数学哲学 | 穿越古代智慧与现代争论的旅程

注：本文为 “ 数学哲学” 相关合辑。
英文引文，机翻未校。
如有内容异常，请看原文。

---

The Philosophy of Mathematics: A Journey Through Ancient Wisdom and Modern Debates

数学哲学：穿越古代智慧与现代争论的旅程

Gurkamal Singh
Volume 16, Issue 3, July-September 2025

Abstract

摘要

When I first encountered the philosophy of mathematics as an undergraduate, I was struck by how such seemingly concrete and certain knowledge could give rise to such profound philosophical puzzles. How can numbers exist if we can’t touch them? Why does mathematics work so well in describing the physical world? These questions have captivated thinkers for over two millennia, and they continue to shape how we understand mathematical knowledge today.
本科阶段初次接触数学哲学时，我深感震撼：这种看似具体且确定的知识，竟能引发如此深刻的哲学谜题。若无法触摸数字，它们何以存在？为何数学能如此精准地描述物理世界？这些问题已吸引思想家们探索逾两千年，至今仍在塑造我们对数学知识的理解。

This paper explores the fascinating world of mathematical philosophy, tracing its development from ancient Greek discoveries to contemporary classroom debates. Through examining major philosophical schools-Platonism, formalism, constructivism, and naturalism-I investigate how different views about the nature of mathematical truth affect everything from research mathematics to how we teach calculus to freshmen. My analysis draws on historical sources, philosophical arguments, and educational research to show that these seemingly abstract questions have very real consequences for mathematical practice and learning.
本文将探索数学哲学的迷人世界，追溯其从古希腊的发现到当代课堂争论的发展历程。通过考察主要哲学流派——柏拉图主义、形式主义、建构主义和自然主义——我将探究关于数学真理本质的不同观点，如何影响从数学研究到大学新生微积分教学的方方面面。我的分析结合历史文献、哲学论证和教育研究，旨在说明这些看似抽象的问题，对数学实践与学习具有切实的影响。

Introduction: Why Philosophy Matters for Mathematics

引言：为何哲学对数学至关重要

Most mathematicians, if pressed, would probably say they don’t think much about philosophy. They’re too busy proving theorems, solving equations, or grading papers. But here’s the thing: whether we realize it or not, every mathematician operates with implicit philosophical assumptions about what mathematics is and how it works. These assumptions shape how we teach, how we do research, and how we think about mathematical truth.
若被追问，大多数数学家可能会表示他们不太关注哲学。他们忙于证明定理、求解方程或批改作业。但关键在于：无论是否意识到，每位数学家都持有关于数学本质及其运作方式的隐性哲学假设。这些假设影响着我们的教学方式、研究方法以及对数学真理的认知。

Consider a simple example. When I teach my calculus students that the derivative of $x^2$ is $2x$, am I revealing a timeless truth about abstract mathematical objects? Or am I teaching them a useful computational procedure that humans have invented? The difference might seem academic, but it actually influences how I approach the lesson, how I respond to student questions, and what I emphasize in my explanations.
举一个简单的例子。当我向微积分学生讲授 $x^2$ 的导数是 $2x$ 时，我是在揭示抽象数学对象的永恒真理？还是在传授人类发明的一种实用计算方法？这种差异看似仅具学术意义，但实际上会影响我的授课思路、对学生问题的回应方式以及解释中的重点内容。

The philosophy of mathematics isn’t just ivory tower speculation-it’s the foundation underlying all mathematical activity. As James Robert Brown puts it in his excellent introduction to the field, mathematics presents us with a unique puzzle: it seems to give us certain, objective knowledge about abstract entities that exist nowhere in space and time[7]. This paradox has generated some of the most sophisticated philosophical thinking in human history.
数学哲学并非象牙塔中的空想——它是所有数学活动的基础。正如詹姆斯·罗伯特·布朗在其优秀的领域导论中所言，数学为我们呈现了一个独特的谜题：它似乎能为我们提供关于抽象实体的确定、客观的知识，而这些实体不存在于任何时空之中[7]。这一悖论催生了人类历史上一些最精妙的哲学思考。

A Brief Journey Through History

历史简史

The Ancient Greeks and the Birth of Mathematical Philosophy

古希腊与数学哲学的诞生

The story really begins with the Pythagoreans in ancient Greece. These weren’t just mathematicians- they were mystics who believed that “all things are numbers.” When they discovered that the diagonal of a unit square has length $\sqrt{2}$, which cannot be expressed as a ratio of integers, it created what might be considered the first foundational crisis in mathematics.
这段历史真正始于古希腊的毕达哥拉斯学派。他们不仅是数学家，更是神秘主义者，坚信“万物皆数”。当他们发现单位正方形的对角线长度为 $\sqrt{2}$，且无法表示为整数之比时，这引发了数学史上可能是第一次基础危机。

Think about how shocking this must have been. Here were people who had built their entire worldview around the idea that reality consists of numerical relationships, and suddenly they found numbers that couldn’t be expressed as simple fractions. According to historical accounts, the discovery was so disturbing that the Pythagoreans tried to keep it secret[23].
试想这一发现当时何等令人震惊。这些人将整个世界观建立在“现实由数值关系构成”的理念之上，却突然发现了无法用简单分数表示的数。据史料记载，这一发现令人极为不安，毕达哥拉斯学派曾试图将其保密[23]。

Plato took these insights and ran with them, developing a sophisticated philosophical system that placed mathematics at the center of human knowledge. In Plato’s famous cave allegory, mathematics represents a crucial stepping stone between the world of shadows (everyday experience) and the world of pure forms (ultimate reality). Mathematical objects like triangles and numbers exist in this abstract realm, and doing mathematics is a way of training our minds to apprehend eternal truths.
柏拉图吸纳了这些观点并加以发展，构建了一套复杂的哲学体系，将数学置于人类知识的核心。在柏拉图著名的洞穴寓言中，数学是连接影子世界（日常经验）与纯粹形式世界（终极现实）的关键阶梯。三角形、数字等数学对象存在于这一抽象领域，而研究数学则是训练心智领悟永恒真理的一种方式。

This Platonic vision was incredibly influential. For over two thousand years, mathematicians have intuitive felt that they’re discovering rather than inventing mathematical truths. When a mathematician proves a theorem, it feels like uncovering something that was already there, waiting to be found.
这一柏拉图式的观点极具影响力。两千多年来，数学家们直觉上认为自己是在发现而非发明数学真理。当数学家证明一个定理时，感觉就像是揭开了某种早已存在、等待被发现的事物。

Medieval Developments and Islamic Contributions

中世纪的发展与伊斯兰世界的贡献

During the medieval period, Islamic scholars made crucial contributions to both mathematics and its philosophy. Al-Khwarizmi’s work on algebra (the very word comes from the Arabic “al-jabr”) wasn’t just computational-it involved sophisticated thinking about the nature of mathematical objects and operations[23].
中世纪时期，伊斯兰学者对数学及其哲学均做出了关键性贡献。花拉子米关于代数的研究（“代数”一词源于阿拉伯语“al-jabr”）不仅涉及计算，还包含了对数学对象及运算本质的深入思考[23]。

What fascinates me about this period is how mathematical philosophy developed within broader theological contexts. Islamic, Jewish, and Christian thinkers all grappled with questions about how mathematical knowledge relates to divine knowledge. If God knows all mathematical truths, does that mean they exist independently of human minds? These questions laid important groundwork for later secular philosophical developments.
这一时期令我着迷的是，数学哲学如何在更广泛的神学背景下发展。伊斯兰、犹太和基督教思想家均探讨了数学知识与神圣知识的关系问题。若上帝知晓所有数学真理，是否意味着这些真理独立于人类心智而存在？这些问题为后来世俗哲学的发展奠定了重要基础。

The Crisis of the Infinite

无穷的危机

The development of calculus by Newton and Leibniz in the 17th century created new philosophical puzzles. What exactly are infinitesimals? How can we make sense of infinite sums and instantaneous rates of change? These weren’t just technical problems-they were conceptual challenges that threatened the logical foundations of mathematical reasoning.
17世纪，牛顿和莱布尼茨发明微积分，引发了新的哲学谜题。无穷小量究竟是什么？我们如何理解无穷级数和瞬时变化率？这些并非单纯的技术问题——它们是威胁数学推理逻辑基础的概念挑战。

Bishop Berkeley, the Irish philosopher, famously criticized the logical basis of calculus, calling infinitesimals “ghosts of departed quantities.” His criticisms weren’t easily dismissed, and they helped motivate the 19th-century project of placing analysis on rigorous foundations. Mathematicians like Cauchy, Weierstrass, and Dedekind developed increasingly sophisticated approaches to limits, continuity, and infinite processes.
爱尔兰哲学家贝克莱主教曾著名地批判微积分的逻辑基础，将无穷小量称为“逝去量的幽灵”。他的批判并非轻易可被忽视，且推动了19世纪将数学分析建立在严格基础上的研究项目。柯西、魏尔斯特拉斯、戴德金等数学家开发了日益精密的方法，以处理极限、连续性和无穷过程。

But here’s what’s interesting from a philosophical perspective: each attempt to solve these foundational problems raised new philosophical questions. When Cantor developed set theory to provide foundations for analysis, he opened up entirely new questions about infinite sets and their properties. When Dedekind constructed the real numbers from rational numbers, he raised questions about the relationship between mathematical construction and mathematical existence.
但从哲学角度来看，有趣的是：每一次试图解决这些基础问题的尝试，都会引发新的哲学疑问。当康托尔发展集合论以奠定分析学基础时，他提出了关于无穷集合及其性质的全新问题。当戴德金从有理数构造实数时，他引发了关于数学构造与数学存在之间关系的探讨。

The Great Philosophical Schools

主要哲学流派

Mathematical Platonism: The Eternal Realm of Numbers

数学柏拉图主义：数字的永恒领域

Modern mathematical Platonism holds that mathematical objects exist independently of human minds and mathematical practices. Numbers, sets, functions, and geometric shapes populate an abstract realm that exists outside space and time. When mathematicians prove theorems, they’re discovering objective facts about this mathematical reality.
现代数学柏拉图主义认为，数学对象独立于人类心智和数学实践而存在。数字、集合、函数和几何图形存在于一个超越时空的抽象领域中。当数学家证明定理时，他们是在发现关于这一数学实在的客观事实。

I have to admit, Platonism has always held a certain appeal for me. When I work through a proof, it really does feel like I’m uncovering something that was already true, not creating something new. The integers seem to have their properties-being even or odd, prime or composite-regardless of whether anyone thinks about them. Mathematical truth seems objective in a way that many other kinds of truth are not.
我必须承认，柏拉图主义对我始终具有某种吸引力。当我推导一个证明时，确实感觉像是在揭示某种早已成立的事实，而非创造新的事物。整数的性质——偶数或奇数、质数或合数——似乎与是否有人思考它们无关。数学真理的客观性，是许多其他类型的真理所不具备的。

The strongest argument for Platonism is what philosophers call the “indispensability argument,” developed by W.V.O. Quine and Hilary Putnam. The basic idea is this: mathematics is indispensable to our best scientific theories, and we should believe in the existence of entities that are indispensable to successful scientific practice. Since physics, chemistry, and other sciences can’t do without mathematics, we have good reason to think mathematical objects really exist.
柏拉图主义最有力的论证是哲学家们所说的“不可或缺性论证”，由W.V.O.奎因和希拉里·普特南提出。其核心观点是：数学对我们最优秀的科学理论而言不可或缺，而我们应当相信那些对成功的科学实践不可或缺的实体的存在。由于物理学、化学和其他科学都离不开数学，我们有充分理由认为数学对象确实存在。

But Platonism faces serious challenges. The biggest is what philosophers call the “epistemic problem”: if mathematical objects exist in an abstract realm completely separate from physical reality, how could we possibly know anything about them? Knowledge seems to require some kind of causal interaction between knower and known, but abstract objects can’t cause anything. Paul Benacerraf crystallized this problem in his influential 1973 paper “Mathematical Truth,” and philosophers are still wrestling with it today.
但柏拉图主义面临着严峻挑战。最大的挑战是哲学家们所说的“认识论问题”：若数学对象存在于与物理实在完全分离的抽象领域，我们如何可能对它们有任何了解？知识似乎需要认知者与被认知者之间存在某种因果互动，但抽象对象无法产生任何因果作用。保罗·贝纳塞拉夫在其1973年的影响力论文《数学真理》中明确阐述了这一问题，至今哲学家们仍在致力于解决它。

Formalism: Mathematics as Symbol Manipulation

形式主义：作为符号操作的数学

Formalism emerged in the early 20th century as a response to foundational crises in mathematics. The basic idea, developed most systematically by David Hilbert, is that mathematics should be understood as the manipulation of formal symbols according to precise rules, without regard to what those symbols might mean.
形式主义于20世纪初应运而生，是对数学基础危机的回应。其核心思想由大卫·希尔伯特最为系统地发展，认为数学应被理解为根据精确规则进行的形式符号操作，而无需关注这些符号可能具有的含义。

According to Hilbert’s program, mathematics consists of finite combinatorial procedures applied to meaningless formal expressions. Mathematical truth reduces to provability within formal systems, and mathematical existence means consistency-if you can write down axioms for something without deriving a contradiction, then that something exists mathematically.
根据希尔伯特纲领，数学由应用于无意义形式表达式的有限组合程序构成。数学真理可还原为形式系统内的可证明性，而数学存在即意味着一致性——若你能为某事物写下公理且未推导出矛盾，则该事物在数学意义上存在。

I find formalism intellectually appealing in some ways. It promises to eliminate messy metaphysical questions about abstract objects while preserving mathematical practice. Instead of worrying about whether numbers really exist, we can focus on developing formal systems and studying their properties. But then Kurt Gödel came along and threw a wrench in the works. His incompleteness theorems, proved in 1931, showed that any consistent formal system capable of expressing basic arithmetic must contain true but unprovable statements. This meant that mathematical truth can’t be reduced to formal provability-there are mathematical facts that transcend what can be proven in any given formal system. Gödel’s results were devastating for Hilbert’s original program, but they didn’t kill formalism entirely. Contemporary formalists have developed more nuanced positions that acknowledge the limitations revealed by Gödel while maintaining that formal methods remain central to mathematical practice.
我认为形式主义在某些方面具有智识上的吸引力。它承诺在保留数学实践的同时，消除关于抽象对象的复杂形而上学问题。我们无需担忧数字是否真正存在，而是可以专注于发展形式系统并研究其性质。但库尔特·哥德尔的出现打乱了这一计划。他于1931年证明的不完备性定理表明，任何能够表达基本算术的一致形式系统，都必然包含真但不可证明的命题。这意味着数学真理无法还原为形式可证明性——存在超越任何给定形式系统可证明范围的数学事实。哥德尔的结果对希尔伯特的原始纲领是毁灭性的，但并未彻底终结形式主义。当代形式主义者发展出了更微妙的立场，既承认哥德尔揭示的局限性，又坚持形式方法仍是数学实践的核心。

Constructivism: Building Mathematics from the Ground Up

建构主义：从零开始构建数学

Constructivism, developed by the Dutch mathematician L.E.J. Brouwer, takes a radically different approach. According to Brouwerian intuitionism, mathematical objects don’t exist independently of human mathematical activity. Instead, they’re mental constructions created through specific constructive procedures.
建构主义由荷兰数学家L.E.J.布劳威尔提出，采取了一种截然不同的方法。根据布劳威尔的直觉主义，数学对象并非独立于人类数学活动而存在，而是通过特定的构造程序创造的心智建构物。

For a constructivist, to assert that a mathematical object exists, you must provide a method for constructing it. This leads to some surprising consequences. Constructivists reject certain classical theorems that depend on non-constructive proof methods. They also reject classical logic itself- specifically, the law of excluded middle-in cases where constructive proof is unavailable.
对建构主义者而言，要断言一个数学对象存在，必须提供其构造方法。这会导致一些令人意外的结果。建构主义者拒绝某些依赖非构造性证明方法的经典定理。在无法提供构造性证明的情况下，他们还会拒绝经典逻辑本身——尤其是排中律。

I have to confess that I’ve always found constructivism intellectually challenging. On one hand, it seems to take seriously the human, creative aspect of mathematical activity. Mathematics isn’t about discovering pre-existing truths but about the actual process of mathematical construction and proof.
我必须承认，建构主义在智识上对我而言始终具有挑战性。一方面，它似乎重视数学活动中人类的、创造性的方面。数学并非关于发现预先存在的真理，而是关于数学构造与证明的实际过程。

On the other hand, constructivism requires giving up some beautiful and useful mathematics. Many of the most elegant results in analysis and topology depend on non-constructive methods. As a working mathematician, it’s hard to accept that these results should be rejected on philosophical grounds.
另一方面，建构主义要求放弃一些优美且实用的数学内容。分析学和拓扑学中的许多最精妙的结果都依赖于非构造性方法。作为一名从业数学家，很难接受这些结果因哲学理由而被否定。

There’s also a practical problem: constructive mathematics is often much more complicated than classical mathematics. Simple existence statements that can be proved easily using classical methods may require elaborate constructions in the constructive setting.
此外还有一个实际问题：建构性数学通常比经典数学复杂得多。使用经典方法可轻松证明的简单存在性命题，在建构性框架下可能需要复杂的构造过程。

Naturalism: Looking at Mathematics as It Really Is

自然主义：审视数学的本来面目

Mathematical naturalism, advocated by philosophers like Penelope Maddy, takes a different approach entirely. Instead of asking abstract questions about the ultimate nature of mathematical objects, naturalists focus on mathematical practice as it actually occurs.
数学自然主义由彭妮洛普·马迪等哲学家倡导，采取了一种完全不同的方法。自然主义者不追问关于数学对象终极本质的抽象问题，而是关注实际发生的数学实践。

The naturalist says: look, mathematicians have been doing mathematics successfully for centuries. Instead of imposing external philosophical constraints, why don’t we examine how mathematics actually works and let that guide our philosophical theorizing?
自然主义者认为：看，数学家们已经成功地从事数学研究数百年了。与其施加外部的哲学约束，不如考察数学实际如何运作，并以此指导我们的哲学理论建构？

This approach has led to detailed studies of mathematical methodology, the sociology of mathematical communities, and the historical development of mathematical concepts. Rather than asking whether numbers exist, naturalists ask: what role do number-theoretic claims play in successful mathematical practice?
这种方法催生了对数学方法论、数学社群社会学以及数学概念历史发展的详细研究。自然主义者不追问数字是否存在，而是问：数论命题在成功的数学实践中扮演着何种角色？

I find naturalism attractive because it takes mathematical practice seriously. Instead of trying to force mathematics into preconceived philosophical categories, it attempts to understand mathematics on its own terms. But I sometimes wonder whether naturalism can really avoid the traditional philosophical questions. Even if we focus on mathematical practice, don’t we still need to ask what makes that practice successful?
我认为自然主义具有吸引力，因为它重视数学实践。它不试图将数学强行纳入预设的哲学范畴，而是试图以数学自身的方式理解数学。但我有时会疑惑，自然主义是否真的能回避传统的哲学问题。即使我们关注数学实践，难道不需要追问是什么让这种实践取得成功吗？

The Problem of Mathematical Knowledge

数学知识的问题

How Do We Know Mathematical Truths?

我们如何知晓数学真理？

One of the most puzzling aspects of mathematics is how we come to know mathematical truths. Mathematical knowledge seems to have several distinctive features that set it apart from empirical knowledge: it appears necessary (mathematical truths couldn’t be otherwise), certain (once proven, mathematical results seem permanently established), universal (mathematical truths hold everywhere), and a priori (accessible through reason rather than sensory experience).
数学最令人困惑的方面之一，是我们如何获得数学真理的知识。数学知识似乎具有几个区别于经验知识的独特特征：它具有必然性（数学真理不可能是其他样子）、确定性（一旦被证明，数学结果似乎就永久确立）、普遍性（数学真理在任何地方都成立）以及先验性（可通过理性而非感官经验获得）。

But how is such knowledge possible? Immanuel Kant argued that mathematical knowledge is both a priori (knowable through reason alone) and synthetic (providing substantive information about reality). This combination seemed problematic to many philosophers-how can reason alone give us knowledge about the world?
但这种知识如何可能？伊曼努尔·康德认为，数学知识既是先验的（仅通过理性即可知晓），又是综合的（提供关于实在的实质性信息）。这种组合对许多哲学家而言似乎是矛盾的——理性 alone 何以能为我们提供关于世界的知识？

Kant’s solution involved his famous “Copernican revolution” in philosophy. He argued that space and time are not features of things-in-themselves but rather forms of human intuition-ways our minds necessarily structure experience. Mathematical knowledge is possible because we’re not discovering independent truths about external reality, but rather uncovering the necessary structure of our own cognitive apparatus. This is a fascinating solution, but it faces serious challenges. The development of non-Euclidean geometries in the 19th century suggested that geometric truths aren’t as necessary as Kant supposed. Einstein’s use of non-Euclidean geometry in general relativity showed that empirical considerations can lead us to revise our geometric beliefs.
康德的解决方案涉及他著名的哲学“哥白尼革命”。他认为，空间和时间并非物自体的特征，而是人类直观的形式——即我们的心智必然构建经验的方式。数学知识之所以可能，是因为我们并非在发现关于外部实在的独立真理，而是在揭示我们自身认知结构的必然形式。这是一个迷人的解决方案，但它面临着严峻挑战。19世纪非欧几何的发展表明，几何真理并不像康德所认为的那样具有必然性。爱因斯坦在广义相对论中对非欧几何的运用，则表明经验因素可以促使我们修正几何信念。

The Role of Intuition and Insight

直觉与洞见的作用

When I’m working on a mathematical problem, I often experience moments of sudden insight-seeing why a theorem must be true, or recognizing the pattern that will lead to a proof. These moments of mathematical intuition seem crucial to mathematical discovery, but they’re philosophically puzzling. What exactly is mathematical intuition? Different philosophical positions offer different accounts. Platonists might interpret intuition as a form of intellectual perception that provides access to abstract mathematical reality. It’s like having a special sense organ for mathematical facts. Constructivists would understand intuition differently-as the mental activity through which mathematical objects are constructed. When I have an intuitive insight about a mathematical problem, I’m not perceiving pre-existing mathematical facts but rather engaging in the creative activity of mathematical construction. Formalists might be skeptical of intuition altogether, viewing it as psychologically interesting but philosophically irrelevant. What matters for mathematics is not intuitive insight but formal proof and logical rigor. Recent work in cognitive science has begun to shed light on mathematical intuition from an empirical perspective. Studies of mathematical expertise suggest that experienced mathematicians develop sophisticated pattern recognition abilities that allow them to see mathematical structure quickly and accurately. This research suggests that mathematical intuition might be understood as a form of highly trained perceptual skill rather than mystical access to abstract realms.
当我致力于解决一个数学问题时，常常会经历突如其来的洞见——明白某个定理为何必然成立，或识别出能导向证明的模式。这些数学直觉的时刻似乎对数学发现至关重要，但在哲学上却令人困惑。数学直觉究竟是什么？不同的哲学立场给出了不同的解释。柏拉图主义者可能将直觉解释为一种理智感知，使人能够通达抽象的数学实在。这就像拥有一个专门感知数学事实的感官。建构主义者则会以不同的方式理解直觉——将其视为构建数学对象的心智活动。当我对某个数学问题产生直觉洞见时，我并非在感知预先存在的数学事实，而是在从事数学构造的创造性活动。形式主义者可能完全怀疑直觉的意义，认为它在心理上有趣但在哲学上无关紧要。对数学而言，重要的不是直觉洞见，而是形式证明和逻辑严谨性。认知科学领域的最新研究已开始从经验角度阐明数学直觉。对数学专长的研究表明，有经验的数学家会发展出复杂的模式识别能力，使他们能够快速、准确地把握数学结构。这项研究表明，数学直觉或许应被理解为一种高度训练的感知技能，而非对抽象领域的神秘通达。

Mathematics and Human Psychology

数学与人类心理学

One area that fascinates me is the relationship between mathematical thinking and general human cognition. Are mathematical abilities special, or do they emerge from more general cognitive capacities? There’s growing evidence that some basic mathematical intuitions-about number, quantity, and spatial relationships-are present very early in human development and may be shared with other species. This suggests that at least some mathematical thinking might be grounded in our biological heritage rather than being purely cultural constructions. But higher mathematics clearly goes far beyond these basic intuitions. The development of abstract concepts like infinite sets, complex numbers, or higher-dimensional spaces required centuries of cultural evolution. How do humans manage to think about these abstract mathematical objects? One possibility is that we use metaphorical thinking, grounding abstract mathematical concepts in more concrete spatial and temporal intuitions. Lakoff and Núñez have argued that all mathematical thinking involves metaphorical mappings from concrete domains to abstract ones. When we think about numbers as points on a line, or functions as machines that transform inputs into outputs, we’re using spatial and mechanical metaphors to understand abstract mathematical relationships.
有一个领域令我着迷，那就是数学思维与人类一般认知之间的关系。数学能力是特殊的，还是源于更一般的认知能力？越来越多的证据表明，一些基本的数学直觉——关于数字、数量和空间关系的直觉——在人类发展早期就已存在，且可能与其他物种共享。这表明，至少部分数学思维可能植根于我们的生物遗传，而非纯粹的文化建构。但高等数学显然远超这些基本直觉。无穷集合、复数或高维空间等抽象概念的发展，经历了数百年的文化演进。人类如何能够思考这些抽象的数学对象？一种可能性是，我们运用隐喻思维，将抽象的数学概念建立在更具体的空间和时间直觉之上。拉科夫和努涅斯认为，所有数学思维都涉及从具体领域到抽象领域的隐喻映射。当我们将数字视为直线上的点，或将函数视为将输入转化为输出的机器时，我们正是在运用空间和机械隐喻来理解抽象的数学关系。

Mathematics and Reality: The Applicability Problem

数学与实在：适用性问题

The Unreasonable Effectiveness of Mathematics

数学的不合理有效性

One of the most striking features of mathematics is its extraordinary effectiveness in describing and predicting natural phenomena. Eugene Wigner called this “the unreasonable effectiveness of mathematics in the natural sciences,” and it remains one of the deepest puzzles in the philosophy of mathematics.
数学最显著的特征之一，是其在描述和预测自然现象方面的非凡有效性。尤金·维格纳将其称为“数学在自然科学中不合理的有效性”，这至今仍是数学哲学中最深刻的谜题之一。

Consider some examples: complex numbers, originally introduced to solve algebraic equations, turned out to be essential for quantum mechanics. Non-Euclidean geometries, initially regarded as mathematical curiosities, provided the framework for Einstein’s general relativity. Group theory, developed as pure mathematics, became crucial for understanding particle physics.
举几个例子：复数最初被引入是为了解决代数方程，后来却被证明对量子力学至关重要。非欧几何起初被视为数学 curiosities，却为爱因斯坦的广义相对论提供了框架。群论作为纯数学发展而来，后来成为理解粒子物理学的关键。

How can abstract mathematical structures, developed independently of empirical investigation, prove so useful for understanding physical reality? This effectiveness seems too remarkable to be mere coincidence, but it’s hard to explain why it should be expected.
那些独立于经验研究发展起来的抽象数学结构，何以能如此有效地帮助我们理解物理实在？这种有效性如此显著，难以仅仅归因于巧合，但又很难解释为何它是可预期的。

Different philosophical positions offer different explanations. Platonists might argue that mathematical structures exist independently and that physical reality somehow exemplifies or instantiates these structures. The effectiveness of mathematics reflects genuine structural similarities between abstract mathematical reality and concrete physical reality.
不同的哲学立场给出了不同的解释。柏拉图主义者可能会认为，数学结构独立存在，而物理实在以某种方式体现或例示了这些结构。数学的有效性反映了抽象数学实在与具体物理实在之间真正的结构相似性。

Formalists might emphasize the flexibility of formal systems for modeling diverse phenomena. Mathematical formalism provides a rich toolkit of representational resources that can be adapted to represent almost any kind of structure or pattern.
形式主义者可能会强调形式系统在建模各种现象方面的灵活性。数学形式主义提供了丰富的表征资源工具包，可适用于表示几乎任何类型的结构或模式。

Naturalists might focus on the historical co-evolution of mathematical concepts and scientific applications. Mathematical ideas don’t develop in isolation from practical concerns-they emerge partly in response to the needs of physics, engineering, and other applied fields.
自然主义者可能会关注数学概念与科学应用的历史协同进化。数学思想并非脱离实际关切而发展——它们的出现部分是为了回应物理学、工程学和其他应用领域的需求。

Mathematical Modeling and Representation

数学建模与表征

The application of mathematics to empirical domains typically involves mathematical modeling-using mathematical structures to represent features of physical systems. But the relationship between mathematical models and physical reality is complex and philosophically puzzling.
数学在经验领域的应用通常涉及数学建模——即运用数学结构来表示物理系统的特征。但数学模型与物理实在之间的关系是复杂且具有哲学困惑的。

Mathematical models typically involve idealizations and simplifications that sacrifice literal accuracy for mathematical tractability. We model falling objects as point masses, treat gases as collections of perfectly elastic spheres, and represent complex biological systems with systems of differential equations. These models are obviously false as literal descriptions of reality, yet they’re enormously useful for prediction and control.
数学模型通常包含理想化和简化，为了数学上的易处理性而牺牲字面准确性。我们将下落物体建模为质点，将气体视为完全弹性球的集合，并用微分方程组表示复杂的生物系统。这些模型作为对实在的字面描述显然是虚假的，但它们在预测和控制方面却极为有用。

This raises questions about what makes a mathematical model good or successful. Is it accuracy of representation? Predictive power? Mathematical elegance? Practical utility? Different criteria can pull in different directions, and there’s no obvious way to reconcile them.
这引发了一个问题：是什么使得一个数学模型是好的或成功的？是表征的准确性？预测能力？数学的优美性？还是实际效用？不同的标准可能指向不同的方向，且尚无明显的方法调和它们。

I think mathematical modeling reveals something important about the nature of mathematical knowledge. Mathematics doesn’t provide a mirror of reality but rather a toolkit for constructing useful representations of aspects of reality. The effectiveness of mathematics reflects not metaphysical correspondence but rather the flexibility and power of mathematical representational resources.
我认为数学建模揭示了关于数学知识本质的重要内容。数学并非提供实在的镜像，而是提供了一套用于构建实在各方面有用表征的工具包。数学的有效性并非反映形而上学的对应关系，而是反映了数学表征资源的灵活性和力量。

Contemporary Philosophical Developments

当代哲学发展

Structuralism and the Focus on Patterns

结构主义与对模式的关注

One of the most significant developments in recent philosophy of mathematics has been the rise of structuralism. According to structuralist approaches, mathematics is primarily concerned with structural relationships rather than with the intrinsic nature of mathematical objects.
近年来数学哲学最重大的发展之一是结构主义的兴起。根据结构主义的观点，数学主要关注结构关系，而非数学对象的内在本质。

Stewart Shapiro and others have developed sophisticated versions of mathematical structuralism that attempt to preserve the objectivity of mathematics while avoiding problematic ontological commitments to abstract objects. Instead of asking whether the number 2 exists, structuralists ask about the structural position that plays the “2-role” in various mathematical contexts.
斯图尔特·夏皮罗等人发展了复杂的数学结构主义版本，试图在保留数学客观性的同时，避免对抽象对象的有问题的本体论承诺。结构主义者不追问数字 2 是否存在，而是追问在各种数学语境中扮演“2-角色”的结构位置。

I find structuralism appealing because it seems to capture something important about mathematical practice. When I’m proving a theorem about groups, I’m not really concerned with what groups are made of-I’m interested in their structural properties. The same structural insights apply whether I’m thinking about groups of symmetries, groups of permutations, or groups of matrices.
我认为结构主义具有吸引力，因为它似乎捕捉到了数学实践的一个重要方面。当我证明一个关于群的定理时，我并不真正关心群是由什么构成的——我感兴趣的是它们的结构性质。无论我思考的是对称群、置换群还是矩阵群，同样的结构洞见都适用。

Category theory has emerged as a natural mathematical framework for structuralist philosophy. Instead of focusing on sets and membership relations, category theory emphasizes morphisms and structural relationships. Some mathematicians and philosophers have proposed category theory as a foundation for mathematics that’s more natural than set theory for capturing the structural character of mathematical thinking.
范畴论已成为结构主义哲学的自然数学框架。范畴论不关注集合和隶属关系，而是强调态射和结构关系。一些数学家和哲学家提出，范畴论作为数学基础，比集合论更自然地捕捉了数学思维的结构特征。

The Practice Turn in Philosophy of Mathematics

数学哲学的实践转向

Another significant development has been increased attention to mathematical practice as it actually occurs. Instead of focusing on abstract foundational questions, philosophers have begun studying how mathematics is actually done by working mathematicians.
另一项重大发展是，人们越来越关注实际发生的数学实践。哲学家们不再专注于抽象的基础问题，而是开始研究从业数学家实际如何从事数学研究。

This “practice turn” has involved detailed case studies of mathematical development, ethnographic investigation of mathematical communities, and analysis of mathematical discourse. The goal is to understand mathematics from the inside rather than imposing external philosophical categories.
这种“实践转向”涉及对数学发展的详细案例研究、对数学社群的民族志调查以及对数学话语的分析。其目标是从内部理解数学，而非施加外部的哲学范畴。

This work has revealed the importance of non-deductive elements in mathematical reasoning, including analogy, visualization, and heuristic methods. It has also highlighted the social dimensions of mathematical knowledge-how mathematical communities develop standards, evaluate arguments, and transmit understanding.
这项研究揭示了非演绎元素在数学推理中的重要性，包括类比、可视化和启发式方法。它还强调了数学知识的社会维度——数学社群如何制定标准、评估论证和传递理解。

I think this empirical approach to philosophy of mathematics is extremely valuable. Traditional philosophical discussions often seemed disconnected from actual mathematical practice. By paying careful attention to how mathematics actually works, philosophers can develop more adequate and realistic accounts of mathematical knowledge.
我认为这种对数学哲学的经验研究方法极具价值。传统的哲学讨论往往似乎与实际的数学实践脱节。通过密切关注数学实际如何运作，哲学家们能够发展出更充分、更符合实际的数学知识解释。

Computational Approaches and Artificial Intelligence

计算方法与人工智能

The development of computer science and artificial intelligence has created new perspectives on mathematical knowledge and reasoning. Automated theorem proving systems can now discover and verify mathematical proofs, raising questions about the nature of mathematical understanding and insight.
计算机科学和人工智能的发展，为数学知识和推理带来了新的视角。自动定理证明系统现在能够发现和验证数学证明，这引发了关于数学理解和洞见本质的问题。

If computers can prove theorems, what’s special about human mathematical ability? Some philosophers have argued that computational approaches support mechanistic accounts of mathematical reasoning- perhaps mathematical thinking is just sophisticated information processing.
如果计算机能够证明定理，那么人类的数学能力有何特别之处？一些哲学家认为，计算方法支持对数学推理的机械论解释——或许数学思维只是复杂的信息处理。

But I’m not convinced that computational success necessarily undermines traditional accounts of mathematical knowledge. Computers can perform many cognitive tasks without truly understanding what they’re doing. A theorem-proving program might generate valid proofs without having genuine mathematical insight.
但我并不认为计算上的成功必然会削弱传统的数学知识解释。计算机可以执行许多认知任务，却无需真正理解它们在做什么。一个定理证明程序可能生成有效的证明，却没有真正的数学洞见。

Still, computational approaches have led to important developments in mathematical practice. Computer-assisted proofs, like the proof of the four-color theorem, have become increasingly common and sophisticated. These developments raise questions about mathematical certainty and the nature of mathematical proof.
尽管如此，计算方法已在数学实践中带来了重要发展。计算机辅助证明（如四色定理的证明）已变得越来越普遍和复杂。这些发展引发了关于数学确定性和数学证明本质的问题。

Philosophy and Mathematical Education

哲学与数学教育

How Philosophy Shapes Teaching

哲学如何塑造教学

My interest in the philosophy of mathematics was sparked partly by questions about mathematical education. How should we teach mathematics? What does it mean for students to understand mathematical concepts? These pedagogical questions turn out to be intimately connected with philosophical questions about the nature of mathematical knowledge.
我对数学哲学的兴趣部分源于关于数学教育的问题。我们应该如何教授数学？学生理解数学概念意味着什么？这些教学问题被证明与关于数学知识本质的哲学问题密切相关。

Jo Boaler’s fascinating comparative study of two English secondary schools illustrates this connection beautifully[5]. She studied two schools with similar student populations but very different approaches to mathematics education. “Amber Hill” used traditional methods emphasizing procedural skills and individual practice. “Phoenix Park” used project-based methods emphasizing problem-solving and collaborative investigation.
乔·博勒对英国两所中学的迷人比较研究，完美地说明了这种联系[5]。她研究了两所学生群体相似但数学教育方法截然不同的学校。“琥珀山学校”采用传统方法，强调程序技能和个人练习。“凤凰公园学校”采用项目式方法，强调问题解决和协作探究。

The results were striking and counterintuitive. Despite spending much more time on direct instruction and skill practice, Amber Hill students performed worse on assessments requiring flexible problem-solving. Phoenix Park students, who spent most of their time on open-ended projects, were better able to apply their mathematical knowledge in novel situations.
结果令人震惊且违反直觉。尽管琥珀山学校的学生在直接教学和技能练习上花费了更多时间，但在需要灵活解决问题的评估中表现更差。凤凰公园学校的学生大部分时间用于开放式项目，却更能在新情境中应用他们的数学知识。

What explains these differences? Boaler argues that the different teaching approaches created different kinds of mathematical knowledge. Amber Hill students learned mathematics as a collection of discrete procedures to be applied in specific contexts. Phoenix Park students learned mathematics as a toolkit for investigating quantitative relationships in complex situations.
如何解释这些差异？博勒认为，不同的教学方法造就了不同类型的数学知识。琥珀山学校的学生将数学视为一组在特定情境中应用的离散程序。凤凰公园学校的学生将数学视为一套用于探究复杂情境中数量关系的工具包。

These different forms of mathematical knowledge reflect different philosophical assumptions about what mathematics is. Traditional approaches that emphasize procedural fluency align with formalist conceptions of mathematics as rule-following. Reform approaches that emphasize conceptual understanding and problem-solving align with constructivist conceptions of mathematics as sensemaking activity.
这些不同形式的数学知识反映了关于数学本质的不同哲学假设。强调程序流畅性的传统方法，与将数学视为遵循规则的形式主义观点一致。强调概念理解和问题解决的改革方法，与将数学视为意义建构活动的建构主义观点一致。

The Role of Proof in Mathematical Education

证明在数学教育中的作用

One area where philosophical assumptions have particularly strong educational implications concerns the role of proof in mathematical education. What should students learn about mathematical proof? When should proof be introduced? What kinds of proof are appropriate for different educational levels?
哲学假设对教育具有特别强烈影响的一个领域，是证明在数学教育中的作用。学生应该学习关于数学证明的哪些内容？应该在何时引入证明？哪些类型的证明适合不同的教育阶段？

Different philosophical positions suggest different answers to these questions. Formalist approaches might emphasize the logical structure of mathematical arguments, focusing on valid inference patterns and formal proof techniques. Students would learn to construct rigorous arguments that meet the standards of formal mathematical discourse.
不同的哲学立场对这些问题给出了不同的答案。形式主义方法可能强调数学论证的逻辑结构，关注有效的推理模式和形式证明技术。学生将学习构建符合形式数学话语标准的严谨论证。

Platonist approaches might emphasize proof as a method for discovering mathematical truths. Students would learn that proofs don’t create mathematical facts but rather reveal pre-existing relationships among mathematical objects.
柏拉图主义方法可能强调证明作为发现数学真理的方法。学生将学习到，证明并非创造数学事实，而是揭示数学对象之间预先存在的关系。

Constructivist approaches might emphasize proof as mathematical construction. Students would learn that proving a theorem involves constructing a mathematical object or establishing a constructive procedure.
建构主义方法可能强调证明作为数学构造。学生将学习到，证明一个定理涉及构建一个数学对象或确立一个构造程序。

In my own teaching, I’ve found that students often struggle with the concept of proof because they’re not clear about what proofs are supposed to accomplish. Are we trying to convince skeptics? Discover hidden truths? Follow formal rules? Different answers lead to different pedagogical approaches.
在我自己的教学中，我发现学生常常难以理解证明的概念，因为他们不清楚证明应该达成什么目的。我们是想说服怀疑者？发现隐藏的真理？还是遵循形式规则？不同的答案会导向不同的教学方法。

I’ve had the most success when I frame proof as mathematical explanation-helping students see why mathematical statements are true rather than just verifying that they are true. This approach seems to resonate with students’ natural desire to understand rather than merely accept mathematical claims.
当我将证明界定为数学解释时，取得了最大的成功——帮助学生理解数学命题为何为真，而非仅仅验证它们为真。这种方法似乎与学生天生渴望理解而非仅仅接受数学主张的愿望产生了共鸣。

Mathematical Understanding and Meaning

数学理解与意义

Questions about mathematical understanding connect philosophical issues with practical educational concerns. What does it mean for a student to understand a mathematical concept? How is mathematical understanding related to computational skill, conceptual knowledge, and problem-solving ability?
关于数学理解的问题将哲学议题与实际教育关切联系起来。学生理解一个数学概念意味着什么？数学理解与计算技能、概念知识和问题解决能力之间有何关系？

These questions don’t have simple answers, and different philosophical positions suggest different approaches. Platonist assumptions might emphasize understanding as intellectual apprehension of objective mathematical structures. Students who understand mathematical concepts have gained access to mind-independent mathematical reality.
这些问题没有简单的答案，不同的哲学立场给出了不同的方法。柏拉图主义假设可能强调理解作为对客观数学结构的理智把握。理解数学概念的学生，已经获得了对独立于心智的数学实在的通达。

Constructivist assumptions might emphasize understanding as successful mental construction of mathematical concepts. Students understand mathematical ideas when they can actively reconstruct them through their own mathematical activity.
建构主义假设可能强调理解作为对数学概念的成功心智建构。当学生能够通过自己的数学活动主动重构数学思想时，他们就理解了这些思想。

Social constructivist approaches might emphasize understanding as participation in mathematical discourse communities. Students understand mathematical concepts when they can participate meaningfully in mathematical conversations and activities.
社会建构主义方法可能强调理解作为参与数学话语社群。当学生能够有意义地参与数学对话和活动时，他们就理解了数学概念。

In my experience, all of these perspectives capture something important about mathematical understanding. Students need computational fluency, conceptual insight, and the ability to participate in mathematical discourse. The challenge is helping them develop all these capacities in an integrated way.
根据我的经验，所有这些视角都捕捉到了数学理解的一个重要方面。学生需要计算流畅性、概念洞见以及参与数学话语的能力。挑战在于帮助他们以一种整合的方式发展所有这些能力。

Technology and Mathematical Philosophy

技术与数学哲学

The increasing role of technology in mathematical education raises philosophical questions about the nature of mathematical knowledge and understanding. If computers can perform mathematical calculations and even prove theorems, what aspects of mathematical knowledge remain distinctively human?
技术在数学教育中日益增长的作用，引发了关于数学知识和理解本质的哲学问题。如果计算机能够进行数学计算甚至证明定理，那么数学知识中哪些方面仍然是人类独有的？

Some educators argue that technology liberates students from computational drudgery, allowing them to focus on higher-order mathematical thinking. Others worry that reliance on technology might diminish students’ understanding of fundamental mathematical concepts.
一些教育工作者认为，技术将学生从计算的苦差中解放出来，使他们能够专注于高阶数学思维。另一些人则担心，对技术的依赖可能会削弱学生对基本数学概念的理解。

These concerns reflect deeper philosophical questions about the relationship between computational skill and mathematical understanding. If mathematical knowledge is primarily procedural, then technological assistance might threaten genuine understanding. If mathematical knowledge is primarily conceptual, then technology might enhance rather than diminish mathematical understanding.
这些担忧反映了关于计算技能与数学理解之间关系的更深层次的哲学问题。如果数学知识主要是程序性的，那么技术辅助可能会威胁到真正的理解。如果数学知识主要是概念性的，那么技术可能会增强而非削弱数学理解。

I think the key is helping students understand both the capabilities and limitations of technological tools. Calculators and computer algebra systems are powerful aids to mathematical reasoning, but they don’t replace the need for mathematical insight and judgment. Students need to learn when to use technology and when to work by hand, when to trust computational results and when to be skeptical.
我认为关键在于帮助学生理解技术工具的能力和局限性。计算器和计算机代数系统是数学推理的强大辅助，但它们无法替代对数学洞见和判断力的需求。学生需要学习何时使用技术，何时手动计算；何时信任计算结果，何时保持怀疑。

Implications for Mathematical Research and Practice

对数学研究与实践的启示

Research Mathematics and Philosophical Assumptions

数学研究与哲学假设

The philosophy of mathematics influences not only education but also mathematical research and practice. Different philosophical positions suggest different approaches to mathematical investigation, different standards of mathematical rigor, and different conceptions of mathematical progress.
数学哲学不仅影响教育，还影响数学研究和实践。不同的哲学立场给出了不同的数学研究方法、不同的数学严谨性标准以及不同的数学进步观念。

Consider the ongoing debate about computer-assisted proofs. Some mathematicians are uncomfortable with proofs that depend on extensive computer verification, arguing that such proofs don’t provide genuine mathematical understanding. Others embrace computational methods as natural extensions of traditional mathematical reasoning.
以关于计算机辅助证明的持续争论为例。一些数学家对依赖大量计算机验证的证明感到不安，认为此类证明无法提供真正的数学理解。另一些人则将计算方法视为传统数学推理的自然延伸。

These disagreements reflect deeper philosophical differences about the nature of mathematical proof and knowledge. Those who emphasize the explanatory role of proof may be skeptical of computer-assisted arguments that verify results without providing insight into why they’re true. Those who emphasize the verificatory role of proof may be more willing to accept computational methods.
这些分歧反映了关于数学证明和知识本质的更深层次的哲学差异。那些强调证明的解释作用的人，可能会对仅验证结果而不提供其为何为真的洞见的计算机辅助论证持怀疑态度。那些强调证明的验证作用的人，可能更愿意接受计算方法。

Similarly, different attitudes toward non-constructive proof methods reflect different philosophical assumptions about mathematical existence and truth. Mathematicians who are comfortable with the axiom of choice and other non-constructive principles typically embrace a more Platonist conception of mathematical reality. Those who prefer constructive methods often have more constructivist philosophical leanings.
同样，对非构造性证明方法的不同态度，反映了关于数学存在和真理的不同哲学假设。那些接受选择公理和其他非构造性原则的数学家，通常持有更倾向于柏拉图主义的数学实在观。那些偏好构造性方法的数学家，往往具有更倾向于建构主义的哲学立场。

Mathematical Communication and Community

数学交流与社群

The philosophy of mathematics also influences how mathematicians communicate their results and participate in mathematical communities. Different philosophical assumptions suggest different approaches to mathematical exposition, different standards of rigor, and different conceptions of mathematical authority.
数学哲学还影响数学家如何交流他们的成果以及参与数学社群。不同的哲学假设给出了不同的数学阐述方法、不同的严谨性标准以及不同的数学权威观念。

The sociology of mathematical knowledge suggests that mathematical communities develop local standards and practices that may vary across different mathematical subfields and cultural contexts. What counts as an acceptable proof in one area of mathematics might not be acceptable in another area. This raises interesting questions about mathematical objectivity and universality. If mathematical knowledge is objective and universal, why do different mathematical communities sometimes develop different standards and practices? How do we reconcile the apparent objectivity of mathematical results with the evident subjectivity of mathematical judgment?
数学知识社会学表明，数学社群会发展出局部标准和实践，这些标准和实践在不同的数学分支和文化背景中可能有所不同。在数学的一个领域被视为可接受的证明，在另一个领域可能不被接受。这引发了关于数学客观性和普遍性的有趣问题。如果数学知识是客观且普遍的，为什么不同的数学社群有时会发展出不同的标准和实践？我们如何调和数学结果表面上的客观性与数学判断明显的主观性？

I think the answer involves recognizing that mathematical objectivity operates at a different level than mathematical practice. Mathematical results may be objective even if the processes by which they’re discovered, evaluated, and communicated involve subjective and social elements.
我认为答案在于认识到，数学客观性与数学实践运作于不同的层面。即使数学结果的发现、评估和交流过程涉及主观和社会因素，数学结果本身仍可能是客观的。

The Future of Mathematical Philosophy

数学哲学的未来

Where is the philosophy of mathematics headed? I see several promising directions for future research. First, there’s growing interest in empirical approaches to mathematical philosophy that draw on cognitive science, developmental psychology, and the sociology of knowledge. These approaches promise to ground philosophical theorizing in empirical evidence about how mathematical thinking actually works.
数学哲学的发展方向是什么？我认为未来的研究有几个充满希望的方向。首先，人们对借鉴认知科学、发展心理学和知识社会学的数学哲学经验研究方法的兴趣日益浓厚。这些方法有望将哲学理论建构建立在关于数学思维实际如何运作的经验证据之上。

Second, there’s increased attention to the diversity of mathematical practices across different cultures and historical periods. This work challenges universalist assumptions about mathematical knowledge and suggests more pluralistic approaches to mathematical philosophy.
其次，人们越来越关注不同文化和历史时期数学实践的多样性。这项工作挑战了关于数学知识的普遍主义假设，并提出了更具多元主义的数学哲学方法。

Third, there’s growing recognition that philosophy of mathematics needs to engage seriously with mathematical practice as it actually occurs. Rather than focusing exclusively on foundational questions, philosophers are increasingly studying how mathematics actually works in research, education, and applications.
第三，人们日益认识到，数学哲学需要认真关注实际发生的数学实践。哲学家们不再仅仅专注于基础问题，而是越来越多地研究数学在研究、教育和应用中的实际运作方式。

Finally, there’s renewed interest in the relationship between mathematics and other areas of human knowledge and culture. How does mathematical thinking relate to artistic creativity, moral reasoning, and political organization? These interdisciplinary questions promise to enrich our understanding of mathematics as a human activity.
最后，人们重新关注数学与人类知识和文化其他领域之间的关系。数学思维与艺术创造力、道德推理和政治组织之间有何关系？这些跨学科问题有望丰富我们对数学作为人类活动的理解。

Personal Reflections and Conclusions

个人反思与结论

Writing this paper has reinforced my conviction that the philosophy of mathematics is not just an academic exercise but a vital inquiry into the nature of human knowledge and understanding. The questions that philosophers of mathematics grapple with-about existence, truth, knowledge, and meaning-are among the most fundamental questions we can ask about our intellectual lives.
撰写本文强化了我的信念：数学哲学不仅仅是一项学术练习，更是对人类知识和理解本质的重要探究。数学哲学家们所探讨的问题——关于存在、真理、知识和意义的问题——是我们对自身智识生活所能提出的最根本问题之一。

These questions also have practical importance. As I’ve tried to show throughout this paper, philosophical assumptions about mathematical knowledge directly influence mathematical education, research practice, and applications. We can’t avoid philosophical questions by focusing exclusively on technical mathematical work-our philosophical assumptions shape our mathematical practice whether we acknowledge them or not.
这些问题也具有实际重要性。正如我在本文中试图表明的，关于数学知识的哲学假设直接影响数学教育、研究实践和应用。我们无法通过仅仅专注于技术性的数学工作来回避哲学问题——无论我们是否承认，我们的哲学假设都在塑造我们的数学实践。

At the same time, I’ve come to appreciate the complexity and difficulty of these philosophical questions. After surveying major philosophical positions and their arguments, I’m struck by how each position captures important insights while facing serious challenges. Platonism preserves mathematical objectivity but struggles with epistemic problems. Formalism provides precision but may sacrifice meaning. Constructivism ensures mathematical existence through construction but restricts mathematical methods. Naturalism avoids metaphysical controversies but may undermine mathematical objectivity.
同时，我也开始欣赏这些哲学问题的复杂性和难度。在考察了主要的哲学立场及其论证之后，我深感震撼的是，每种立场都捕捉到了重要的洞见，同时又面临着严峻的挑战。柏拉图主义保留了数学客观性，但在认识论问题上陷入困境。形式主义提供了精确性，但可能牺牲了意义。建构主义通过构造确保了数学存在，但限制了数学方法。自然主义避免了形而上学争议，但可能削弱了数学客观性。

Rather than viewing these positions as mutually exclusive alternatives, I’m increasingly convinced that different approaches may be appropriate for different aspects of mathematical practice. The complexity and diversity of mathematical activity may resist reduction to any single philosophical framework.
我越来越相信，不应将这些立场视为相互排斥的选择，而应认为不同的方法可能适用于数学实践的不同方面。数学活动的复杂性和多样性可能无法被还原为任何单一的哲学框架。

Perhaps the most important lesson from studying the philosophy of mathematics is intellectual humility. The questions that philosophers and mathematicians have been grappling with for centuries remain genuinely open and difficult. This suggests that these questions address deep and important features of human knowledge and reality.
研究数学哲学带来的最重要启示或许是智识上的谦逊。哲学家和数学家们数百年來一直在探讨的问题，至今仍然是真正开放且困难的。这表明这些问题触及了人类知识和实在的深刻而重要的特征。

The philosophy of mathematics also reveals the extraordinary richness and complexity of mathematical thinking. Far from being a simple matter of rule-following or mechanical computation, mathematical activity involves creativity, insight, judgment, and understanding. Mathematical knowledge emerges through the complex interaction of individual cognition, social communication, and cultural development.
数学哲学还揭示了数学思维非凡的丰富性和复杂性。数学活动绝非简单的遵循规则或机械计算，而是涉及创造力、洞见、判断力和理解力。数学知识通过个体认知、社会交流和文化发展的复杂互动而产生。

As mathematics continues to evolve through new discoveries, technological developments, and applications, philosophical reflection on the nature of mathematical knowledge becomes increasingly important. The questions addressed in this paper-about existence, truth, knowledge, and practice-will remain central to understanding mathematics as a human activity.
随着数学通过新发现、技术发展和应用不断演进，对数学知识本质的哲学反思变得日益重要。本文所探讨的问题——关于存在、真理、知识和实践的问题——将继续是理解数学作为人类活动的核心。

The journey through mathematical philosophy has also deepened my appreciation for the collaborative nature of intellectual inquiry. Philosophers and mathematicians have worked together for centuries to develop increasingly sophisticated understanding of mathematical knowledge. This collaborative enterprise continues today as new generations of scholars contribute their insights to ongoing conversations.
数学哲学的探索之旅也加深了我对智识探究协作本质的理解。数百年來，哲学家和数学家们一直合作，以发展对数学知识日益复杂的理解。如今，这一协作事业仍在继续，新一代学者为持续的对话贡献着他们的洞见。

Looking ahead, I’m optimistic about the future of mathematical philosophy. The integration of philosophical analysis with empirical investigation of mathematical practice promises to yield insights that are both philosophically sophisticated and practically relevant. By maintaining connections between theoretical investigation and mathematical practice, the philosophy of mathematics can continue to contribute to our understanding of human knowledge and mathematical activity.
展望未来，我对数学哲学的前景持乐观态度。哲学分析与数学实践经验研究的结合，有望产生既具有哲学深度又具有实际意义的洞见。通过保持理论探究与数学实践之间的联系，数学哲学能够继续为我们理解人类知识和数学活动做出贡献。

In the end, the philosophy of mathematics reveals mathematics not as a collection of abstract truths or formal procedures, but as a remarkable human achievement that continues to shape our understanding of knowledge, reality, and ourselves. Whether we’re teaching calculus to freshmen, proving theorems in graduate school, or applying mathematics to solve practical problems, we’re participating in this extraordinary intellectual tradition that spans cultures and centuries.
最终，数学哲学揭示了数学并非抽象真理或形式程序的集合，而是一项非凡的人类成就，它持续塑造着我们对知识、实在和自身的理解。无论我们是在向大学新生教授微积分、在研究生院证明定理，还是应用数学解决实际问题，我们都是在参与这一跨越文化和世纪的非凡智识传统。

The questions that originally drew me to the philosophy of mathematics-How can numbers exist if we can’t touch them? Why does mathematics work so well in describing the world?-remain as fascinating and important as ever. While we may never have complete answers to these questions, the ongoing inquiry into their depths continues to enrich our understanding of mathematics and its place in human knowledge.
那些最初吸引我进入数学哲学领域的问题——若无法触摸数字，它们何以存在？为何数学能如此出色地描述世界？——仍然像以往一样迷人且重要。尽管我们可能永远无法对这些问题给出完整的答案，但对它们的深入探究将继续丰富我们对数学及其在人类知识中地位的理解。

This exploration has convinced me that every mathematician, whether primarily interested in research, teaching, or applications, can benefit from engaging with philosophical questions about mathematical knowledge. These questions illuminate assumptions that often remain implicit in mathematical practice, and considering them explicitly can enhance both our mathematical work and our understanding of what makes mathematical activity so distinctive and valuable.
这一探索使我确信，每位数学家——无论其主要兴趣在于研究、教学还是应用——都能从思考关于数学知识的哲学问题中获益。这些问题阐明了数学实践中常常隐含的假设，而明确思考这些假设能够提升我们的数学工作，以及我们对数学活动为何如此独特和有价值的理解。

As I continue my own mathematical journey, I’m grateful for the opportunity to engage with these profound questions about the nature of mathematical knowledge. The philosophy of mathematics doesn’t provide easy answers, but it does provide conceptual tools for thinking more carefully and systematically about the extraordinary phenomenon of mathematical understanding. In a world increasingly shaped by mathematical and technological developments, such reflection becomes not just intellectually interesting but practically essential.
在我继续自己的数学之旅时，我很感激有机会探讨这些关于数学知识本质的深刻问题。数学哲学并未提供简单的答案，但它确实提供了概念工具，使我们能够更仔细、更系统地思考数学理解这一非凡现象。在一个日益被数学和技术发展所塑造的世界中，这种反思不仅具有智识上的趣味性，而且具有实际的必要性。

References

参考文献

1. Aristotle. Metaphysics. Translated by W.D. Ross. In The Complete Works of Aristotle, edited by Jonathan Barnes. Princeton University Press, 1984.
   亚里士多德. 《形而上学》. W.D. 罗斯 译. 收录于《亚里士多德全集》，乔纳森·巴恩斯 编. 普林斯顿大学出版社, 1984.
2. Benacerraf, Paul. “Mathematical Truth.” Journal of Philosophy 70, no. 19 (1973): 661-679.
   保罗·贝纳塞拉夫. 《数学真理》. 《哲学杂志》第 70 卷第 19 期 (1973): 661-679.
3. Benacerraf, Paul, and Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings. 2nd ed. Cambridge University Press, 1983.
   保罗·贝纳塞拉夫、希拉里·普特南 编. 《数学哲学：精选读本》. 第 2 版. 剑桥大学出版社, 1983.
4. Berkeley, George. The Analyst: A Discourse Addressed to an Infidel Mathematician. 1734.
   乔治·贝克莱. 《分析者：致一位不信教数学家的谈话》. 1734.
5. Boaler, Jo. Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and Their Impact on Student Learning. Revised and Expanded Edition. Lawrence Erlbaum Associates, 2002.
   乔·博勒. 《体验学校数学：传统与改革教学方法及其对学生学习的影响》. 修订扩展版. 劳伦斯·埃尔鲍姆联合出版社, 2002.
6. Brouwer, L.E.J. “Intuitionism and Formalism.” Bulletin of the American Mathematical Society 20 (1912): 81-96.
   L.E.J. 布劳威尔. 《直觉主义与形式主义》. 《美国数学会通报》第 20 卷 (1912): 81-96.
7. Brown, James Robert. Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge, 1999.
   詹姆斯·罗伯特·布朗. 《数学哲学：证明与图像世界导论》. 罗德里奇出版社, 1999.
8. Cantor, Georg. Contributions to the Founding of the Theory of Transfinite Numbers. Translated by Philip E.B. Jourdain. Open Court, 1915.
   格奥尔格·康托尔. 《超穷数理论基础》. 菲利普·E.B. 若尔丹 译. 开放法院出版社, 1915.
9. Dedekind, Richard. Essays on the Theory of Numbers. Translated by Wooster Woodruff Beman. Open Court, 1901.
   理查德·戴德金. 《数论随笔》. 伍斯特·伍德拉夫·比曼 译. 开放法院出版社, 1901.
10. Dummett, Michael. Elements of Intuitionism. Oxford University Press, 1977.
    迈克尔·达米特. 《直觉主义原理》. 牛津大学出版社, 1977.
11. Euclid. The Elements. Translated by T.L. Heath. Dover Publications, 1956.
    欧几里得. 《几何原本》. T.L. 希思 译. 多佛出版社, 1956.
12. Feferman, Solomon. In the Light of Logic. Oxford University Press, 1998.
    所罗门·费弗曼. 《在逻辑的光辉下》. 牛津大学出版社, 1998.
13. Ferreirs, José. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser, 1999.
    何塞·费雷罗斯. 《思想的迷宫：集合论史及其在现代数学中的作用》. 伯克霍夫出版社, 1999.
14. Ferreirs, José, and Jeremy J. Gray, eds. The Architecture of Modern Mathematics: Essays in History and Philosophy. Oxford University Press, 2006.
    何塞·费雷罗斯、杰里米·J. 格雷 编. 《现代数学的架构：历史与哲学论文集》. 牛津大学出版社, 2006.
15. Field, Hartry. Science Without Numbers: A Defence of Nominalism. Princeton University Press, 1980.
    哈特里·菲尔德. 《没有数字的科学：唯名论辩护》. 普林斯顿大学出版社, 1980.
16. Fink, Karl. A Brief History of Mathematics. Open Court Publishing Company, 1900.
    卡尔·芬克. 《数学简史》. 开放法院出版公司, 1900.
17. Frege, Gottlob. The Foundations of Arithmetic. Translated by J.L. Austin. Northwestern University Press, 1980.
    戈特洛布·弗雷格. 《算术基础》. J.L. 奥斯汀 译. 西北大学出版社, 1980.
18. Gödel, Kurt. “What Is Cantor’s Continuum Problem?” American Mathematical Monthly 54 (1947): 515-525.
    库尔特·哥德尔. 《康托尔的连续统问题是什么？》. 《美国数学月刊》第 54 卷 (1947): 515-525.
19. Hacking, Ian. Why Is There Philosophy of Mathematics At All? Cambridge University Press, 2014.
    伊恩·哈金. 《为何会有数学哲学？》. 剑桥大学出版社, 2014.
20. Hersh, Reuben. What Is Mathematics, Really? Oxford University Press, 1997.
    鲁本·赫什. 《数学究竟是什么？》. 牛津大学出版社, 1997.
21. Hilbert, David. Foundations of Geometry. Translated by Leo Unger. Open Court, 1971.
    大卫·希尔伯特. 《几何基础》. 利奥·昂格尔 译. 开放法院出版社, 1971.
22. Kant, Immanuel. Critique of Pure Reason. Translated by Norman Kemp Smith. Macmillan, 1929.
    伊曼努尔·康德. 《纯粹理性批判》. 诺曼·坎普·史密斯 译. 麦克米伦出版社, 1929.
23. Kitcher, Philip. The Nature of Mathematical Knowledge. Oxford University Press, 1983.
    菲利普·基切尔. 《数学知识的本质》. 牛津大学出版社, 1983.
24. Krantz, Steven G. How to Teach Mathematics. 2nd ed. American Mathematical Society, 1999.
    史蒂文·G. 克兰茨. 《如何教授数学》. 第 2 版. 美国数学会, 1999.
25. Lakatos, Imre. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press, 1976.
    伊姆雷·拉卡托斯. 《证明与反驳：数学发现的逻辑》. 剑桥大学出版社, 1976.
26. Lakoff, George, and Rafael E. Núñez. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books, 2000.
    乔治·拉科夫、拉斐尔·E. 努涅斯. 《数学从何而来：具身心智如何创造数学》. 基础图书出版社, 2000.
27. Maddy, Penelope. Realism in Mathematics. Oxford University Press, 1990.
    彭妮洛普·马迪. 《数学实在论》. 牛津大学出版社, 1990.
28. Parsons, Charles. Mathematical Thought and Its Objects. Cambridge University Press, 2008.
    查尔斯·帕森斯. 《数学思维及其对象》. 剑桥大学出版社, 2008.
29. Plato. Republic. Translated by Benjamin Jowett. In The Collected Dialogues of Plato, edited by Edith Hamilton and Huntington Cairns. Princeton University Press, 1961.
    柏拉图. 《理想国》. 本杰明·乔伊特 译. 收录于《柏拉图对话集》，伊迪丝·汉密尔顿、亨廷顿·凯恩斯 编. 普林斯顿大学出版社, 1961.
30. Putnam, Hilary. Philosophy and Our Mental Life. In Philosophical Papers, Volume 2: Mind, Language and Reality. Cambridge University Press, 1975.
    希拉里·普特南. 《哲学与我们的心智生活》. 收录于《哲学论文集》第 2 卷：《心智、语言与实在》. 剑桥大学出版社, 1975.
31. Quine, W.V.O. “Two Dogmas of Empiricism.” Philosophical Review 60 (1951): 20-43.
    W.V.O. 奎因. 《经验主义的两个教条》. 《哲学评论》第 60 卷 (1951): 20-43.
32. Russell, Bertrand. The Principles of Mathematics. Cambridge University Press, 1903.
    伯特兰·罗素. 《数学原理》. 剑桥大学出版社, 1903.
33. Shapiro, Stewart. Philosophy of Mathematics: Structure and Ontology. Oxford University Press, 1997.
    斯图尔特·夏皮罗. 《数学哲学：结构与本体论》. 牛津大学出版社, 1997.
34. Steiner, Mark. The Applicability of Mathematics as a Philosophical Problem. Harvard University Press, 1998.
    马克·斯坦纳. 《数学的适用性作为一个哲学问题》. 哈佛大学出版社, 1998.
35. Tappenden, Jamie. “Extending Knowledge and ‘Fruitful Concepts’: Fregean Themes in the Philosophy of Mathematics.” Noûs 29 (1995): 427-467.
    杰米·塔彭登. 《扩展知识与“富有成果的概念”：数学哲学中的弗雷格主题》. 《努斯》第 29 卷 (1995): 427-467.
36. Tiles, Mary. Mathematics and the Image of Reason. Routledge, 1991.
    玛丽·泰尔斯. 《数学与理性形象》. 罗德里奇出版社, 1991.
37. Weyl, Hermann. Philosophy of Mathematics and Natural Science. Princeton University Press, 1949.
    赫尔曼·外尔. 《数学与自然科学哲学》. 普林斯顿大学出版社, 1949.
38. Wigner, Eugene P. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Communications on Pure and Applied Mathematics 13 (1960): 1-14.
    尤金·P. 维格纳. 《数学在自然科学中不合理的有效性》. 《纯粹与应用数学通讯》第 13 卷 (1960): 1-14.

---

另一篇

• 数学哲学 | 解开哲学与数学之间复杂的联系-优快云博客
  https://blog.youkuaiyun.com/u013669912/article/details/154802939

---

via：

• The Philosophy of Mathematics: A Journey Through Ancient Wisdom and Modern Debates
  Gurkamal Singh - 7923.pdf
  https://www.ijsat.org/papers/2025/3/7923.pdf