What Can We Gain by Losing Infinity?
Kristina Armitage/ Quanta Magazine
克里斯蒂娜·阿米蒂奇/广达杂志
D oron Zeilberger is a mathematician who believes that all things come to an end.
多伦·蔡尔伯格 (Doron Zeilberger) 是一位数学家，他相信万物都会终结。
That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers.
正如我们是有限的存在一样，自然也有边界——因此数字也有边界。
Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks.
望向窗外，其他人将现实视为一片连续的广阔空间，时时刻刻无情地向前流动，而蔡尔伯格则看到了一个滴答作响的宇宙。
It is a discrete machine.
它是一台离散机器。
In the smooth motion of the world around him, he catches the subtle blur of a flip-book.
在他周围世界的平稳运动中，他捕捉到了一本翻页书的微妙模糊。
To Zeilberger, believing in infinity is like believing in God.
对于蔡尔伯格来说，相信无限就像相信上帝一样。
It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena.
这是一个诱人的想法，它可以满足我们的直觉并帮助我们理解各种现象。
But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is.
但问题是我们无法真正观察到无穷大，因此我们无法真正说出它是什么。
Equations define lines that carry on off the chalkboard, but to where?
方程式定义了黑板上的线条，但延伸到哪里呢？
Proofs are littered with suggestive ellipses.
证据中充满了暗示性的省略。
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.
罗格斯大学的长期教授、组合学领域的著名人物 Zeilberger 认为，这些方程和证明既“非常丑陋”又错误。
It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.
他说，这“完全是无稽之谈”，每一个音节都用沙哑的声音吐出，似乎因为表达自己的观点而疲惫不堪。
Most mathematicians take the notion of infinity for granted — it’s deeply rooted in math’s most fundamental assumptions.
大多数数学家都认为无穷大的概念是理所当然的——它深深植根于数学最基本的假设中。
But a small group of researchers hopes to banish infinity completely.
但一小群研究人员希望完全消除无穷大。
As a matter of practicality, infinity can be scrubbed out, he contends.
他认为，从实用性角度来看，无穷大是可以被消除的。
“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.
“你其实并不需要它。”例如，数学家可以构建一种没有无穷大的微积分形式，将无穷小的极限完全排除在外。
Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits.
曲线可能看起来很平滑，但它们隐藏着细粒度的粗糙度；计算机在数字有限的情况下可以很好地处理数学。
(Zeilberger lists his own computer, which he named “Shalosh B.
（Zeilberger 列出了他自己的计算机，他将其命名为“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.
Ekhad，”作为他论文的合作者。）随着无穷大被消除，唯一失去的是“根本不值得做”的数学，蔡尔伯格说。
Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense.
大多数数学家会说恰恰相反——蔡尔伯格完全是胡说八道。
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.
不仅因为无穷对于我们对宇宙的描述如此有用和如此自然，而且因为将数字集（如整数）视为实际的无限对象是数学的核心，嵌入在其最基本的规则和假设中。
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.
至少，即使数学家不想将无穷大视为一个实际实体，他们也承认序列、形状和其他数学对象有无限增长的潜力。
Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.
理论上，两条平行线可以永远延伸；总是可以将另一个数字添加到数轴的末尾。
Zeilberger disagrees.
蔡尔伯格不同意。
To him, what matters is not whether something is possible in principle, but whether it is actually feasible.
对他来说，重要的不是原则上是否可行，而是实际上是否可行。
What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well.
在实践中，这意味着不仅无穷大值得怀疑，而且极大的数字也值得怀疑。
Consider “Skewes’ number,” $latex e^{e^{e^{79}}}$.
考虑“倾斜数”$latex e^{e^{e^{79}}}$。
This is an exceptionally large number, and no one has ever been able to write it out in decimal form.
这是一个非常大的数字，没有人能够以十进制形式写出它。
So what can we really say about it?
那么我们到底能说什么呢？
Is it an integer?
是整数吗？
Is it prime?
是总理吗？
Can we find such a number anywhere in nature?
我们能在自然界中的任何地方找到这样的数字吗？
Could we ever write it down?
我们能把它写下来吗？
Perhaps, then, it is not a number at all.
那么，也许它根本就不是一个数字。
Doron Zeilberger is perhaps the most vocal proponent of banishing infinity from mathematics.
多伦·蔡尔伯格（Doron Zeilberger）也许是从数学中消除无穷大最有力的支持者。
“Infinity may or may not exist; God may or may not exist,” he said.
“无限可能存在，也可能不存在；上帝可能存在，也可能不存在，”他说。
“But in mathematics, there should not be any place, neither for infinity nor God.”
“但在数学中，无论是无穷还是上帝，都不应该有任何地位。”
This raises obvious questions, such as where, exactly, we will find the end point.
这提出了明显的问题，例如我们到底会在哪里找到终点。
Zeilberger can’t say.
齐尔伯格不能说。
Nobody can.
没有人可以。
Which is the first reason that many dismiss his philosophy, known as ultrafinitism.
这是许多人驳斥他的哲学（即超有限主义）的第一个原因。
“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 , a philosopher at Columbia University.
哥伦比亚大学哲学家贾斯汀·克拉克-多恩 (Justin Clarke-Doane) 表示：“当你第一次向某人提出超有限论的想法时，听起来像是庸医——就像‘我认为存在最大的数字’之类的。”
“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins , a set theorist at the University of Notre Dame.
“许多数学家认为整个提议很荒谬，”圣母大学的集合论学家乔尔·大卫·哈姆金斯说。
Ultrafinitism is not polite talk at a mathematical society dinner.
超有限论并不是数学协会晚宴上的礼貌谈话。
Few (one might say an ultrafinite number) work on it.
很少有人（有人可能会说是超有限数）对此进行研究。
Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void.
像泽尔伯格这样的正式成员愿意大声说出自己的观点的人就更少了。
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.
这不仅是因为超有限论是逆向的，还因为它提倡一种从根本上更小的数学，在这种数学中，某些重要的问题不能再被问到。
And yet it gives Hamkins and others a good deal to think about.
然而，这给了哈姆金斯和其他人很多值得思考的地方。
From one angle, ultrafinitism can be seen as a more realistic mathematics.
从一个角度来看，超有限论可以被视为一种更现实的数学。
It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe.
数学能更好地反映人们可以创造和验证的极限；它甚至可能更好地反映物理宇宙。
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.
虽然我们可能倾向于认为空间和时间是永恒扩张和可分的，但超有限论者会认为，这些假设是科学越来越受到质疑的——就像泽尔伯格可能会说的那样，科学将怀疑带到了上帝的家门口。
“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.
“我们所描述的世界需要彻底诚实，”克拉克-多恩说，他于 2025 年 4 月召集了一次罕见的专家聚会来探索超有限主义想法。
“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.”
“如果事物的数量是有限的，那么我们最好也使用一种数学方法，而不仅仅是在一开始就假设事物的数量是无限的。”对他来说，“这确实应该成为数学哲学菜单的一部分。”
Justin Clarke-Doane recently organized a conference where ultrafinitists could discuss and debate their ideas.
贾斯汀·克拉克-多恩最近组织了一次会议，超有限主义者可以在会上讨论和辩论他们的想法。
He thinks that ultrafinitism “should be part of the menu in the philosophy of math.”
他认为超有限论“应该成为数学哲学菜单的一部分”。
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.
然而，要让数学家认真对待它，超有限论者首先需要就他们所讨论的内容达成一致——将哈姆金斯所说的听起来像是“咆哮”的论点变成官方理论。
Mathematics is steeped in formal systems and common frameworks.
数学植根于形式系统和通用框架。
Ultrafinitism, meanwhile, lacks such structure.
与此同时，超有限论则缺乏这样的结构。
It is one thing to tackle problems piecemeal.
零碎解决问题是一回事。
It is quite another to rewrite the logical foundations of mathematics itself.
重写数学本身的逻辑基础是另一回事。
“I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said.
“我不认为超有限主义被驳回的原因是人们有充分的理由反对它，”克拉克-多恩说。
“The feeling is that, oh, well, it’s hopeless.”
“感觉是，哦，好吧，这是绝望的。”
That’s a problem that some ultrafinitists are still trying to address.
这是一些超有限主义者仍在试图解决的问题。
Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is.
与此同时，蔡尔伯格准备放弃数学理想，转而选择本质上混乱的数学——就像世界一样。
He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website .
与其说他是一个基础理论的人，不如说他是一个有观点的人，他在自己的网站上列出了 195 个观点。
“I cannot be a tenured professor without doing this crackpot stuff,” he said.
“如果不做这些疯狂的事情，我就不可能成为一名终身教授，”他说。
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.
但有一天，他补充说，数学家们回头看时会发现，这个疯子就像那些质疑神灵和迷信的人一样，是正确的。
“Luckily, heretics are no longer burned at the stake.”
“幸运的是，异教徒不再被烧死在火刑柱上。”
Aristotle saw infinity as something that you could move toward but never reach.
亚里士多德认为无限是一种你可以接近但永远无法到达的东西。
“The fact that the process of dividing never comes to an end ensures that this activity exists potentially,” he wrote.
“分裂过程永远不会结束，这一事实确保了这种活动的潜在存在，”他写道。
“But not that the infinite exists separately.” For millennia, this “potential” version of infinity reigned supreme.
“但这并不是说无限是单独存在的。”几千年来，这种“潜力”版本的无限一直占据着至高无上的地位。
But in the late 1800s, Georg Cantor and other mathematicians showed that the infinite really can exist.
但在 1800 年代末，乔治·康托 (Georg Cantor) 和其他数学家证明了无限确实可以存在。
Cantor’s approach was to treat a series of numbers, such as the integers, as a complete infinite set.
康托尔的方法是将一系列数字（例如整数）视为一个完整的无限集。
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.
这种方法对于创建数学基础理论（称为策梅洛-弗兰克尔集合论）至关重要，数学家至今仍在使用该理论。
Infinity, he showed, is an actual object.
他表明，无限是一个真实的物体。
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.
此外，它可以有多种不同的尺寸；通过操纵和比较这些不同的无穷大，数学家可以证明令人惊讶的事实，这些事实表面上似乎与无穷大毫无关系。
While few mathematicians spend much time in the realm of the higher infinite, “nowadays, almost every mathematician is an actualist,” Hamkins said.
虽然很少有数学家在更高的无限领域花费大量时间，但“如今，几乎每个数学家都是现实主义者，”哈姆金斯说。
Infinity is assumed by default.
默认情况下假定无穷大。
But this foundation of modern math has inspired fierce arguments since it was first proposed.
但这一现代数学基础自首次提出以来就引发了激烈的争论。
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.
一个原因是，接受关于无穷大的核心假设可以让你构造出奇怪的悖论：例如，可以将一个球分成五个部分，并用它们创建五个新球，每个球的体积等于第一个球的体积。
Another objection is more philosophical.
另一种反对意见更具哲学性。
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.
在康托揭秘之后的几十年里，一些数学家认为，你不能简单地断言数学结构的存在——你必须通过心理建构的过程来证明它的存在。
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.
例如，在这种“直觉主义”哲学中，圆周率与其说是一个具有无限不重复十进制扩展的数字，不如说是一个代表生成数字的算法过程的符号。
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.
如果事物的数量是有限的，那么我们最好也使用一种数学方法，而不仅仅是在一开始就假设事物的数量是无限的。
Justin Clarke-Doane, Columbia University
贾斯汀·克拉克-多恩，哥伦比亚大学
But intuitionism only requires that a given mental construction be possible in theory: It prohibits actual infinity but permits potential infinity.
但直觉主义只要求给定的心理构造在理论上是可能的：它禁止实际的无限，但允许潜在的无限。
Some mathematicians still weren’t satisfied with this.
一些数学家对此仍然不满意。
They remained troubled by Skewes’ number and other values so large they could never be written down.
他们仍然对斯科维斯的数字和其他数值如此之大以至于永远无法写下来感到困扰。
And so they sought to take intuitionist ideas to an extreme.
因此，他们试图将直觉主义思想发挥到极致。
“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 , a philosopher at the University of Oxford.
牛津大学哲学家奥弗拉·马吉多 (Ofra Magidor) 表示：“如果你想，在这种观点中将会存在哪些数字，这些数字必须是我们可以在实践中构建的数字，”而不仅仅是理论上构建。
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.
一种新版本的直觉主义——将这些实际限制牢记于心——在 20 世纪 60 年代和 70 年代在苏联数学家和诗人亚历山大·叶赛宁-沃尔平 (Alexander Esenin-Volpin) 的著作中具体化。
Esenin-Volpin was known first and foremost as a political dissident.
叶赛宁-沃尔平首先被认为是一位政治异见人士。
For leading protests and spreading anti-Soviet rhetoric and poetry, he was institutionalized.
由于领导抗议活动并传播反苏联言论和诗歌，他被制度化。
“He said, ‘I’m a human being.
“他说，‘我是一个人。
I have fundamental rights,’” said Rohit Parikh , 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.
我有基本权利。”纽约城市大学逻辑学家罗希特·帕里克 (Rohit Parikh) 说，20 世纪 70 年代苏联迫使叶赛宁-沃尔平移民后，他在家中接待了他。
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.
叶赛宁-沃尔平是一位奇怪的房客，他整夜在帕里克的阁楼上踱步，用妻子心爱的陶瓷作为烟灰缸，同时研究一种奇怪的理论，该理论不仅拒绝潜在的无穷大，甚至拒绝极大的数字——那些无法在人的头脑中构建的数字。
Alexander Esenin-Volpin was a Soviet dissident, mathematician, and poet who was imprisoned several times for his human rights activism.
亚历山大·叶赛宁-沃尔平是苏联持不同政见者、数学家和诗人，因其人权活动而多次入狱。
The logician Harvey Friedman once asked Esenin-Volpin to pinpoint a cutoff for what makes a number too large.
逻辑学家哈维·弗里德曼 (Harvey Friedman) 曾经要求叶赛宁-沃尔平 (Esenin-Volpin) 确定一个界限，以判断数字过大的原因。
Given an expression like 2 n , at what value of n do numbers stop?
给定一个像 2 n 这样的表达式，数字在 n 的值是多少时停止？
Was 2 0 actually a number?
2 0 实际上是一个数字吗？
What about 2 1 , 2 2 , and so on, up to 2 100 ?
那么 2 1 、 2 2 等等，直到 2 100 呢？
Esenin-Volpin responded to each number in turn.
叶赛宁-沃尔平依次对每个数字做出回应。
Yes, 2 1 existed.
是的，2 1 存在。
Yes, 2 2 did.
是的，2 2 做到了。
But each time, he waited longer to reply.
但每一次，他都会等待更长的时间才能回复。
The dialogue soon grew interminable.
对话很快变得无休无止。
Esenin-Volpin had made his point.
叶赛宁-沃尔平已经表达了他的观点。
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.
正如帕里克和其他人后来所说，数量的限制植根于证明其存在所需的有限资源，例如时间。
Or available computer memory, or the physical length of a proof.
或者可用的计算机内存，或者证明的物理长度。
“Most ultrafinitists have the view that the distinction between the finite and infinite is inherently vague,” Clarke-Doane said.
“大多数超有限主义者都认为有限和无限之间的区别本质上是模糊的，”克拉克-多恩说。
For Esenin-Volpin, a condition may be true for n , and for n + 1 — until it is not.
对于 Esenin-Volpin，条件可能对 n 和 n + 1 为真 — 直到它不成立为止。
A child grows and grows, until one day they’re no longer a child.
孩子一天天长大，直到有一天他们不再是孩子。
One need not specify a specific end point.
无需指定具体终点。
The important thing is that the end is in there, somewhere.
重要的是结局就在那里，在某个地方。
Esenin-Volpin’s work was a call for a new kind of mathematics that could, in some sense, tolerate vagueness.
叶赛宁-沃尔平的工作呼吁建立一种新的数学，在某种意义上可以容忍模糊性。
Ultrafinitists have since picked up where he left off, exploring how to make his vague, borderline-nonsensical mathematics solid.
此后，超有限论者继续他停下来的地方，探索如何使他模糊的、近乎荒谬的数学变得坚实。
One day, Edward Nelson woke up and realized that infinity might not be real.
有一天，爱德华·纳尔逊醒来，意识到无穷可能并不真实。
It left him in an existential crisis.
这让他陷入了生存危机。
One morning in 1976, the Princeton mathematician Edward Nelson woke up and experienced a crisis of faith.
1976 年的一天早晨，普林斯顿大学数学家爱德华·尼尔森 (Edward Nelson) 一觉醒来，经历了一场信仰危机。
“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 , “leaving me like an infant in my crib reduced to counting on my fingers.”
几十年后，他回忆道：“我瞬间感受到了一个压倒性的存在，他因为我相信无限数字世界的真实存在而判定我傲慢，让我像婴儿床上的婴儿一样只能靠手指数数。”
Mathematics has basic rules, or axioms.
数学有基本规则或公理。
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.
纳尔逊知道，即使是使简单算术成为可能的基本公理也包含关于无穷大的假设——例如，我们总是可以在一个数字上加 1 以创建新的数字。
He wanted to start over, to construct a new set of rules that would forbid infinity entirely.
他想重新开始，构建一套新的规则，完全禁止无穷大。
What would mathematics look like if it could be built up from only these new axioms?
如果数学只能由这些新公理建立起来，那么数学会是什么样子？
Remarkably weak, it turned out.
事实证明，他的实力非常弱。
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 + b always equals b + a .
纳尔逊研究了各种消除无穷大的公理，发现如果他使用其中任何一个公理来尝试进行基本算术，就不可能证明像 a + b 始终等于 b + a 这样简单的陈述。
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 100 1,000 .
像求幂这样的基本运算不再总是可能的：您可能能够构造数字 100 或数字 1,000，但不能构造数字 100 1,000 。
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.
数学家工具包中最强大的技术之一——一种称为归纳法的方法，即如果你能证明一个陈述对于一个数字是正确的，那么它对于所有数字都必须是正确的——完全丢失了。
To Nelson, this weakness represented a glimmer of truth.
对于纳尔逊来说，这个弱点代表了一丝真理。
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.
他希望表明，数学家认为理所当然的更强大的算术公理（无限允许的“皮亚诺公理”）从根本上来说是有缺陷的——它们可能会导致矛盾。
“I believe that many of the things we regard as being established in mathematics will be overthrown,” he once said.
“我相信，许多我们认为在数学中已确立的东西将会被推翻，”他曾经说过。
Nelson was unable to overthrow them, however.
然而，纳尔逊无法推翻他们。
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.
2003年，他宣布他用自己较弱的公理发现了皮亚诺公理中的不一致之处，但这个引人注目的结果很快就被揭穿了。
Rohit Parikh’s ultrafinitist ideas have had applications in theoretical computer science.
罗希特·帕里克 (Rohit Parikh) 的超有限主义思想已在理论计算机科学中得到应用。
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.
尼尔森的更有限的算术——以及帕里克和其他人开发的相关形式的非标准算术——确实在计算机领域被证明是有用的，研究人员希望了解哪些算法可以有效地证明什么和不能有效地证明什么。
These ultrafinitist approaches to mathematics have been translated into the language of computational efficiency and used to probe the limits of algorithms’ capabilities.
这些超有限主义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。
To Nelson, mathematics is all about “the truth you choose to believe” — the axioms that you decide are the right ones.
对纳尔逊来说，数学就是“你选择相信的真理”——你认为正确的公理。
That’s true even if you’ve chosen to believe the default axioms.
即使您选择相信默认公理，情况也是如此。
Of course, the ultrafinitist, as the heretic without stable foundations, has a lot more to prove.
当然，超有限论者作为没有稳定基础的异端，还有很多东西需要证明。
In April 2025, a motley crew gathered in New York City for a conference at Columbia University on abolishing the infinite.
2025 年 4 月，一群形形色色的人聚集在纽约市，参加在哥伦比亚大学举行的关于废除无限的会议。
They included physicists, philosophers, logicians, and mathematicians.
他们包括物理学家、哲学家、逻辑学家和数学家。
There were card-carrying ultrafinitists like Zeilberger; set theorists, who believe in all sorts of infinity; and the merely curious.
有像泽尔伯格这样的名副其实的超有限主义者；集合理论家，他们相信各种无限；以及仅仅是好奇的人。
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.
会议组织者克拉克-多恩回忆道，结果是“对每个人来说都是一次耐心的锻炼”。一般来说，哲学家习惯于在课堂上激烈地表达不同意见，然后聚在一起喝啤酒。
Mathematicians aren’t.
数学家则不然。
Usually, if they disagree, it means somebody royally messed up.
通常，如果他们不同意，就意味着有人彻底搞砸了。
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.
显而易见的是，超有限主义普遍理论的进展一直处于停滞状态，部分原因是该运动没有明确的动机，也没有任何单一的方法来决定其基本逻辑应该是什么样子。
Perhaps, then, fixating on the ground rules, like Nelson did, isn’t the right approach.
那么，也许像尼尔森那样专注于基本规则并不是正确的方法。
“I think it’s a waste,” Parikh told me.
“我认为这是一种浪费，”帕里克告诉我。
“You have to use the formalism as a binocular and pay more attention to what you are seeing.
“你必须把形式主义当作双筒望远镜，更加关注你所看到的东西。
If you start studying the binoculars themselves, you’ve lost the game.”
如果你开始研究双筒望远镜本身，你就输了。”
Where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks.
在其他人看来，现实是一片连续不断的广阔空间，时时刻刻无情地向前流动，而蔡尔伯格则看到了一个滴答作响的宇宙。
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.
蔡尔伯格很高兴通过（可能是扭曲的）镜子看到事物，即使他必须在一个无限活跃且存在的世界中这样做。
He doesn’t hope to rebuild a mathematics without infinity from scratch.
他不希望从头开始重建一个没有无穷大的数学。
He can work from the top down instead.
相反，他可以自上而下地工作。
Take, for example, real analysis, which deals with how real numbers and functions behave.
以实分析为例，它涉及实数和函数的行为方式。
Zeilberger calls it a “ degenerate case ” of discrete analysis (which studies the behavior of distinct objects rather than continuous ones).
蔡尔伯格将其称为离散分析的“退化案例”（研究不同对象而不是连续对象的行为）。
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.
他说，你可以用数字的“离散项链”来取代实数的连续景观，这些数字被微小的——但不是无限小的——价值差异分开。
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.
然后，您可以使用它来重写微积分和微分方程（现在称为“差分”方程）的规则，以消除其中对无穷大的微妙使用。
The going is tough, he acknowledges, but doable, especially with the help of a computer.
他承认，虽然过程很艰难，但还是可行的，尤其是在计算机的帮助下。
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.
他说，虽然结果可能看起来不如经典数学那么优雅，但它更美丽，因为它反映了他认为真实的物理现实。
For Jean Paul Van Bendegem , a philosopher of mathematics at the Free University of Brussels, the journey into ultrafinitism began not with numbers, but with elementary school geometry.
对于布鲁塞尔自由大学的数学哲学家让·保罗·范·本德海姆 (Jean Paul Van Bendegem) 来说，超有限主义之旅不是从数字开始的，而是从小学几何学开始的。
He watched his math teacher draw a line on the chalkboard that supposedly extended infinitely.
他看着数学老师在黑板上画了一条据说无限延伸的线。
“To where?” he recalled asking.
“去哪儿？”他记得他曾这样问过。
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?
如果右侧向一个方向无限远，左侧向另一个方向无限远，它们会到达同一个地方吗？
Or did different infinities lurk off the edges of the board?
或者棋盘边缘是否潜藏着不同的无穷大？
His teacher told him to stop asking questions.
他的老师告诉他不要再问问题。
Jean-Paul van Bendegem developed a finite version of geometry in which points and curves have width.
Jean-Paul van Bendegem 开发了一种有限版本的几何，其中点和曲线都有宽度。
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.
范·本德海姆（Van Bendegem）后来成为超有限逻辑领域的顶尖学者，他后来通过考虑直线或曲线具有宽度并且既有限又有限可整的几何结构来解决这些问题。
It can be broken up into an array of points that, though incredibly small, are not infinitely so.
它可以被分解成一系列点，这些点虽然小得令人难以置信，但并不是无限的。
Any structure one then builds with these points, lines, and curves must also be finite, providing a discrete analogue of classical geometry.
用这些点、线和曲线构建的任何结构也必须是有限的，提供经典几何的离散模拟。
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.
尽管这些工具仍然有限，但在过去的几十年里，它们已经得到了深入的探索——不仅仅是为了超有限论，而且因为理清事物的形状对于发展有限物理学很重要。
While we often imagine the physical universe as both endlessly vast and endlessly divisible, physicists themselves question this assumption.
虽然我们经常想象物理宇宙既无限广阔又无限可分，但物理学家自己却质疑这一假设。
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.
存在一些基本限制，例如普朗克尺度（有时称为宇宙的像素大小），超过该限制，距离的概念就失去了意义。
And when infinity does crop up in physicists’ equations, it can be problematic, something they want to avoid.
当无穷大确实出现在物理学家的方程中时，它可能会产生问题，这是他们想要避免的。
“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 , a physicist at Johns Hopkins University who has experimented with finitistic models of quantum mechanics .
约翰·霍普金斯大学的物理学家肖恩·卡罗尔（Sean Carroll）曾对量子力学的有限模型进行过实验，他说：“要预测在一个无限增长并自我重复的宇宙中会发生什么，以及类似的事情，是非常非常困难的。”
“The way that most cosmologists deal with that problem is by pretending it’s not there.”
“大多数宇宙学家处理这个问题的方式是假装它不存在。”
For Nicolas Gisin , 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.
对于德国不来梅建筑大学和日内瓦大学的量子物理学家尼古拉斯·吉辛来说，直觉主义数学提供了一种思考物理学核心谜团之一的方法：在大尺度上，物理系统的行为是确定性的、可预测的。
But in the quantum realm, randomness reigns; a particle comes with multiple quantum states, collapsing to just one of them in unpredictable ways.
但在量子领域，随机性占主导地位。一个粒子具有多种量子态，并以不可预测的方式坍缩成其中一种。
Physicists have been trying to understand the source of this mismatch for the past century.
在过去的一个世纪里，物理学家一直试图理解这种不匹配的根源。
Nicolas Gisin proposed that one of the greatest mysteries in physics might be due to incorrect assumptions about infinity.
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。
Gisin posits that it’s due to a faulty assumption.
吉辛认为这是由于错误的假设造成的。
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.
他说，研究人员隐含地相信，从宇宙诞生之初，粒子的量子态就可以通过无限多位数的实数来无限精确地定义。
But, according to Gisin, using the real numbers is a mistake.
但是，根据吉辛的说法，使用真实数字是一个错误。
If you use intuitionist mathematics instead, then it becomes clear that determinism is but an artifact of having unrealistically perfect information.
如果你使用直觉主义数学，那么很明显，决定论只不过是拥有不切实际的完美信息的产物。
The large-scale, deterministic behavior of physical systems naturally becomes imprecise and unpredictable, dissolving the divide between the classical and quantum realms.
物理系统的大规模、确定性行为自然会变得不精确和不可预测，从而消除了经典领域和量子领域之间的鸿沟。
Gisin’s theory has proved intriguing to other physicists, in part because it could help resolve paradoxes about phenomena like the Big Bang.
事实证明，吉辛的理论引起了其他物理学家的兴趣，部分原因是它可以帮助解决大爆炸等现象的悖论。
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.
但值得注意的是，他的工作并没有废除亚里士多德意义上的潜在无限，即可能达到的事物。
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.
按照直觉主义数学家花费时间和精力计算更大或更精确数字的传统，吉辛允许创建越来越多的信息。
Someday, the universe will contain perfect, infinitely precise information.
有一天，宇宙将包含完美、无限精确的信息。
But it doesn’t matter, because that someday will never come.
但这并不重要，因为那一天永远不会到来。
“The potential infinity here is really waiting infinite time, which has nothing to do with reality,” Gisin said.
“这里的潜在无限实际上是等待无限的时间，这与现实无关，”吉辛说。
The important thing is that infinity is no longer the default assumption.
重要的是无穷大不再是默认假设。
The physicist Sean Carroll is intrigued by the possibility that the universe might be finite.
物理学家肖恩·卡罗尔对宇宙可能是有限的可能性很感兴趣。
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.
这些基于物理的对无限的挑战往往会让超有限主义数学家感到高兴，他们将它们视为他们的数学是对现实的更真实描述的证据。
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.
在 2025 年会议上，卡罗尔关于宇宙是否真的是无限的或用他的话来说“只是相当大”的演讲使他成为哥伦比亚大学礼堂中的名人。
But the burden of proof, he cautions, remains with the infinity doubters.
但他警告说，举证责任仍由无限怀疑者承担。
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.
如果你能以某种方式通过实验证明物理宇宙确实是有限的，那么即使是更高无限的最热心支持者也可能会花点时间停下来反思。
They would probably even wonder about the consistency of set theory, given the towers of actual infinities that it allows.
考虑到集合论允许的实际无穷大塔，他们甚至可能想知道集合论的一致性。
That’s a healthy thing to do from time to time, anyway.
无论如何，时不时地这样做是一件健康的事情。
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.
即使这种情况发生，研究和使用无穷大的集合理论家仍然有权不受干扰地继续他们的工作——可以说，也许这就是物理和数学必须相互分离的地方。
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.
数学和物理学并不要求描述相同的事物（尽管许多人相信它是），并且无穷可能在某种更大的柏拉图意义上继续存在。
But if those experiments proved the opposite — that infinity does exist in nature — the ultrafinitist would have far less room to negotiate.
但如果这些实验证明了相反的情况——自然界中确实存在无限——超有限论者的谈判空间就会小得多。
“It would be hard to be an ultrafinitist if the actual physical world had infinities in it,” Carroll said.
卡罗尔说：“如果实际的物理世界有无穷大，那么就很难成为一个超有限论者。”
“I feel bad for the ultrafinitists because people dismiss it without understanding it,” Carroll told me later.
“我为超有限论者感到难过，因为人们在不理解它的情况下就忽视了它，”卡罗尔后来告诉我。
“But on the other hand, the ultrafinitists don’t do a good enough job of marketing their product.”
“但另一方面，超有限主义者在产品营销方面做得不够好。”
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.
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，尼尔森所寻求的那种理论——一套正式的规则，就像现代数学的基础一样，排除了无穷大，但足够强大，可以做有用的数学。
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.
克拉克-多恩说，想法并不缺乏，尽管可能缺乏愿意将自己的早期职业生涯投入到发展这些想法的研究生。
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.
对他来说，纽约的聚会是变化的标志，人们有足够的好奇心重新审视它，并且不太害怕潜在的强烈反对。
“People are talking about the view and actively trying to think about how to put the view on a serious foundation,” he said.
“人们正在谈论这种观点，并积极思考如何将这种观点建立在严肃的基础上，”他说。
Most mathematicians live outside all this.
大多数数学家生活在这一切之外。
Formal theories encompassing the totality of mathematics do not concern them.
涵盖整个数学的形式理论与他们无关。
They are interested in what works, in solving specific problems and building proofs.
他们对有效的方法、解决具体问题和构建证明感兴趣。
Foundational questions — do numbers exist beyond physical reality?
基本问题——数字是否存在于物理现实之外？
Is math a process of invention or discovery?
数学是发明或发现的过程吗？
— can feel a little cringe, the sort of thing mathematicians only do when they wake up one day in a state of crisis.
——会感到有点畏缩，这种事情数学家只有在有一天在危机状态中醒来时才会做。
But the working mathematician might find common ground with Zeilberger, who is similarly unbothered by the arguments of set theorists and philosophers.
但这位工作数学家可能会与蔡尔伯格找到共同点，蔡尔伯格同样不受集合论学家和哲学家的论点困扰。
His is a method of ruthless practicality, of taking apart mathematics piece by piece and asking what is necessary.
他的方法是一种无情的实用性方法，将数学一点一点地拆开并询问有什么必要。
Perhaps, he says, we have assumed too much, made infinity too much of a default, put faith in illusions.
他说，也许我们假设得太多，把无限当作默认值，过于相信幻想。
One doesn’t need to declare oneself an ultrafinitist to get some satisfaction from that, to add it to the menu of truthful options.
人们不需要宣称自己是超有限主义者就能从中获得一些满足，将其添加到真实选项的菜单中。
Zeilberger is fond of quoting himself in a BBC documentary from 2010 — what he considers his 15 seconds of fame.
Zeilberger 喜欢在 2010 年 BBC 的一部纪录片中引用自己的话——他认为这就是他 15 秒的成名之处。
“Infinity may or may not exist; God may or may not exist,” he said.
“无限可能存在，也可能不存在；上帝可能存在，也可能不存在，”他说。
“But in mathematics, there should not be any place, neither for infinity nor God.” He was replying, asynchronously, to Hugh Woodin , a leading set theorist and one of the most intrepid explorers of the higher infinite, who had said he felt sorry for Zeilberger, unable to look up at the sky and grasp the beauty of the infinite expanse.
“但在数学中，无论是无穷还是上帝，都不应该有任何地位。”他正在异步地回复休·伍丁（Hugh Woodin），休·伍丁是一位领先的集合理论家，也是高等无限领域最勇敢的探索者之一，他说他为泽尔伯格感到难过，因为他无法仰望天空并领会无限广阔的美丽。
“I feel sorry for him that he needs the opiate of infinity to keep him going,” Zeilberger said.
“我为他感到遗憾，因为他需要无限的鸦片来让他继续前进，”泽尔伯格说。
“There’s so much beauty in the trees and in the ground.
“树上和地上都有很多美丽的东西。
You don’t have to look toward fiction.”
你不必寻找小说。”
“So we both feel sorry for each other,” he said.
“所以我们都为彼此感到难过，”他说。
Sorry that the other should feel imprisoned in a world of his chosen faith.
很抱歉，对方会感到自己被囚禁在他所选择的信仰的世界中。
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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。
Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.
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。
Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.
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。