数学的终极问题：上帝是一位数学家吗？

注：本文为 “数学的终极问题” 相关合辑。 英文引文，机翻未校。 如有内容异常，请看原文。

God is a Mathematician

上帝是一位数学家

Article by Sunil Mehrotra 作者 苏尼尔・梅赫罗特拉

Mathematics is abstract, symbolic, structured and precise. It is true everywhere and always and mathematical laws cannot be violated, ever. Math sounds a lot like the attributes of God—eternal, omnipresent and omnipotent. According to theoretical physicist Michio Koku, “The mind of God we believe is cosmic music, the music of strings resonating through 11‑dimensional hyperspace. That is the mind of God.”ⁱ Vern Poythress, who teaches New Testament in Cambridge University and has two doctorates, a PhD in mathematics from Harvard and a doctorate in Divinity, argues in his book Redeeming Mathematics: A God Centered Approachⁱⁱ, that “the harmony of abstract mathematics, the physical world of things and our thinking depends on the existence of a Christian God.” Srinivas Ramanujan, on whose life the book and the movie “The Man who knew Infinity”ⁱⁱⁱ are based is known to have said that “an equation to me has no meaning unless it represents a thought of God”. 数学是抽象的、符号化的、结构化的且精确的。它在任何地方、任何时候都成立，数学规律永远无法被违背。数学听起来与上帝的诸多属性十分相似——永恒、无所不在、无所不能。理论物理学家加来道雄认为：“我们所相信的上帝之心，是宇宙的音乐，是弦在 11 维超空间中共振的乐章。这便是上帝之心。” 弗恩・波伊思雷斯在剑桥大学教授新约，拥有两个博士学位：一个是哈佛大学的数学博士，一个是神学博士，他在其著作《救赎数学：一种以上帝为中心的进路》中提出：“抽象数学、物质世界与人类思维之间的和谐，依赖于基督教之上帝的存在。” 以其生平为蓝本创作了书籍与电影《知无涯者》的斯里尼瓦瑟・拉马努金曾说过：“对我而言，一个方程若不表达上帝的思想，便毫无意义。”

The structures of the universe, from the tiniest (subatomic) to the largest (cosmic), can be visualized as networks or webs of connections. And, these networks are interlocking, pulsating particles, exchanging, sharing and transforming energy from one form to another. Physics today is understood through mathematics. Scientists have long used mathematics to describe the physical properties of the universe, but physicist Max Tegmark^iv goes even further and believes that the universe itself is math. In Tegmark’s view, everything in the universe — humans included — is part of a mathematical structure. All matter is made up of particles, which have properties such as charge and spin, but these properties are purely mathematical, he says. And space itself has properties such as dimensions, but is still ultimately a mathematical structure. 从最微小的（亚原子）到最宏大的（宇宙），宇宙的结构都可以被视作相互联结的网络或网状系统。这些网络由相互咬合、脉动的粒子构成，它们进行能量的交换、共享与形式转化。当今的物理学通过数学来理解。长期以来，科学家用数学描述宇宙的物理性质，而物理学家马克斯・泰格马克则更进一步，认为宇宙本身就是数学。在泰格马克看来，宇宙中的一切——包括人类——都属于某种数学结构。他提出，所有物质都由粒子组成，粒子具有电荷、自旋等性质，而这些性质纯粹是数学性质。空间本身具有维度等属性，但归根结底也仍是一种数学结构。

Mathematics, numbers, symbols, information and energy are different ways by which physicists have attempted to describe the universe. Modern theories in physics are abstract and mystifying to most. For many, faith in the divine origin of the universe provides more certitude than modern physics does. Faith gives one certainty which physics is unable to do, this is the appeal of faith for many. Certainty in an uncertain world is comforting. 数学、数字、符号、信息与能量，是物理学家用来描述宇宙的不同方式。现代物理理论对大多数人而言抽象而费解。对许多人来说，相信宇宙具有神圣起源，比现代物理能带来更多确定性。信仰能给予人物理学无法提供的确定性，这正是信仰对许多人的吸引力所在。在不确定的世界中拥有确定感，会让人感到安心。

Scientific knowledge has an asymptotic relationship to truth or Truth. Scientists are getting closer to the Truth but, I suspect, will never reach it. Scientists are like Adam reaching out to touch the hand of God but not making it, as depicted in the fresco^v on the ceiling of the Sistine Chapel. 科学知识与真理（或绝对真理）之间呈渐近关系。科学家正在不断接近真理，但我认为他们永远无法抵达。科学家就如同西斯廷教堂天顶壁画中所描绘的亚当，伸手想要触碰上帝之手，却始终未能触及。

Physicists are peering into the outer reaches of the cosmos and probing deep into the inner sanctum of atoms, discovering realms that are beyond the reach of our senses. For most of us, these realms are difficult to comprehend, because we cannot see, touch or feel them. No one has seen a quark or been able to visualize Einstein’s four‑dimensional space and time. Hence, to some, modern physics is incomprehensible, abstract, hard to relate to, and indistinguishable from a myth. 物理学家既眺望宇宙的极远边界，也深入原子的内部秘境，发现了诸多超出人类感官范围的领域。对我们大多数人而言，这些领域难以理解，因为我们无法看见、触摸或感知它们。没有人见过夸克，也无法直观想象爱因斯坦的四维时空。因此，对一些人来说，现代物理学晦涩难懂、抽象疏离，与神话几乎无法区分。

Thankfully, we do not rely just on our senses to understand the universe; if we did, we would still be in the dark ages. 值得庆幸的是，我们并非只依靠感官来理解宇宙。倘若如此，人类仍将停留在黑暗时代。

Physicists are looking for a single theory or, as Michio Koku states, “an equation about six inches long,” which can explain all phenomenon, from the largest (cosmos) to the tiniest (subatomic particles). The holy grail in physics is to find a theory that reconciles general relativity and quantum physics. Science is in search of ultimate unity, the God Particle^vi, as Nobel Laureate Leon Lederman wrote in his book of the same name. Particle physicists keep building bigger and bigger particle accelerators, like the one at CERN^vii, in search of the God particle. At CERN, in the Large Hadron Collider, energy at the point of collision of the protons approaches the energy moments after the Big Bang, in the hope of finding the God particle. 物理学家正在寻找一种大一统理论，或者如加来道雄所言，一个“约 6 英寸长的方程”，它能够解释从最宏大（宇宙）到最微小（亚原子粒子）的一切现象。物理学的圣杯，是找到一种能调和广义相对论与量子物理的理论。科学界正在追寻终极的统一，即“上帝粒子”，诺贝尔奖得主利昂・莱德曼在其同名著作中如此命名。粒子物理学家不断建造越来越大的粒子加速器，例如欧洲核子研究中心（CERN）的加速器，以寻找上帝粒子。在欧洲核子研究中心的大型强子对撞机中，质子碰撞瞬间的能量接近宇宙大爆炸后瞬间的能量，以期找到上帝粒子。

The truth is that the Truth might not be a particle. Truth might not be a thing. It might be an abstraction, like an “idea in the mind of God”, as some have suggested, or perhaps as Max Tegmark posits, “There’s something very mathematical about our Universe, and the more carefully we look, the more math we seem to find. …… So, the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Everything in our world is purely mathematical – including you.” 事实是，绝对真理或许并非一种粒子。真理或许并非某个“事物”。它可能是一种抽象存在，正如一些人所提出的，是“上帝心中的一个理念”；或者如马克斯・泰格马克所主张的：“我们的宇宙具有极强的数学属性，我们观察得越细致，就会发现越多的数学规律……因此，结论是：如果你相信存在独立于人类之外的客观现实，那么你也必须相信，我们的物理现实就是一种数学结构。我们世界中的一切——包括你在内——都纯粹是数学的。”

Theologists, scientists and philosophers seem to agree that Reality, absolute truth, or God, if you will, is an abstract reality. Not a reality that can be detected by our senses or known through our intellect. In this view mathematics are an expression of the mind of God. She is a mathematician! 神学家、科学家与哲学家似乎达成了共识：实在、绝对真理，或者你愿意称之为上帝，是一种抽象的实在。它并非能被人类感官探测或仅凭理智理解的实在。在这种观点下，数学是上帝思想的表达。上帝，是一位数学家！

Further proof is in sacred geometries found in nature. 更多的证据，存在于自然界中显现的神圣几何之中。

Sunil Mehrotra 苏尼尔・梅赫罗特拉

Sunil Mehrotra has an MS in Aeronautical Engineering from Purdue University and an MBA from The University of Chicago. He has held senior executive positions at Fortune 500 companies and has been a CEO and founder of two start‑ups. He has also taught at Pepperdine University in Malibu, California. You can read more about his book, Shiva’s Dance: A Scientist Dances with the Sages, at https://shivasdance.org. *苏尼尔・梅赫罗特拉拥有普渡大学航空工程硕士学位与芝加哥大学工商管理硕士学位。他曾在财富 500 强企业担任高级管理职位，是两家初创公司的首席执行官与创始人，并曾在加利福尼亚州马里布的佩珀代因大学任教。

另一个观点

Mathematics declaring the glory of God

数学彰显上帝的荣耀

Verbum et Ecclesia ISSN: (Online) 2074-7705, (Print) 1609-9982 AOSIS Original Research 《道与教会》期刊 国际标准连续出版物号：（网络版）2074-7705，（印刷版）1609-9982 非洲开放科学倡议出版机构 原创研究 Author: Volker Kessler¹,² 作者： 沃尔克·克斯勒¹，² Affiliations: ¹Gesellschaft für Bildung und Forschung in Europa, Gummersbach, Germany ²Department of Philosophy, Practical and Systematic Theology, College of Human Sciences, University of South Africa, Pretoria, South Africa 作者单位： ¹ 德国古默斯巴赫 欧洲教育与研究协会 ² 南非比勒陀利亚 南非人文科学大学 哲学、实践与系统神学系 Corresponding author: Volker Kessler, volker.kessler@gbfe.eu 通讯作者： 沃尔克·克斯勒 邮箱：volker.kessler@gbfe.eu Dates: 日期： Received: 18 Nov. 2021 / 收稿日期： Accepted: 08 Mar. 2022 / 录用日期 Published: 19 Apr. 2022 / 出版日期

How to cite this article: 本文引用格式：

Kessler, V., 2022, ‘Mathematics declaring the glory of God’, Verbum et Ecclesia 43(1), a2432. https://doi.org/10.4102/ve.v43i1.2432

克斯勒，V.，2022，《数学彰显上帝的荣耀》，《道与教会》，43（1），a2432。 Copyright: © 2022. The Author. Licensee: AOSIS. This work is licensed under the Creative Commons Attribution License. 著作权说明： ©2022 本文作者。出版授权方：非洲开放科学倡议出版机构。本作品采用知识共享署名许可协议授权。

This article discussed the question ‘Does God speak through the language of mathematics?’ For centuries, mathematicians with different religious backgrounds would have answered this question in the affirmative. Due to changes in mathematics from the 19th century onwards, this question cannot be answered as easily as it used to be. If one regards mathematical concepts as creations of the human mind, it is difficult to argue that mathematical formulae exist in a divine mind. The article argued that there were traces of the divine in mathematics. Six kinds of traces were explained: (1) the existence of indisputable truth, (2) the existence of beauty, (3) the importance of community, (4) rational speaking about infinity, (5) the discovery that speaking about unseen and abstract objects is reasonable and (6) the unreasonable effectiveness of mathematics. In practice, traces (1), (2) and (6) are probably the most convincing. 本文探讨了“上帝是否通过数学的语言言说？”这一问题。数个世纪以来，不同宗教背景的数学家都会对这一问题给出肯定的答案。19 世纪以来，数学领域发生了诸多变革，这一问题也不再像以往那样能轻易作答。若将数学概念视作人类思维的创造物，便难以论证数学公式存在于神圣的思维之中。本文提出，数学中存在着神圣的踪迹，并阐释了六种踪迹：(1)不容置疑的真理的存在；(2)美的存在；(3)共同体的重要性；(4)对无限进行理性阐释；(5)发现对不可见的抽象客体进行言说具有合理性；(6)数学超乎常理的有效性。实际上，其中第(1)､(2)､(6)种踪迹或许最具说服力。

Intradisciplinary and/or interdisciplinary implications: This article is very much interdisciplinary as it combines mathematics and theology, especially the philosophy of mathematics and systematic theology. 学科内及跨学科意义：本文具有高度的跨学科性，融合了数学与神学两大领域，尤其结合了数学哲学与系统神学的相关研究。

Keywords: philosophy of mathematics; general revelation; truth; beauty; infinity; unreasonable effectiveness. 关键词：数学哲学；普遍启示；真理；美；无限；超乎常理的有效性

Introduction

引言

This article is about how traces of the divine can be found in the language of mathematics. The title alludes to Psalm 19:1: ‘The heavens declare the glory of God’, which is part of the Jewish and Christian Bible. 本文旨在探讨如何在数学的语言中发现神圣的踪迹。文章标题化用了犹太教与基督教经典《圣经》中《诗篇》19 篇 1 节的内容：“诸天述说上帝的荣耀”。

Obviously, mathematics is some kind of language (Livio 2009:239–241). Mathematics is often called the ‘language of nature’, which goes back to a saying by the Italian astronomer Galileo Galilei (1564–1642) (Mukunda 2015:347). Isaac Newton called his famous work on physics The Mathematical Principles of Natural Philosophy because he was convinced that natural philosophy has to be expressed in the language of mathematics. 显然，数学是一种语言（利维奥，2009：239-241）。数学常被称作“自然的语言”，这一说法可追溯至意大利天文学家伽利略·伽利莱（1564-1642）（穆昆达，2015：347）。艾萨克·牛顿将其经典的物理学著作命名为《自然哲学的数学原理》，因他坚信，自然哲学必须以数学的语言来表达。

The question to be investigated in this article is as follows: Does God speak through the language of mathematics? Two centuries ago, it would have been easy to argue for a positive answer. Due to changes in mathematics and the growing influence of atheism, today’s answer has to be different from the answers given two centuries ago. In this article, I will first explain the changes in mathematics and their implications for the question to be investigated. Secondly, I will discuss six examples of indications of divine elements in mathematics. I am writing from the perspective of a mathematician and Christian theologian. Thus, I will focus on doctrines of the Christian faith although some implications will also hold for other faith systems. 本文探讨的核心问题为：上帝是否通过数学的语言言说？两个世纪前，人们能轻易为这一问题给出肯定的论证。而随着数学的变革与无神论影响的不断扩大，如今的答案必然与两个世纪前有所不同。本文中，笔者将首先阐释数学领域的变革及其对这一研究问题的意义；其次，探讨数学中存在神圣元素的六大佐证。笔者以数学家与基督教神学家的双重视角展开写作，因此将聚焦于基督教的教义，尽管部分研究结论也适用于其他宗教体系。

In this article, I will not treat mathematical theistic proofs such as Gödel ¹ or Meyer (1987). I do not think that mathematics alone can say anything about God’s existence. Mathematics is a formal construct, and as such it does not say anything about the real nature of things.² Instead I will argue that mathematics provides an opportunity to discover some properties that are usually associated with God. I will list several aspects of mathematics that I regard as traces or hints leading one to reflect upon God and divine attributes. 本文不探讨诸如哥德尔 ¹ 或迈耶（1987）所提出的数学神存在证明。笔者认为，仅凭数学无法对上帝的存在作出任何论断。数学是一种形式化的建构，其本身无法揭示事物的真实本质 ²。笔者将论证，数学为人们发现那些通常与上帝相关的属性提供了契机，并列举数学中若干可被视作踪迹或线索的方面，引导人们思考上帝与神圣的属性。

This article develops some ideas that I presented in two previous articles (Kessler 2018, 2019). In particular, I take into account some reactions to these articles.³ 本文进一步发展了笔者在两篇前期论文中提出的观点（克斯勒，2018、2019），尤其结合了学界对这两篇论文的相关回应³。

When the answer seemed to be straightforward

答案曾一目了然的时代

Mathematics and the mind of God – before the big changes

数学与上帝的思维——重大变革之前

In the past, strong links between mathematics and the divine have always been assumed; see, for example, the historical study Koetsier and Bergmann (eds. 2005). For many, mathematical knowledge used to be ‘an aspect of spiritual knowledge, knowledge in the mind of God’ (Hersh 1997:236). Jonas (1966) and Livio (2009) raised the same question: ‘Is God a mathematician?’ ⁴ 在过去，人们始终认为数学与神圣之间存在着紧密的联系，例如可参见科齐尔与伯格曼主编的相关历史研究（2005）。对许多人而言，数学知识曾是“灵性知识的一部分，是存在于上帝思维中的知识”（赫什，1997：236）。约纳斯（1966）与利维奥（2009）都提出过同一个问题：“上帝是数学家吗？” ⁴

In the Classic Greek tradition, theology and mathematics were close to each other (Phillips 2009). Pythagoras (ca. 570–510 BC) was convinced that numbers and formulae had an inherent spiritual meaning, and his school at Croton was ‘more a religious brotherhood than an academy’ (Phillips 2009:5). Pythagoras and Plato both regarded mathematics as a spiritual path leading to the divine, but they came to different conclusions. For Plato (428–348 BC), mathematical studies were a preparation for the contemplation of the divine principles, whereas for the Pythagoreans mathematics actually was God (Livio 2009:28). 在古希腊传统中，神学与数学联系紧密（菲利普斯，2009）。毕达哥拉斯（约公元前 570-前 510 年）坚信，数字与公式蕴含着内在的灵性意义，他在克罗托内创办的学园“与其说是一所学术机构，不如说是一个宗教兄弟会”（菲利普斯，2009：5）。毕达哥拉斯与柏拉图均将数学视作通往神圣的灵性之路，却得出了不同的结论。对柏拉图（公元前 428-前 348 年）而言，数学研究是默想神圣法则的准备；而对毕达哥拉斯学派而言，数学本身就是上帝（利维奥，2009：28）。

To give an example of the close link between mathematics and theology some centuries ago, we look at the beliefs of some important mathematicians of the 17th century, the start of the scientific age and of modern mathematics. In the Europe of the 17th century, it was quite common to believe in the Christian God, and this is also true for the most brilliant European mathematicians of the 17th century. It is, of course, debatable that mathematician was most important at a given time; for example, Stewart (2017) selects 25 mathematicians throughout history. Nevertheless there are good reasons to regard the following six persons as the most important mathematicians of the 17th century: (1) René Descartes, France (1596–1650), who invented analytic geometry; his compatriots (2) Pierre de Fermat (1607–1665) and (3) Blaise de Pascal (1623–1662), both of whom (among others) laid the foundation of probability theory; (4) Sir Isaac Newton, England (1642–1726) and (5) Gottfried Wilhelm Leibniz, Germany (1646–1716), both of whom invented calculus independently of each other and (6) Jacob Bernoulli, Switzerland (1654–1705), after whom the ‘law of large numbers’ is named. It is interesting to note that five of them saw a strong connection between their mathematics and their Christian faith. (We do not have a statement from Fermat on this.) 为佐证数个世纪前数学与神学的紧密联系，我们来看看 17 世纪一众重要数学家的信仰。17 世纪是科学时代与现代数学的开端，彼时的欧洲，信仰基督教上帝是一种普遍现象，17 世纪欧洲最杰出的数学家们亦不例外。当然，关于某一时期哪位数学家最为重要，历来众说纷纭，例如斯图尔特（2017）便遴选出了历史上的 25 位杰出数学家。尽管如此，我们仍有充分的理由将以下六位数学家视作 17 世纪最具影响力的数学家：（1）法国的勒内·笛卡尔（1596-1650），解析几何的发明者；其法国同胞（2）皮埃尔·德·费马（1607-1665）与（3）布莱士·帕斯卡（1623-1662），二人共同为概率论奠定了基础；（4）英国的艾萨克·牛顿爵士（1642-1726）与（5）德国的戈特弗里德·威廉·莱布尼茨（1646-1716），二人各自独立发明了微积分；（6）瑞士的雅各布·伯努利（1654-1705），“大数定律”便是以他的名字命名。值得注意的是，这六位数学家中，有五位都认为自己的数学研究与基督教信仰之间存在着紧密的联系（目前尚未发现费马对此的相关论述）。

Descartes provided two theistic proofs in his Meditations (Descartes 1976). Pascal provided a famous ‘wager’ in which he gives a probabilistic argument for choosing to believe in God (discussed in Heller 2018:42–44). Newton himself believed strongly in a Designer who worked through mathematical laws (Davies 1992:76). ‘The most beautiful system of sun, planets and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being’ (Newton, quoted in Livio 2009:114). According to the Leibniz specialist Martin (1960), Leibniz regarded mathematical theorems as ‘primarily and continuously thought by God’, and when a mathematician discovers them, ‘this knowing is a repetition of the primary divine knowing’ (quoted in Hersh 1997:126). Bernoulli was a strong Calvinist and reflected on the theological implications of his discoveries in probability theory (Heller 2018:48–55). 笛卡尔在其《第一哲学沉思集》中提出了两种神存在证明（笛卡尔，1976）。帕斯卡提出了著名的“帕斯卡赌注”，以概率论的视角论证了信仰上帝的合理性（海勒对此有相关探讨，2018：42-44）。牛顿本人坚信，存在一位通过数学法则施展其创造的造物主（戴维斯，1992：76），他曾言：“太阳、行星与彗星构成的这一最美妙的体系，只能诞生于一位智慧而全能的存在的谋划与主宰。”（牛顿，转引自利维奥，2009：114）。莱布尼茨研究专家马丁（1960）指出，莱布尼茨将数学定理视作“上帝首要且持续的思考之物”，而数学家对定理的发现，“这种认知是对神圣本源认知的重现”（转引自赫什，1997：126）。伯努利是坚定的加尔文教徒，他深入思考了自己在概率论领域的发现所蕴含的神学意义（海勒，2018：48-55）。

In a world where mathematical truths were seen as part of the divine mind, it was easy to argue that God reveals himself through mathematics. The quotation from Edward Everett (1794–1865) gives a poetic description of this viewpoint: 在那个将数学真理视作神圣思维一部分的时代，人们很容易论证上帝通过数学彰显自身。爱德华·埃弗雷特（1794-1865）的一段话诗意地诠释了这一观点：

In the pure mathematics, we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven. (quoted in Hersh 1997:9) 在纯粹的数学中，我们凝视着绝对的真理，这些真理在晨星一同歌唱之前，便已存在于神圣的思维之中；即便当最后一颗星辰从天际陨落，它们也将永存于那神圣的思维里。（转引自赫什，1997：9）

Similar statements are known from mathematicians with a different religious background. For example, the remarkable Indian mathematician S. Ramanujan (1887–1920) once noted: ‘An equation means nothing to me unless it expresses a thought of God’ (quoted in Kanigel 1991: pos 221). 不同宗教背景的数学家也有过类似的表述。例如，印度杰出的数学家 S·拉马努金（1887-1920）曾说：“若一个方程无法表达上帝的思想，那它对我而言便毫无意义。”（转引自卡尼格尔，1991：第 221 段）。

Discovery of non-Euclidean geometry and its implications

非欧几何的发现及其意义

As the above examples show, the validity of Euclidean geometry used to be regarded as indisputable and was often taken as an analogy for certainty in theological issues. ‘Geometry served from the time of Plato as proof that certainty is possible in human knowledge – including religious certainty’ (Hersh 1997:137). This reasoning was challenged by the discovery of non-Euclidean geometry in the 19th century by Gauss (Germany), Lobachevsky (Russia), Bolyai (Hungary) and finally Riemann (Germany). 如上述例子所示，欧几里得几何的有效性曾被视作不容置疑的真理，且常被用作类比，论证神学问题中存在确定的答案。“自柏拉图时代起，几何学便成为证明人类的认知中存在确定性的依据——这其中也包括宗教认知的确定性。”（赫什，1997：137）。而 19 世纪，高斯（德国）、罗巴切夫斯基（俄罗斯）、鲍耶（匈牙利）以及黎曼（德国）相继发现了非欧几何，这一论证逻辑受到了挑战。

The angle-sum theorem teaches that the angles in a triangle add up to 180°. This theorem was often used as an example of an absolutely certain statement. For example, Descartes (1976:59) claimed in the fifth meditation that the existence of God is as certain as the fact that three angles of a triangle are equal to two right angles. Spinoza regarded this theorem as indubitable (Hersh 1997:121). David Hume noted that Euclidean geometry was as solid as the rock of Gibraltar (Livio 2009:151), and for Kant, who regarded the concept of space as a priori, the space was indisputably Euclidean (Kant 1966:88; Livio 2009:152). 三角形内角和定理指出，三角形的内角和为

180

∘

180^\\circ

180∘。这一定理常被用作绝对确定命题的范例。例如，笛卡尔在其《第一哲学沉思集》的第五个沉思中提出，上帝的存在与三角形的内角和等于两个直角这一事实同样确定（笛卡尔，1976：59）。斯宾诺莎将这一定理视作不容置疑的真理（赫什，1997：121）。大卫·休谟认为，欧几里得几何如同直布罗陀巨岩一般坚不可摧（利维奥，2009：151）；而康德将空间概念视作先验的存在，在他看来，空间必然是欧几里得式的（康德，1966：88；利维奥，2009：152）。

But this theorem does not hold in non-Euclidean geometry. The sum of the angles in a triangle can be more than 180° (elliptic or Riemannian geometry) or less than 180° (hyperbolic geometry).⁵ 但这一定理在非欧几何中并不成立。在非欧几何中，三角形的内角和可大于

180

∘

180^\\circ

180∘（椭圆几何或黎曼几何），也可小于

180

∘

180^\\circ

180∘（双曲几何）⁵。

This discovery led to a re-thinking of the foundations of mathematics. The famous French mathematician and physicist Henri Poincaré (1854–1912) made the consequences very clear: The geometrical axioms are therefore neither synthetic à priori intuitions⁶ nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free … (Poincaré 1905:50) 这一发现促使人们重新思考数学的基础。法国著名数学家、物理学家亨利·庞加莱（1854-1912）清晰地阐明了这一发现的影响： 因此，几何公理既非先天综合直观⁶，亦非实验事实，而是一种约定。我们在诸多可能的约定中作出选择时，会以实验事实为指引，但这种选择始终是自由的……（庞加莱，1905：50）

The new situation was that the mathematicians would offer different sorts of geometry from which the physicists had to choose the geometry, which would be most appropriate for studying the physical world. This new situation challenged the understanding of mathematics as a divine science. 此后，数学家提出了多种几何体系，物理学家则需要从中选择最适合研究物理世界的几何体系。这一新的局面，对将数学视作神圣科学的传统认知提出了挑战。

The fact that one could select a different set of axioms and construct a different type of geometry raised for the first time the suspicion that mathematics is, after all, a human invention, rather than a discovery of truths that exist independently of the human mind. (Livio 2009:159) 人们可以选择不同的公理体系，构建出不同的几何体系，这一事实首次让人们产生了这样的质疑：数学终究是人类的发明，而非对独立于人类思维之外的真理的发现（利维奥，2009：159）。

At this point, it became obvious that intuition alone is not sufficient for mathematical certainty. Cantor’s set theory was a candidate for the foundation of mathematics. But then paradoxes in the set theory were discovered by Zermelo in 1899 and by Russell in 1901 independently of each other. This finally led to three different schools of thought on how to lay a good foundation for mathematics: logicism, formalism and intuitionism (Shapiro 2000:107–197). It turned out that each of these three major schools has its deficiencies; none can fulfil the promises of their inventors. Intuitionism is almost dead, and Gödel destroyed ‘all hope for a consistent and complete axiomatization of mathematics’ (Mendelson 1987:175). 至此，人们明确认识到，仅凭直观无法实现数学的确定性。康托尔的集合论曾被视作奠定数学基础的候选理论，然而策梅洛于 1899 年、罗素于 1901 年各自独立发现了集合论中的悖论。这一发现最终催生了三大数学基础学派：逻辑主义、形式主义与直觉主义（夏皮罗，2000：107-197）。事实证明，这三大主流学派均存在缺陷，没有一个学派能实现其创立者的初衷。直觉主义如今几近消亡，而哥德尔的理论则彻底打破了“为数学建立一致且完备的公理化体系的所有希望”（门德尔松，1987：175）。

Actually, the question of the foundation of mathematics has no clear answer to this day (Hersh 1997, Shapiro 2000). As a consequence, the vast majority of working mathematicians do not spend too much time on the question of foundations. 事实上，数学基础问题至今仍无明确答案（赫什，1997；夏皮罗，2000）。因此，绝大多数一线数学家并不会在数学基础问题上耗费过多精力。

Pragmatic Platonism afterwards

此后的实用柏拉图主义

Although Platonism is no longer considered an up-to-date epistemology or philosophy of science, it is still alive among working mathematicians. A crucial question to mathematicians is ‘is mathematics created or discovered?’ In 1940, the renowned British mathematician G.F. Hardy (1877–1947) wrote about the ‘mathematical reality’, admitting that there is no sort of agreement on the nature of this reality (Hardy 2001:123). Hardy (p. 123) believed ‘that the mathematical reality lies outside us, that our function it to discover or observe it’. Similarly, the German-Austrian logician Kurt Gödel (1906–1978) argued in 1944: Classes and concepts may … be considered as real objects … It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. (quoted in Shapiro 2000:10) 尽管柏拉图主义已不再被视作前沿的认识论或科学哲学理论，但在一线数学家中，这一思想仍有一席之地。对数学家而言，一个核心问题是：“数学是被创造的，还是被发现的？”1940 年，英国著名数学家 G·F·哈代（1877-1947）在探讨“数学实在”时坦言，学界对这一实在的本质始终未能达成共识（哈代，2001：123）。哈代认为，“数学实在存在于我们之外，我们的任务是去发现或观察它”（同上，第 123 页）。无独有偶，德裔奥地利逻辑学家库尔特·哥德尔（1906-1978）在 1944 年提出： 类与概念可以被视作真实的客体……在我看来，假定这类客体的存在，与假定物理实体的存在同样合理，我们也有同样充分的理由相信它们的存在。（转引自夏皮罗，2000：10）

A small survey done by Pamela Aschbacher, the spouse of an American mathematician shows that this is probably the thinking of most working mathematicians. Aschbacher (2015:17) asked the colleagues of her husband whether math was something like an unseen star, out there to be discovered, or something created by us. Except in one case, she received the answer: ‘Oh, it’s definitely the “truths” of the world waiting to be discovered’ (p. 17). This attitude is also implicit in mathematical parlance, because most mathematicians would say: ‘I discovered this mathematical law’. The underlying assumption is that it was already there. 美国一位数学家的配偶帕梅拉·阿希巴赫开展的一项小型调查表明，这一观点或许代表了绝大多数一线数学家的想法。阿希巴赫询问其丈夫的同事，数学究竟是如同遥远天际中未被发现的星辰，等待人们去探寻，还是由人类创造的产物（2015：17）。除一人之外，所有人都给出了这样的答案：“显然，数学是世间等待被发现的真理。”（同上，第 17 页）。这种态度也隐含在数学的专业表述中，多数数学家都会说“我发现了这一数学定律”，其背后的预设是，这一定律本就存在。

Harrison (2017:481) writes that ‘virtually everyone in (the mathematical) community assumes a realist interpretation of ontology’. This statement ‘virtually everyone’ might be a bit too strong. For example, the American mathematician Reuben Hersh (1927–2020) does not share this realist interpretation and quotes several mathematicians on his side (Hersh 1997:182–232). Actually Hersh strongly objects to the fact that mainstream mathematicians still stick to Platonism: ‘Why do mathematicians believe something so unscientific, so far-fetched as an independent immaterial timeless world of mathematical truth?’ (Hersh 1997:11).⁷ ‘The trouble with today’s Platonism is that it gives up God, but wants to keep mathematics a thought in the mind of God’ (p. 135). 哈里森（2017：481）写道：“数学界几乎所有人都秉持本体论的实在论解释。”“几乎所有人”这一表述或许略显绝对。例如，美国数学家鲁本·赫什（1927-2020）便不认同这种实在论解释，还援引了多位持相同观点的数学家的论述（赫什，1997：182-232）。事实上，赫什强烈反对主流数学家对柏拉图主义的坚守，他提出质疑：“为何数学家会相信数学真理存在于一个独立、非物质、永恒的世界之中，这一观点既不科学，又荒诞不经。”（赫什，1997：11）⁷。他还指出：“如今的柏拉图主义的问题在于，它抛弃了上帝，却仍想将数学视作存在于上帝思维中的事物。”（同上，第 135 页）。

As rightly noted by Hersh (1997:42), the working mathematician usually oscillates between Platonism and formalism: 赫什（1997：42）的观点十分中肯，他指出一线数学家的思想往往在柏拉图主义与形式主义之间摇摆：

The working mathematician is a Platonist on weekdays, a formalist on weekends. On weekdays, when doing mathematics, he’s a Platonist, convinced he’s dealing with an objective reality whose properties he’s trying to determine. On weekends, if challenged to give a philosophical account of this reality, it’s easiest to pretend he doesn’t believe in it. He plays formalist, and pretends mathematics is a meaningless game. (Hersh 1997:39f.)⁸ 一线数学家在工作日是柏拉图主义者，在周末则成了形式主义者。工作日做研究时，他坚信自己所研究的是客观的实在，致力于探寻其属性，是坚定的柏拉图主义者；而到了周末，若被要求从哲学角度阐释这一实在，最省事的做法便是佯装自己并不相信它的存在，转而站在形式主义的立场，将数学视作一场无意义的游戏。（赫什，1997：39-40）⁸

Hersh has a different view in mathematics. He sees mathematics as part of human culture and history and regards mathematics as a ‘socio-historical reality’ (Hersh 1997:17), comparable, for example, with institutions like the Supreme Court. 赫什则对数学有着不同的理解。他将数学视作人类文化与历史的一部分，认为数学是一种“社会历史实在”（赫什，1997：17），可与最高法院这类社会机构相类比。

My own position

笔者的观点

I think that Livio presents a well-balanced answer to the question of whether mathematics is created or discovered. Our mathematics is a combination of inventions and discoveries… Humans commonly invent mathematical concepts and discover the relations among those concepts. (Livio 2009:238, 242) 笔者认为，利维奥对“数学是被创造还是被发现”这一问题给出了最为公允的答案： 我们的数学，是发明与发现的结合……人类创造数学概念，而后发现这些概念之间的关联。（利维奥，2009：238、242）

A mathematician might invent a mathematical concept like a ‘prime ideal’or a ‘crossed-product order’ (Kessler 1994). 数学家可能会创造出“素理想”或“交叉积序”这类数学概念（克斯勒，1994）。

But once the concept and its axioms are in place, the mathematician is bound by this chosen concept. It is a bit comparable to the writing of a novel. An author can freely choose a character for a book, but once the character is chosen the author has to stick to it. The axioms of a mathematical object can be freely chosen, invented, but then the theorems about this object are discovered. 但一旦数学概念及其公理体系确立，数学家的研究便会受限于这一既定概念。这一点与小说创作颇有相似之处：作者可以自由塑造小说中的人物，但人物形象一旦确立，作者的创作便需符合其性格设定。数学客体的公理可以由人类自由选择、创造，但关于该客体的定理，则是人们去发现的结果。

Without digging deeper into the ontology of mathematics, I just want to state that I have much sympathy for Popper’s model of the three worlds. There is the physical world (world 1), the mental or psychological world (world 2), which is our inner world, inaccessible from outside and there is a third world (world 3), consisting of ‘the products of the human mind’ (Popper 1980:44) such as concepts, languages and theorems. At the moment when Fermat made public the formula, which was in his mind (world 2) and became later known as ‘Fermat’s Last Theorem’, this formula could be discussed and analysed by the mathematical community as part of world 3. Singh (1997) tells the story of these discussions. 笔者不打算深入探讨数学的本体论问题，仅想表明，笔者十分认同波普尔的三个世界理论。波普尔提出，世界分为三个层次：物理世界（世界 1）；精神或心理世界（世界 2），即人类的内在世界，具有不可被外部窥探的特性；以及由“人类思维的产物”（波普尔，1980：44）构成的第三世界（世界 3），概念、语言、定理皆属于这一世界。当费马将其脑海中（世界 2）的公式公之于众时，这一后来被称作“费马大定理”的公式便成为了世界 3 的一部分，供数学界探讨与分析。辛格（1997）详细记述了学界对这一定理的探讨历程。

Already in 1918, the German logician Gottlob Frege (1848–1925) published a similar idea in his essay Der Gedanke: That seems to be the result: Thoughts are neither things of the external world nor conceptions. A third realm must be recognised. (Frege 1986:43)⁹ 早在 1918 年，德国逻辑学家戈特洛布·弗雷格（1848-1925）便在其论文《思想》中提出了类似的观点： 结论似乎是：思想既非外部世界的事物，亦非主观的表象，我们必须承认存在第三个领域。（弗雷格，1986：43）⁹

A thought in Frege’s sense is something, which is either true or false.¹⁰ What Frege called ‘a third realm’ – the term ‘Drittes Reich’ got a totally different meaning two decades later! – was later turned into ‘world 3’ by the Austrian-British philosopher Karl Popper (1980:157). At first glance, Popper’s world 3 might look a bit Platonic (Hersh 1997: 220), but there are important differences. World 3 is a human creation, that is, it consists of the products of the human mind, whereas the Platonic ideal forms were regarded as eternal.¹¹ 在弗雷格的理论中，思想是具有真假值的事物¹⁰。弗雷格所称的“第三个领域”——二十年后，“第三帝国”这一词汇被赋予了全然不同的含义——后来被奥裔英国哲学家卡尔·波普尔发展为“世界 3”（波普尔，1980：157）。初看之下，波普尔的世界 3 理论似乎带有柏拉图主义的色彩（赫什，1997：220），但二者存在本质区别：世界 3 是人类的创造物，由人类思维的产物构成；而柏拉图所提出的理念形式，则被视作永恒的存在¹¹。

Please note that the conclusions in the next section do not presuppose the existence of world 3. I hope that my arguments will also convince readers who do not share my view on the ontology of mathematics. 需说明的是，下一部分的结论并不以世界 3 的存在为前提，笔者希望相关论证也能让那些与笔者在数学本体论问题上持不同观点的读者信服。

How can one detect divine elements in mathematics?

如何在数学中发现神圣的元素？

How can it be argued today that God speaks through the language of mathematics? Although a pragmatic Platonism is still alive among mathematicians, I will not build on it in the following section, because I think that mathematics is a combination of inventions and discoveries, see above.¹² 如今，我们该如何论证上帝通过数学的语言言说？尽管实用柏拉图主义在数学家中仍有影响，但笔者在下文中不会以这一理论为基础，因笔者认为，数学是发明与发现的结合，这一点前文已作阐释¹²。

I do argue that some features of mathematics hint at divine attributes. I am not saying that one can detect God within mathematics, but that one can at least find traces of the divine. Reflections by mathematicians on their work (like in Cassaza et al. 2015) and the empirical study Witz (2007) provide some empirical evidence for the fact that one might detect divine elements in mathematics. 笔者认为，数学的某些特征暗示着神圣的属性。笔者并非提出人们能在数学中直接感知上帝，而是认为，人们至少能在数学中发现神圣的踪迹。数学家对其研究的反思（如卡萨扎等人，2015）以及维茨开展的实证研究（2007），均为“数学中存在神圣元素”这一观点提供了实证依据。

Mathematics shows that there is indisputable truth

数学彰显了不容置疑的真理的存在

Let us start with a personal remark by the journalist Masha Gessen. In her work on the mathematician Grigori Perelman, she also shares her own experiences with math clubs in the former Soviet Union. ‘To a Soviet child, the after-school math club was a miracle’ (Gessen 2009:21). The ordinary school system followed the ideal of uniformity and taught the doctrines of the communist party. By contrast, the math clubs encouraged individuality and creativity. One could argue with others, and statements could be proved! Each student could receive merits by solving a math problem, regardless of his or her gender, ethnic background, political conviction or whatever. ‘(I)t felt like love, truth, hope and justice all handed to me at once’ (p. 22). Gessen mentions four transcendentals – love, truth, hope and justice – which we consider to be divine attributes, at least in the Jewish-Christian tradition. Thus, mathematics obviously can lead to the discovery that there are transcendentals. 我们先来看记者玛莎·格森的一段个人感悟。格森在撰写数学家格里戈里·佩雷尔曼的传记时，也分享了自己在前苏联数学俱乐部的经历：“对一个苏联孩子而言，课外数学俱乐部宛如一个奇迹。”（格森，2009：21）。彼时的常规学校教育奉行统一化的理念，灌输共产党的教义；而数学俱乐部则鼓励个性与创新，人们可以相互辩论，命题可以被证明！每个学生都能凭借解出数学难题获得认可，无论其性别、种族、政治立场如何。“那一刻，我仿佛同时拥有了爱、真理、希望与正义。”（同上，第 22 页）。格森提及了爱、真理、希望、正义这四种超验的价值，至少在犹太教与基督教传统中，这些价值被视作神圣的属性。由此可见，数学显然能引导人们发现超验价值的存在。

Most importantly, in mathematics, one can learn that there is an indisputable truth. A mathematical formula is either true or false. Every mathematician would agree that Fermat’s Last Theorem actually was true even before Andrew Wiles proved it in 1994 (Singh 1997). There is no need for Platonism to share this point of view. It is the belief in the objectivity of mathematical discourse, without committing to the belief that mathematical objects exist in a Platonic realm (Livio 2009:243). 最重要的是，人们能从数学中认识到，存在着不容置疑的真理。一个数学公式非真即假。所有数学家都会认同，即便在安德鲁·怀尔斯 1994 年证明费马大定理之前，这一定理本身就是成立的（辛格，1997）。认同这一观点，并不需要以柏拉图主义为前提，它仅仅是对数学话语客观性的坚信，而非认定数学客体存在于柏拉图的理念世界中（利维奥，2009：243）。

The detection of indisputable truth might lead people to look for the source of truth, which is linked to God, at least according to monotheistic religions (see, e.g. Job 28:23 in the Jewish-Christian Bible, Jn 14:6 in the New Testament). 对不容置疑的真理的发现，可能会引导人们探寻真理的本源。至少在一神教的传统中，真理的本源与上帝相关（例如可参见犹太教与基督教经典《圣经》中的《约伯记》28 章 23 节、《约翰福音》14 章 6 节）。

Mathematics shows that there is beauty

数学彰显了美的存在

In 1907, the British mathematician Bertrand Russell (1872–1970) wrote ‘Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere’ (Russell 2014:49). His compatriot Hardy stated in 1940 that beauty is the first test for mathematics, because ‘there is no permanent place in the world for ugly mathematics’ (Hardy 2001:85). In 1952, the German mathematician Helmut Hasse (1898–1979) lectured on ‘mathematics as science, art and power’ (Hasse 1952). There he declared that beauty leads to mathematical truth (:19). He further stated that truth is a necessary condition for real mathematics but is not sufficient. There must also be beauty and harmony (:26). Sir Michael Atijah (1929–2019) was a strong advocate for beauty in mathematics.¹³ He was convinced that mathematicians ‘serve two masters, Beauty and Truth’ (Atijah 2015:29; see also Atijah 1973). In the abovementioned survey, Aschbacher also asked her husband’s colleagues why they were drawn to mathematics. A typical answer was ‘drawn towards it by its beauty and elegance’ (Aschbacher 2015:18). Heller (2018:229) regards mathematics to be ‘the most beautiful product of human rationality’. Zeki et al. (2014) report on a ‘beauty competition’ in which mathematicians rated formulae for beauty. The winner was the so-called Euler’s identity (Kessler 2018:4–5). More information about the aesthetical dimension of mathematics can be found in the collective volume (Sinclair, Pimm & Higginson 2006) or in the special issue (Journal of Humanistic Mathematics 2016). Mathematical beauty can be found in the result itself, in the proof of the result, or as a guide to the right formula. 1907 年，英国数学家伯特兰·罗素（1872-1970）写道：“正确看待的数学，不仅拥有真理，还拥有至高无上的美——一种冷峻而庄重的美。”（罗素，2014：49）。其同胞哈代在 1940 年提出，美是评判数学的首要标准，因为“丑陋的数学在世界上没有容身之地”（哈代，2001：85）。1952 年，德国数学家赫尔穆特·哈塞（1898-1979）发表了题为《作为科学、艺术与力量的数学》的演讲（哈塞，1952），他在演讲中提出，美是通往数学真理的指引（同上，第 19 页），并进一步指出，真理是真正的数学的必要条件，而非充分条件，真正的数学还必须兼具美与和谐（同上，第 26 页）。迈克尔·阿蒂亚爵士（1929-2019）极力推崇数学中的美¹³，他坚信，数学家“侍奉两位主人：美与真理”（阿蒂亚，2015：29；亦可参见阿蒂亚，1973）。在上述阿希巴赫开展的调查中，她还询问了其丈夫的同事被数学吸引的原因，典型的回答是：“被数学的美与优雅所吸引。”（阿希巴赫，2015：18）。海勒将数学视作“人类理性最美丽的产物”（2018：229）。泽基等人（2014）记录了一场特殊的“美之竞赛”，数学家们为数学公式的美感打分，最终欧拉恒等式拔得头筹（克斯勒，2018：4-5）。关于数学的美学维度，可参见辛克莱、皮姆与希金森主编的论文集（2006），或《人文数学期刊》的相关特刊（2016）。数学之美既体现在数学结论本身，也体现在结论的证明过程中，还能作为指引，帮助人们找到正确的公式。

When people experience beauty in mathematics, they might start wondering about the source of this beauty. The feeling might be comparable to arriving at the top of a high mountain, enjoying the beauty of the view. At such moments one is reminded of Psalm 19:1: ‘The heavens declare the glory of God’.¹⁴ Such a feeling might also come to one experiencing the beauty of maths. It is the ‘shuddering before the beautiful’ (Chandrasekhar 1987:541), an emotion similar to spiritual emotion called ‘mysterium tremendum’ by the German theologian Rudolf Otto (1936:12). This ‘mysterium tremendum’ contains elements of awefulness and overpoweringness (majestas) (:14, 20). 当人们在数学中感受到美时，便会开始探寻这份美的本源。这种感受，如同登上高山之巅，欣赏眼前的壮丽景色，彼时，人们会想起《诗篇》19 篇 1 节的内容：“诸天述说上帝的荣耀。”¹⁴ 体验数学之美时，人们也会产生这样的感受。这是一种“面对美时的震撼”（钱德拉塞卡，1987：541），与德国神学家鲁道夫·奥托提出的“神秘的超验性”这一灵性情感相似（奥托，1936：12）。这种“神秘的超验性”，蕴含着令人敬畏与折服的神圣威严（奥托，1936：14、20）。

Mathematics shows that there is beauty and glory, and the Bible links these attributes to God (e.g. Psalm 19:1, Isaiah 33:17, cf. Von Balthasar’s seven volumes on the glory of God, Von Balthasar 1984). 数学彰显了美与荣耀的存在，而《圣经》将这些属性与上帝相联结（例如《诗篇》19 篇 1 节、《以赛亚书》33 篇 17 节，亦可参见冯·巴尔塔萨的七卷本著作《上帝的荣耀》，冯·巴尔塔萨，1984）。

Mathematics shows the importance of community

数学彰显了共同体的重要性

Because human beings are social beings, there are many human activities demonstrating the importance of community. And each science has its own scientific community. But I think that the previous aspect of mathematical beauty (3.2) has an interesting side effect. 人类是社会性的存在，诸多人类活动都能体现出共同体的重要性，每一门科学也都有其专属的科学共同体。但笔者认为，前文所探讨的数学之美这一特质，还带来了一个有趣的衍生效应。

Mathematics is a product of a community contributing together to create beauty. Usually, a piece of art or a symphony is the product of an individual. And in the case of a cathedral where many people have contributed at different times (632 years for the Cologne Cathedral!), they at least have to work at the same location. In mathematics, an invisible community consisting of many people in different places and at different times jointly contribute to the beauty of mathematics. In the case of Fermat’s Last Theorem, it took 358 years (Singh 1997). 数学是共同体携手共创的美的结晶。一件艺术品、一首交响曲，通常出自单一创作者之手；即便是像科隆大教堂这样，由不同时代的人们历时 632 年共同建造的建筑，创作者们至少也身处同一地域。而数学的美，由一个跨越时空的无形共同体共同铸就，身处不同地域、不同时代的数学家，都为这份美贡献着力量。仅费马大定理的证明，便凝聚了 358 年的集体智慧（辛格，1997）。

In his personal account, Boas (2015:256) stresses the communal aspect of mathematics: ‘After sitting at the feet of these gurus for a year, I was a lifelong convert to the religion of mathematics’. 博阿斯在其个人自述中，着重强调了数学的共同体属性：“在这些数学大师的门下学习一年后，我便终身皈依了数学这一信仰。”（博阿斯，2015：256）。

This communal contribution to beauty has interesting counterparts in Jewish-Christian theology. The Song of Songs in the Jewish Bible has also been interpreted as a metaphor for a marriage between God and His people. This metaphor is taken up in the New Testament calling the church ‘Christ’s bride’ (Eph 5:31–32, 2 Cor 11:2). Note that this metaphor does not refer to a mystic union between God and an individual; the whole community of believers is involved. According to this Jewish-Christian metaphor, God is looking for community. 这种为缔造美而凝聚的共同体精神，在犹太教-基督教神学中也能找到有趣的对应。犹太教经典中的《雅歌》，常被解读为上帝与其子民之间结合的隐喻；这一隐喻也被《新约》沿用，将教会称作“基督的新妇”（《以弗所书》5 章 31-32 节、《哥林多后书》11 章 2 节）。值得注意的是，这一隐喻所指的并非上帝与单个信徒之间的神秘结合，而是上帝与整个信徒共同体的联结。从这一犹太教-基督教的隐喻中可以看出，上帝亦在寻求共同体的联结。

In addition, there is the Christian doctrine of the Trinity. According to this doctrine, there is already community within God; thus community could be considered a divine attribute, at least in Christian theology (see the wonderful icon The Trinity, painted in 1410 by Andrei Rublev). 此外，基督教的三位一体教义也蕴含着共同体的内涵。该教义提出，上帝的本体之中便存在着共同体，因此，至少在基督教神学中，共同体可被视作一种神圣的属性（可参见安德烈·鲁布廖夫于 1410 年创作的经典圣像画《三位一体》）。

Mathematics shows that one can speak about infinity in reasonable terms

数学表明人们能对无限进行理性阐释

There has been a long dispute about the understanding of infinity (Hilbert 1925). Is there an actual infinity? The German mathematician Georg Cantor (1845–1918) provided a new understanding of infinity by demonstrating that there are infinities of different sizes (Hilbert 1925:167). For example, the set of integers and the set of rational numbers are both infinite and countable, but the set of real numbers is infinite and uncountable, thus ‘much larger’ than the set of rational numbers. Cantor introduced transfinite numbers in order to distinguish infinite sets of different sizes. 学界对无限的理解，历来争议不断（希尔伯特，1925）。实无限是否存在？德国数学家格奥尔格·康托尔（1845-1918）为人们理解无限提供了全新的视角，他证明了无限存在不同的层级（希尔伯特，1925：167）。例如，整数集与有理数集都是无限的、可数的，而实数集则是无限的、不可数的，其“规模”远大于有理数集。康托尔还提出了超穷数的概念，用以区分不同大小的无限集合。

This discovery shows that it is reasonable to speak about infinity, thus demonstrating that speaking about an infinite God is not senseless per se. 这一发现表明，人们对无限的探讨具有理性基础，这也意味着，对无限的上帝的阐释，其本身并非毫无意义。

Studying different degrees of infinity might also show the mathematician that there is something, which is provably higher than him or herself. We only know a subsection of all infinite sets, and this is not only because our IQ might be too small, but because the structure of the system of thought has limitations.¹⁵ Thus the mathematician might come to the conclusion that there is somebody higher than him or herself (see Rm 1:20). 对不同层级无限的研究，也会让数学家认识到，存在着一种可被证明的、高于人类自身的存在。人类对无限集合的认知，仅为冰山一角，这并非仅仅因为人类的智力有限，更源于人类思维体系的结构本身存在局限性¹⁵。因此，数学家可能会得出这样的结论：存在着高于人类自身的至高存在（参见《罗马书》1 章 20 节）。

Mathematics shows that reasoning about unseen and abstract objects can make sense

数学表明对不可见的抽象客体的推理具有合理性

In her essay, Victoria Harrison argues that ‘realism about mathematical objects can provide a model for thinking about realism within theology’ (Harrison 2017:479). From a strictly empirical point of view, talking about God does not make much sense. Experiences might be interpreted as God speaking or acting, but our senses ‘cannot provide direct knowledge of God’ (p. 490). Theology shares this deficiency with mathematics because mathematical knowledge cannot be generated by our senses. Mathematical formulae are not part of the physical world. Still, mathematical language has proven to be meaningful and rational. This might serve as a hint that talking about theological objects like God might also be meaningful and rational. 维多利亚·哈里森在其论文中提出，“对数学客体的实在论解读，可为神学中的实在论思考提供范本”（哈里森，2017：479）。从严格的经验主义视角来看，对上帝的探讨并无太多实际意义。人们或许会将某些体验解读为上帝的言说或作为，但人类的感官“无法直接认知上帝”（同上，第 490 页）。神学的这一特质与数学相通，因为数学知识同样无法由人类的感官产生，数学公式并非物质世界的组成部分。即便如此，数学语言已被证明是富有意义且具有理性的，这或许暗示着，对上帝这类神学客体的探讨，同样可以是富有意义且具备理性的。

The unreasonable effectiveness of mathematics

数学超乎常理的有效性

The sections above dealt with mathematics only, looking for traces of the divine in the pure world of mathematics. This section will also look at the physical world and how mathematics is used to describe it. Physics Nobel laureate Eugene Wigner (1902–1995) once said: ‘The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve’ (quoted in Livio 2009:3). Wigner dubbed this ‘the unreasonable effectiveness of mathematics’ (4). Livio (2009:203) points out that this unreasonable effectiveness has two aspects. There is an active aspect when natural scientists develop a mathematical tool to describe a phenomenon they have observed. It is quite plausible that mathematical reasoning can lead to new knowledge about the physical world through this process. 前文的探讨仅围绕纯数学展开，在纯粹的数学世界中探寻神圣的踪迹。本节将结合物质世界，探讨数学如何被用于描述物质世界。诺贝尔物理学奖得主尤金·维格纳（1902-1995）曾说：“数学语言竟能如此恰切地表述物理定律，这一奇迹是一份我们既无法理解、也不配拥有的珍贵礼物。”（转引自利维奥，2009：3）。维格纳将这一现象称作“数学超乎常理的有效性”（同上，第 4 页）。利维奥（2009：203）指出，这种超乎常理的有效性体现在两个方面。其一为主动层面：自然科学家为描述所观察到的现象，研发相应的数学工具，通过这一过程，数学推理能够为人类带来关于物质世界的新认知，这一点是合乎情理的。

But there is also a passive aspect, which is quite mysterious. In the history of science, it has happened several times that abstract mathematical theories were developed for pure mathematical reasons and later on were successfully applied to the natural sciences. Just to give three examples: Firstly, the complex number

i

i

i as a root of

(

−

1

)

(-1)

(−1) was invented in order to solve algebraic equations; nowadays it is heavily used in physics and electrical engineering. Secondly, the so-called Riemannian geometry was introduced by Riemann in a brilliant lecture in Göttingen on June 10, 1854; half a century later, the physicist Albert Einstein used the Riemannian geometry to formulate his general theory of relativity. Thirdly, the ‘story of knot theory demonstrates beautifully the unexpected power of mathematics’ (Livio 2009: 217). The mathematical theory of knots was born in 1771; today it has significant applications in quantum field theory and in studying DNA. 其二为被动层面，这一层面的现象则充满了神秘色彩。在科学史上，多次出现这样的情况：数学家出于纯数学研究的目的提出抽象的数学理论，而后这些理论被成功应用于自然科学领域。仅举三例：第一，作为

−

1

-1

−1平方根的虚数

i

i

i，最初是为求解代数方程而被提出，如今却在物理学与电气工程领域得到广泛应用；第二，黎曼于 1854 年 6 月 10 日在哥廷根大学的一场经典演讲中提出了黎曼几何，半个世纪后，物理学家阿尔伯特·爱因斯坦运用黎曼几何构建了广义相对论；第三，“纽结论的发展历程，完美印证了数学超乎预期的力量”（利维奥，2009：217），纽结论于 1771 年诞生，如今已在量子场论与 DNA 研究领域发挥着重要作用。

The issue of chance was long seen as a theological problem. How can the existence of chance be combined with a rational and omniscient God? In his book on the Philosophy of Chance, Polish cosmologist and theologian Michael Heller argue that chance is not irrational and ‘does not destroy the mathematicality of the world’ (Heller 2018:152). ‘The world turns out to be mathematical … also in its own random and probabilistic behaviours’ (p. 79). Heller (p. 143) also refers to the ‘effective’ characterisation of the physical universe by mathematical structure, even in the area of probabilities. 偶然性问题长期以来被视作一个神学难题：偶然性的存在，如何与理性、全知的上帝相调和？波兰宇宙学家、神学家米哈伊尔·海勒在其著作《偶然性的哲学》中提出，偶然性并非非理性的存在，它“不会破坏世界的数学本质”（海勒，2018：152）。“即便在随机与概率性的行为中，世界的本质依然是数学的”（同上，第 79 页）。海勒（同上，第 143 页）还指出，即便在概率领域，数学结构仍能对物理宇宙作出“有效的”描述。

Einstein raised the question: ‘How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?’ (quoted in Livio 2009:1). It is hard to give a rational explanation for this ‘unreasonable effectiveness of mathematics’.¹⁶ Wigner called it a ‘gift’. It could also be called a miracle, which brings us to theology again. 爱因斯坦曾提出这样的疑问：“数学作为人类思维的产物，独立于经验之外，为何能如此完美地契合物理实在的客体？”（转引自利维奥，2009：1）。人们很难为这种“数学超乎常理的有效性”给出一个理性的解释¹⁶。维格纳将其称作一份“礼物”，而这份礼物也可被视作一个奇迹，这一结论又将我们的探讨带回了神学领域。

Creation theology as understood in the Judeo-Christian tradition offers an explanatory model for the phenomenon of ‘the unreasonable effectiveness’. God created the world with wisdom (Pr 8:22–31) and He created human beings in His own image (Gn 1:26). Thus, human beings are enabled to discover the laws that God put into His creation. I am not saying that the study of mathematics will automatically lead people to believe in creation theology. But obviously, one can find traces in mathematics that might point to the Creator-God. 犹太教-基督教传统中的创造神学，为“数学超乎常理的有效性”这一现象提供了一套解释范式。上帝以智慧创造了世界（《箴言》8 章 22-31 节），并按照自己的形象造人（《创世记》1 章 26 节）。正因如此，人类才拥有了发现上帝赋予其创造物的法则的能力。笔者并非提出，数学研究必然会让人们相信创造神学，而是认为，人们显然能在数学中发现指向造物主上帝的踪迹。

Conclusion

结论

The question of this article was ‘Does God speak through the language of mathematics?’ Being a mathematician and a Christian theologian, I am convinced that the answer should be ‘yes’. But I am also aware that the evidence for this answer is not as obvious as it was several centuries ago. The changes in mathematics made it necessary to rethink the evidence for this answer. We have learned that mathematical objects are created by human beings. Thus, it is not possible to argue that mathematical formulae exist in a divine mind. However, I think that there are hints in mathematics that seem to point towards attributes of God. 本文的核心问题是：“上帝是否通过数学的语言言说？”作为一名数学家与基督教神学家，笔者坚信答案是肯定的。但笔者也认识到，如今支撑这一答案的证据，已不再像数个世纪前那般显而易见。数学领域的变革，使得我们有必要重新审视这些证据。我们已然认识到，数学客体是人类的创造物，因此，无法再论证数学公式存在于神圣的思维之中。但笔者认为，数学中仍存在诸多线索，指向上帝的神圣属性。

I have mentioned six traces of the divine: (1) the existence of indisputable truth, (2) the existence of beauty, (3) the importance of community, (4) rational speaking about infinity, (5) the discovery that speaking about unseen and abstract objects is reasonable and (6) the unreasonable effectiveness of mathematics. I think that each of these arguments has a certain weight, but I do not think that they have the same weight. If one looks at the current discussion, arguments (1), (2) and (6) are probably the most convincing. 笔者在文中提出了神圣的六种踪迹：(1) 不容置疑的真理的存在；(2) 美的存在；(3) 共同体的重要性；(4) 对无限进行理性阐释；(5) 发现对不可见的抽象客体进行言说具有合理性；(6) 数学超乎常理的有效性。笔者认为，这六大论证均具有一定的说服力，但并非同等重要。从当下的学界探讨来看，第 (1)､(2)､(6) 种论证或许是最具说服力的。

Acknowledgements

致谢

Competing interests 利益冲突 The author declares that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. 作者声明，在撰写本文的过程中，不存在任何可能造成不当影响的财务或个人关系。

Author’s contributions 作者贡献 V.K. is the sole author of this article. 本文由 V·克斯勒独立撰写完成。

Ethical considerations 伦理考量 This article followed all ethical standards for research without direct contact with human or animal subjects. 本文的研究未直接涉及人类或动物受试对象，完全遵循相关研究伦理准则。

Funding information 基金资助 This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. 本研究未获得任何公共、商业或非营利性资助机构的专项资助。

Data availability 数据可用性 Data sharing is not applicable to this article as no new data were created or analysed in this study. 本文未生成或分析新的研究数据，因此不涉及数据共享相关事宜。

Disclaimer 免责声明 The views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. 本文所表达的观点与见解均为作者个人观点，并不一定代表作者所属机构的官方政策或立场。

References

参考文献

Aschbacher, P., 2015, ‘What are mathematicians really like? Observations of a spouse’, in P. Cassaza, S. Krantz & R. Ruden (eds.), I, Mathematician, pp. 16–28, The Mathematical Association of America, Washington, DC. 阿希巴赫,P.,2015,《数学家究竟是怎样的人?一位配偶的观察》,载于卡萨扎,P.､克兰茨,S.､鲁登,R.(主编),《我,数学家》,第 16-28 页,美国数学协会,华盛顿哥伦比亚特区｡

Atiyah, M., 1973, ‘Mathematics: Art and science’, Bulletin of the American Mathematical Society 43(1), 87–88. https://doi.org/10.1090/S0273-0979-05-01095-5 阿蒂亚,M.,1973,《数学:艺术与科学》,《美国数学会通报》,43(1),第 87-88 页｡

Atiyah, M., 2015, ‘Mathematics: Art and science’, in P. Cassaza, S. Krantz & R. Ruden (eds.), I, Mathematician, pp. 29–30, The Mathematical Association of America, Washington, DC. 阿蒂亚,M.,2015,《数学:艺术与科学》,载于卡萨扎,P.､克兰茨,S.､鲁登,R.(主编),《我,数学家》,第 29-30 页,美国数学协会,华盛顿哥伦比亚特区｡

Boas, H.P., 2015, ‘Why I became a Mathematician: A personal account’, in P. Cassaza, S. Krantz & R. Ruden (eds.), I, Mathematician, pp. 255–256, The Mathematical Association of America, Washington, DC. 博阿斯,H·P.,2015,《我为何成为数学家:个人自述》,载于卡萨扎,P.､克兰茨,S.､鲁登,R.(主编),《我,数学家》,第 255-256 页,美国数学协会,华盛顿哥伦比亚特区｡

Bromand, J. & Kreis, G. (eds.), 2011, Gottesbeweise von Anselm bis Gödel, Suhrkamp, Berlin. 布罗曼德,J.､克雷斯,G.(主编),2011,《从安瑟伦到哥德尔的神存在证明》,苏尔坎普出版社,柏林｡

Cassaza, P., Krantz, S. & Ruden, R. (eds.), 2015, I, Mathematician, The Mathematical Association of America, Washington, DC. 卡萨扎,P.､克兰茨,S.､鲁登,R.(主编),2015,《我,数学家》,美国数学协会,华盛顿哥伦比亚特区｡

Chandrasekhar, S., 1987, Truth and beauty: Aesthetics and motivations in science, Chicago University Press, Chicago, IL. 钱德拉塞卡,S.,1987,《真理与美:科学中的美学与动机》,芝加哥大学出版社,伊利诺伊州芝加哥市｡

Davies, P., 1992, The mind of God: The scientific basis for a rational world, Simon & Schuster, New York, NY. 戴维斯,P.,1992,《上帝的思维:理性世界的科学基础》,西蒙与舒斯特出版社,纽约州纽约市｡

Descartes, R., 1976, Meditationen über die Grundlagen der Philosophie, Felix Meiner, Hamburg. 笛卡尔,R.,1976,《第一哲学沉思集》,费利克斯·迈纳出版社,汉堡｡

Frege, G., 1986, ‘Der Gedanke’, in G. Frege (ed.), Logische Untersuchungen, 3rd edn., pp. 30–53, Vandenhoeck & Ruprecht, Göttingen. 弗雷格,G.,1986,《思想》,载于弗雷格,G.(主编),《逻辑研究》(第 3 版),第 30-53 页,范登霍克与鲁普雷希特出版社,哥廷根｡

Gessen, M., 2009, Perfect rigor: A genius and the mathematical breakthrough, Houghton Mifflin Harcourt, New York, NY. 格森,M.,2009,《完美的严谨:一位天才与数学的突破》,霍顿·米夫林·哈考特出版社,纽约州纽约市｡

Hardy, G.H., 2001, A mathematician’s apology, Cambridge University Press, Cambridge. 哈代,G·H.,2001,《一个数学家的辩白》,剑桥大学出版社,剑桥｡

Harrison, V.S., 2017, ‘Mathematical objects and the object of the theology’, Religious Studies 53(4), 479–496. https://doi.org/10.1017/S0034412516000238 哈里森,V·S.,2017,《数学客体与神学的客体》,《宗教研究》,53(4),第 479-496 页｡

Hasse, H., 1952, Mathematik als Wissenschaft, Kunst und Macht, Verlag für Angewandte Wissenschaften, Wiesbaden. 哈塞,H.,1952,《作为科学､艺术与力量的数学》,应用科学出版社,威斯巴登｡

Heller, M., 2018, Philosophy of chance, 2nd edn., Copernicus Center, Kraków. 海勒,M.,2018,《偶然性的哲学》(第 2 版),哥白尼研究中心,克拉科夫｡

Hersh, R., 1997, What is mathematics, really?, Jonathan Cape, London. 赫什,R.,1997,《数学究竟是什么?》,乔纳森·凯普出版社,伦敦｡

Hilbert, D., 1925, ‘Über das Unendliche’, Mathematische Annalen 95, 161–190. https://doi.org/10.1007/BF01206605 希尔伯特,D.,1925,《论无限》,《数学年刊》,95,第 161-190 页｡

Jonas, H., 1966, ‘Is God a mathematician?’ in H. Jonas (ed.), The phemomenon of life: Toward a philosophical biology, pp. 64–93, Harper & Row, New York, NY. 约纳斯,H.,1966,《上帝是数学家吗?》,载于约纳斯,H.(主编),《生命现象:走向哲学生物学》,第 64-93 页,哈珀与罗出版社,纽约州纽约市｡

Journal of Humanistic Mathematics, 2016, ‘Special issue on the nature and experience of mathematical beauty’, Journal of Humanistic Mathematics 6(1), 3–7. https://doi.org/10.5642/jhummath.201601.03 《人文数学期刊》,2016,《数学之美的本质与体验》(特刊),《人文数学期刊》,6(1),第 3-7 页｡

Kanigel, R. 1991, The man who knew infinity, Abacus, London. 卡尼格尔,R.,1991,《知无涯者:拉马努金传》,算盘出版社,伦敦｡

Kant, I., 1966, Kritik der reinen Vernunft, Reclam, Stuttgart. 康德,I.,1966,《纯粹理性批判》,莱克拉姆出版社,斯图加特｡

Kessler, V., 1994, ‘Crossed product orders and non-commutative arithmetic’, Journal of Number Theory 46(3), 255–302. https://doi.org/10.1006/jnth.1994.1015 克斯勒,V.,1994,《交叉积序与非交换算术》,《数论期刊》,46(3),第 255-302 页｡

Kessler, V., 2018, ‘God becomes beautiful … in mathematics’, HTS Teologiese Studies / Theological Studies 74(1), 4886. https://doi.org/10.4102/hts.v74i1.4886 克斯勒,V.,2018,《上帝在数学中彰显其美……》,《神学研究》,74(1),4886｡

Kessler, V., 2019, ‘Spirituality in mathematics’, Journal for the Study of Spirituality 9(1), 49–61. https://doi.org/10.1080/20440243.2019.1581384 克斯勒,V.,2019,《数学中的灵性》,《灵性研究期刊》,9(1),第 49-61 页｡

Koetsier, T. & Bergmann, L. (eds.), 2005, Mathematics and the divine: A historical study, Elsevier, Amsterdam. 科齐尔,T.､伯格曼,L.(主编),2005,《数学与神圣:一项历史研究》,爱思唯尔出版社,阿姆斯特丹｡

Livio, M., 2009, Is God a mathematician?, Simon & Schuster, New York, NY. 利维奥,M.,2009,《上帝是数学家吗?》,西蒙与舒斯特出版社,纽约州纽约市｡

Martin, G., 1960, Leibniz: Logik und Metaphysik, Kölner Universitätsverlag, Köln. 马丁,G.,1960,《莱布尼茨:逻辑与形而上学》,科隆大学出版社,科隆｡

Mendelson, E., 1987, Introduction to mathematical logic, 3rd edn., Wadsworth & Brooks/Cole, Monterey. 门德尔松,E.,1987,《数理逻辑导论》(第 3 版),沃兹沃思与布鲁克斯/科尔出版社,蒙特雷｡

Meyer, R.K., 1987, ‘God exists!', Noûs 21(3), 345–361. https://doi.org/10.2307/2215186 迈耶,R·K.,1987,《上帝存在!》,《诺努斯》,21(3),第 345-361 页｡

Mukunda, N., 2015, ‘Mathematics as the language of nature – A historical view’, Proceedings of the Indian National Science Academy 81(2), 437–356. https://doi.org/10.16943/ptinsa/2015/v81i2/48092 穆昆达,N.,2015,《作为自然语言的数学:一种历史视角》,《印度国家科学院院刊》,81(2),第 347-356 页｡

Otto, R., 1936, The idea of the holy: An inquiry into the non-rational factor in the idea of the divine and its relation to the rational, Oxford University Press, Oxford. 奥托,R.,1936,《神圣的观念:对神圣观念中非理性因素及其与理性关系的探究》,牛津大学出版社,牛津｡

Penrose, R., 1991, The emperor’s new mind: Concerning computers, minds, and the laws of physics, Penguin Books, New York, NY. 彭罗斯,R.,1991,《皇帝新脑:有关电脑､人脑与物理定律》,企鹅出版社,纽约州纽约市｡

Phillips, S., 2009, The mathematical connection between religion and sciences, Anthony Rowe, Chippenham. 菲利普斯,S.,2009,《宗教与科学之间的数学联结》,安东尼·罗出版社,奇彭纳姆｡

Poincaré, H., 1905, Science and hypothesis, Walter Scott, London. 庞加莱,H.,1905,《科学与假设》,沃尔特·斯科特出版社,伦敦｡

Popper, K., 1980, Three worlds, The Tanner lecture on Human values, viewed 11 January 2021, from https://tannerlectures.utah.edu/_resources/documents/a-to-z/p/popper80.pdf. 波普尔,K.,1980,《三个世界》,坦纳人类价值讲座,2021 年 1 月 11 日查阅,自 https://tannerlectures.utah.edu/_resources/documents/a-to-z/p/popper80.pdf｡

Russell, B., 2014, Mysticism and logic and other essays, Martino Publishing, Mansfield Centre, Mansfield. 罗素,B.,2014,《神秘主义与逻辑及其他论文》,马蒂诺出版社,曼斯菲尔德中心｡

Shapiro, S., 2000, Thinking about mathematics: The philosophy of mathematics, Oxford University Press, Oxford. 夏皮罗,S.,2000,《思考数学:数学哲学》,牛津大学出版社,牛津｡

Sinclair, N., Pimm, D. & Higginson, W. (eds.), 2006, Mathematics and the aesthetic: New approaches to an ancient affinity, Canadian Mathematical Society No. 25, Springer, New York, NY. 辛克莱,N.､皮姆,D.､希金森,W.(主编),2006,《数学与美学:对一种古老联结的新探讨》,加拿大数学会丛书(第 25 卷),斯普林格出版社,纽约州纽约市｡

Singh, S., 1997, Fermat’s last theorem: The story of a riddle that confounded the world’s greatest minds for 358 years, Fourth Estate, London. 辛格,S.,1997,《费马大定理:一个困惑了世间智者 358 年的谜题》,第四等级出版社,伦敦｡

Smit, P.-B. & De Heide, N., 2021, ‘God, the beautiful and mathematics: A response’, HTS Teologiese Studies / Theological Studies 77(4), a6208. https://doi.org/10.4102/hts.v77i4.6208 斯密特,P·B.､德·海德,N.,2021,《上帝､美与数学:一则回应》,《神学研究》,77(4),a6208｡

Stewart, I., 2017, Significant figures: Lives and works of trailblazing mathematicians, Profile Books, London. 斯图尔特,I.,2017,《伟大的数学家:从芝诺到庞加莱》,普罗菲勒出版社,伦敦｡

Von Balthasar, H.U., 1984, The glory of the Lord: A theological aesthetics, Seven volumes, T&T Clark, Edinburgh. 冯·巴尔塔萨,H·U.,1984,《上帝的荣耀:神学美学》(七卷本),T&T 克拉克出版社,爱丁堡｡

Witz, K.G., 2007, Spiritual aspirations connected with mathematics: The experience of American university students, Edwin Mellen Press, Lewiston. 维茨,K·G.,2007,《与数学相关的灵性追求:美国大学生的体验》,埃德温·梅伦出版社,刘易斯顿｡

Zeki, S., Romaya, J.P., Benincasa, D.M. & Atiyah, M., 2014, ‘The experience of mathematical beauty and its neural correlates’, Frontiers in Human Neuroscience 8(68), 1–12. https://doi.org/10.3389/fnhum.2014.00068 泽基,S.､罗玛雅,J·P.､贝宁卡萨,D·M.､阿蒂亚,M.,2014,《数学之美的体验及其神经关联》,《人类神经科学前沿》,8(68),第 1-12 页｡

数学的终极问题：上帝是一位数学家吗？

一、引言：数学的"无理由有效性"

数学在描述自然现象时展现出的非凡效力，构成了科学哲学中最深刻的谜题之一。英国物理学家詹姆斯·金斯（James Jeans, 1877–1946）曾言："宇宙看上去是由一位理论数学家设计的。"这一观察指向一个根本性问题：数学为何能够如此精确地刻画物理实在？

阿尔伯特·爱因斯坦于 1934 年提出的疑问更为尖锐：

“数学，这个独立于经验的人类思维的产物，为何能如此完美地符合物理实在中的对象？”

此即著名的"数学无理由的有效性"（unreasonable effectiveness of mathematics）问题，由诺贝尔物理学奖得主尤金·维格纳（Eugene Wigner, 1902–1995）于 1960 年系统阐述。保罗·狄拉克（Paul Dirac）亦曾表述类似洞见：

"上帝用美丽的数学创造世界。

二、数学有效性的双重面向

2.1 主动层面：从现象到规律

物理学家主动运用数学工具构建理论，这一过程本身即令人惊异。艾萨克·牛顿从苹果落地、月球轨道与潮汐现象中抽象出普适的数学定律；詹姆斯·克拉克·麦克斯韦（James Clerk Maxwell, 1831–1879）以四个简洁的方程概括了电磁学的全部已知现象；爱因斯坦的广义相对论则以精确的数学结构揭示时空的几何本质。

2.2 被动层面：纯粹数学的意外应用

更为神秘的是，数学家出于纯粹理论兴趣发展的抽象结构，往往在数十年甚至数百年后被发现恰好描述物理实在。

典型案例：

反讽案例：英国数学家戈弗雷·哈罗德·哈代（Godfrey Harold Hardy, 1877–1947）曾自豪地宣称其纯数学研究"今天没有、将来也不会给世界带来丝毫影响"。然而，其发展的数论方法不仅构成了群体遗传学中哈代-温伯格定律（Hardy-Weinberg law）的基础，更在 1973 年由克利福德·柯克斯（Clifford Cocks）应用于公钥密码学——直接否定了哈代关于"数论不会用于战争目的"的断言。

三、三个世界的谜题

牛津大学数学物理学家罗杰·彭罗斯（Roger Penrose）在其著作《皇帝的新脑》与《通向实在之路》中提出了一个分析框架，将实在划分为三个 distinct 领域：

这三个世界之间存在神秘的关联：

彭罗斯对此的结论是谦逊的：“毫无疑问，并不真正存在三个世界，而是只有一个世界。并且直到目前为止，对于这个真实世界的本质，我们对它的认识甚至连肤浅也谈不上。”

四、发现还是发明？本体论之争

数学哲学的根本分歧在于：数学对象是独立于人类心智的客观存在（发现论），还是人类思维的构造（发明论）？

4.1 柏拉图主义：数学作为发现

法国数学家阿兰·孔涅（Alain Connes, 1982 年菲尔兹奖、2001 年克拉福德奖得主）明确主张：

“根据我的观察，质数组成的世界，远比我们周围的物质世界稳定。数学家的工作可以与探险家发现世界相媲美。”

对这一观点的支持包括：

• 数学真理的必然性：$3^2+4^2=5^2$
• 数学结论的永恒性：欧几里得几何定理历经两千余年依然有效，而物理学理论不断被修正；
• 数学的可错性（fallibilism）仅限于证明过程的严谨性，而非结论本身的真伪。

数学科普作家马丁·加德纳（Martin Gardner）以生动比喻支持此说：“如果森林中有

2

2

2 只恐龙与另外

2

2

2 只恐龙相遇，不管周围是否有人类在观察，那儿都会有

4

4

4 只恐龙。”

4.2 构造主义：数学作为发明

反对意见同样有力。英国数学家迈克尔·阿蒂亚爵士（Sir Michael Atiyah, 1966 年菲尔兹奖、2004 年阿贝尔奖得主）质疑：

“假如宇宙是一维空间，或者甚至是离散的，很难想象几何学在这个一维空间中是如何孕育发展的。”

阿蒂亚主张：“通过理想化和抽象物理世界中的那些基本要素，人类创造了数学。”

认知科学家乔治·莱考夫（George Lakoff）与拉斐尔·努涅斯（Rafael Núñez）在《数学来自哪里》（Where Mathematics Comes From）中提出：数学源于"我们的身体、大脑，以及我们在这个世界中每天的经历"。

4.3 关键问题

这一分歧引出深层问题：

• 若数学为人类发明，其普遍性如何保证？外星文明是否会发展出不同的数学？
• 若数学独立存在，人类如何跨越时空限制接触这一永恒领域？
• 数学与语言、艺术的本质区别何在？

卡尔·萨根（Carl Sagan）认为质数可作为跨文明的通用信号；史蒂芬·沃尔夫拉姆（Stephen Wolfram）则提出人类数学仅是众多可能形式中的一种。

五、数学与其他知识体系的差异

在自然科学中，旧理论常被新观测与新理论取代。例如 19 世纪原子模型在短时间内被修正，牛顿力学在高速与微观尺度下被相对论与量子力学拓展。

而数学呈现出不同的演进模式：

• 已被严格证明的数学结论，通常不会被后续研究否定；
• 新数学成果可以兼容、拓展旧成果，而非简单抛弃；
• 公元前 250 年阿基米德证明的球表面积公式，在当代仍被使用。

伊恩·斯图尔特（Ian Stewart）指出：

“在数学领域里，谬误一词表示先前以为是正确的、而后来却发现有错误并被纠正的结论。”

这些结论之所以被判定为谬误，是由于数学内部更严格的检验，而非外部学科的冲击。

六、历史视角：从毕达哥拉斯到现代

数学与实在的关系问题并非现代独有。托马斯·霍布斯（Thomas Hobbes, 1588–1679）在《利维坦》中将几何学视为理性论证的典范：

“在几何学（这是迄今为止唯一令上帝满意并恩赐给人类的学问）中，人们首先确定名称的含义（他们把确定含义称为’定义’），并把它们作为认知的起点。”

这一传统可追溯至毕达哥拉斯学派的"万物皆数"与柏拉图的理型论。然而，历史亦显示数学观的演变：从古希腊的确定性追求，到 19 世纪非欧几何带来的危机，再到 20 世纪公理化方法与形式主义的发展。

七、结论：未解之谜的持续意义

"上帝是数学家吗？"这一问题——无论是神学还是隐喻意义上的——指向人类知识的基础结构。数学无理由的有效性表明，在人类的抽象思维与自然的深层秩序之间存在某种深刻关联，其本质至今未被充分理解。

正如维格纳所言，数学与物理的契合是"上天赐予我们的绝妙礼物"——我们既未真正理解这份礼物，亦受之有愧。这一谜题将继续激发数学家、物理学家与哲学家的思考，提醒我们：在知识的边界，惊奇本身即是理解的起点。

参考文献

本文内容基于马里奥·利维奥（Mario Livio）著作《最后的数学问题》（Is God a Mathematician?）的思想脉络整理。

利维奥为哈勃太空望远镜科学研究所天体物理学家，数学与科学史研究者，曾获皮亚诺奖与国际毕达哥拉斯数学畅销书奖。

延伸阅读：图灵数学史系列简介

reference

• God is a Mathematician – Science and Nonduality (SAND) https://scienceandnonduality.com/article/god-is-a-mathematician/
• Mathematics declaring the glory of God https://scielo.org.za/pdf/vee/v43n1/12.pdf
• 回答数学的终极问题：上帝是一位数学家吗？ https://mp.weixin.qq.com/s/MGdJW4P5EN7nkbPE6FHvvA