field of study that would eventually bring about a rewriting of the entire subject.
它催生了一个新的研究领域，最终导致整个学科的重写。
How did the concept of infinity make the leap from a blasphemous idea to a foundational principle of modern mathematics?
无穷大的概念是如何从亵渎神明的想法一跃成为现代数学的基本原理的？
It was a hard-fought battle, rife with pride, betrayal, and dissolution.
这是一场艰苦的战斗，充满了骄傲、背叛和解体。
In the first part of a two-part episode, we set the scene at the dawn of modern math.
在两部分的第一部分中，我们将场景设定在现代数学的黎明时期。
Now Goos, a 35-year-old mathematician and journalist, had come to Halle — a five-hour train ride from his home in Mainz — to look at some letters from Cantor’s estate.
现在，35 岁的数学家兼记者古斯来到哈勒——距离他在美因茨的家有 5 个小时的火车车程——查看来自康托庄园的一些信件。
He’d seen a scan of one and was pretty sure he knew what the others would say.
他看过其中一张的扫描图，并且非常确定他知道其他人会说什么。
But he wanted to see them in person.
但他想亲自见到他们。
Richter — who, like Cantor, had spent her entire career here, first as a research mathematician and then, after retiring, as a lecturer on the history of mathematics — gestured for Goos to sit.
里克特——和康托尔一样，在这里度过了她的整个职业生涯，首先是一名研究数学家，然后在退休后担任数学史讲师——示意古斯坐下。
She lifted a thin blue binder from the scattered piles of books and papers on her desk.
她从桌上散落的书籍和文件堆中拿起一本薄薄的蓝色活页夹。
Inside were dozens of plastic sheet protectors, each one containing an old, handwritten letter.
里面有几十个塑料保护膜，每个保护膜里都有一封旧的手写信。
Goos began flipping through, contemplating the letters with the relish of an archaeologist entering a long-lost tomb.
古斯开始翻阅这些信件，就像考古学家进入一座失落已久的坟墓一样津津有味。
Then he reached a particular page and froze.
然后他读到某一页时愣住了。
He struggled to catch his breath.
他努力喘口气。
It wasn’t the handwriting.
这不是笔迹。
At this point in his research on Cantor, he’d become accustomed to the strange, nearly indecipherable Gothic script known as kurrentschrift , which Germans used until around 1900.
在他对康托尔的研究中，他已经习惯了一种奇怪的、几乎难以辨认的哥特文字，称为 kurrentschrift，德国人一直使用这种文字直到 1900 年左右。
It wasn’t the signature.
这不是签名。
He knew that the German mathematician Richard Dedekind had been a key player in Cantor’s quest to understand infinity and solidify math’s foundations, and that the two had exchanged many letters.
他知道德国数学家理查德·戴德金 (Richard Dedekind) 是康托探索无穷大和巩固数学基础的关键人物，两人交换了很多书信。
It was the date: November 30, 1873.
日期是：1873 年 11 月 30 日。
He’d never seen this letter before.
他以前从未见过这封信。
No one had.
没有人有过。
It was believed to be lost, destroyed in the tumult of World War II or perhaps by Cantor himself.
人们相信它在第二次世界大战的混乱中丢失或被毁坏，也可能是被康托本人毁坏的。
This was the letter that had the power to rewrite Cantor’s legacy.
这封信有能力改写康托的遗产。
The letter that proved once and for all that Cantor’s famous 1874 paper, the one that would go on to reshape all of mathematics, had been an act of plagiarism.
这封信一劳永逸地证明了康托尔 1874 年发表的那篇后来重塑所有数学的著名论文是一种抄袭行为。
Cantor was born in St. Petersburg, Russia, in 1845.
康托尔 1845 年出生于俄罗斯圣彼得堡。
When he was 11 years old, his father got sick, and the family moved to Germany to escape the dangerous Russian winters.
当他 11 岁时，他的父亲生病了，全家搬到了德国，以躲避危险的俄罗斯冬天。
Cantor would live there his entire life and eventually lose any trace of an accent.
坎托一生都住在那里，最终失去了任何口音的痕迹。
But he never felt totally comfortable in his adoptive home.
但他在收养的家里从来没有感到完全舒服。
As Cantor’s father continued to decline, he placed all his hopes on the eldest of his six children.
随着康托尔父亲的身体日渐衰弱，他把所有的希望都寄托在了六个孩子中的老大身上。
For Cantor’s confirmation, his father wrote the 15-year-old boy a letter, warning him that many promising talents are defeated by those who resist their ideas — that only an unshakable religious conviction would prevent him from becoming another “so-called ruined genius.” In order to meet his potential as a “shining star on the horizon of science,” he would have to persevere in the face of his detractors.
为了得到康托的确认，他的父亲给这个15岁的男孩写了一封信，警告他，许多有前途的天才都会被那些抵制他们想法的人所击败——只有不可动摇的宗教信仰才能阻止他成为另一个“所谓的毁掉的天才”。为了发挥他作为“科学地平线上一颗闪亮的明星”的潜力，他必须面对批评者坚持不懈。
Cantor carried his father’s letter with him for the rest of his life.
康托尔一生都带着父亲的信。
He internalized its heroic view of intellectual defiance, and he soon found a place to direct his own talent: mathematics.
他内化了其对知识的反抗的英雄主义观点，并且很快就找到了发挥自己才能的地方：数学。
As he put it, math was the field “toward which an unknown, secret voice calls him.” At 18, when his father died, he used his inheritance to enroll in the University of Berlin, one of the great capitals of mathematics.
正如他所说，数学是“一个未知的秘密声音向他召唤的领域”。 18 岁时，他的父亲去世，他利用继承的遗产入读数学伟大之都之一的柏林大学。
There, a conflict was beginning to simmer.
在那里，一场冲突开始酝酿。
Georg Cantor constantly sought to make his mark in mathematics.
乔治·康托 (Georg Cantor) 不断寻求在数学领域留下自己的印记。
But by the 1890s, his ambition had led him into a state of major depression.
但到了 1890 年代，他的野心让他陷入了严重的抑郁状态。
The issue was infinity.
问题是无限的。
Mathematicians had invented the abstraction millennia ago to deal with the problem that, for any number you name, you can always name a bigger one.
数学家在几千年前就发明了这一抽象概念，以解决这样的问题：对于你说出的任何数字，你总是可以说出一个更大的数字。
But infinity came with its own problems.
但无穷也有其自身的问题。
The ancient Greek philosopher Zeno used it to concoct all sorts of paradoxes.
古希腊哲学家芝诺用它来炮制各种悖论。
When infinity entered the picture, straightforward concepts like size and addition seemed to break down.
当无穷大出现时，像大小和加法这样简单的概念似乎就崩溃了。
Infinity also presented a religious challenge.
无限也提出了宗教挑战。
Christian theology decreed that God must be greater than any of his creations — the only true infinity, bigger than any number.
基督教神学规定，上帝必须比他的任何创造物都伟大——唯一真正的无限，比任何数字都大。
If everyday mathematicians could control this unquantifiable quantity, it would be an affront to God, and thus the authority of the Church.
如果日常数学家能够控制这个无法量化的量，这将是对上帝的侮辱，从而也是对教会权威的侮辱。
For thousands of years, mathematicians evaded these hazards by agreeing that infinity is just a useful trick, not a valid mathematical entity.
几千年来，数学家们都认为无穷大只是一个有用的技巧，而不是一个有效的数学实体，从而避免了这些危险。
As the great mathematician Carl Friedrich Gauss put it in a letter from 1831, infinity was nothing more than a “façon de parler” — a figure of speech.
正如伟大数学家卡尔·弗里德里希·高斯 (Carl Friedrich Gauss) 在 1831 年的一封信中所说，无穷大只不过是一种“façon de parler”——一种修辞手法。
But within a few decades, infinity became harder to ignore.
但几十年内，无穷变得更难以忽视。
Mathematicians were starting to revisit their most fundamental concepts, hoping to make them more precise.
数学家开始重新审视他们最基本的概念，希望使它们更加精确。
Even their understanding of what numbers were, they began to realize, lay on shaky foundations.
他们开始意识到，甚至他们对数字的理解也建立在不稳固的基础上。
Until then, they had only thought about numbers as the answers they got when they solved equations in algebra: whole numbers, fractions, square roots.
在那之前，他们只将数字视为解决代数方程时得到的答案：整数、分数、平方根。
Now some of them wanted to explore how these different species all related to one another, and whether there were other species out there to discover.
现在，他们中的一些人想探索这些不同的物种如何相互关联，以及是否还有其他物种有待发现。
Richard Dedekind spent decades trying to understand math’s most basic objects.
理查德·戴德金花费了数十年的时间试图理解数学最基本的对象。
One of his most famous papers was titled: “What Are Numbers and What Should They Be?”
他最著名的论文之一的标题是：“数字是什么以及它们应该是什么？”
Among these explorers was a quiet German mathematician named Richard Dedekind.
这些探险家中有一位安静的德国数学家，名叫理查德·戴德金。
In 1858, he found a way to rigorously define the real numbers — any number that appears on the number line.
1858 年，他找到了一种严格定义实数（数轴上出现的任何数字）的方法。
But he didn’t share his finding.
但他没有分享他的发现。
A slow and methodical thinker, he preferred to discuss his results with others until he was sure he was right.
作为一个缓慢而有条理的思考者，他更喜欢与其他人讨论他的结果，直到他确信自己是对的。
In 1870, meanwhile, Cantor, unaware of Dedekind’s work, had finished his graduate studies and was starting to examine practical questions about how certain equations behave.
与此同时，1870 年，康托尔完成了研究生学业，并开始研究有关某些方程如何表现的实际问题，他并不知道戴德金的工作。
He wasn’t yet interested in philosophical questions about the nature of numbers, but his work led him to come up with his own definition of the real numbers.
他对有关数字本质的哲学问题还不感兴趣，但他的工作使他提出了自己对实数的定义。
In early 1872, Dedekind and Cantor independently published their results.
1872年初，戴德金德和康托尔独立发表了他们的结果。
They had both done something radical: They’d redefined the number line.
他们都做了一些激进的事情：他们重新定义了数轴。
Before their papers, mathematicians had assumed that even though the number line might look like a continuous object, if you zoomed in far enough, you’d eventually find gaps.
在发表论文之前，数学家们假设，尽管数轴看起来像是一个连续的物体，但如果放大得足够远，最终还是会发现间隙。
Take the stretch of the number line between zero and 1.
将数轴延伸到 0 到 1 之间。
It contains infinitely many fractions: For any two fractions, you can always zoom in to find another one between them.
它包含无限多个分数：对于任意两个分数，您始终可以放大以找到它们之间的另一个分数。
But no matter how much you zoom in, there are certain numbers, like $latex \sqrt{2}$, that you’ll never reach.
但无论你放大多少，都有一些数字是你永远无法达到的，比如 $latex \sqrt{2}$。
There are gaps — the infinity is broken.
有间隙——无限被打破。
In their 1872 papers, though, Cantor and Dedekind had found a way to construct a number line that was complete.
然而，在 1872 年的论文中，康托尔和戴德金找到了一种构造完整数轴的方法。
No matter how much you zoomed in on any given stretch of it, it remained an unbroken expanse of infinitely many real numbers, continuously linked.
无论你把它的任何给定范围放大多少，它仍然是一个由无限多个实数组成的不间断的区域，并且连续相连。
Suddenly, the monstrosity of infinity, long feared by mathematicians, could no longer be relegated to some unreachable part of the number line.
突然之间，数学家们长期以来所恐惧的无穷大的怪物，再也不能被归入数轴上某个遥不可及的部分了。
It hid within its every crevice.
它隐藏在每一个缝隙里。
That summer, both Cantor and Dedekind spent their holiday in the scenic lakefront village of Gersau, Switzerland, where they crossed paths for the first time and took a long walk together to discuss their ideas.
那年夏天，康托尔和戴德金德在瑞士盖尔绍风景秀丽的湖畔村庄度假，在那里他们第一次相遇，一起散步很长一段时间，讨论他们的想法。
A print from the 1890s of the Swiss village of Gersau, where Cantor and Dedekind met by chance and quickly became friends.
一幅 1890 年代瑞士盖尔绍村庄的照片，坎托和戴德金在那里偶然相遇，并很快成为了朋友。
To any spectator, the two men strolling along the lakeshore would have seemed an odd match.
对于任何旁观者来说，这两个人沿着湖岸漫步都会显得很奇怪。
At 27, Cantor was tall, broad-shouldered, and boisterous.
27 岁的康托身材高大，肩膀宽阔，性格活泼。
He reveled in attention from his peers, but beneath it all, he was deeply anxious about how they perceived him.
他陶醉于同侪的关注，但在这一切之下，他对他们如何看待他深感焦虑。
It made him work fast, trying to publish quickly and often.
这让他工作速度加快，试图快速、频繁地发表文章。
Dedekind, on the other hand, was 13 years older than Cantor but much shorter and more reserved.
另一方面，戴德金比康托大 13 岁，但个子矮得多，也更内向。
And he shared none of Cantor’s urgency to publish.
他并不像康托那样急于出版。
In fact, over the course of his life, he would publish relatively little.
事实上，在他的一生中，他发表的文章相对较少。
But they hit it off at once.
但他们一拍即合。
In letters they wrote later, both reminisced again and again about that beautiful day spent discussing math by the lake.
在他们后来写的信中，两人一次又一次地回忆起在湖边讨论数学的美好一天。
They’d found in each other a partner, a friend.
他们在彼此身上找到了伴侣、朋友。
Ever since he can remember, Demian Goos has deeply cared about the rules.
从他记事起，德米安·古斯就非常关心规则。
In 2008, when he was 17 years old, he and his family moved from Germany, where he’d grown up, to Argentina, where his mother was from.
2008年，当他17岁的时候，他和家人从他长大的德国搬到了他母亲的家乡阿根廷。
There, Goos decided to take up refereeing.
在那里，古斯决定从事裁判工作。
“I enjoyed playing soccer with friends,” he said — but even more than he enjoyed playing, “I always felt annoyed by injustice in sports.
“我喜欢和朋友一起踢足球，”他说，但比他更喜欢踢球更重要的是，“我总是对体育运动中的不公正感到恼火。
When I was watching a match and they got a call wrong, I wanted to contribute to getting it right.”
当我观看一场比赛时，他们接到了错误的判罚，我想为纠正错误做出贡献。”
It gave him the chance to turn his convictions into action.
这给了他将信念转化为行动的机会。
For the next 15 years — during his time as an undergraduate, graduate student, and postdoctoral researcher and lecturer in mathematics at the National University of Rosario — he refereed professional games for a major regional soccer tournament.
在接下来的 15 年里，他在罗萨里奥国立大学担任本科生、研究生、博士后研究员和数学讲师期间，担任了重大地区足球锦标赛的职业比赛裁判。
One time, he recalled, a fan in the crowd flashed a machete at him, an oblique threat.
他回忆说，有一次，人群中的一名球迷向他挥舞着一把砍刀，这是一种间接的威胁。
But when the fan’s team committed a foul on the following play, Goos didn’t flinch.
但当球迷的球队在接下来的比赛中犯规时，古斯并没有退缩。
He simply took a deep breath and pulled out his red card.
他只是深吸了一口气，拿出了红牌。
“Refereeing was a really formative experience,” he said.
“当裁判是一次真正的成长经历，”他说。
“I don’t back down when people try to intimidate me.”
“当人们试图恐吓我时，我不会退缩。”
Demian Goos, a German Argentinian mathematician and journalist, has always felt most at home in Argentina.
德裔阿根廷数学家兼记者德米安·古斯 (Demian Goos) 在阿根廷一直感到宾至如归。
Here, he drinks mate, a traditional herbal tea from South America.
在这里，他喝的是马黛茶，一种来自南美的传统花草茶。
Though he enjoyed mathematics research, Goos was most drawn to the stories behind the theorems.
尽管古斯喜欢数学研究，但他最感兴趣的是定理背后的故事。
He spent his free time reading about the history of mathematical ideas and dramatically recounting the stories he’d learned for his peers at the university café, which they took to calling his “office.” As a postdoc, he sometimes took his students outside to illustrate math concepts, like optimization algorithms or chaotic systems, through interpretive dance.
他利用业余时间阅读数学思想的历史，并戏剧性地向同学们讲述他在大学咖啡馆里学到的故事，他们称之为他的“办公室”。作为一名博士后，他有时会带着学生到外面通过解释性舞蹈来说明数学概念，例如优化算法或混沌系统。
Many students enjoyed it, he said, but some professors warned him against using such unorthodox methods.
他说，许多学生喜欢它，但一些教授警告他不要使用这种非正统的方法。
“They probably thought they could scare me,” Goos said.
“他们可能认为他们可以吓到我，”古斯说。
“They hadn’t heard the machete story.”
“他们没有听说过砍刀的故事。”
In 2020, while still a postdoc, he became ill and had to travel back to Germany frequently for treatment.
2020年，当他还是博士后时，他就生病了，不得不频繁返回德国接受治疗。
A couple years later, he moved back full time.
几年后，他全职搬回来。
Once he finished his postdoc and was in better health, he decided it was time to leave academia and pursue his love for storytelling.
当他完成博士后并且健康状况有所改善后，他决定是时候离开学术界并追求他对讲故事的热爱了。
And so, in early 2023, he started a science journalism fellowship at the Free University of Berlin, where he focused on developing a podcast.
因此，2023 年初，他在柏林自由大学设立了科学新闻奖学金，专注于开发播客。
He wanted to tell the most gripping stories in the history of mathematics.
他想讲述数学史上最扣人心弦的故事。
And he had an idea of where to start.
他知道从哪里开始。
“Since I’m the emotional guy I am, I focused on the most emotional story ever,” he said — the story of how infinity became real and led to the birth of set theory, which offered a new foundation for all of modern math.
“因为我是一个情绪化的人，所以我专注于有史以来最情绪化的故事，”他说——这个故事讲述了无穷大如何成为现实并导致集合论的诞生，这为所有现代数学提供了新的基础。
“It pushes our understanding of mathematics to its limit,” Goos said.
“它将我们对数学的理解推向了极限，”古斯说。
“You have to say goodbye to mathematical intuition and just be open and receptive to all the shenanigans that you will encounter there.”
“你必须告别数学直觉，对你将在那里遇到的所有恶作剧保持开放和接受的态度。”
Goos produces a podcast about the history of math and science.
古斯制作了一个关于数学和科学史的播客。
It was his work on this podcast that led him to uncover new evidence in the story of what really happened between Cantor and Dedekind.
正是他在这个播客中的工作让他在康托和戴德金之间真实发生的故事中发现了新的证据。
He had learned in school that Cantor was the sole founder of set theory — and that it all started with a proof he published in 1874.
他在学校了解到康托尔是集合论的唯一创始人，而这一切都始于他于 1874 年发表的证明。
In that proof, Cantor showed that there are different sizes of infinity, putting to bed the notion that infinity was merely a piece of mathematical trickery.
在该证明中，康托证明了无穷大有不同的大小，从而推翻了无穷大只是数学诡计的观念。
Goos began research for a podcast about Cantor’s discovery.
古斯开始研究关于康托尔发现的播客。
But he soon found that the true story was more complicated than he’d been told.
但他很快发现真实的故事比他听说的要复杂。
“My approach originally was to tell the story everybody tells.
“我最初的方法是讲每个人都讲的故事。
It’s a beautiful story,” he said.
这是一个美丽的故事，”他说。
“But it’s a wrong story.
“但这是一个错误的故事。
It’s not really what happened.”
这并不是真正发生的事情。”
The true story was that Cantor wasn’t a lone genius.
真实的故事是，康托并不是一个孤独的天才。
He had a partner — at least for a time.
他有一个伴侣——至少在一段时间内是这样。
Whenever Cantor met like-minded mathematicians, he was known to court them eagerly.
每当康托遇到志同道合的数学家时，他都会热切地向他们求爱。
He would show up at a collaborator’s residence at daybreak, excited to discuss some new idea he’d had, sometimes waiting for hours until they woke up.
他会在黎明时出现在合作者的住所，兴奋地讨论他的一些新想法，有时会等待几个小时直到他们醒来。
So it was with Dedekind.
戴德金也是如此。
After their 1872 encounter in Gersau, Cantor took every opportunity to ask the older mathematician for advice.
1872 年他们在盖尔绍相遇后，康托利用一切机会向这位年长的数学家寻求建议。
In November 1873, Cantor began an exchange that would forever alter the course of human knowledge.
1873 年 11 月，康托开始了一次永远改变人类知识进程的交流。
“Allow me to put a question to you,” he wrote to Dedekind in a hastily penned letter.
“请允许我问你一个问题，”他在一封仓促写下的信中写给戴德金。
“It has a certain theoretical interest for me, but I cannot answer it myself; perhaps you can.”
“这对我来说有一定的理论兴趣，但我自己无法回答；也许你可以。”
Cantor had found an outlet for the zealous drive his father had instilled: the infinite nature of the number line.
康托尔为他父亲灌输的热情找到了一个出口：数轴的无限本质。
“He had a very strong sense of mission,” said José Ferreirós , a historian and philosopher of mathematics at the University of Seville in Spain.
“他有非常强烈的使命感，”西班牙塞维利亚大学历史学家和数学哲学家何塞·费雷罗斯 (José Ferreirós) 说。
“He was convinced that the introduction of actual infinity was going to change not only mathematics, but science in general.” To Cantor, this kind of infinity didn’t contradict God’s supremacy.
“他坚信，实际无穷的引入不仅会改变数学，而且会改变整个科学。”对于康托尔来说，这种无限性并不与上帝的至高无上相矛盾。
It just meant that rather than being remote and unknowable, God was everywhere, residing between all things.
它只是意味着上帝不是遥远的、不可知的，而是无处不在的，存在于万物之间。
DEDEKIND TO CANTOR 14 July 1873 “ I have often and with great pleasure recalled the beautiful day in Gersau and Beckenried when I had the pleasure of making your personal acquaintance.
戴德金致康托尔 1873 年 7 月 14 日 “我经常非常高兴地回忆起在盖尔绍和贝肯里德度过的美好的一天，当时我有幸结识了您。
”
”
He began studying the real numbers as a single, infinite package, asking questions no one had thought to ask before.
他开始研究实数作为一个单一的、无限的包，提出以前没有人想到过的问题。
Was there a difference between the infinity signaled by the three dots in 1, 2, 3, …
1、2、3……中的三个点所表示的无穷大之间有区别吗？
, and the one built into the mysterious continuum of the number line?
，以及内置于神秘的数轴连续统中的那个？
In other words, were there more real numbers than whole numbers?
换句话说，实数比整数多吗？
On its face, the question seemed nonsensical.
从表面上看，这个问题似乎毫无意义。
What would it even mean for these infinite sets to be different sizes?
这些无限集的大小不同意味着什么？
He asked Dedekind whether the two sets of numbers could be put in “one-to-one correspondence” — a pairing of every real number with its own distinct whole number.
他问戴德金德这两组数字是否可以“一一对应”——每个实数与其自己不同的整数配对。
He’d managed to do this, he wrote, for a different set: He’d proved that the rational numbers (numbers that can be written as a fraction) could each be assigned a unique whole number, without leaving any numbers left over.
他写道，他已经成功地为不同的集合做到了这一点：他证明了有理数（可以写成分数的数字）可以为每个有理数分配一个唯一的整数，而不会留下任何数字。
That is, even though there appeared to be far more rational numbers than whole numbers, the two sets were actually the same size.
也就是说，尽管有理数的数量看起来比整数多得多，但这两个集合实际上大小相同。
Both were therefore what mathematicians would later call “countable.”
因此，两者都是数学家后来所说的“可数”。
But Cantor couldn’t figure out how to compare the whole numbers to the real numbers in the same way.
但康托无法弄清楚如何以同样的方式将整数与实数进行比较。
Dedekind quickly replied that neither could he — but that he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted.
戴德金很快回答说他也不能——但他已经证明了代数数（解决代数问题时得到的数）是可以计算的。
“I would not have written all this,” Dedekind wrote to Cantor in closing, “if I did not consider it possible that one or the other remark might be useful to you.”
戴德金在给康托尔的最后信中写道：“如果我没有考虑到其中的某一句话可能对你有用，我就不会写下这一切。”
From there, the mathematical volley continued.
从那时起，数学竞赛继续进行。
Energized by Dedekind’s progress, Cantor spent the following days plugging away at the remaining question — the real numbers.
在戴德金的进步的鼓舞下，康托尔在接下来的几天里一直致力于解决剩下的问题——真实数字。
Could he finally show that, unlike the algebraic numbers, they were a bigger infinity than the whole numbers?
他最终能否证明，与代数数不同，它们是比整数更大的无穷大？
On December 7, 1873, he wrote to Dedekind that he thought he’d finally succeeded: “But if I should be deceiving myself, I should certainly find no more indulgent judge than you.” He laid out his proof.
1873 年 12 月 7 日，他写信给戴德金，他认为自己终于成功了：“但如果我欺骗自己，我肯定找不到比你更宽容的法官了。”他列出了他的证据。
But it was unwieldy, convoluted.
但它很笨拙、令人费解。
Dedekind replied with a way to simplify Cantor’s proof, building a clearer argument without losing any rigor or accuracy.
戴德金给出了一种简化康托尔证明的方法，在不失去任何严谨性或准确性的情况下构建更清晰的论证。
Meanwhile Cantor, before he’d received Dedekind’s letter, sent him a similar idea for how to streamline the proof, though he hadn’t worked out the details the way Dedekind had.
与此同时，康托在收到戴德金的信之前，向他发送了一个关于如何简化证明的类似想法，尽管他没有像戴德金那样弄清楚细节。
Cantor considered what he had in hand: two sets, both infinite, but one somehow larger than the other.
康托考虑了他手头的东西：两组，都是无限的，但不知何故，一组比另一组大。
The implications were revolutionary.
其影响是革命性的。
He began to dream of not one infinity, but an entire hierarchy of them.
他开始梦想的不是一个无限，而是一整套的层次结构。
And if infinities could be so concretely compared, then they had to be real, not just figures of speech.
如果无穷大可以如此具体地进行比较，那么它们一定是真实的，而不仅仅是修辞手法。
His proof, he realized, had the potential to shake the math world to its core.
他意识到，他的证明有可能从根本上动摇数学世界。
But not without angering some of its most prominent figures.
但这并非没有激怒一些最著名的人物。
One of those figures was Leopold Kronecker, a mathematical ideologue who detested infinity.
其中一位人物是利奥波德·克罗内克 (Leopold Kronecker)，他是一位厌恶无穷的数学理论家。
He didn’t believe in the number line’s packed nooks and crannies.
他不相信数轴上充满了角落和缝隙。
According to the mathematician Ferdinand von Lindemann, who proved that π isn’t algebraic — you can never pose an ordinary algebra problem where π is the answer — Kronecker once told him his work was worthless, since such “transcendental” numbers didn’t exist.
数学家费迪南德·冯·林德曼（Ferdinand von Lindemann）证明了 π 不是代数——你永远无法提出一个以 π 为答案的普通代数问题——克罗内克曾经告诉他，他的工作毫无价值，因为这样的“超越”数字并不存在。
To Leopold Kronecker, infinity had no place in mathematics.
对于利奥波德·克罗内克来说，无穷大在数学中没有地位。
When Cantor challenged this belief, Kronecker set out to destroy his reputation and stop him from publishing.
当康托尔挑战这一信念时，克罗内克开始破坏他的声誉并阻止他出版。
Kronecker was also a major gatekeeper in the world of math.
克罗内克也是数学世界的主要看门人。
He was on the editorial board of Crelle’s Journal , one of the world’s preeminent math publications.
他是世界上最杰出的数学出版物之一《Crelle’s Journal》的编辑委员会成员。
And he never hesitated to use his enormous influence to push his reactionary agenda.
他毫不犹豫地利用自己的巨大影响力来推行他的反动议程。
Often, he would decide which results would reach other mathematicians quickly — or at all.
通常，他会决定哪些结果会很快（或者根本不会）传给其他数学家。
Cantor, after discussing his work with his mentor Karl Weierstrass, wanted to publish the findings in Crelle.
康托在与导师卡尔·威尔斯特拉斯讨论了他的工作后，想在《克雷尔》上发表研究结果。
There, he figured, he’d be able to bring infinity into the mainstream.
他认为，在那里，他能够将无限带入主流。
To reveal the mind of God to the entire world.
向全世界揭示神的心意。
To become a shining star on math’s horizon.
成为数学地平线上一颗闪亮的明星。
Cantor’s sense of mission, that “secret voice” within him, began to swell.
康托的使命感，即他内心的“秘密声音”，开始膨胀。
Cantor had a good relationship with Kronecker.
康托尔与克罗内克关系良好。
But several years before, Dedekind had beaten Kronecker to a major result, and Kronecker’s dislike for him was well known.
但几年前，戴德金击败了克罗内克，取得了重大成绩，而克罗内克对他的厌恶是众所周知的。
If Cantor submitted a paper co-authored with Kronecker’s nemesis — a paper that openly declared that multiple sizes of infinity exist — it might never get published.
如果康托提交了一篇与克罗内克的宿敌共同撰写的论文——一篇公开宣称存在多种无穷大尺寸的论文——它可能永远不会发表。
The first was to build a mathematical Trojan horse.
第一个是构建数学特洛伊木马。
Weierstrass had been most excited about the proof that algebraic numbers are countable.
韦尔斯特拉斯对代数数可数的证明最为兴奋。
(He would later use that result to prove a theorem of his own.) So Cantor chose a misleading title that only mentioned algebraic numbers.
（他后来用这个结果来证明他自己的定理。）因此康托选择了一个仅提到代数数的误导性标题。
CANTOR TO DEDEKIND 29 November 1873 “ Allow me to put a question to you.
康托致戴德金 1873 年 11 月 29 日 “请允许我向您提出一个问题。
It has a certain theoretical interest for me, but I cannot answer it myself; perhaps you can, and would be so good as to write me about it.
它对我有一定的理论兴趣，但我自己无法回答；也许你可以，并且愿意给我写信告诉我这件事。
It is as follows.
如下。
”
”
But he saw that proof — Dedekind’s proof — as a decoy, a wedge he could use to pry open the forbidden gates of infinity.
但他认为这个证明——戴德金的证明——是一个诱饵，一个他可以用来撬开无限的禁忌之门的楔子。
Writing his paper, Cantor put the proof about algebraic numbers first.
康托在写论文时首先提出了代数数的证明。
Below it, he added his own proof that the real numbers cannot be counted — Dedekind’s simplified version of it, that is.
在其下方，他添加了自己的证明，证明实数无法计算——即戴德金的简化版本。
Cantor downplayed this second section’s true import.
康托淡化了第二部分的真正重要性。
“He deliberately chose a wording that would not sound suspicious to Kronecker and all those who hated infinity,” Goos said.
“他故意选择了一种不会让克罗内克和所有讨厌无穷的人听起来可疑的措辞，”古斯说。
Cantor’s second decision was to claim full authorship for himself.
康托尔的第二个决定是声称自己拥有全部作者权。
He carefully erased every trace of his collaborator’s contribution, including stray uses of terms that anyone in the know would recognize as Dedekind’s.
他小心翼翼地抹掉了合作者贡献的每一个痕迹，包括任何知情者都会认出戴德金的术语的误用。
In classic Cantor fashion, he slapped the paper together within a day and submitted it to Crelle .
按照经典的康托风格，他在一天内完成了论文并将其提交给克雷尔。
The following morning, Christmas Day 1873, he posted a letter to Dedekind, letting him know that Weierstrass had convinced him to publish.
第二天早上，即 1873 年圣诞节，他给戴德金寄了一封信，让他知道维尔斯特拉斯说服他发表这篇文章。
“As you will see,” he wrote, “your remarks, which I value highly, and your manner of putting some of the points were of great assistance to me.”
“正如你将看到的，”他写道，“我非常重视你的言论，以及你提出某些观点的方式对我有很大帮助。”
The first evidence of Cantor’s deception was uncovered in the early 20th century by another great German mathematician.
康托尔欺骗的第一个证据是在 20 世纪初由另一位伟大的德国数学家发现的。
Emmy Noether was a Dedekind acolyte.
艾米·诺特是戴德金的助手。
She would often wax poetic about his mathematical prescience.
她经常对他的数学先见之明赞叹不已。
As she liked to tell her students, “Everything is already in Dedekind.” In 1930, she was collecting all of his mathematical work into a four-volume publication when she happened on some of the letters he’d kept from his correspondence with Cantor.
正如她喜欢告诉她的学生的那样，“一切都已经在戴德金那里了。” 1930 年，她正在将他所有的数学著作整理成四卷本，这时她偶然发现了他保留的与康托通信中的一些信件。
She partnered with the French philosopher Jean Cavaillès to gather and publish them as well.
她还与法国哲学家让·卡瓦耶 (Jean Cavaillès) 合作收集并出版了这些文献。
The renowned mathematician Emmy Noether helped collect the first evidence of Cantor’s wrongdoing.
著名数学家艾美·诺特帮助收集了康托不当行为的第一个证据。
It had been over a decade since Dedekind and Cantor had died.
戴德金德和康托去世已经十多年了。
Noether and Cavaillès spent the next few years tracking down letters from Dedekind’s estate.
诺特和卡瓦耶斯在接下来的几年里一直在寻找来自戴德金遗产的信件。
In 1933, after Adolf Hitler’s rise to power, Noether, who was Jewish, fled from Germany to the U.S., where she died two years later from cancer.
1933 年，阿道夫·希特勒上台后，犹太人诺特从德国逃到美国，两年后在美国因癌症去世。
But Cavaillès completed their project in 1937.
但 Cavaillès 于 1937 年完成了他们的项目。
The correspondence as it was presented in the book was strange.
书中所呈现的信件很奇怪。
It began with a flurry of letters starting shortly after Cantor and Dedekind met in 1872.
一切始于 1872 年康托和戴德金会面后不久开始的一系列信件。
The letters, from Dedekind’s estate, included only those that he’d received, not ones he’d sent to Cantor.
这些来自戴德金庄园的信件只包括他收到的信件，不包括他寄给康托的信件。
Then the correspondence suddenly ended in January 1874, and several years of silence followed.
随后，通信于 1874 年 1 月突然结束，随后几年陷入沉默。
When the exchange resumed in 1877, Dedekind’s own letters to Cantor now appeared as well.
当 1877 年恢复交流时，戴德金自己写给康托尔的信也出现了。
Dedekind had apparently decided to keep a copy of everything he was sending to his fellow mathematician.
戴德金显然决定保留他发送给他的数学同事的所有内容的副本。
There was also a note Dedekind seemed to have written to himself after he saw Cantor’s 1874 publication in Crelle .
还有一张便条似乎是戴德金在看到康托尔 1874 年在克雷尔出版的出版物后写给自己的。
In it, he recounted how he’d sent Cantor the first proof in the paper and the revised version of the second — only to see them both appear “almost word for word” in print just a few months later under Cantor’s name alone.
在其中，他讲述了他如何将论文中的第一个证明和第二个证明的修订版发送给康托，结果几个月后，它们都“几乎逐字逐句”地以康托的名义出现在印刷品中。
Dedekind never went public with this claim, and Noether and Cavaillès didn’t comment on it.
戴德金从未公开表达过这一说法，诺特和卡瓦耶斯也没有对此发表评论。
“I think for them it was a very conscious decision not to say anything and just to let the letters speak for themselves,” said Ferreirós, the historian in Seville.
塞维利亚的历史学家费雷罗斯说：“我认为对他们来说，不说什么、让信件自己说话是一个非常有意识的决定。”
“That was the honor code of the time.”
“这就是当时的荣誉准则。”
No one else called attention to it either — at least not in print.
也没有其他人引起人们的注意——至少在印刷品上没有。
The earliest biographies of Cantor, written by his mathematical disciples, simply lauded his genius.
康托尔最早的传记由他的数学弟子撰写，只是简单地赞扬了他的天才。
Video : For thousands of years, mathematicians avoided the concept of infinity.
视频：数千年来，数学家一直回避无穷大的概念。
Then, in 1874, Georg Cantor published a four-page paper that changed mathematics forever.
然后，1874 年，乔治·康托 (Georg Cantor) 发表了一篇四页纸的论文，永远改变了数学。
Hidden inside was a shocking claim: some infinities are larger than others.
其中隐藏着一个令人震惊的主张：一些无穷大比其他无穷大。
Today, Cantor is celebrated as the father of set theory and one of the great mathematical revolutionaries.
今天，康托被誉为集合论之父和伟大的数学革命家之一。
But newly uncovered letters reveal a far more complicated story — one involving collaboration, secrecy, and plagiarism.
但新发现的信件揭示了一个更为复杂的故事——一个涉及合作、保密和剽窃的故事。
Decades later, historians such as Ivor Grattan-Guinness revisited the Cantor-Dedekind exchange.
几十年后，艾弗·格拉坦-吉尼斯等历史学家重新审视了康托-戴德金的交流。
Grattan-Guinness tried desperately to track down the letters Dedekind had sent to Cantor in 1873 — the ones that might prove Cantor’s wrongdoing.
格拉坦-吉尼斯拼命地试图追查戴德金在 1873 年寄给康托尔的信件——这些信件可能会证明康托尔的不当行为。
They had supposedly been left in Cantor’s office at the University of Halle after his death, but now they were nowhere to be found.
据称，在康托去世后，它们被留在哈雷大学的办公室里，但现在却无处可寻。
Most likely, Grattan-Guinness concluded, they’d been lost during World War II, or in the destruction that followed once American and Soviet forces occupied Halle in 1945.
格拉坦-吉尼斯总结道，它们很可能是在第二次世界大战期间丢失的，或者是在 1945 年美国和苏联军队占领哈勒后遭受的破坏中丢失的。
Without the letters, Grattan-Guinness and his contemporaries decided not to accuse Cantor of ethical misconduct.
没有这些信件，格拉坦-吉尼斯和他的同时代人决定不指控康托的道德不当行为。
Some decided that he’d acted with Dedekind’s permission; others excused his choice, since he’d begun the exchange and come up with the first version of the more significant second proof.
一些人认为他的行为得到了戴德金的许可；另一些人认为他的行为得到了戴德金的许可。其他人原谅了他的选择，因为他已经开始交流并提出了更重要的第二个证明的第一个版本。
But when Goos learned of this history while working on his podcast in 2024, he was outraged.
但当古斯在 2024 年制作播客时了解到这段历史时，他感到愤怒。
He could only find one piece of writing that explicitly discussed Cantor’s wrongdoing.
他只能找到一篇文章明确讨论康托的不当行为。
In a 1993 paper, Ferreirós accused Cantor of stealing and publishing Dedekind’s work without credit.
在 1993 年的一篇论文中，费雷罗斯指责康托窃取并未经署名地出版了戴德金的作品。
But other Cantor biographers immediately pushed back on Ferreirós’ narrative, arguing that it was too extreme an interpretation of what had happened.
但其他康托传记作者立即反驳了费雷罗斯的叙述，认为这对所发生的事情的解释过于极端。
Besides, without Dedekind’s missing letter, there was no real proof of the supposed crime — only Dedekind’s note, written afterward.
此外，如果没有戴德金丢失的信，就没有所谓犯罪的真正证据——只有戴德金后来写的字条。
How could anyone be so sure its claims were true?
怎么会有人如此确信它的说法是真实的呢？
José Ferreirós, a historian and philosopher of mathematics at the University of Seville, was the first person to accuse Cantor of plagiarism in print.
塞维利亚大学历史学家和数学哲学家何塞·费雷罗斯是第一个指控康托在印刷品上抄袭的人。
It remained an obscure debate among historians of mathematics, and Cantor’s lone-genius mystique endured.
这在数学史学家中仍然是一个模糊的争论，而康托的孤独天才的神秘感也一直存在。
Goos wanted to tell the real story on his podcast — and to back it up.
古斯想在他的播客上讲述真实的故事 - 并支持它。
He saw one way to do that.
他看到了一种方法可以做到这一点。
But it was a long shot.
但这是一个漫长的过程。
“They were always saying the letters were lost after the war,” he said.
“他们总是说这些信件在战后丢失了，”他说。
That bothered him.
这让他很烦恼。
“There is a lot that was lost without any doubt, but it doesn’t mean that nothing else survived.”
“毫无疑问，很多东西都消失了，但这并不意味着没有其他东西幸存下来。”
Many great historians had searched for the letters and come up short.
许多伟大的历史学家都曾搜寻过这些信件，但都一无所获。
Goos had just started his research.
古斯刚刚开始他的研究。
But could all the experts have missed something?
但所有专家都可能错过了什么吗？
Cantor’s paper in Crelle didn’t make a huge splash, as mathematicians largely missed what he’d hidden between the lines.
康托尔在克雷尔的论文并没有引起巨大的轰动，因为数学家们很大程度上错过了他隐藏在字里行间的东西。
But he’d landed his first blow in what would become a lifelong assault on the mathematical status quo.
但他的第一击是对数学现状的终生攻击。
He’d published a proof, in math’s foremost journal, that infinity came in different sizes.
他在数学最重要的期刊上发表了证明，证明无穷大有不同的大小。
It would eventually force mathematicians to rethink the foundations of the field — to decide on its most fundamental rules and their consequences.
它最终将迫使数学家重新思考该领域的基础——决定其最基本的规则及其后果。
Meanwhile, Dedekind stopped replying to Cantor.
与此同时，戴德金停止回复康托尔。
For nearly three years, they didn’t correspond at all.
近三年来，他们根本没有通信。
Then, for reasons that aren’t entirely clear, Dedekind cautiously reengaged.
然后，出于尚不完全清楚的原因，戴德金谨慎地重新介入。
But this time, he kept a draft of every letter he sent: a record for safekeeping.
但这一次，他保留了他寄出的每封信的草稿：一份妥善保管的记录。
The two began discussing infinity again.
两人又开始讨论无限。
Cantor was planning to follow up on his work on infinity’s many sizes, and he wanted advice.
坎托计划继续他在无穷大的多种尺寸方面的工作，他需要建议。
His letters were now more supplicating, Dedekind’s warier.
他的信现在更加恳求，戴德金更加谨慎。
But the correspondence was productive, and Cantor soon submitted a new, more daring paper to Crelle — this time, without a disguise.
但通信很有成效，坎托很快向克雷尔提交了一篇新的、更大胆的论文——这一次，没有任何掩饰。
Kronecker revolted.
克罗内克起义了。
He used all his influence in the Berlin circle to delay the review process as much as possible.
他利用自己在柏林圈子里的一切影响力，尽可能拖延审查进程。
But after several months, Weierstrass and others interceded on Cantor’s behalf, and the paper eventually appeared in the journal.
但几个月后，维尔斯特拉斯和其他人为康托求情，这篇论文最终出现在期刊上。
DEDEKIND TO CANTOR 30 November 1873 “ I would not have written all of this if I didn’t think that one or another of these remarks might be useful to you.
DEDEKIND TO CANTOR 1873 年 11 月 30 日 “如果我不认为这些评论中的某一点或另一点可能对您有用，我就不会写下这一切。
If that is not the case, I apologize: the immediate, fresh engagement with your question took me away from my duties for a few hours and has made me talkative.
如果情况并非如此，我深表歉意：对你的问题的立即、新鲜的参与使我离开了我的职责几个小时，并且让我变得健谈。
”
”
Once again, ideas from Dedekind’s letters to Cantor appeared in the paper without credit.
戴德金德写给康托尔的信中的想法再次出现在报纸上，但没有署名。
Once again, Dedekind cut off their correspondence.
戴德金再次切断了他们的通信。
Cantor would perhaps come to regret this break with one of his only intellectual allies.
康托尔也许会对与他唯一的知识盟友之一的决裂感到遗憾。
He’d been struggling to turn the mathematical backwater of Halle into an epicenter of the field that was growing out of his work: set theory.
他一直在努力将哈雷的数学死水区变成他的工作中不断发展的领域的中心：集合论。
His best bet for achieving this was to hire Dedekind.
实现这一目标的最佳选择是聘请戴德金。
In 1882, he tried to recruit him, as if nothing had happened.
1882年，他试图招募他，就像什么都没发生一样。
Dedekind politely declined.
戴德金礼貌地拒绝了。
As Cantor continued to publish results on infinity, Kronecker worked to turn the mathematical community against him.
当康托尔继续发表关于无穷大的结果时，克罗内克努力让数学界反对他。
He called Cantor a “corruptor of the youth” and a “renegade.” When Cantor, trying to leave Halle, applied for a position at the more prestigious University of Berlin in 1883, Kronecker — a professor there — blocked his appointment.
他称康托尔是“年轻人的腐蚀者”和“叛徒”。 1883 年，当康托尔试图离开哈雷，向更有声望的柏林大学申请职位时，克罗内克——那里的教授——阻止了他的任命。
Other mathematicians, including some of Cantor’s friends, began to discourage him from publishing, too.
其他数学家，包括康托的一些朋友，也开始劝阻他发表论文。
Cantor took all of this resistance personally.
康托将所有这些阻力视为个人的。
“He has this longing for approval,” Goos said.
“他非常渴望得到认可，”古斯说。
“But it’s the very nature of doing things differently from everybody else that they won’t like it.” In 1884, Cantor was hospitalized due to a major depressive episode.
“但他们不会喜欢以与其他人不同的方式做事，这就是他们的本质。” 1884年，坎托因严重抑郁症住院。
Over time, he grew more and more isolated.
随着时间的推移，他变得越来越孤立。
“There was a pattern,” Ferreirós said.
“有一种模式，”费雷罗斯说。
“Many of his relationships with colleagues ended on bad terms.”
“他与同事的许多关系都以不愉快的方式结束。”
Cantor often appears self-assured and powerful in his later photos.
康托在他后来的照片中经常显得自信而有力。
But beneath his confident exterior, he was lonely and anxious.
但在他自信的外表下，却是孤独和焦虑。
UAHW, Rep.
UAHW，众议员。
40/VI, Nr.
40/VI，Nr。
1, Bild 75
1、图片75
Eventually, Cantor fell victim to the opposition his father had warned him about.
最终，康托成为他父亲警告他的反对派的受害者。
When he was repeatedly denied the academic posts and honors he felt he deserved, his sense of mission gave way to resentment.
当他一再被剥夺他认为应得的学术职位和荣誉时，他的使命感让位于怨恨。
His depression returned, and he was hospitalized several times over the next two decades.
他的抑郁症再次出现，在接下来的二十年里他多次住院。
In 1917, he was finally committed to a sanatorium, where he wrote his wife regularly, begging her to let him come home.
1917年，他最终被送进疗养院，在那里他定期给妻子写信，恳求她让他回家。
He died the following year.
次年他去世了。
Cantor had been forced to the margins.
康托尔被逼到了边缘。
But gradually his ideas began to gain traction among a new generation of mathematicians.
但渐渐地，他的想法开始受到新一代数学家的关注。
They saw in Cantor’s work the potential to rewrite all of mathematics from the ground up.
他们在康托的工作中看到了从头开始重写所有数学的潜力。
Goos, too, was looking to do some rewriting.
古斯也希望进行一些重写。
In his 2024 podcast, he covered the conflict between Cantor and Dedekind.
在他的 2024 年播客中，他报道了康托尔和戴德金之间的冲突。
But having failed to unearth any new evidence, he found it difficult to shift the debate.
但由于未能发现任何新证据，他发现很难扭转争论。
He turned to other projects.
他转向其他项目。
Still, he couldn’t let the story go.
尽管如此，他还是不能放弃这个故事。
He continued to dig for clues about the lost letters in his free time.
他利用空闲时间继续挖掘丢失信件的线索。
“I don’t really think there is one book left that I don’t have,” he said.
“我真的不认为我还剩下一本书，”他说。
He tracked down original sources and scoured university archives whenever he could.
他尽可能地追踪原始资料并搜寻大学档案。
“I’m talking really about primary sources that are mentioned once in a single line in a single article,” he said.
“我真正谈论的是在一篇文章中单行提到过一次的主要来源，”他说。
That’s how, in the summer of 2024, he stumbled on a partial scan of what looked like a letter from Dedekind to Cantor.
就这样，2024 年夏天，他偶然发现了一封看起来像戴德金写给康托尔的信的部分扫描件。
It was on a webpage titled “Georg-Cantor-Vereinigung” — the Georg Cantor Association.
它位于一个名为“Georg-Cantor-Vereinigung”（乔治·康托协会）的网页上。
“It’s a group of people who try to keep Cantor’s memory alive,” Goos said.
“这是一群人试图让坎托的记忆永存，”古斯说。
The letter was from 1877, long after the conflict, so Dedekind’s draft of it was already in the historical record.
这封信是 1877 年写的，距离冲突已经过去很久了，所以戴德金的草稿已经被载入史册。
But there had never been any record of the copy he’d sent to Cantor.
但从来没有任何关于他寄给康托的副本的记录。
Goos tried contacting various members of the organization but got no reply.
古斯尝试联系该组织的多名成员，但没有得到回复。
CANTOR TO DEDEKIND 7 December 1873 “ In the last days I have had the time to pursue more thoroughly the conjecture I spoke to you about; only today do I believe myself to have finished with the thing; but if I should be deceiving myself, I should certainly find no more indulgent judge than you.
康托致戴德金 1873 年 12 月 7 日 “在过去的日子里，我有时间更彻底地研究我对你所说的猜想；直到今天，我才相信自己已经完成了这件事；但如果我欺骗自己，我肯定找不到比你更宽容的法官了。
”
”
Months later, he returned to the webpage.
几个月后，他又回到了该网页。
But this time, he noticed that below the scan, the site mentioned a 2009 donation of letters from an heir.
但这一次，他注意到在扫描下方，该网站提到了 2009 年一位继承人捐赠的信件。
He tracked down who that heir could possibly be, and after poring over many family trees and other documents, he finally came across a Dr. Angelika Vahlen — Cantor’s great-granddaughter, who appeared to be living in Halle.
他追踪到了那个继承人可能是谁，在仔细研究了许多家谱和其他文件后，他终于找到了安吉莉卡·瓦伦博士——康托尔的曾孙女，她似乎住在哈雷。
When he called her, she told him that she knew nothing about mathematics (she was an archaeologist, in fact), but that she’d wanted to make whatever letters she possessed available to historians for study.
当他打电话给她时，她告诉他，她对数学一无所知（事实上，她是一名考古学家），但她想将她拥有的所有信件提供给历史学家研究。
She had given them to the University of Halle (today formally known as Martin Luther University Halle-Wittenberg), and they’d ended up with the president of the Cantor Association, a math professor named Karin Richter.
她把它们交给了哈勒大学（今天的正式名称是哈勒维滕贝格马丁路德大学），最后它们得到了康托协会主席、一位名叫卡琳·里希特的数学教授。
Goos tracked Richter down.
古斯追踪到了里克特。
Arrived at her office in March 2025.
2025 年 3 月到达她的办公室。
Opened the thin blue binder she handed him.
打开她递给他的蓝色薄活页夹。
He’d been expecting to see the later letter from Dedekind that was posted on the Cantor Association’s website.
他一直期待着看到康托协会网站上发布的戴德金德后来的来信。
It would be like his other pursuits of original sources — a good way to verify what was already known and perhaps glean some new insights.
这就像他对原始资料的其他追求一样——这是验证已知信息并或许收集一些新见解的好方法。
But here before him was the letter he’d been hoping to find for over a year.
但摆在他面前的是他一年多来一直希望找到的那封信。
He was sure of it.
他对此深信不疑。
Although Dedekind’s meticulous, ornate handwriting was somehow even more indecipherable than Cantor’s uneven scrawl, Goos could see that the pages were peppered with the phrase algebraischen Zahlen : “algebraic numbers.” And at the bottom, unmistakable, was the sign-off: “With warmest regards, your most devoted R.
尽管戴德金一丝不苟、华丽的笔迹在某种程度上比康托尔参差不齐的潦草字迹更难以辨认，但古斯可以看到书页上布满了短语 algebraischen Zahlen：“代数数”。底部明确无误地写着结束语：“致以最诚挚的问候，您最忠诚的 R.
Dedekind — Braunschweig, 30 November, 1873.”
戴德金 — 不伦瑞克，1873 年 11 月 30 日。”
Did Richter even know what she had?
里克特知道她有什么吗？
He asked for scans.
他要求进行扫描。
Richter said she’d think about it.
里克特说她会考虑一下。
The letter from Dedekind to Cantor, dated November 30, 1873, that went missing for more than a century.
戴德金于 1873 年 11 月 30 日写给康托尔的信，现已失踪一个多世纪。
In it, Dedekind provides a proof that the set of algebraic numbers is the same size as the set of whole numbers — a result that Cantor later plagiarized.
在其中，戴德金提供了一个证明，证明代数数集与整数集的大小相同——这一结果后来被康托尔剽窃。
On the train ride back home — his second five-hour trip of the day — Goos contemplated the discovery he was sitting on.
在回家的火车上——这是他当天第二次五个小时的旅程——古斯思考着他坐在上面的发现。
He knew the situation was delicate.
他知道情况很微妙。
He’d experienced German mathematicians’ dismissiveness when he brought up Cantor’s betrayal of Dedekind.
当他提起康托尔对戴德金的背叛时，他经历过德国数学家的不屑一顾。
“Pride is something that Germans don’t often feel comfortable with,” Goos said.
“德国人常常感到不自在，”古斯说。
“But we are proud of Cantor.” It would be hard to find a bigger Cantor fan than Richter, and she hadn’t seemed eager to share the scans.
“但我们为坎托感到骄傲。”很难找到比里希特更喜欢康托的人了，而且她似乎并不急于分享扫描结果。
When he called Richter’s phone number two days later, his hopes sank.
两天后，当他拨通里克特的电话号码时，他的希望破灭了。
It was no longer in service.
它不再服役。
“How do you tell somebody this?
“你怎么告诉别人这个？
You know, I talked to this lady, she didn’t seem really happy to share these letters, so I called her, and her phone doesn’t exist anymore,” he said.
你知道，我和这位女士谈过，她似乎不太乐意分享这些信件，所以我打电话给她，但她的电话已经不存在了，”他说。
“Come on, Demian, come on!” He berated himself for being too polite to pull out his phone in Richter’s office and start taking photos.
“来吧，德米安，来吧！”他责备自己太有礼貌，没有在里克特的办公室里拿出手机开始拍照。
He spent the next month contacting everyone he knew in Halle, begging them to find a way to get to Richter.
接下来的一个月，他联系了哈勒认识的每个人，恳求他们找到找到里克特的方法。
“I’m starting to think I’m going crazy,” he said.
“我开始觉得我疯了，”他说。
“Does she even exist?” Finally, one of Richter’s colleagues told him when and where she’d be giving her next lecture.
“她真的存在吗？”最后，里希特的一位同事告诉他她下一次演讲的时间和地点。
He made the 10-hour round trip again in April, and Richter explained that she’d changed phone providers.
4 月份，他再次进行了 10 个小时的往返，里希特解释说她已经更换了电话提供商。
She handed him a single scan and transcription.
她递给他一份扫描件和抄本。
It was just one letter, but it was the one that mattered.
这只是一封信，但却是最重要的一封信。
Another month later, another trek, and Richter handed over another Dedekind letter, this one from the summer of 1873.
又一个月后，里希特再次长途跋涉，交出了另一封戴德金的信，这封信是 1873 年夏天写的。
Goos didn’t know if he could afford any more of these trips: “I’m not rich,” he said.
古斯不知道他是否还能负担得起更多这样的旅行：“我并不富有，”他说。
It was time, he decided, to let the world know what he’d found .
他决定，是时候让全世界知道他的发现了。
Goos reading the scan of the missing letter.
古斯正在阅读丢失信件的扫描件。
Today, Cantor’s renown far exceeds Dedekind’s.
如今，康托尔的声誉远远超过了戴德金。
Both made major contributions to the foundations of mathematics.
两人都对数学基础做出了重大贡献。
But Cantor is often credited with wrangling infinity and inventing set theory, the language that all of modern math is now written in.
但康托尔经常因争论无穷大和发明集合论而受到赞誉，集合论是现在所有现代数学的语言。
His reputation as one of history’s greatest mathematicians has been bolstered by biographies and popular books, and he’s one of the few mathematicians known to people outside the world of math.
他作为历史上最伟大的数学家之一的声誉得到了传记和流行书籍的支撑，他是数学界之外为数不多的人们所熟知的数学家之一。
There are no English-language biographies of Dedekind.
戴德金没有英文传记。
His Wikipedia page is a quarter the length of his erstwhile friend’s.
他的维基百科页面只有他以前朋友的四分之一。
Among mathematicians — largely thanks to Noether’s efforts — he retains a reputation as a lesser-known visionary.
在数学家中——很大程度上要归功于诺特的努力——他保留了一个鲜为人知的梦想家的声誉。
“The more I learn about Dedekind, the more impressed I am,” said Joel David Hamkins , a set theorist and philosopher at the University of Notre Dame.
“我对戴德金了解得越多，印象就越深刻，”圣母大学集合论学家和哲学家乔尔·戴维·哈姆金斯 (Joel David Hamkins) 说。
“Cantor proved all these great theorems, but Dedekind was probably the greater mathematician.”
“康托证明了所有这些伟大的定理，但戴德金可能是更伟大的数学家。”
The real story behind Cantor’s 1874 paper has been sitting in the open for 90 years.
康托尔 1874 年论文背后的真实故事已经公开了 90 年。
But it isn’t the kind of story people like to tell.
但这不是人们喜欢讲的那种故事。
“Every branch of science needs a hero,” Ferreirós said.
“科学的每个分支都需要英雄，”费雷罗斯说。
“Chemistry has Lavoisier, mechanics has Newton, relativity has Einstein.
“化学有拉瓦锡，力学有牛顿，相对论有爱因斯坦。
There’s always this one, only one.
总是有这个，只有一个。
But that’s always a lie.”
但这始终是谎言。”
CANTOR TO DEDEKIND 25 December 1873 “ As you will see, your comments, which I value highly, and your manner of putting some of the points were of great assistance to me.
康托致戴德金 1873 年 12 月 25 日 “正如你所看到的，我非常重视你的评论，以及你提出某些观点的方式对我有很大帮助。
”
”
Since challenging the lie, Goos has met resistance.
自从挑战谎言以来，古斯就遇到了阻力。
When he’s shared his discovery of the lost letter, mathematicians have questioned its importance — especially in Germany.
当他分享他发现丢失的信件时，数学家们质疑了它的重要性——尤其是在德国。
He’s had trouble getting people to see why it matters.
他很难让人们明白为什么它很重要。
Their reaction echoes historians’ response to Ferreirós’ paper 30 years ago.
他们的反应与 30 年前历史学家对费雷罗斯论文的反应相呼应。
But it does matter.
但这确实很重要。
Math is often viewed as the science that lives at the safest distance from the real world and its imperfections.
数学通常被视为与现实世界及其不完美保持最安全距离的科学。
Its truths are absolute.
它的真理是绝对的。
It values beauty and elegance above all other things.
它重视美丽和优雅高于一切。
What matters is the work, the world being explored.
重要的是工作和正在探索的世界。
Everything else, including authorship and credit, is secondary.
其他一切，包括作者身份和信用，都是次要的。
But this masks the reality of how the pursuit of scientific truth works.
但这掩盖了追求科学真理的现实。
“Math is a collective enterprise,” Ferreirós said.
“数学是一项集体事业，”费雷罗斯说。
“Even in the case of set theory, you don’t have this wonderful example of a single guy inventing the whole thing.”
“即使在集合论的情况下，你也没有一个人发明整个理论的精彩例子。”
It also masks the fact that math is done by people.
它还掩盖了数学是由人完成的事实。
It’s impossible to divorce egos and opinions and personal flaws from the work itself.
将自我、观点和个人缺陷与工作本身分开是不可能的。
“Wonderful,” Goos likes to reply to those mathematicians who dismiss Cantor’s misconduct.
“太棒了，”古斯喜欢这样回答那些驳斥康托不当行为的数学家。
“The next paper you write, make it anonymous.
“你写的下一篇论文，请匿名。
Then we’ll see if it’s about the science.” When it comes to their own work, mathematicians are very concerned with credit.
然后我们会看看这是否与科学有关。”当谈到自己的工作时，数学家非常关心信用。
Many of them have a near-encyclopedic knowledge of who came up with which theorem, and who won which awards.
他们中的许多人对谁提出了哪个定理、谁获得了哪些奖项有着近乎百科全书的知识。
The revelation about Cantor’s result doesn’t undermine his legacy.
康托结果的曝光并没有损害他的遗产。
He was still the first person to prove that there are more real numbers than whole ones, which is what ultimately opened up infinity to study.
他仍然是第一个证明实数比整数更多的人，这最终开启了无限研究的大门。
“It’s really the second theorem that’s important, in my view,” Hamkins said.
“在我看来，第二个定理确实很重要，”哈姆金斯说。
And the original proof of that theorem wasn’t Dedekind’s.
该定理的原始证明并不是戴德金的。
But it’s still important to recognize Dedekind’s role in one of math’s greatest discoveries, and Cantor’s decision not to credit him.
但认识戴德金在数学最伟大的发现之一中所扮演的角色以及康托尔决定不承认他的决定仍然很重要。
Ultimately, Cantor’s choices only reduce him from hero to human — a more honest picture.
最终，康托的选择只会让他从英雄变成普通人——这是一幅更诚实的画面。
“Cantor was a man who did not easily connect to other people,” Richter said.
“坎托是一个不容易与其他人建立联系的人，”里克特说。
“It was very, very hard for Cantor.”
“这对康托来说非常非常困难。”
“He was very young, very passionate and enthusiastic,” Ferreirós said.
“他非常年轻，非常充满激情和热情，”费雷罗斯说。
“And he made a big mistake.”
“而且他犯了一个大错误。”
This, in the end, is the better story — because it’s true.
归根结底，这是一个更好的故事——因为它是真实的。
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《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。
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辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。
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版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。