内容来源：https://www.quantamagazine.org/the-man-who-stole-infinity-20260225/

内容总结：

尘封百年的数学公案告破：康托尔“窃取”无穷理论关键证明

去年3月12日，当数学家兼记者德米安·古斯走进德国哈勒大学卡琳·里希特教授的办公室时，他未曾想到自己将揭开数学史上一段被掩盖的真相。在一本蓝色活页夹中，他发现了被认为早已遗失的关键信件——这封信件证实，集合论奠基人格奥尔格·康托尔1874年那篇划时代的论文，实则是剽窃了数学家理查德·戴德金的思想成果。

“无穷”之争与失落的信件

150年前，康托尔在哈勒大学发表了《数学四千年历史上最重要的论文之一》，首次提出“无穷”可作为严谨的数学对象进行研究，并证明无穷存在不同大小，由此撼动了数学根基，开创了集合论。传统叙事将康托尔塑造为孤独的天才革命者。

然而，历史学者早已发现疑点：康托尔与戴德金自1872年相识后频繁通信，戴德金在信中提供了关键证明思路。康托尔1874年论文中的两大核心证明——代数数的可数性，以及实数不可数性（即存在比自然数更大的无穷）——均直接源于戴德金的贡献。康托尔却将论文以个人名义发表，未提及戴德金。

戴德金曾私下记录此事，但从未公开指控。相关信件在二战期间被认为已毁，使得这段公案始终缺乏铁证。

执着追寻：从足球裁判到历史侦探

古斯的发现并非偶然。这位曾担任职业足球裁判的学者，以“不向胁迫低头”的准则投身数学史研究。在制作关于无穷理论的播客时，他察觉到官方叙事的不妥，并决心追查。

通过追踪康托尔曾孙女的捐赠线索，古斯最终在哈勒大学找到了戴德金写于1873年11月30日的亲笔信。信中清晰展示了戴德金对关键证明的贡献。历经波折获得扫描件后，尘封的真相得以重见天日。

被掩盖的合作与孤独的天才

史料显示，康托尔当时面临巨大压力。其理论遭到以利奥波德·克罗内克为首的数学权威强烈反对，后者掌控顶级期刊《克雷尔杂志》。为让论文顺利发表，康托尔采取了两项策略：一是将论文标题聚焦于争议较小的“代数数”，将实数不可数的革命性结论“隐藏”在内；二是彻底抹去戴德金的贡献，独自署名发表。

此举导致两位数学家关系破裂。戴德金中断通信多年，此后始终谨慎保存信件副本。康托尔则在与主流数学界的孤立对抗中，屡遭学术打压，长期受抑郁困扰，最终在精神病院去世。

重塑历史：从“英雄”到“人”

古斯的发现并未否定康托尔的伟大。他仍是第一个成功证明实数集大于自然数集、从而打开无穷层次研究大门的数学家。然而，戴德金作为关键合作者的角色必须得到承认。

“每个科学领域都需要英雄，但这总是一个谎言。”数学史家费雷罗斯指出。数学是集体事业，康托尔的故事提醒我们，即使是最抽象的真理探索，也无法脱离人性的复杂——既有天才的远见与激情，也有野心的驱使与过失。

这段更真实、更完整的历史，或许比孤独天才的神话更具启示意义。它告诉我们，数学的殿堂由无数双手共同建造，而诚实与协作，才是科学精神最坚实的基石。

中文翻译：

窃取无穷之人

去年3月12日，当德米安·古斯跟随卡琳·里希特走进她的办公室时，他首先注意到的是那尊半身像。它矗立在房间角落的高台上，描绘的是一位面容坚毅、神情肃穆的光头老绅士。古斯没有看到那个困扰了他一年多的焦虑、孤独男人的丝毫痕迹。

相反，这是历史眼中的格奥尔格·康托尔。一位思想巨擘：坚定、意志顽强，决心在同行们的喧嚣反对中掀起一场数学革命。

正是在这里，在德国的哈雷大学，康托尔于150年前发动了他的革命。就在这里，1874年，他发表了数学四千年历史中最重要的论文之一。那篇论文使一个长期以来被视为数学毒瘤、不惜一切代价也要回避的概念变得清晰具体：无穷。它迫使数学家质疑一些他们最根深蒂固的假设，动摇了数学的根基。它催生了一个新的研究领域，最终将导致整个数学学科的重写。

如今，35岁的数学家兼记者古斯来到了哈雷——从他位于美因茨的家坐火车需要五个小时——目的是查看康托尔遗产中的一些信件。他见过其中一封的扫描件，并且相当确定其他信件的内容。但他想亲眼看看。

里希特——和康托尔一样，她整个职业生涯都在这里度过，先是作为研究数学家，退休后则担任数学史讲师——示意古斯坐下。她从桌上散乱的书堆和文件中拿起一个薄薄的蓝色活页夹。里面是几十个塑料文件保护袋，每个都装着一封古老的手写信件。

古斯开始翻阅，带着考古学家进入失落已久的墓穴般的兴致审视着这些信件。然后，他翻到某一页，僵住了。他努力平复呼吸。

问题不在于笔迹。在他研究康托尔的这个阶段，他已经习惯了那种奇怪、几乎难以辨认的哥特体手写字体（kurrentschrift），德国人一直使用到大约1900年。

问题也不在于签名。他知道德国数学家理查德·戴德金在康托尔理解无穷、巩固数学基础的努力中扮演了关键角色，两人曾有过大量书信往来。

问题在于日期：1873年11月30日。

他从未见过这封信。没有人见过。人们认为它已经遗失，毁于二战的动荡，或者可能是康托尔本人销毁的。

这正是那封有力量改写康托尔遗产的信件。那封最终证明，康托尔那篇著名的、后来重塑了整个数学的1874年论文，是一次剽窃行为的信件。

思想的交汇

康托尔1845年出生于俄罗斯圣彼得堡。11岁时，他的父亲生病，全家为了躲避危险的俄罗斯严冬搬到了德国。康托尔在那里度过了一生，最终口音全无。但他从未在他的收养地感到完全自在。

随着康托尔父亲的健康状况持续恶化，他将所有希望寄托在了六个孩子中的长子身上。在康托尔的坚信礼上，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被抵制他们思想的人所击败——只有不可动摇的宗教信仰才能防止他成为另一个“所谓被毁掉的天才”。为了发挥他作为“科学地平线上一颗闪亮之星”的潜力，他必须在诋毁者面前坚持不懈。

康托尔余生都随身携带着父亲的信。他将信中关于知识反抗的英雄主义观点内化于心，并很快找到了施展才华的方向：数学。正如他所说，数学是“一个未知的、秘密的声音召唤他前往”的领域。18岁那年，父亲去世，他用遗产进入了数学重镇之一——柏林大学。

在那里，一场冲突开始酝酿。

问题在于无穷。数学家们在数千年前发明了这个抽象概念，以应对“对于你命名的任何数字，你总能命名一个更大的数字”这个问题。但无穷带来了它自身的问题。古希腊哲学家芝诺用它炮制了各种各样的悖论。当无穷介入时，大小和加法等直白的概念似乎都失效了。

无穷也提出了宗教上的挑战。基督教神学规定，上帝必须比他的任何造物都伟大——是唯一真正的无穷，大于任何数字。如果日常数学家能够控制这个不可量化的量，那将是对上帝的冒犯，从而也是对教会权威的冒犯。

数千年来，数学家们通过一致认为无穷只是一个有用的技巧，而非有效的数学实体，来规避这些危险。正如伟大的数学家卡尔·弗里德里希·高斯在1831年的一封信中所说，无穷不过是一种“说话方式”——一种修辞手法。

但几十年内，无穷变得难以忽视。

数学家们开始重新审视他们最基本的概念，希望使其更加精确。他们开始意识到，甚至他们对于“数字是什么”的理解，也建立在摇摇欲坠的基础之上。

在那之前，他们只把数字视为解代数方程时得到的答案：整数、分数、平方根。现在，他们中的一些人想要探索这些不同种类的数字如何相互关联，以及是否存在其他有待发现的种类。

在这些探索者中，有一位名叫理查德·戴德金的沉默寡言的德国数学家。1858年，他找到了一种严格定义实数（数轴上的任何数字）的方法。但他没有分享他的发现。作为一个思维缓慢而缜密的思想家，他更喜欢与他人讨论他的结果，直到确信自己是正确的。

与此同时，在1870年，康托尔在不知晓戴德金工作的情况下，完成了研究生学业，并开始研究关于某些方程行为的实际问题。他那时对关于数字本质的哲学问题还不感兴趣，但他的工作促使他提出了自己对实数的定义。

1872年初，戴德金和康托尔各自独立发表了他们的成果。

他们都做了一些激进的事情：他们重新定义了数轴。

在他们的论文之前，数学家们假设，即使数轴看起来像一个连续的对象，但如果你放大得足够远，最终会找到间隙。

以数轴上0到1之间的这段为例。它包含无穷多个分数：对于任意两个分数，你总可以放大，在它们之间找到另一个分数。但无论你放大多少，总有一些数字，比如$latex \sqrt{2}$，是你永远无法达到的。存在间隙——无穷被打破了。

然而，在他们1872年的论文中，康托尔和戴德金找到了一种构建一个完整数轴的方法。无论你在数轴的任何给定段落上放大多少，它仍然是一片由无穷多个实数组成的、连续相连的、不间断的广阔领域。

突然间，数学家们长期恐惧的无穷这个怪物，再也无法被放逐到数轴某个遥不可及的部分。它隐藏在它的每一个缝隙中。

那年夏天，康托尔和戴德金都在风景如画的瑞士湖畔村庄格尔绍度假，他们第一次相遇，并一起散步很长时间，讨论他们的想法。

彩印照片收藏

在任何旁观者看来，这两个沿着湖岸散步的人似乎并不般配。27岁的康托尔身材高大，肩膀宽阔，精力充沛。他陶醉于同行的关注，但在这一切之下，他对自己在同行眼中的形象深感焦虑。这促使他工作迅速，试图频繁快速地发表成果。另一方面，戴德金比康托尔年长13岁，但身材矮小得多，也更加内敛。他完全没有康托尔那种发表的紧迫感。事实上，在他的一生中，他发表的成果相对较少。

但他们一见如故。在他们后来写的信中，两人都一再回忆起在湖边讨论数学的那个美好日子。他们在彼此身上找到了伙伴，朋友。

但这并未持久。

追寻一个故事

德米安·古斯自记事起就非常在乎规则。2008年，17岁的他和家人从他长大的德国搬到了他母亲的故乡阿根廷。在那里，古斯决定从事裁判工作。“我喜欢和朋友踢足球，”他说——但比踢球更甚的是，“我总是对体育中的不公正感到恼火。当我看比赛时，如果他们判罚错了，我就想出力把它纠正过来。”

这给了他机会将信念转化为行动。在接下来的15年里——在他作为本科生、研究生、博士后研究员以及在罗萨里奥国立大学担任数学讲师期间——他为一个重要的地区足球锦标赛担任职业比赛的裁判。他回忆说，有一次，人群中一名球迷向他挥舞砍刀，这是一种隐晦的威胁。但当该球迷的球队在接下来的比赛中犯规时，古斯没有退缩。他只是深吸一口气，掏出了红牌。

“裁判工作是一次真正塑造性格的经历，”他说。“当人们试图恐吓我时，我不会退缩。”

尽管他喜欢数学研究，但古斯最被吸引的是定理背后的故事。他利用空闲时间阅读数学思想史，并在大学咖啡馆里为同行们生动地讲述他学到的故事，他们戏称那里是他的“办公室”。作为博士后，他有时会带学生到户外，通过即兴舞蹈来阐释数学概念，比如优化算法或混沌系统。他说，许多学生很喜欢，但一些教授警告他不要使用这种非正统的方法。“他们可能以为能吓住我，”古斯说。“他们没听过砍刀的故事。”

2020年，还在做博士后期间，他生病了，不得不频繁回德国治疗。几年后，他全职搬了回来。完成博士后工作且健康状况好转后，他决定是时候离开学术界，去追求他对讲故事的热爱了。于是，在2023年初，他在柏林自由大学开始了科学新闻学奖学金项目，专注于开发播客。他想讲述数学史上最引人入胜的故事。

而且他知道从哪里开始。

“鉴于我是那种感性的人，我专注于有史以来最感性的故事，”他说——关于无穷如何变得真实并导致集合论诞生的故事，集合论为所有现代数学提供了新的基础。“它将我们对数学的理解推向了极限，”古斯说。“你必须告别数学直觉，只是开放地、接纳地去面对你在那里将遇到的所有‘把戏’。”

他在学校学到，康托尔是集合论的唯一创始人——而这一切始于他1874年发表的一个证明。在那个证明中，康托尔展示了存在不同大小的无穷，终结了无穷仅仅是数学伎俩的观念。

古斯开始为关于康托尔发现的播客进行研究。但他很快发现，真实的故事比他被告知的要复杂得多。

“我最初的方法是讲述每个人都在讲的故事。那是个美丽的故事，”他说。“但那是错误的故事。那不是真正发生的事情。”

特洛伊木马

真实的故事是，康托尔并非孤身天才。他有一个伙伴——至少曾经有过。

每当康托尔遇到志同道合的数学家，他都会急切地争取他们。他会在黎明时分出现在合作者的住所，兴奋地讨论他想到的新点子，有时会等上几个小时直到他们醒来。对戴德金也是如此。1872年在格尔绍相遇后，康托尔抓住一切机会向这位年长的数学家请教。

1873年11月，康托尔开始了一段将永远改变人类知识进程的交流。“请允许我向您提出一个问题，”他在一封匆忙写就的信中给戴德金写道。“它对我有一定的理论意义，但我自己无法回答；也许您可以。”

康托尔为他父亲灌输给他的狂热驱动力找到了一个出口：数轴的无穷本质。“他有非常强烈的使命感，”西班牙塞维利亚大学的数学史学家兼哲学家何塞·费雷罗斯说。“他深信，实无穷的引入不仅将改变数学，还将改变整个科学。”对康托尔来说，这种无穷并不与上帝的至高无上相矛盾。它只是意味着，上帝并非遥不可及、不可知，而是无处不在，存在于万物之间。

他开始将实数作为一个单一的、无穷的整体来研究，提出了一些以前没人想过要问的问题。由1, 2, 3, … 这三个点表示的无穷，与内嵌于数轴神秘连续统中的无穷，有区别吗？换句话说，实数比整数多吗？

表面上看，这个问题似乎毫无意义。这些无穷集合大小不同，这到底意味着什么？

康托尔想弄清楚。

他问戴德金，这两组数字是否可以建立“一一对应”——将每个实数与一个唯一的整数配对。他写道，他已经为另一组数字做到了这一点：他证明了有理数（可以写成分数的数）中的每一个都可以分配一个唯一的整数，不会有任何数字剩余。也就是说，尽管看起来有理数比整数多得多，但这两个集合实际上大小相同。因此，两者后来都被数学家称为“可数的”。

但康托尔无法想出同样的方法来比较整数和实数。戴德金很快回复说，他也不能——但他已经证明代数数（作为代数问题解得到的数字）是可数的。“如果我不认为这个或那个评论可能对您有用，”戴德金在信末写道，“我就不会写这些了。”

从那时起，数学上的交锋继续着。受到戴德金进展的鼓舞，康托尔在接下来的几天里埋头研究剩下的问题——实数。他最终能否证明，与代数数不同，实数是一个比整数更大的无穷？

1873年12月7日，他写信给戴德金，认为他终于成功了：“但如果我是在欺骗自己，我肯定找不到比您更宽容的法官了。”他阐述了他的证明。但这个证明笨拙、复杂。戴德金回复了一个简化康托尔证明的方法，构建了一个更清晰的论证，同时不失严谨或准确性。与此同时，康托尔在收到戴德金的信之前，给他寄去了一个关于如何精简证明的类似想法，尽管他没有像戴德金那样完善细节。

康托尔思考着他手中的成果：两个集合，都是无穷的，但其中一个在某种意义上比另一个大。其影响是革命性的。他开始梦想的不是一个无穷，而是整个无穷的层级结构。如果无穷可以如此具体地比较，那么它们必须是真实的，而不仅仅是修辞手法。

他意识到，他的证明有潜力从根本上撼动数学界。但这也必然会激怒一些最杰出的人物。

其中之一就是利奥波德·克罗内克，一个憎恨无穷的数学意识形态家。他不相信数轴上挤满了角落和缝隙。根据证明了π不是代数数（你永远无法提出一个以π为答案的普通代数问题）的数学家费迪南德·冯·林德曼的说法，克罗内克曾告诉他，他的工作毫无价值，因为这样的“超越”数并不存在。

克罗内克也是数学界的主要把关人。他是世界顶级数学出版物之一《克雷尔杂志》的编委会成员。他从不犹豫利用自己的巨大影响力来推行他的反动议程。他常常决定哪些成果能迅速——或者根本能——到达其他数学家手中。

康托尔在与导师卡尔·魏尔斯特拉斯讨论了他的工作后，想在《克雷尔杂志》上发表这些发现。他认为，在那里，他能够将无穷带入主流。向全世界揭示上帝的思想。成为数学地平线上一颗闪亮的星。

康托尔的使命感，他内心的那个“秘密声音”，开始膨胀。

康托尔与克罗内克关系良好。但几年前，戴德金在一个重要成果上击败了克罗内克，克罗内克对他的厌恶众所周知。如果康托尔提交一篇与克罗内克的宿敌合著的论文——一篇公开宣称存在多种大小无穷的论文——它可能永远无法发表。

于是他做了两个决定。

第一个是建造一个数学特洛伊木马。

魏尔斯特拉斯对代数数是可数的证明最为兴奋。（他后来会用这个结果来证明他自己的一个定理。）因此，康托尔选择了一个具有误导性的标题，只提到了代数数。

但他将那个证明——戴德金的证明——视为一个诱饵，一个他可以用来撬开被禁止的无穷大门的楔子。在撰写论文时，康托尔将关于代数数的证明放在前面。在它下面，他添加了自己的证明，即实数不可数——也就是戴德金简化后的版本。康托尔淡化了这第二部分真正的重要性。“他故意选择了一种不会让克罗内克和所有憎恨无穷的人起疑的措辞，”古斯说。

康托尔的第二个决定是声称自己是唯一的作者。他小心翼翼地抹去了合作者贡献的所有痕迹，包括任何知情者都能认出是戴德金用语的零星术语。

以典型的康托尔风格，他在一天之内拼凑好论文，提交给了《克雷尔杂志》。第二天早上，1873年圣诞节，他给戴德金寄了一封信，让他知道魏尔斯特拉斯说服了他发表。“正如您将看到的，”他写道，“您的评论，我高度重视，以及您表述某些观点的方式，对我帮助极大。”

书写故事

康托尔欺骗行为的第一个证据在20世纪初由另一位伟大的德国数学家发现。埃米·诺特是戴德金的追随者。她常常诗意地赞美他的数学远见。正如她喜欢告诉学生的：“一切都已经在戴德金那里了。”1930年，她正在将戴德金所有的数学工作收集成四卷出版物，这时她偶然发现了他保存的一些与康托尔通信的信件。她与法国哲学家让·卡瓦耶合作，也收集并出版了这些信件。

此时戴德金和康托尔都已去世十多年。诺特和卡瓦耶花了接下来几年时间追踪戴德金遗产中的信件。1933年，阿道夫·希特勒上台后，身为犹太人的诺特从德国逃往美国，两年后因癌症去世。但卡瓦耶在1937年完成了他们的项目。

书中呈现的通信很奇怪。它始于1872年康托尔和戴德金相遇后不久的一连串信件。这些来自戴德金遗产的信件，只包括他收到的信，而不是他寄给康托尔的信。然后通信在1874年1月突然中断，随后是几年的沉默。当交流在1877年恢复时，戴德金自己写给康托尔的信也出现了。戴德金显然决定保留他寄给这位数学家同行的所有信件的副本。

还有一张便条，似乎是戴德金在看到康托尔1874年在《克雷尔杂志》上发表的文章后写给自己看的。其中，他叙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寄给了康托尔——结果却在仅仅几个月后看到它们“几乎逐字逐句”地以康托尔一人的名义出现在印刷品上。

戴德金从未公开提出这一主张，诺特和卡瓦耶也没有对此发表评论。“我认为对他们来说，这是一个非常刻意的决定，什么都不说，只是让信件自己说话，”塞维利亚的历史学家费雷罗斯说。“那是当时的荣誉准则。”

也没有其他人注意到这一点——至少没有在出版物中。康托尔最早的传记由他的数学门徒撰写，只是颂扬他的天才。

几十年后，像艾弗·格拉坦-吉尼斯这样的历史学家重新审视了康托尔-戴德金的通信。格拉坦-吉尼斯拼命试图追踪戴德金在1873年寄给康托尔的信件——那些可能证明康托尔不当行为的信件。据称这些信件在康托尔去世后留在了他在哈雷大学的办公室，但现在却无处可寻。格拉坦-吉尼斯得出结论，它们很可能在二战期间丢失了，或者在1945年美国和苏联军队占领哈雷后的破坏中遗失了。

由于没有这些信件，格拉坦-吉尼斯和他的同时代人决定不指控康托尔有伦理不当行为。一些人认为他是得到了戴德金的许可；另一些人则为他的选择开脱，因为是他发起了交流，并且想出了更重要的第二个证明的第一个版本。

但当古斯在2024年制作播客期间了解到这段历史时，他感到愤怒。他只找到了一篇明确讨论康托尔不当行为的文章。在1993年的一篇论文中，费雷罗斯指控康托尔剽窃并发表了戴德金的工作而未给予署名。但其他康托尔传记作者立即反驳了费雷罗斯的叙述，认为这是对所发生事件的过于

英文来源：

The Man Who Stole Infinity
When Demian Goos followed Karin Richter into her office on March 12 of last year, the first thing he noticed was the bust. It sat atop a tall pedestal in the corner of the room, depicting a bald, elderly gentleman with a stoic countenance. Goos saw no trace of the anxious, lonely man who had obsessed him for over a year.
Instead, this was Georg Cantor as history saw him. An intellectual giant: steadfast, strong-willed, determined to bring about a mathematical revolution over the clamorous objections of his peers.
It was here, at the University of Halle in Germany, that Cantor launched his revolution 150 years ago. Here, in 1874, he published one of the most important papers in math’s 4,000-year history. That paper crystallized a concept that had long been viewed as a mathematical malignancy to be shunned at all costs: infinity. It forced mathematicians to question some of their longest-held assumptions, rocking mathematics to its very foundations. And it gave rise to a new field of study that would eventually bring about a rewriting of the entire subject.
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. He’d seen a scan of one and was pretty sure he knew what the others would say. But he wanted to see them in person.
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. She lifted a thin blue binder from the scattered piles of books and papers on her desk. Inside were dozens of plastic sheet protectors, each one containing an old, handwritten letter.
Goos began flipping through, contemplating the letters with the relish of an archaeologist entering a long-lost tomb. Then he reached a particular page and froze. He struggled to catch his breath.
It wasn’t the handwriting. 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.
It wasn’t the signature. 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.
It was the date: November 30, 1873.
He’d never seen this letter before. No one had. It was believed to be lost, destroyed in the tumult of World War II or perhaps by Cantor himself.
This was the letter that had the power to rewrite Cantor’s legacy. 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.
A Meeting of Minds
Cantor was born in St. Petersburg, Russia, in 1845. When he was 11 years old, his father got sick, and the family moved to Germany to escape the dangerous Russian winters. Cantor would live there his entire life and eventually lose any trace of an accent. But he never felt totally comfortable in his adoptive home.
As Cantor’s father continued to decline, he placed all his hopes on the eldest of his six children. 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.
Cantor carried his father’s letter with him for the rest of his life. He internalized its heroic view of intellectual defiance, and he soon found a place to direct his own talent: mathematics. 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.
There, a conflict was beginning to simmer.
The issue was infinity. 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. But infinity came with its own problems. The ancient Greek philosopher Zeno used it to concoct all sorts of paradoxes. When infinity entered the picture, straightforward concepts like size and addition seemed to break down.
Infinity also presented a religious challenge. Christian theology decreed that God must be greater than any of his creations — the only true infinity, bigger than any number. If everyday mathematicians could control this unquantifiable quantity, it would be an affront to God, and thus the authority of the Church.
For thousands of years, mathematicians evaded these hazards by agreeing that infinity is just a useful trick, not a valid mathematical entity. 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.
But within a few decades, infinity became harder to ignore.
Mathematicians were starting to revisit their most fundamental concepts, hoping to make them more precise. Even their understanding of what numbers were, they began to realize, lay on shaky foundations.
Until then, they had only thought about numbers as the answers they got when they solved equations in algebra: whole numbers, fractions, square roots. 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.
Among these explorers was a quiet German mathematician named Richard Dedekind. In 1858, he found a way to rigorously define the real numbers — any number that appears on the number line. But he didn’t share his finding. A slow and methodical thinker, he preferred to discuss his results with others until he was sure he was right.
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. 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.
In early 1872, Dedekind and Cantor independently published their results.
They had both done something radical: They’d redefined the number line.
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.
Take the stretch of the number line between zero and 1. It contains infinitely many fractions: For any two fractions, you can always zoom in to find another one between them. But no matter how much you zoom in, there are certain numbers, like $latex \sqrt{2}$, that you’ll never reach. There are gaps — the infinity is broken.
In their 1872 papers, though, Cantor and Dedekind had found a way to construct a number line that was complete. 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.
Suddenly, the monstrosity of infinity, long feared by mathematicians, could no longer be relegated to some unreachable part of the number line. It hid within its every crevice.
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.
Photochrom Print Collection
To any spectator, the two men strolling along the lakeshore would have seemed an odd match. At 27, Cantor was tall, broad-shouldered, and boisterous. He reveled in attention from his peers, but beneath it all, he was deeply anxious about how they perceived him. It made him work fast, trying to publish quickly and often. Dedekind, on the other hand, was 13 years older than Cantor but much shorter and more reserved. And he shared none of Cantor’s urgency to publish. In fact, over the course of his life, he would publish relatively little.
But they hit it off at once. In letters they wrote later, both reminisced again and again about that beautiful day spent discussing math by the lake. They’d found in each other a partner, a friend.
It wouldn’t last.
In Pursuit of a Story
Ever since he can remember, Demian Goos has deeply cared about the rules. 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. There, Goos decided to take up refereeing. “I enjoyed playing soccer with friends,” he said — but even more than he enjoyed playing, “I always felt annoyed by injustice in sports. When I was watching a match and they got a call wrong, I wanted to contribute to getting it right.”
It gave him the chance to turn his convictions into action. 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. One time, he recalled, a fan in the crowd flashed a machete at him, an oblique threat. But when the fan’s team committed a foul on the following play, Goos didn’t flinch. He simply took a deep breath and pulled out his red card.
“Refereeing was a really formative experience,” he said. “I don’t back down when people try to intimidate me.”
Zack Savitsky for Quanta Magazine
Though he enjoyed mathematics research, Goos was most drawn to the stories behind the theorems. 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. Many students enjoyed it, he said, but some professors warned him against using such unorthodox methods. “They probably thought they could scare me,” Goos said. “They hadn’t heard the machete story.”
In 2020, while still a postdoc, he became ill and had to travel back to Germany frequently for treatment. A couple years later, he moved back full time. Once he finished his postdoc and was in better health, he decided it was time to leave academia and pursue his love for storytelling. And so, in early 2023, he started a science journalism fellowship at the Free University of Berlin, where he focused on developing a podcast. He wanted to tell the most gripping stories in the history of mathematics.
And he had an idea of where to start.
“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. “It pushes our understanding of mathematics to its limit,” Goos said. “You have to say goodbye to mathematical intuition and just be open and receptive to all the shenanigans that you will encounter there.”
Zack Savitsky for Quanta Magazine
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. 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.
Goos began research for a podcast about Cantor’s discovery. But he soon found that the true story was more complicated than he’d been told.
“My approach originally was to tell the story everybody tells. It’s a beautiful story,” he said. “But it’s a wrong story. It’s not really what happened.”
The Trojan Horse
The true story was that Cantor wasn’t a lone genius. He had a partner — at least for a time.
Whenever Cantor met like-minded mathematicians, he was known to court them eagerly. 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. So it was with Dedekind. After their 1872 encounter in Gersau, Cantor took every opportunity to ask the older mathematician for advice.
In November 1873, Cantor began an exchange that would forever alter the course of human knowledge. “Allow me to put a question to you,” he wrote to Dedekind in a hastily penned letter. “It has a certain theoretical interest for me, but I cannot answer it myself; perhaps you can.”
Cantor had found an outlet for the zealous drive his father had instilled: the infinite nature of the number line. “He had a very strong sense of mission,” said José Ferreirós, a historian and philosopher of mathematics at the University of Seville in Spain. “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. It just meant that rather than being remote and unknowable, God was everywhere, residing between all things.
He began studying the real numbers as a single, infinite package, asking questions no one had thought to ask before. Was there a difference between the infinity signaled by the three dots in 1, 2, 3, … , and the one built into the mysterious continuum of the number line? In other words, were there more real numbers than whole numbers?
On its face, the question seemed nonsensical. What would it even mean for these infinite sets to be different sizes?
Cantor wanted to find out.
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. 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. That is, even though there appeared to be far more rational numbers than whole numbers, the two sets were actually the same size. Both were therefore what mathematicians would later call “countable.”
But Cantor couldn’t figure out how to compare the whole numbers to the real numbers in the same way. 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. “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.”
From there, the mathematical volley continued. Energized by Dedekind’s progress, Cantor spent the following days plugging away at the remaining question — the real numbers. Could he finally show that, unlike the algebraic numbers, they were a bigger infinity than the whole numbers?
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. But it was unwieldy, convoluted. Dedekind replied with a way to simplify Cantor’s proof, building a clearer argument without losing any rigor or accuracy. 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.
Cantor considered what he had in hand: two sets, both infinite, but one somehow larger than the other. The implications were revolutionary. He began to dream of not one infinity, but an entire hierarchy of them. And if infinities could be so concretely compared, then they had to be real, not just figures of speech.
His proof, he realized, had the potential to shake the math world to its core. But not without angering some of its most prominent figures.
One of those figures was Leopold Kronecker, a mathematical ideologue who detested infinity. He didn’t believe in the number line’s packed nooks and crannies. 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.
Public Domain
Kronecker was also a major gatekeeper in the world of math. He was on the editorial board of Crelle’s Journal, one of the world’s preeminent math publications. And he never hesitated to use his enormous influence to push his reactionary agenda. Often, he would decide which results would reach other mathematicians quickly — or at all.
Cantor, after discussing his work with his mentor Karl Weierstrass, wanted to publish the findings in Crelle. There, he figured, he’d be able to bring infinity into the mainstream. To reveal the mind of God to the entire world. To become a shining star on math’s horizon.
Cantor’s sense of mission, that “secret voice” within him, began to swell.
Cantor had a good relationship with Kronecker. But several years before, Dedekind had beaten Kronecker to a major result, and Kronecker’s dislike for him was well known. 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.
So he made two decisions.
The first was to build a mathematical Trojan horse.
Weierstrass had been most excited about the proof that algebraic numbers are countable. (He would later use that result to prove a theorem of his own.) So Cantor chose a misleading title that only mentioned algebraic numbers.
But he saw that proof — Dedekind’s proof — as a decoy, a wedge he could use to pry open the forbidden gates of infinity. Writing his paper, Cantor put the proof about algebraic numbers first. Below it, he added his own proof that the real numbers cannot be counted — Dedekind’s simplified version of it, that is. Cantor downplayed this second section’s true import. “He deliberately chose a wording that would not sound suspicious to Kronecker and all those who hated infinity,” Goos said.
Cantor’s second decision was to claim full authorship for himself. 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.
In classic Cantor fashion, he slapped the paper together within a day and submitted it to Crelle. The following morning, Christmas Day 1873, he posted a letter to Dedekind, letting him know that Weierstrass had convinced him to publish. “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.”
Writing the Story
The first evidence of Cantor’s deception was uncovered in the early 20th century by another great German mathematician. Emmy Noether was a Dedekind acolyte. She would often wax poetic about his mathematical prescience. 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. She partnered with the French philosopher Jean Cavaillès to gather and publish them as well.
Ian Dagnall Computing/Alamy
It had been over a decade since Dedekind and Cantor had died. Noether and Cavaillès spent the next few years tracking down letters from Dedekind’s estate. 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. But Cavaillès completed their project in 1937.
The correspondence as it was presented in the book was strange. It began with a flurry of letters starting shortly after Cantor and Dedekind met in 1872. The letters, from Dedekind’s estate, included only those that he’d received, not ones he’d sent to Cantor. Then the correspondence suddenly ended in January 1874, and several years of silence followed. When the exchange resumed in 1877, Dedekind’s own letters to Cantor now appeared as well. Dedekind had apparently decided to keep a copy of everything he was sending to his fellow mathematician.
There was also a note Dedekind seemed to have written to himself after he saw Cantor’s 1874 publication in Crelle. 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.
Dedekind never went public with this claim, and Noether and Cavaillès didn’t comment on it. “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. “That was the honor code of the time.”
No one else called attention to it either — at least not in print. The earliest biographies of Cantor, written by his mathematical disciples, simply lauded his genius.
Zack Savitsky for Quanta Magazine
Decades later, historians such as Ivor Grattan-Guinness revisited the Cantor-Dedekind exchange. Grattan-Guinness tried desperately to track down the letters Dedekind had sent to Cantor in 1873 — the ones that might prove Cantor’s wrongdoing. 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. 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.
Without the letters, Grattan-Guinness and his contemporaries decided not to accuse Cantor of ethical misconduct. 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.
But when Goos learned of this history while working on his podcast in 2024, he was outraged. He could only find one piece of writing that explicitly discussed Cantor’s wrongdoing. In a 1993 paper, Ferreirós accused Cantor of stealing and publishing Dedekind’s work without credit. But other Cantor biographers immediately pushed back on Ferreirós’ narrative, arguing that it was too extreme an interpretation of what had happened. Besides, without Dedekind’s missing letter, there was no real proof of the supposed crime — only Dedekind’s note, written afterward. How could anyone be so sure its claims were true?
Universidad de Sevilla
It remained an obscure debate among historians of mathematics, and Cantor’s lone-genius mystique endured.
Goos wanted to tell the real story on his podcast — and to back it up. He saw one way to do that. But it was a long shot.
“They were always saying the letters were lost after the war,” he said. That bothered him. “There is a lot that was lost without any doubt, but it doesn’t mean that nothing else survived.”
Many great historians had searched for the letters and come up short. Goos had just started his research. But could all the experts have missed something?
A Lonely Existence
Cantor’s paper in Crelle didn’t make a huge splash, as mathematicians largely missed what he’d hidden between the lines. But he’d landed his first blow in what would become a lifelong assault on the mathematical status quo. He’d published a proof, in math’s foremost journal, that infinity came in different sizes. It would eventually force mathematicians to rethink the foundations of the field — to decide on its most fundamental rules and their consequences.
Meanwhile, Dedekind stopped replying to Cantor. For nearly three years, they didn’t correspond at all. Then, for reasons that aren’t entirely clear, Dedekind cautiously reengaged. But this time, he kept a draft of every letter he sent: a record for safekeeping.
The two began discussing infinity again. Cantor was planning to follow up on his work on infinity’s many sizes, and he wanted advice. His letters were now more supplicating, Dedekind’s warier. But the correspondence was productive, and Cantor soon submitted a new, more daring paper to Crelle — this time, without a disguise.
Kronecker revolted. He used all his influence in the Berlin circle to delay the review process as much as possible. But after several months, Weierstrass and others interceded on Cantor’s behalf, and the paper eventually appeared in the journal.
Once again, ideas from Dedekind’s letters to Cantor appeared in the paper without credit. Once again, Dedekind cut off their correspondence.
Cantor would perhaps come to regret this break with one of his only intellectual allies. 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. His best bet for achieving this was to hire Dedekind. In 1882, he tried to recruit him, as if nothing had happened. Dedekind politely declined.
As Cantor continued to publish results on infinity, Kronecker worked to turn the mathematical community against him. 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. Other mathematicians, including some of Cantor’s friends, began to discourage him from publishing, too.
Cantor took all of this resistance personally. “He has this longing for approval,” Goos said. “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. Over time, he grew more and more isolated. “There was a pattern,” Ferreirós said. “Many of his relationships with colleagues ended on bad terms.”
UAHW, Rep. 40/VI, Nr. 1, Bild 75
Eventually, Cantor fell victim to the opposition his father had warned him about. When he was repeatedly denied the academic posts and honors he felt he deserved, his sense of mission gave way to resentment. His depression returned, and he was hospitalized several times over the next two decades. In 1917, he was finally committed to a sanatorium, where he wrote his wife regularly, begging her to let him come home. He died the following year.
Cantor had been forced to the margins. But gradually his ideas began to gain traction among a new generation of mathematicians. They saw in Cantor’s work the potential to rewrite all of mathematics from the ground up.
A Lucky Find
Goos, too, was looking to do some rewriting. In his 2024 podcast, he covered the conflict between Cantor and Dedekind. But having failed to unearth any new evidence, he found it difficult to shift the debate. He turned to other projects.
Still, he couldn’t let the story go.
He continued to dig for clues about the lost letters in his free time. “I don’t really think there is one book left that I don’t have,” he said. He tracked down original sources and scoured university archives whenever he could. “I’m talking really about primary sources that are mentioned once in a single line in a single article,” he said.
That’s how, in the summer of 2024, he stumbled on a partial scan of what looked like a letter from Dedekind to Cantor. It was on a webpage titled “Georg-Cantor-Vereinigung” — the Georg Cantor Association. “It’s a group of people who try to keep Cantor’s memory alive,” Goos said. The letter was from 1877, long after the conflict, so Dedekind’s draft of it was already in the historical record. But there had never been any record of the copy he’d sent to Cantor. Goos tried contacting various members of the organization but got no reply.
Months later, he returned to the webpage. But this time, he noticed that below the scan, the site mentioned a 2009 donation of letters from an heir. 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.
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. 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.
Goos tracked Richter down. Arrived at her office in March 2025. Opened the thin blue binder she handed him.
He’d been expecting to see the later letter from Dedekind that was posted on the Cantor Association’s website. 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.
But here before him was the letter he’d been hoping to find for over a year. He was sure of it. 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. Dedekind — Braunschweig, 30 November, 1873.”
Did Richter even know what she had? He asked for scans. Richter said she’d think about it.
Braunschweig University Archive (G98 No. 4)
On the train ride back home — his second five-hour trip of the day — Goos contemplated the discovery he was sitting on. He knew the situation was delicate. He’d experienced German mathematicians’ dismissiveness when he brought up Cantor’s betrayal of Dedekind. “Pride is something that Germans don’t often feel comfortable with,” Goos said. “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.
When he called Richter’s phone number two days later, his hopes sank. It was no longer in service. “How do you tell somebody this? 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. “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.
He spent the next month contacting everyone he knew in Halle, begging them to find a way to get to Richter. “I’m starting to think I’m going crazy,” he said. “Does she even exist?” Finally, one of Richter’s colleagues told him when and where she’d be giving her next lecture. He made the 10-hour round trip again in April, and Richter explained that she’d changed phone providers. She handed him a single scan and transcription. It was just one letter, but it was the one that mattered.
Another month later, another trek, and Richter handed over another Dedekind letter, this one from the summer of 1873. Goos didn’t know if he could afford any more of these trips: “I’m not rich,” he said. It was time, he decided, to let the world know what he’d found.
Zack Savitsky for Quanta Magazine
A Truer Legacy
Today, Cantor’s renown far exceeds Dedekind’s.
Both made major contributions to the foundations of mathematics. But Cantor is often credited with wrangling infinity and inventing set theory, the language that all of modern math is now written in. 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.
There are no English-language biographies of Dedekind. His Wikipedia page is a quarter the length of his erstwhile friend’s. Among mathematicians — largely thanks to Noether’s efforts — he retains a reputation as a lesser-known visionary. “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. “Cantor proved all these great theorems, but Dedekind was probably the greater mathematician.”
The real story behind Cantor’s 1874 paper has been sitting in the open for 90 years. But it isn’t the kind of story people like to tell. “Every branch of science needs a hero,” Ferreirós said. “Chemistry has Lavoisier, mechanics has Newton, relativity has Einstein. There’s always this one, only one. But that’s always a lie.”
Since challenging the lie, Goos has met resistance. When he’s shared his discovery of the lost letter, mathematicians have questioned its importance — especially in Germany. He’s had trouble getting people to see why it matters. Their reaction echoes historians’ response to Ferreirós’ paper 30 years ago.
But it does matter. Math is often viewed as the science that lives at the safest distance from the real world and its imperfections. Its truths are absolute. It values beauty and elegance above all other things. What matters is the work, the world being explored. Everything else, including authorship and credit, is secondary.
But this masks the reality of how the pursuit of scientific truth works. “Math is a collective enterprise,” Ferreirós said. “Even in the case of set theory, you don’t have this wonderful example of a single guy inventing the whole thing.”
It also masks the fact that math is done by people. It’s impossible to divorce egos and opinions and personal flaws from the work itself. “Wonderful,” Goos likes to reply to those mathematicians who dismiss Cantor’s misconduct. “The next paper you write, make it anonymous. Then we’ll see if it’s about the science.” When it comes to their own work, mathematicians are very concerned with credit. Many of them have a near-encyclopedic knowledge of who came up with which theorem, and who won which awards.
The revelation about Cantor’s result doesn’t undermine his legacy. 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. “It’s really the second theorem that’s important, in my view,” Hamkins said. And the original proof of that theorem wasn’t Dedekind’s.
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. Ultimately, Cantor’s choices only reduce him from hero to human — a more honest picture. “Cantor was a man who did not easily connect to other people,” Richter said. “It was very, very hard for Cantor.”
“He was very young, very passionate and enthusiastic,” Ferreirós said. “And he made a big mistake.”
This, in the end, is the better story — because it’s true.