超有限论，一种否定无限的数学思想|克拉克|宇宙|数学思想|物理学|科学|超有限论|逻辑学

失去无限性我们能获得什么？

格雷戈里·巴伯

2026年4月29日

超有限论，一种否定无限的哲学，长期以来被视为数学异端邪说。但它也在数学及其他领域产生了新的见解。

多伦·泽尔伯格是一位数学家，他认为万物终有尽头。正如我们是有限的存在，自然也有其边界——因此，数字亦是如此。看看窗外，在其他人眼中，现实是一片连续不断的广阔，一刻接一刻地不可阻挡地向前流动，而泽尔伯格却认为并非如此。（在新标签页中打开）他看到的是一个运转的宇宙，一个独立的机器。在他周围世界流畅的运动中，他捕捉到了翻页书的细微模糊。

在泽尔伯格看来，相信无穷大就像相信上帝一样。这是一个诱人的想法，它迎合了我们的直觉，帮助我们理解各种各样的现象。但问题在于，我们无法真正观测到无穷大，因此也无法真正定义它。方程式定义了延伸到黑板之外的线条，但它们延伸到哪里呢？证明中充斥着意味深长的省略号。泽尔伯格——罗格斯大学的资深教授，组合数学领域的著名人物——认为，这些方程式和证明既“丑陋不堪”，又是错误的。他用沙哑的声音喘息着，仿佛已经筋疲力尽，说道：“这完全是胡说八道。”

他认为，从实际角度来看，无穷大是可以剔除的。“你其实并不需要它。”例如，数学家可以构建一种不涉及无穷大的微积分形式，彻底排除无穷小极限。曲线看起来或许光滑，但其中隐藏着细微的粗糙；计算机在有限的数字范围内也能很好地处理数学运算。（泽尔伯格将他自己的计算机——他称之为“Shalosh B. Ekhad”——列为论文的合作者。）泽尔伯格说，剔除无穷大之后，唯一失去的只是“根本不值得做的”数学。

大多数数学家会持相反的观点——认为泽尔伯格才是胡说八道。这不仅是因为无穷的概念对于我们描述宇宙非常有用且自然，还因为将数集（例如整数集）视为实际的、无限的对象，正是数学的核心所在，它根植于数学最基本的规则和假设之中。

至少，即便数学家们不愿将无穷视为一个实际存在的实体，他们也承认数列、形状和其他数学对象具有无限增长的潜力。理论上，两条平行线可以无限延伸；数轴的末端总可以添加另一个数字。

多伦·泽尔伯格或许是主张将无穷从数学中剔除的最积极倡导者。“无穷可能存在，也可能不存在；上帝可能存在，也可能不存在，”他说。“但在数学中，无论是无穷还是上帝，都不应该有任何容身之地。”

承蒙多伦·泽尔伯格惠允

泽尔伯格对此持不同意见。对他而言，重要的不是某件事在理论上是否可能，而是它在实践中是否可行。这意味着，不仅无穷大值得怀疑，极其巨大的数字也同样如此。以“斯奎斯数”为例，$latex e^{e^{e^{79}}}$。这是一个极其巨大的数字，至今无人能够将其写成十进制形式。那么，我们究竟能对它做出怎样的判断呢？它是整数吗？它是质数吗？我们能在自然界中找到这样的数字吗？我们有可能把它写出来吗？或许，它根本就不是一个数字。

这就引出了一个显而易见的问题，比如我们究竟会在哪里找到终点。泽尔伯格无法回答。事实上，没有人能回答。这正是许多人对他的哲学——被称为“超有限论”——嗤之以鼻的首要原因。“当你第一次向别人介绍超有限论时，听起来就像是江湖骗术——比如‘我认为存在一个最大数’之类的，”贾斯汀·克拉克-多恩说道。他是哥伦比亚大学的一位哲学家

“很多数学家都觉得这个提议荒谬至极，”乔尔·戴维·汉金斯说道。（在新标签页中打开）他是圣母大学的一位集合论学家。超有限论在数学学会的晚宴上可不是个好话题。研究它的人寥寥无几（或许可以说少得可怜）。像泽尔伯格这样愿意公开表达自己观点的正式会员更是凤毛麟角。这不仅是因为超有限论本身就带有反主流色彩，更是因为它提倡一种本质上更小的数学，在这种数学中，某些重要的问题将无法再被提出。

然而，这确实给汉金斯等人提供了许多值得思考的问题。从某种角度来看，超有限论可以被视为一种更现实的数学。它更好地反映了人类创造和验证能力的局限性；它甚至可能更好地反映了物理宇宙。虽然我们可能倾向于认为空间和时间是永恒扩张且可分割的，但超有限论者会认为，这些假设正日益受到科学的质疑——正如泽尔伯格可能会说的那样，科学甚至让对上帝的怀疑也受到了质疑。

“我们所描述的世界必须从头到尾都诚实可靠，”克拉克-多恩说道。他于2025年4月召集了一批专家，罕见地探讨了超有限论的思想。“如果事物的数量可能只有有限个，那么我们最好也使用一种不会一开始就假定事物数量无限多的数学。” 在他看来，“这当然应该成为数学哲学的一部分。”

贾斯汀·克拉克-多恩最近组织了一场会议，让超有限论者们能够讨论和辩论他们的观点。他认为超有限论“应该成为数学哲学的一部分”。詹妮弗·麦克唐纳

然而，要想让数学家们认真对待超有限论，首先超有限论者需要就他们所讨论的内容达成共识——正如汉金斯所说，他们需要将那些听起来像是“虚张声势”的论点转化为正式的理论。数学深谙形式体系和通用框架之道，而超有限论却缺乏这样的结构。

逐个解决问题是一回事，而改写数学本身的逻辑基础则是另一回事。“我认为超有限论被摒弃的原因并非人们提出了有力的反驳论点，”克拉克-多恩说，“而是人们觉得，唉，这根本没希望。”

这是某些极端有限论者仍在试图解决的问题。

与此同时，泽尔伯格准备放弃数学理想，转而接受一种本质上混乱的数学——就像世界本身一样。与其说他是一位基础理论家，不如说他是一位观点家，他在自己的网站上列出了195条观点。（在新标签页中打开）“如果不做这些异想天开的事，我就当不了终身教授，”他说。但他补充道，总有一天，数学家们会回过头来看，发现这个异想天开的人，就像过去那些质疑神明和迷信的人一样，是对的。“幸运的是，异端分子不再会被绑在火刑柱上烧死了。”

异议数学

亚里士多德认为无限是你可以努力接近却永远无法企及的东西。他写道：“分割的过程永无止境，这保证了这种活动的潜在存在，但这并不意味着无限本身独立存在。” 数千年来，这种“潜在”的无限概念一直占据主导地位。

但在19世纪末，格奥尔格·康托尔和其他数学家证明了无穷确实存在。康托尔的方法是将一系列数字（例如整数）视为一个完备的无穷集合。这种方法对于创建数学基础理论——策梅洛-弗兰克尔集合论——至关重要，而数学家们至今仍在运用这一理论。他证明，无穷是一个真实存在的对象。此外，它可以呈现多种不同的大小；通过操纵和比较这些不同的无穷，数学家们可以证明一些令人惊讶的真理，而这些真理表面上看起来似乎与无穷毫无关系。虽然很少有数学家会花费大量时间研究高阶无穷，但汉金斯说：“如今，几乎每个数学家都是一个实数主义者。”无穷的存在已成为默认的假设。

但自现代数学的这一基础被提出以来，它就引发了激烈的争论。原因之一是，接受关于无穷的核心假设会让人构建出奇特的悖论：例如，可以将一个球切成五份，然后用这五份再制作五个新球，每个新球的体积都与第一个球的体积相等。

另一种反对意见更偏向哲学层面。在康托尔揭示其原理后的几十年里，一些数学家认为，你不能简单地断言某种数学结构的存在——你必须通过一种思维建构的过程来证明它的存在。例如，在这种“直觉主义”哲学中，π与其说是一个具有无限不循环小数展开式的数字，不如说是一个代表生成数字的算法过程的符号。

“如果事物的数量可能只有有限的，那么我们最好也使用一种数学方法，而不是一开始就假设事物的数量是无限的。”贾斯汀·克拉克-多恩，哥伦比亚大学

但直觉主义只要求某种心理构造在理论上可行：它禁止实际的无穷大，但允许潜在的无穷大。一些数学家对此仍不满意。他们仍然对斯基维斯数以及其他一些大到无法用文字表达的数值感到困惑。于是，他们试图将直觉主义的思想推向极致。

“如果你在思考，在这种观点下哪些数字会存在，那么这些数字必须是我们能够在实践中构建的数字，”而不仅仅是理论上构建的数字，牛津大学的一位哲学家奥弗拉·马吉多尔说道。

20 世纪 60 年代和 70 年代，苏联数学家兼诗人亚历山大·叶赛宁-沃尔平的著作使一种新的直觉主义——一种将这些实际限制铭记于心的直觉主义——逐渐形成。

纽约市立大学的一位逻辑学家指出，叶赛宁-沃尔平移民美国后，曾接待过他，是一位古怪的房客，他会在帕里克家的阁楼里整夜踱步，用他妻子心爱的陶瓷器皿当烟灰缸，同时研究一种奇怪的理论，这种理论不仅否定了潜在的无穷大，甚至否定了极其巨大的数字——那些无法在人脑中构建的数字。

亚历山大·叶赛宁-沃尔平是一位苏联持不同政见者、数学家和诗人。

逻辑学家哈维·弗里德曼曾请叶赛宁-沃尔平指出一个截止点。（在新标签页中打开）什么才算太大？给定一个表达式 2^ n ， n取何值时，数字就停止存在了？2^ 0真的是个数字吗？2 ^1、2 ^ 2等等，直到 2^ 100呢？叶赛宁-沃尔平逐一回答了这个问题。是的，2^ 1存在。是的，2^ 2也存在。但每次他都等待更长时间才作答。对话很快变得没完没了。

埃赛宁-沃尔平已经阐明了他的观点。正如帕里克等人后来所言，数字的局限性根植于证明其存在所需的有限资源，例如时间、计算机内存或证明过程的物理长度。“大多数超有限论者认为，有限与无限之间的区别本质上是模糊的，”克拉克-多恩说道。

在埃赛宁-沃尔平理论中，一个条件可能对n成立，也可能对n + 1 成立——直到它不再成立为止。孩子不断成长，直到有一天他们不再是孩子。我们不必指定一个具体的终点。重要的是，终点存在于其中，在某个地方。

埃赛宁-沃尔平的工作呼吁建立一种新的数学，这种数学在某种意义上能够容忍模糊性。此后，超有限主义者们继承了他的事业，探索如何使他那些模糊不清、近乎荒谬的数学变得严谨可靠。

危机控制

有一天，爱德华·纳尔逊醒来后意识到，无限可能并不存在。这让他陷入了存在论危机。

1976 年的一个早晨，普林斯顿大学数学家爱德华·纳尔逊（在新标签页中打开）他醒来后经历了一场信仰危机。“我感到一瞬间，一种强大的存在感扑面而来，让我意识到自己因为相信存在一个无限的数字世界而显得傲慢自大，”几十年后他回忆道。（在新标签页中打开）“把我像个婴儿一样留在摇篮里，只能掰着手指头数数。”

数学有一些基本规则，或者说公理。纳尔逊知道，即使是那些使我们能够进行简单算术运算的最基本公理，也包含着关于无穷的假设——例如，我们总能通过给一个数加1来得到新的数。他想推翻之前的规则，构建一套完全禁止无穷的规则。如果数学完全由这些新的公理构成，它会是什么样子呢？

事实证明，这些公理体系极其脆弱。纳尔逊研究了各种排除无穷的公理体系，发现如果他用其中任何一套来尝试进行基本的算术运算，就连证明“ a + b永远等于b + a”这样简单的命题都变得不可能。像乘方这样的基本运算也不再总是可行：你或许可以构造出数字 100，或者数字 1000，但却无法构造出数字 100 × 1000。数学家工具箱中最强大的技巧之一——归纳法（它指出，如果你能证明某个命题对一个数成立，那么它对所有数也成立）——就此彻底失效了。

在纳尔逊看来，这种弱点代表着真理的一丝曙光。他希望证明，数学家们习以为常的那些更强大的算术公理（允许无穷存在的“皮亚诺公理”）从根本上存在缺陷——它们会导致矛盾。“我相信，许多我们视为数学既定事实的东西终将被推翻，”他曾这样说道。

然而，纳尔逊未能推翻他们。2003年，他宣布自己利用较弱的公理找到了皮亚诺公理中的矛盾之处，但这一轰动性的结果很快就被驳斥了。

罗希特·帕里克的超有限主义思想已在理论计算机科学中得到应用。劳伦·弗莱什曼

纳尔逊更为有限的算术——以及帕里克等人发展的相关非标准算术形式——在计算机领域确实发挥了作用。研究人员希望了解算法能够高效地证明哪些结论，又不能证明哪些结论。这些超有限主义的数学方法已被转化为计算效率的语言，并用于探索算法能力的极限。

在纳尔逊看来，数学的本质在于“你选择相信的真理”——你认定哪些公理是正确的。即便你选择相信的是默认公理，这个道理依然成立。当然，作为缺乏稳固根基的异端，极端有限论者还有更多需要证明的地方。

耐心练习

2025年4月，一群形形色色的人齐聚纽约，参加哥伦比亚大学举办的一场关于“废除无限”的会议。他们中有物理学家、哲学家、逻辑学家和数学家。其中既有像泽尔伯格那样坚定的超有限论者；也有相信各种无限存在的集合论学者；还有一些纯粹出于好奇的人。会议组织者克拉克-多恩回忆说，结果是“对每个人来说都是一次耐心考验”。一般来说，哲学家们习惯于在课堂上激烈辩论，然后聚在一起喝啤酒。数学家们则不然。通常情况下，如果他们意见不合，那就意味着有人犯了严重的错误。

显而易见的是，构建普适的超有限论理论进展缓慢，部分原因是该运动缺乏明确的动机，也缺乏确定其基本逻辑的统一方法。因此，像纳尔逊那样执着于基本规则或许并非正确的方法。“我认为这是浪费时间，”帕里克告诉我，“你必须把形式主义当作双筒望远镜，更关注你所看到的。如果你开始研究双筒望远镜本身，你就输了。”

其他人将现实视为一个连续的广阔空间，一刻又一刻地不可阻挡地向前流动，而泽尔伯格则将现实视为一个滴答作响的宇宙。

泽尔伯格乐于透过（或许扭曲的）镜子来看待事物，即便他必须在一个无穷概念依然鲜活存在的世界里这样做。他并不希望从零开始重建一套没有无穷概念的数学体系。他可以自上而下地进行研究。以实分析为例，实分析研究的是实数和实函数的性质。泽尔伯格称之为“退化情形” 。（在新标签页中打开）离散分析（研究的是离散对象而非连续对象的行为）的原理是，你可以用一串由微小（但并非无穷大）数值差异构成的“离散数字项链”来取代连续的实数域。他说道，你可以利用这个项链重写微积分和微分方程（现在称为“差分”方程）的规则，从而消除其中哪怕是最细微的无穷大运算。他承认，这条路走得很艰难，但并非不可能，尤其是在计算机的帮助下。他认为，虽然最终的结果可能不如经典数学那样优雅，但却更加优美，因为它反映了他所认为的物理现实的真实面貌。

作为布鲁塞尔自由大学的一位数学哲学家，让·保罗·范·本德根对超有限论的探索并非始于数字，而是始于小学几何。他看着数学老师在黑板上画了一条线，这条线似乎无限延伸。“延伸到哪里？”他回忆起自己当时的提问。如果右边向一个方向无限延伸，左边向另一个方向无限延伸，它们最终会到达同一个地方吗？还是说，黑板边缘隐藏着不同的无穷？他的老师让他别再问问题了。

让-保罗·范·本德根发展出一种有限几何学，其中点和曲线都有宽度。

范·本德根后来成为超有限论逻辑领域的领军学者，他通过构建一种几何学来解决这些问题。在这种几何学中，直线或曲线具有宽度，并且既是有限的，又是有限可分的。它可以被分割成一系列点，这些点虽然极其微小，但并非无限小。任何用这些点、直线和曲线构建的结构也必须是有限的，从而提供了一种离散的经典几何学类比。尽管这些工具仍然存在局限性，但在过去的几十年里，人们对它们进行了深入的研究——这不仅是为了超有限论本身，更是因为厘清事物的形状对于发展有限物理学至关重要。

我们常常将物理宇宙想象成无限广阔且无限可分割，但物理学家们自己却对这种假设提出质疑。宇宙存在一些根本性的极限，例如普朗克尺度（有时被称为宇宙的像素大小），超过这个尺度，距离的概念就失去了意义。当无穷大出现在物理学家的方程式中时，它会带来问题，而这正是他们想要避免的。“要预测一个无限增长、不断重复的宇宙会发生什么，真的非常非常难，”肖恩·卡罗尔说道。（在新标签页中打开）约翰·霍普金斯大学的一位物理学家，曾对量子力学的有限模型进行过实验。（在新标签页中打开）“大多数宇宙学家处理这个问题的方式就是假装它不存在。”

不来梅康斯特拉特大学和日内瓦大学的量子物理学家尼古拉斯·吉辛认为，直觉主义数学提供了一种思考物理学核心谜题之一的方法：在大尺度上，物理系统的行为是确定性的、可预测的。但在量子领域，随机性占据主导地位；一个粒子具有多个量子态，并以不可预测的方式坍缩到其中之一。过去一个世纪以来，物理学家一直在试图理解这种不匹配的根源。

尼古拉斯·吉辛提出，物理学中最大的谜团之一可能是由于对无穷大的错误假设造成的。

吉辛认为这是由于一个错误的假设造成的。他指出，研究人员默认认为，从宇宙诞生之初，粒子的量子态就可以用无限多位的实数来无限精确地定义。但吉辛认为，使用实数是错误的。如果改用直觉主义数学，就会发现决定论不过是拥有不切实际的完美信息所造成的假象。物理系统的大规模确定性行为自然会变得不精确且不可预测，从而消弭了经典领域和量子领域之间的界限。吉辛的理论引起了其他物理学家的兴趣，部分原因是它有助于解决诸如宇宙大爆炸等现象的悖论。

但值得注意的是，他的研究并非否定了潜在的无限性，即亚里士多德意义上的无限性，也就是某种潜在可达到的事物。吉辛秉承直觉主义数学家通过时间和努力计算更大或更精确数字的传统，允许创造越来越多的信息。总有一天，宇宙将包含完美且无限精确的信息。但这无关紧要，因为那一天永远不会到来。“这里的潜在无限性实际上是指无限的等待时间，这与现实无关，”吉辛说道。重要的是，无限性不再是默认的假设。

物理学家肖恩·卡罗尔对宇宙可能是有限的这种可能性很感兴趣。

这些基于物理学的对无限性的挑战往往令那些极端有限论的数学家们欣喜不已，他们以此为证，证明他们的数学理论更能真实地描述现实。在2025年的会议上，卡罗尔关于宇宙究竟是真正的无限，还是如他所说“仅仅相当大”的演讲，使他在哥伦比亚大学的校园里声名鹊起。但他告诫说，举证的责任仍然在于那些对无限性持怀疑态度的人。如果你能通过实验证明物理宇宙实际上是有限的，即使是最坚定的“更高层次的无限”支持者也可能会停下来思考片刻。考虑到集合论允许存在如此之多的实际无限，他们甚至可能会质疑集合论的自洽性。无论如何，时不时地进行这样的思考是件好事。

即便这种情况真的发生，研究和运用无穷概念的集合论学者仍然有权继续他们的工作，而不会受到影响——他们甚至可以说，这或许正是物理学和数学必须分道扬镳的地方。数学和物理学并不一定描述相同的事物（尽管许多人认为它们必须如此），而无穷概念或许会在某种更广义的柏拉图式意义上继续存在下去。

但如果这些实验证明结果恰恰相反——无限确实存在于自然界——那么极端有限论者的回旋余地就小得多。“如果真实的物理世界真的存在无限，那么极端有限论者就很难立足了，”卡罗尔说道。

重新命名 Ultrafinite

“我为那些极端有限论者感到难过，因为人们在不了解他们的情况下就对他们不屑一顾，”卡罗尔后来告诉我。“但另一方面，极端有限论者在推销他们的理论方面做得不够好。”

在数学领域，更好的营销活动可能看起来像一个连贯的理论，就像纳尔逊所追求的那种理论——一套形式化的规则，就像现代数学的基础规则一样，它排除了无穷大，但又足够强大，可以进行有用的数学运算。

克拉克-多恩说，想法并不匮乏，但或许缺乏愿意将职业生涯早期投入到这些想法发展中的研究生。在他看来，纽约的这次聚会标志着一种转变，表明人们足够好奇，愿意重新审视这个问题，并且并不惧怕潜在的负面影响。“人们正在讨论这种观点，并积极思考如何将其建立在坚实的基础之上，”他说。

大多数数学家都置身事外。涵盖数学全貌的形式理论与他们无关。他们感兴趣的是行之有效的方法，是解决具体问题并构建证明。基础性问题——数字是否存在于物理现实之外？数学是一个发明的过程还是一个发现的过程？——对他们来说可能有点尴尬，只有当数学家某天突然陷入危机时才会去思考这类问题。

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巴拿赫-塔斯基与无限克隆悖论

What Can We Gain by Losing Infinity?

By GREGORY BARBER

April 29, 2026

Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond.

Kristina Armitage/Quanta Magazine

Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger(opens a new tab) sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.

To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.

As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.

Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.

At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.

Doron Zeilberger is perhaps the most vocal proponent of banishing infinity from mathematics. “Infinity may or may not exist; God may or may not exist,” he said. “But in mathematics, there should not be any place, neither for infinity nor God.”

Courtesy of Doron Zeilberger

Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” eee79. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.

This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane(opens a new tab), a philosopher at Columbia University.

“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins(opens a new tab), a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.

And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe. While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.

“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”

Justin Clarke-Doane recently organized a conference where ultrafinitists could discuss and debate their ideas. He thinks that ultrafinitism “should be part of the menu in the philosophy of math.”

Jennifer McDonald

For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.

It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”

That’s a problem that some ultrafinitists are still trying to address.

Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website(opens a new tab). “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”

Dissident Mathematics

Aristotle saw infinity as something that you could move toward but never reach. “The fact that the process of dividing never comes to an end ensures that this activity exists potentially,” he wrote. “But not that the infinite exists separately.” For millennia, this “potential” version of infinity reigned supreme.

Why Math’s Final Axiom Proved So Controversial

SET THEORY

Why Math’s Final Axiom Proved So Controversial

APRIL 29, 2026

But in the late 1800s, Georg Cantor and other mathematicians showed that the infinite really can exist. Cantor’s approach was to treat a series of numbers, such as the integers, as a complete infinite set. This approach would become essential in the creation of the foundational theory of mathematics, known as Zermelo-Fraenkel set theory, that mathematicians still use today. Infinity, he showed, is an actual object. Moreover, it can come in many different sizes; by manipulating and comparing these different infinities, mathematicians can prove surprising truths that on their face seem to have nothing to do with infinity at all. While few mathematicians spend much time in the realm of the higher infinite, “nowadays, almost every mathematician is an actualist,” Hamkins said. Infinity is assumed by default.

But this foundation of modern math has inspired fierce arguments since it was first proposed. One reason is that accepting a core assumption about infinity allows you to construct strange paradoxes: It becomes possible, for instance, to carve a ball into five parts and use them to create five new balls, each with a volume equal to that of the first.

Another objection is more philosophical. In the decades after Cantor’s revelations, some mathematicians argued that you cannot simply assert the existence of a mathematical structure — you must prove that it exists through a process of mental construction. In this “intuitionist” philosophy, for example, pi is less a number with an infinite non-repeating decimal expansion, and more a symbol that represents an algorithmic process for generating digits.

If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.

Justin Clarke-Doane, Columbia University

But intuitionism only requires that a given mental construction be possible in theory: It prohibits actual infinity but permits potential infinity. Some mathematicians still weren’t satisfied with this. They remained troubled by Skewes’ number and other values so large they could never be written down. And so they sought to take intuitionist ideas to an extreme.

“If you’re thinking, which numbers are going to exist in this view, those are going to have to be numbers that we can in practice construct,” not just theoretically construct, said Ofra Magidor(opens a new tab), a philosopher at the University of Oxford.

A new version of intuitionism — one that took these practical constraints to heart — crystallized in the 1960s and ’70s, with the work of Alexander Esenin-Volpin, a Soviet mathematician and poet.

Esenin-Volpin was known first and foremost as a political dissident. For leading protests and spreading anti-Soviet rhetoric and poetry, he was institutionalized. “He said, ‘I’m a human being. I have fundamental rights,’” said Rohit Parikh(opens a new tab), a logician at the City University of New York who hosted Esenin-Volpin in his home after the Soviets forced him to emigrate in the 1970s. Esenin-Volpin was a strange houseguest, who would pace around Parikh’s attic all night and use his wife’s beloved ceramics as an ashtray while working on a strange theory that rejected not only potential infinity but even extremely large numbers — those that couldn’t be constructed in a person’s mind.

Alexander Esenin-Volpin was a Soviet dissident, mathematician, and poet who was imprisoned several times for his human rights activism.

Irene Caesar

The logician Harvey Friedman once asked Esenin-Volpin to pinpoint a cutoff(opens a new tab) for what makes a number too large. Given an expression like 2n, at what value of n do numbers stop? Was 20 actually a number? What about 21, 22, and so on, up to 2100? Esenin-Volpin responded to each number in turn. Yes, 21 existed. Yes, 22 did. But each time, he waited longer to reply. The dialogue soon grew interminable.

Esenin-Volpin had made his point. As Parikh and others would later put it, the limits of numbers were rooted in the limited resources needed to demonstrate their existence, like time. Or available computer memory, or the physical length of a proof. “Most ultrafinitists have the view that the distinction between the finite and infinite is inherently vague,” Clarke-Doane said.

For Esenin-Volpin, a condition may be true for n, and for n + 1 — until it is not. A child grows and grows, until one day they’re no longer a child. One need not specify a specific end point. The important thing is that the end is in there, somewhere.

Esenin-Volpin’s work was a call for a new kind of mathematics that could, in some sense, tolerate vagueness. Ultrafinitists have since picked up where he left off, exploring how to make his vague, borderline-nonsensical mathematics solid.

Crisis Control

One day, Edward Nelson woke up and realized that infinity might not be real. It left him in an existential crisis.

Mariana Cook

One morning in 1976, the Princeton mathematician Edward Nelson(opens a new tab) woke up and experienced a crisis of faith. “I felt the momentary overwhelming presence of one who convicted me of arrogance for my belief in the real existence of an infinite world of numbers,” he reflected decades later(opens a new tab), “leaving me like an infant in my crib reduced to counting on my fingers.”

Mathematics has basic rules, or axioms. Nelson knew that even the bare-bones axioms that make it possible to do simple arithmetic contain assumptions about infinity — for instance, that we can always add 1 to a number to create new numbers. He wanted to start over, to construct a new set of rules that would forbid infinity entirely. What would mathematics look like if it could be built up from only these new axioms?

Remarkably weak, it turned out. Nelson studied various sets of axioms that banished infinity and found that if he used any of them to try to do basic arithmetic, it became impossible to prove something as simple as the statement that a + balways equals b + a. Elementary operations like exponentiation were no longer always possible: You might be able to construct the number 100, or the number 1,000, but not the number 1001,000. One of the most powerful techniques in a mathematician’s tool kit — a method known as induction, which says that if you can prove that a statement is true for one number, then it must be true for them all — was lost entirely.

To Nelson, this weakness represented a glimmer of truth. He hoped to show that the more powerful axioms of arithmetic that mathematicians took for granted (the infinity-permitting “Peano axioms”) were fundamentally flawed — that they could lead to contradictions. “I believe that many of the things we regard as being established in mathematics will be overthrown,” he once said.

Nelson was unable to overthrow them, however. In 2003, he announced that he’d used his weaker axioms to find an inconsistency in the Peano axioms, but the splashy result was quickly debunked.

Rohit Parikh’s ultrafinitist ideas have had applications in theoretical computer science.

Lauren Fleishman

Nelson’s more limited arithmetic — as well as related forms of nonstandard arithmetic developed by Parikh and others — did prove useful in the realm of computers, where researchers want to understand what algorithms can efficiently prove and what they can’t. These ultrafinitist approaches to mathematics have been translated into the language of computational efficiency and used to probe the limits of algorithms’ capabilities.

To Nelson, mathematics is all about “the truth you choose to believe” — the axioms that you decide are the right ones. That’s true even if you’ve chosen to believe the default axioms. Of course, the ultrafinitist, as the heretic without stable foundations, has a lot more to prove.

Exercises in Patience

In April 2025, a motley crew gathered in New York City for a conference at Columbia University on abolishing the infinite. They included physicists, philosophers, logicians, and mathematicians. There were card-carrying ultrafinitists like Zeilberger; set theorists, who believe in all sorts of infinity; and the merely curious. The result was, recalled Clarke-Doane, the conference organizer, “an exercise in patience for everyone.” Philosophers, in general, are used to disagreeing vehemently in the classroom and then gathering over a beer. Mathematicians aren’t. Usually, if they disagree, it means somebody royally messed up.

What was clear was that progress toward a universal theory of ultrafinitism has been halting in part because there has been no one clear motivation for the movement, or any singular approach to deciding what its underlying logic should look like. Perhaps, then, fixating on the ground rules, like Nelson did, isn’t the right approach. “I think it’s a waste,” Parikh told me. “You have to use the formalism as a binocular and pay more attention to what you are seeing. If you start studying the binoculars themselves, you’ve lost the game.”

Where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks.

Zeilberger is happy to see things through the (possibly distorted) looking glass, even if he must do so in a world where infinity is very much alive and present. He doesn’t hope to rebuild a mathematics without infinity from scratch. He can work from the top down instead. Take, for example, real analysis, which deals with how real numbers and functions behave. Zeilberger calls it a “degenerate case(opens a new tab)” of discrete analysis (which studies the behavior of distinct objects rather than continuous ones). You can replace the continuous landscape of the reals, he says, with a “discrete necklace” of numbers, separated by tiny — but not infinitesimal — differences in value. You can then use this to rewrite the rules of calculus and differential equations (now called “difference” equations) to remove even subtle uses of infinity from them. The going is tough, he acknowledges, but doable, especially with the help of a computer. And while the result may look less elegant than classical math, it is more beautiful, he says, because it reflects physical reality as he believes it to truly be.

For Jean Paul Van Bendegem(opens a new tab), a philosopher of mathematics at the Free University of Brussels, the journey into ultrafinitism began not with numbers, but with elementary school geometry. He watched his math teacher draw a line on the chalkboard that supposedly extended infinitely. “To where?” he recalled asking. If the right-hand side went infinitely far in one direction and the left-hand side in another, did they arrive at the same place? Or did different infinities lurk off the edges of the board? His teacher told him to stop asking questions.

Jean-Paul van Bendegem developed a finite version of geometry in which points and curves have width.

Inge Kinnet

Van Bendegem, who would become a leading scholar on ultrafinitist logic, later addressed these concerns by considering a geometry in which a line or curve has width and is both finite and finitely divisible. It can be broken up into an array of points that, though incredibly small, are not infinitely so. Any structure one then builds with these points, lines, and curves must also be finite, providing a discrete analogue of classical geometry. While these tools remain limited, they have been explored deeply over the past few decades — not just for the sake of ultrafinitism, but because sorting out the shape of things is important for developing a finite physics.

While we often imagine the physical universe as both endlessly vast and endlessly divisible, physicists themselves question this assumption. There are fundamental limits, such as the Planck scale (sometimes called the pixel size of the universe), beyond which the very idea of distance loses meaning. And when infinity does crop up in physicists’ equations, it can be problematic, something they want to avoid. “To make predictions about what to expect in a universe that grows without bounds and repeats itself and things like that turns out to be really, really hard,” said Sean Carroll(opens a new tab), a physicist at Johns Hopkins University who has experimented with finitistic models of quantum mechanics(opens a new tab). “The way that most cosmologists deal with that problem is by pretending it’s not there.”

For Nicolas Gisin(opens a new tab), a quantum physicist at Constructor University in Bremen, Germany, and the University of Geneva, intuitionist mathematics provides a way to think about one of the core mysteries in physics: At large scales, the behavior of physical systems is deterministic, predictable. But in the quantum realm, randomness reigns; a particle comes with multiple quantum states, collapsing to just one of them in unpredictable ways. Physicists have been trying to understand the source of this mismatch for the past century.

Nicolas Gisin proposed that one of the greatest mysteries in physics might be due to incorrect assumptions about infinity.

Carole Parodi

Gisin posits that it’s due to a faulty assumption. Researchers implicitly believe, he says, that from the start of the universe, a particle’s quantum state can be defined with infinite precision, by real numbers with infinitely many digits. But, according to Gisin, using the real numbers is a mistake. If you use intuitionist mathematics instead, then it becomes clear that determinism is but an artifact of having unrealistically perfect information. The large-scale, deterministic behavior of physical systems naturally becomes imprecise and unpredictable, dissolving the divide between the classical and quantum realms. Gisin’s theory has proved intriguing to other physicists, in part because it could help resolve paradoxes about phenomena like the Big Bang.

But it’s important to note that his work does not abolish potential infinity, in the Aristotelian sense of something that can potentially be reached. In the tradition of the intuitionist mathematician calculating larger or more precise numbers with time and effort, Gisin allows for more and more information to be created. Someday, the universe will contain perfect, infinitely precise information. But it doesn’t matter, because that someday will never come. “The potential infinity here is really waiting infinite time, which has nothing to do with reality,” Gisin said. The important thing is that infinity is no longer the default assumption.

The physicist Sean Carroll is intrigued by the possibility that the universe might be finite.

Larry Canner/Johns Hopkins

These physics-based challenges to the infinite tend to delight ultrafinitist mathematicians, who hold them up as evidence that their mathematics is a truer description of reality. At the 2025 conference, Carroll’s talk on whether the universe is truly infinite or “merely quite large,” as he put it, made him something of a celebrity in the Columbia University halls. But the burden of proof, he cautions, remains with the infinity doubters. If you could somehow prove experimentally that the physical universe is indeed finite, even the most ardent backers of the higher infinite would likely take a moment to pause and reflect. They would probably even wonder about the consistency of set theory, given the towers of actual infinities that it allows. That’s a healthy thing to do from time to time, anyway.

Even if this were to happen, set theorists who study and use infinity would still be within their rights to continue their work unfazed — to say that perhaps this is where physics and math must branch off from each other. It is no requirement that math and physics describe the same things (though many believe it is), and infinity might live on in some larger Platonic sense.

But if those experiments proved the opposite — that infinity does exist in nature — the ultrafinitist would have far less room to negotiate. “It would be hard to be an ultrafinitist if the actual physical world had infinities in it,” Carroll said.

Rebranding the Ultrafinite

“I feel bad for the ultrafinitists because people dismiss it without understanding it,” Carroll told me later. “But on the other hand, the ultrafinitists don’t do a good enough job of marketing their product.”

Within mathematics, a better marketing campaign would probably look like a coherent theory, the kind Nelson sought — a set of formal rules, like those underlying modern math, that excludes infinity but is powerful enough to do useful mathematics.

There is no shortage of ideas, Clarke-Doane said — though there’s perhaps a shortage of graduate students willing to stake their early careers on developing them. To him, the gathering in New York was a sign of change, that people are curious enough to give it another look, and not too scared of the potential backlash. “People are talking about the view and actively trying to think about how to put the view on a serious foundation,” he said.

Most mathematicians live outside all this. Formal theories encompassing the totality of mathematics do not concern them. They are interested in what works, in solving specific problems and building proofs. Foundational questions — do numbers exist beyond physical reality? Is math a process of invention or discovery? — can feel a little cringe, the sort of thing mathematicians only do when they wake up one day in a state of crisis.