数学哲学 | 解开哲学与数学之间复杂的联系

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。

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The Relationship Between Philosophy and Mathematics

哲学与数学的关系

Exploring the Interplay: Unraveling the Complex Bond between Philosophy and Mathematics
探索互动：解开哲学与数学之间复杂的联系

January 13, 2024

Introduction

引言

Philosophy and mathematics have been intertwined disciplines throughout history. Both fields share a common quest for truth and seek to understand fundamental concepts and principles that govern our world. While they pursue different methods of inquiry, the relationship between philosophy and mathematics is symbiotic, with each discipline enriching and influencing the other. In this article, we will explore the historical and conceptual connections between philosophy and mathematics, highlighting their mutual influence and the ways in which they contribute to our understanding of reality.
哲学和数学在历史上一直是相互交织的学科。这两个领域共同追求真理，试图理解支配我们世界的那些基本概念和原则。尽管它们采用不同的研究方法，但哲学与数学之间的关系是共生的，每个学科都在丰富和影响着另一个学科。在本文中，我们将探讨哲学与数学之间的历史和概念联系，强调它们的相互影响以及它们对我们理解现实的贡献。

Historical Connections

历史联系

The historical connections between philosophy and mathematics can be traced back to ancient Greece, where prominent thinkers such as Pythagoras, Plato, and Aristotle made significant contributions to both disciplines. Pythagoras, for instance, is renowned for his mathematical theorem, the Pythagorean theorem, which asserts a geometrical relationship between the sides of a right-angled triangle. However, Pythagoras and his followers saw mathematics not only as a means to solve practical problems but also as a fundamental aspect of reality. They believed that numbers and mathematical concepts held a metaphysical significance, connecting the abstract realm of mathematics to the physical world.
哲学与数学的历史联系可以追溯到古希腊，当时的著名思想家如毕达哥拉斯、柏拉图和亚里士多德对这两个学科都做出了重要贡献。例如，毕达哥拉斯以其数学定理——毕达哥拉斯定理而闻名，该定理断言直角三角形的边之间存在一种几何关系。然而，毕达哥拉斯及其追随者不仅将数学视为解决实际问题的手段，还将其视为现实的一个基本方面。他们相信数字和数学概念具有形而上学的意义，将数学的抽象领域与物理世界联系起来。

Plato, another influential philosopher of ancient Greece, built upon Pythagorean ideas and developed the concept of mathematical idealism. He argued that mathematical entities exist independently of the physical world and are eternal, unchangeable forms. According to Plato, mathematical truths, such as the existence of perfect geometric shapes, reveal a higher reality that transcends the imperfections of the material world. This perspective laid the foundation for the philosophical exploration of mathematics as a means to understand the nature of reality.
柏拉图是古希腊另一位有影响力的哲学家，他继承了毕达哥拉斯的思想，发展了数学理想主义的概念。他认为数学实体独立于物理世界而存在，是永恒的、不可改变的形式。根据柏拉图的说法，数学真理，如完美几何形状的存在，揭示了一种超越物质世界不完美的更高现实。这种观点为将数学作为理解现实本质的手段进行哲学探索奠定了基础。

Conceptual Connections

概念联系

Beyond their historical connections, philosophy and mathematics share conceptual ties that continue to shape contemporary thought. Both disciplines seek to uncover universal truths and employ rigorous reasoning to arrive at logical conclusions. Mathematics, with its emphasis on logical deduction and proof, exemplifies the systematic and precise thinking that philosophy often aspires to achieve. Philosophy, on the other hand, provides the theoretical framework and conceptual tools necessary to interpret and understand the implications of mathematical discoveries.
除了历史联系之外，哲学和数学还存在着概念联系，这种联系继续塑造着当代思想。这两个学科都试图揭示普遍真理，并运用严格的推理来得出逻辑结论。数学以其对逻辑推理和证明的强调，体现了哲学通常努力实现的系统性和精确性思维。另一方面，哲学提供了理论框架和概念工具，以解释和理解数学发现的含义。

The philosophy of mathematics delves into foundational questions about the nature and existence of mathematical objects and the validity of mathematical reasoning. One of the central debates in this field is the conflict between Platonism and nominalism. Platonists argue that mathematical objects exist objectively and independently of human thought, while nominalists assert that mathematics is a human construct and mathematical objects only have existence within our minds. This debate illustrates the philosophical impact of mathematics, as it raises questions about the nature of abstract entities and the relationship between mathematical truths and the external world.
数学哲学深入探讨了关于数学对象的性质和存在以及数学推理的有效性的基础性问题。这个领域的一个核心争论是柏拉图主义和唯名论之间的冲突。柏拉图主义者认为数学对象客观存在且独立于人类思想，而唯名论者则认为数学是人类的构造，数学对象仅存在于我们的思维中。这场争论体现了数学的哲学影响，因为它提出了关于抽象实体的性质以及数学真理与外部世界之间关系的问题。

Mutual Influence

相互影响

The relationship between philosophy and mathematics is not merely one of theoretical abstraction but also of practical interaction and mutual influence. Mathematics provides a powerful tool for philosophers to quantify and formalize their arguments. By using mathematical models, philosophers can analyze complex structures and systems, discern patterns, and evaluate the logical consistency of their ideas. Mathematical logic, in particular, has played an essential role in the development of formal logic, which provides a rigorous method for analyzing and evaluating philosophical arguments.
哲学与数学之间的关系不仅仅是理论上的抽象，还涉及实践中的互动和相互影响。数学为哲学家提供了一个强大的工具，用于量化和形式化他们的论点。通过使用数学模型，哲学家可以分析复杂的结构和系统，识别模式，并评估他们观点的逻辑一致性。特别是数理逻辑，在形式逻辑的发展中发挥了重要作用，形式逻辑为分析和评估哲学论点提供了一种严格的方法。

Conversely, philosophy offers important insights to mathematics by addressing foundational questions and conceptual issues that emerge from mathematical inquiry. The philosophy of mathematics explores the nature of mathematical knowledge, the role of mathematical intuition, and the limits and possibilities of mathematical proof. By engaging with philosophical inquiries, mathematicians gain a deeper understanding of the assumptions and implications of their work, which can lead to new directions and discoveries within mathematics itself.
反过来，哲学通过解决数学研究中出现的基础性问题和概念性问题，为数学提供了重要的见解。数学哲学探讨了数学知识的性质、数学直觉的作用以及数学证明的局限性和可能性。通过参与哲学探究，数学家对自己的工作假设和含义有了更深入的理解，这可能会导致数学本身的新的方向和发现。

Conclusion

结论

The relationship between philosophy and mathematics is a dynamic and symbiotic one. The historical connections between the two disciplines have laid the groundwork for understanding their conceptual interplay. Philosophy contributes to mathematics by providing a platform for questioning fundamental assumptions and exploring the philosophical implications of mathematical discoveries. Mathematics, in turn, enhances philosophy by offering logical tools and quantitative models to analyze and substantiate philosophical arguments. As these two disciplines continue to evolve, their mutual influence will continue to shape our understanding of reality and our approach to unraveling its mysteries.
哲学与数学之间的关系是动态的、共生的。这两个学科之间的历史联系为理解它们的概念互动奠定了基础。哲学通过提供一个质疑基本假设和探索数学发现的哲学含义的平台，为数学做出了贡献。反过来，数学通过提供逻辑工具和定量模型来分析和证实哲学论点，增强了哲学。随着这两个学科的不断发展，它们的相互影响将继续塑造我们对现实的理解以及我们揭开其奥秘的方法。

References

参考文献

1. Mancosu, P., (1996). Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford University Press.
   曼科苏，P.（1996）。《十七世纪的数学哲学与数学实践》。牛津大学出版社。

2. Resnik, M., (1997). Mathematics as a Science of Patterns. Oxford University Press.
   雷斯尼克，M.（1997）。《数学作为一门模式科学》。牛津大学出版社。

3. Shapiro, S., (2000). Thinking about Mathematics: The Philosophy of Mathematics. Oxford University Press.
   沙皮罗，S.（2000）。《思考数学：数学哲学》。牛津大学出版社。

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Philosophy of Mathematics

数学哲学

First published Tue Sep 25, 2007; substantive revision Tue Jan 25, 2022

首次发表于 2007 年 9 月 25 日；2022 年 1 月 25 日进行了实质性修订

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case for the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way: by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics.
如果将数学视为一门科学，那么数学哲学就可以被视为科学哲学的一个分支，与物理学哲学和生物学哲学等学科并列。然而，由于其研究对象的特殊性，数学哲学在科学哲学中占据了特殊的地位。虽然自然科学研究的是位于时空中的实体，但数学研究的对象是否也位于时空之中却并不明显。此外，数学的研究方法与自然科学的研究方法也大不相同。自然科学通过归纳方法获取一般性知识，而数学知识似乎是以不同的方式获得的：从基本原理中进行演绎。数学知识的地位似乎也与自然科学知识的地位不同。自然科学的理论似乎比数学理论更不确定，也更易于修订。由于这些原因，数学为哲学提出了相当独特的难题。因此，哲学家特别关注与数学相关的本体论和认识论问题。

1. Philosophy of Mathematics, Logic, and the Foundations of Mathematics

数学哲学、逻辑学以及数学基础

On the one hand, philosophy of mathematics is concerned with problems that are closely related to central problems of metaphysics and epistemology. At first blush, mathematics appears to study abstract entities. This makes one wonder what the nature of mathematical entities consists in and how we can have knowledge of mathematical entities. If these problems are regarded as intractable, then one might try to see if mathematical objects can somehow belong to the concrete world after all.
一方面，数学哲学关注与形而上学和认识论的核心问题密切相关的问题。乍一看，数学似乎研究的是抽象实体。这使人不禁思考数学实体的本质是什么，以及我们如何能够获得关于数学实体的知识。如果这些问题被视为难以解决，那么人们可能会尝试看看数学对象是否最终可以属于现实世界。

On the other hand, it has turned out that to some extent it is possible to bring mathematical methods to bear on philosophical questions concerning mathematics. The setting in which this has been done is that of mathematical logic when it is broadly conceived as comprising proof theory, model theory, set theory, and computability theory as subfields. Thus the twentieth century has witnessed the mathematical investigation of the consequences of what are at bottom philosophical theories concerning the nature of mathematics.
另一方面，事实证明，在一定程度上可以将数学方法应用于关于数学的哲学问题。这种应用的背景是广义的 数理逻辑，它包括证明论、模型论、集合论和可计算性理论等子领域。因此，20 世纪见证了对数学本质的哲学理论的数学后果的研究。

When professional mathematicians are concerned with the foundations of their subject, they are said to be engaged in foundational research. When professional philosophers investigate philosophical questions concerning mathematics, they are said to contribute to the philosophy of mathematics. Of course the distinction between the philosophy of mathematics and the foundations of mathematics is vague, and the more interaction there is between philosophers and mathematicians working on questions pertaining to the nature of mathematics, the better.
当专业数学家关注他们学科的基础时，他们被认为是在进行基础研究。当专业哲学家研究关于数学的哲学问题时，他们被认为是在为数学哲学做出贡献。当然，数学哲学与数学基础之间的区别是模糊的，哲学家和数学家之间在关于数学本质的问题上互动越多越好。

2. Four schools

四种学派

The general philosophical and scientific outlook in the nineteenth century tended toward the empirical: platonistic aspects of rationalistic theories of mathematics were rapidly losing support. Especially the once highly praised faculty of rational intuition of ideas was regarded with suspicion. Thus it became a challenge to formulate a philosophical theory of mathematics that was free of platonistic elements. In the first decades of the twentieth century, three non-platonistic accounts of mathematics were developed: logicism, formalism, and intuitionism. There emerged in the beginning of the twentieth century also a fourth program: predicativism. Due to contingent historical circumstances, its true potential was not brought out until the 1960s. However it deserves a place beside the three traditional schools that are discussed in most standard contemporary introductions to philosophy of mathematics, such as (Shapiro 2000) and (Linnebo 2017).
19 世纪的哲学和科学总体倾向于经验论：理性主义数学理论中的柏拉图主义元素迅速失去了支持。尤其是曾经备受赞誉的理性直觉能力受到了怀疑。因此，制定一个没有柏拉图主义元素的数学哲学理论成为一个挑战。在 20 世纪的最初几十年里，发展了三种非柏拉图主义的数学理论：逻辑主义、形式主义和直觉主义。20 世纪初还出现了第四个项目：预设主义。由于偶然的历史因素，其真正的潜力直到 20 世纪 60 年代才得以展现。然而，它值得与在大多数标准的当代数学哲学导论中讨论的三个传统学派并列，例如（Shapiro 2000）和（Linnebo 2017）。

2.1 Logicism

逻辑主义

The logicist project consists in attempting to reduce mathematics to logic. Since logic is supposed to be neutral about matters ontological, this project seemed to harmonize with the anti-platonistic atmosphere of the time.
逻辑主义项目旨在将数学归结为逻辑。由于逻辑被认为在本体论问题上是中立的，因此该项目似乎与当时的反柏拉图主义氛围相契合。

The idea that mathematics is logic in disguise goes back to Leibniz. But an earnest attempt to carry out the logicist program in detail could be made only when in the nineteenth century the basic principles of central mathematical theories were articulated (by Dedekind and Peano) and the principles of logic were uncovered (by Frege).
数学是伪装起来的逻辑这一观点可以追溯到莱布尼茨。然而，只有在 19 世纪，当中心数学理论的基本原理被阐述（由戴德金和皮亚诺完成）并且逻辑原理被揭示（由弗雷格完成）时，才能认真地详细实施逻辑主义项目。

Frege devoted much of his career to trying to show how mathematics can be reduced to logic (Frege 1884). He managed to derive the principles of (second-order) Peano arithmetic from the basic laws of a system of second-order logic. His derivation was flawless. However, he relied on one principle which turned out not to be a logical principle after all. Even worse, it is untenable. The principle in question is Frege’s Basic Law V:
弗雷格在其职业生涯中花费了大量时间试图证明数学如何可以归结为逻辑（Frege 1884）。他成功地从一个二阶逻辑系统的基本法则中推导出了（二阶）皮亚诺算术的原理。他的推导是无懈可击的。然而，他依赖了一个最终被证明并非逻辑原则的原则。更糟糕的是，这个原则是不可接受的。这个原则就是弗雷格的 基本法则 V：

{ x ∣ F x } = { x ∣ G x }  if and only if  ∀ x ( F x ≡ G x ) \{x|Fx\}=\{x|Gx\} \text{ if and only if } \forall x(Fx \equiv Gx) {x∣Fx}={x∣Gx} if and only if ∀x(Fx≡Gx)

In words: the set of the Fs is identical with the set of the Gs iff the Fs are precisely the Gs.
用文字来说：F 的集合与 G 的集合相同，当且仅当 F 的对象恰好是 G 的对象。

In a famous letter to Frege, Russell showed that Frege’s Basic Law V entails a contradiction (Russell 1902). This argument has come to be known as Russell’s paradox (see section 2.4).
在一封写给弗雷格的著名信件中，罗素证明了弗雷格的基本法则 V会导致一个矛盾（Russell 1902）。这个论证被称为罗素悖论（见第 2.4 节）。

Russell himself then tried to reduce mathematics to logic in another way. Frege’s Basic Law V entails that corresponding to every property of mathematical entities, there exists a class of mathematical entities having that property. This was evidently too strong, for it was exactly this consequence which led to Russell’s paradox. So Russell postulated that only properties of mathematical objects that have already been shown to exist, determine classes. Predicates that implicitly refer to the class that they were to determine if such a class existed, do not determine a class. Thus a typed structure of properties is obtained: properties of ground objects, properties of ground objects and classes of ground objects, and so on. This typed structure of properties determines a layered universe of mathematical objects, starting from ground objects, proceeding to classes of ground objects, then to classes of ground objects and classes of ground objects, and so on.
罗素本人随后尝试以另一种方式将数学归结为逻辑。弗雷格的基本法则 V意味着，对于每一个数学实体的属性，都存在一个具有该属性的数学实体的类。这显然过于强大，因为它正是导致罗素悖论的后果。因此，罗素提出，只有那些已经被证明存在的数学对象的属性，才能决定一个类。那些隐含地指涉如果存在这样的类就会决定该类的谓词，并不决定一个类。因此，得到了一个分层的属性结构：底层对象的属性，底层对象和底层对象类的属性，等等。这种分层的属性结构决定了一个分层的数学对象宇宙，从底层对象开始，进而到底层对象的类，然后是底层对象和底层对象类的类，如此等等。

Unfortunately, Russell found that the principles of his typed logic did not suffice for deducing even the basic laws of arithmetic. He needed, among other things, to lay down as a basic principle that there exists an infinite collection of ground objects. This could hardly be regarded as a logical principle. Thus the second attempt to reduce mathematics to logic also faltered.
不幸的是，罗素发现他的分层逻辑的原理甚至不足以推导出算术的基本法则。他需要，除了其他事情外，将存在一个无限的底层对象集合作为一个基本原理。这很难被视为一个逻辑原则。因此，第二次将数学归结为逻辑的尝试也失败了。

And there matters stood for more than fifty years. In 1983, Crispin Wright’s book on Frege’s theory of the natural numbers appeared (Wright 1983). In it, Wright breathes new life into the logicist project. He observes that Frege’s derivation of second-order Peano Arithmetic can be broken down in two stages. In a first stage, Frege uses the inconsistent Basic Law V to derive what has come to be known as Hume’s Principle:
这种情况持续了五十多年。1983 年，克里普辛·赖特关于弗雷格自然数理论的书出版了（Wright 1983）。在书中，赖特为逻辑主义项目注入了新的活力。他指出，弗雷格对二阶皮亚诺算术的推导可以分为两个阶段。在第一阶段，弗雷格使用不一致的基本法则 V推导出了后来被称为休谟原理的内容：

The number of the $F$s = the number of the $G$s if and only if $F\approx G$;
$F$ 的数量 = $G$ 的数量，当且仅当 $F\approx G$；

where $F\approx G$ means that the $F$s and the $G$s stand in one-to-one correspondence with each other. (This relation of one-to-one correspondence can be expressed in second-order logic.) Then, in a second stage, the principles of second-order Peano Arithmetic are derived from Hume’s Principle and the accepted principles of second-order logic. In particular, Basic Law V is not needed in the second part of the derivation. Moreover, Wright conjectured that in contrast to Frege’s Basic Law V, Hume’s Principle is consistent. George Boolos and others observed that Hume’s Principle is indeed consistent (Boolos 1987).
其中 $F\approx G$ 表示 $F$ 和 $G$ 之间存在一一对应关系。（这种一一对应关系可以用二阶逻辑来表达。）然后，在第二阶段，从休谟原则和已接受的二阶逻辑原理中推导出二阶皮亚诺算术的原则。特别是在推导的第二部分中，并不需要基本法则 V。此外，赖特推测，休谟原则与弗雷格的基本法则 V 不同，且是一致的。乔治·布尔和其他人观察到，休谟原则确实是一致的（Boolos 1987）。

Wright went on to claim that Hume’s Principle can be regarded as a truth of logic. If that is so, then at least second-order Peano arithmetic is reducible to logic alone. Thus a new form of logicism was born; today this view is known as neo-logicism (Hale & Wright 2001). Most philosophers of mathematics today doubt that Hume’s Principle is a principle of logic. Indeed, even Wright later sought to qualify this claim. Nonetheless, many philosophers of mathematics feel that the introduction of natural numbers through Hume’s Principle is attractive from an ontological and from an epistemological point of view. Linnebo argues that because the left-hand-side of Hume’s Principle merely re-carves the content of its right-hand-side, not much is needed from the world to make Hume’s Principle true. For this reason, he calls natural numbers and mathematical objects that can be introduced in a similar way light mathematical objects (Linnebo 2018).
赖特进而声称，休谟原理可以被视为逻辑的真理。如果是这样，那么至少二阶皮亚诺算术可以单独归结为逻辑。因此，一种新的逻辑主义形式诞生了；如今这种观点被称为新逻辑主义（Hale & Wright 2001）。如今，大多数数学哲学家怀疑休谟原理是否是逻辑的原则。事实上，赖特后来也试图对这一主张进行限定。尽管如此，许多数学哲学家认为，通过休谟原理引入自然数在本体论和认识论上都是有吸引力的。林内博认为，由于休谟原理的左侧只是对右侧内容的重新刻画，因此不需要太多来自世界的东西就能使休谟原理为真。因此，他将自然数以及可以通过类似方式引入的数学对象称为轻量级数学对象（Linnebo 2018）。

Wright’s work has drawn the attention of philosophers of mathematics to the kind of principles of which Basic Law V and Hume’s Principle are examples. These principles are called abstraction principles. At present, philosophers of mathematics attempt to construct general theories of abstraction principles that explain which abstraction principles are acceptable and which are not, and why (Weir 2003; Fine 2002). Also, it has emerged that in the context of weakened versions of second-order logic, Frege’s Basic Law V is consistent. But these weak background theories only allow very weak arithmetical theories to be derived from Basic Law V (Burgess 2005).
赖特的工作吸引了数学哲学家对基本法则 V和休谟原理这类原则的关注。这些原则被称为抽象原则。目前，数学哲学家试图构建关于抽象原则的一般理论，以解释哪些抽象原则是可以接受的，哪些是不可接受的，以及原因何在（Weir 2003；Fine 2002）。此外，已经发现，在弱化的二阶逻辑的背景下，弗雷格的基本法则 V是一致的。但这些弱背景理论只允许从基本法则 V中推导出非常弱的算术理论（Burgess 2005）。

2.2 Intuitionism

直觉主义

Intuitionism originates in the work of the mathematician L.E.J. Brouwer (van Atten 2004), and it is inspired by Kantian views of what objects are (Parsons 2008, chapter 1). According to intuitionism, mathematics is essentially an activity of construction. The natural numbers are mental constructions, the real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction… Mathematical constructions are produced by the ideal mathematician, i.e., abstraction is made from contingent, physical limitations of the real life mathematician. But even the ideal mathematician remains a finite being. She can never complete an infinite construction, even though she can complete arbitrarily large finite initial parts of it. This entails that intuitionism resolutely rejects the existence of the actual (or completed) infinite; only potentially infinite collections are given in the activity of construction. A basic example is the successive construction in time of the individual natural numbers.
直觉主义起源于数学家 L.E.J. 布劳威尔（van Atten 2004）的工作，并受到康德关于对象是什么的观点的启发（Parsons 2008，第 1 章）。根据直觉主义，数学本质上是一种构造活动。自然数是精神构造，实数是精神构造，证明和定理是精神构造，数学意义是精神构造……数学构造是由 理想 数学家产生的，即从现实生活中数学家的偶然的、物理的限制中进行抽象。但即使是理想数学家仍然是一个有限的存在。她永远无法完成一个无限的构造，尽管她可以完成其任意大的有限初始部分。这意味着直觉主义坚决拒绝实际（或已完成的）无限的存在；只有潜在的无限集合在构造活动中被给出。一个基本的例子是自然数的连续构造。

From these general considerations about the nature of mathematics, based on the condition of the human mind (Moore 2001), intuitionists infer to a revisionist stance in logic and mathematics. They find non-constructive existence proofs unacceptable. Non-constructive existence proofs are proofs that purport to demonstrate the existence of a mathematical entity having a certain property without even implicitly containing a method for generating an example of such an entity. Intuitionism rejects non-constructive existence proofs as ‘theological’ and ‘metaphysical’. The characteristic feature of non-constructive existence proofs is that they make essential use of the principle of excluded third
在这些关于数学本质的一般性思考中，基于人类心智的条件（Moore 2001），直觉主义者得出了在逻辑和数学中的修正主义立场。他们认为非构造性存在证明是不可接受的。非构造性存在证明是声称证明具有某种属性的数学实体的存在，而甚至没有隐含地包含生成这样一个实体的实例的方法。直觉主义将非构造性存在证明视为“神学的”和“形而上学的”。非构造性存在证明的特征在于，它们本质上使用了排中律

$$\varphi \vee \neg \varphi$$

or one of its equivalents, such as the principle of double negation
或者其等价形式之一，例如双重否定律

$$\neg \neg \varphi \to \varphi$$

In classical logic, these principles are valid. The logic of intuitionistic mathematics is obtained by removing the principle of excluded third (and its equivalents) from classical logic. This of course leads to a revision of mathematical knowledge. For instance, the classical theory of elementary arithmetic, Peano Arithmetic, can no longer be accepted. Instead, an intuitionistic theory of arithmetic (called Heyting Arithmetic) is proposed which does not contain the principle of excluded third. Although intuitionistic elementary arithmetic is weaker than classical elementary arithmetic, the difference is not all that great. There exists a simple syntactical translation which translates all classical theorems of arithmetic into theorems which are intuitionistically provable.
在经典逻辑中，这些原则是有效的。直觉主义数学的逻辑是通过从经典逻辑中移除排中律（及其等价形式）获得的。这当然导致了数学知识的修订。例如，经典的小学算术理论，即皮亚诺算术，不再被接受。相反，提出了一种直觉主义的算术理论（称为海廷算术），其中不包含排中律。尽管直觉主义小学算术比经典小学算术弱，但差别并不大。存在一种简单的语法转换，可以将所有经典算术定理转换为直觉主义可证明的定理。

In the first decades of the twentieth century, parts of the mathematical community were sympathetic to the intuitionistic critique of classical mathematics and to the alternative that it proposed. This situation changed when it became clear that in higher mathematics, the intuitionistic alternative differs rather drastically from the classical theory. For instance, intuitionistic mathematical analysis is a fairly complicated theory, and it is very different from classical mathematical analysis. This dampened the enthusiasm of the mathematical community for the intuitionistic project. Nevertheless, followers of Brouwer have continued to develop intuitionistic mathematics onto the present day (Troelstra & van Dalen 1988).
在 20 世纪的最初几十年里，数学界的一部分人对直觉主义对经典数学的批判以及它提出的替代方案表示同情。当人们清楚地看到，在高等数学中，直觉主义的替代方案与经典理论有很大的不同，这种情况就发生了变化。例如，直觉主义数学分析是一种相当复杂的理论，它与经典数学分析有很大的不同。这削弱了数学界对直觉主义项目的热情。然而，布劳威尔的追随者一直将直觉主义数学发展到今天（Troelstra & van Dalen 1988）。

2.3 Formalism

形式主义

David Hilbert agreed with the intuitionists that there is a sense in which the natural numbers are basic in mathematics. But unlike the intuitionists, Hilbert did not take the natural numbers to be mental constructions. Instead, he argued that the natural numbers can be taken to be symbols. Symbols are strictly speaking abstract objects. Nonetheless, it is essential to symbols that they can be embodied by concrete objects, so we may call them quasi-concrete objects (Parsons 2008, chapter 1). Perhaps physical entities could play the role of the natural numbers. For instance, we may take a concrete ink trace of the form | to be the number 0, a concretely realized ink trace || to be the number 1, and so on. Hilbert thought it doubtful at best that higher mathematics could be directly interpreted in a similarly straightforward and perhaps even concrete manner.
大卫·希尔伯特同意直觉主义者关于自然数在数学中具有某种基本性的观点。但与直觉主义者不同，希尔伯特并不认为自然数是精神构造。相反，他认为自然数可以被视为 符号。符号严格来说是抽象对象。然而，符号能够被具体对象体现是其本质属性，因此我们可以称它们为 准具体 对象（Parsons 2008，第 1 章）。也许物理实体可以充当自然数的角色。例如，我们可以将一个具体形式为 | 的墨迹视为数字 0，将一个具体实现的墨迹 || 视为数字 1，依此类推。希尔伯特认为，高等数学能否以同样直接且或许甚至是具体的方式进行解释，最多是值得怀疑的。

Unlike the intuitionists, Hilbert was not prepared to take a revisionist stance toward the existing body of mathematical knowledge. Instead, he adopted an instrumentalist stance with respect to higher mathematics. He thought that higher mathematics is no more than a formal game. The statements of higher-order mathematics are uninterpreted strings of symbols. Proving such statements is no more than a game in which symbols are manipulated according to fixed rules. The point of the ‘game of higher mathematics’ consists, in Hilbert’s view, in proving statements of elementary arithmetic, which do have a direct interpretation (Hilbert 1925).
与直觉主义者不同，希尔伯特并不准备对现有的数学知识体系采取修正主义的立场。相反，他对于高等数学采取了一种工具主义的立场。他认为高等数学不过是一种形式化的游戏。高阶数学的陈述是未经解释的符号字符串。证明这样的陈述不过是在固定规则下操纵符号的游戏。在希尔伯特看来，“高等数学的游戏”的目的在于证明初等算术的陈述，而初等算术是有直接解释的（Hilbert 1925）。

Hilbert thought that there can be no reasonable doubt about the soundness of classical Peano Arithmetic — or at least about the soundness of a subsystem of it that is called Primitive Recursive Arithmetic (Tait 1981). And he thought that every arithmetical statement that can be proved by making a detour through higher mathematics, can also be proved directly in Peano Arithmetic. In fact, he strongly suspected that every problem of elementary arithmetic can be decided from the axioms of Peano Arithmetic. Of course solving arithmetical problems in arithmetic is in some cases practically impossible. The history of mathematics has shown that making a “detour” through higher mathematics can sometimes lead to a proof of an arithmetical statement that is much shorter and that provides more insight than any purely arithmetical proof of the same statement.
希尔伯特认为，对于经典皮亚诺算术的可靠性——或者至少对于其一个被称为原始递归算术的子系统的可靠性（Tait 1981）——不应存在合理的怀疑。而且，他认为，任何可以通过高等数学的迂回证明的算术命题，也可以直接在皮亚诺算术中证明。实际上，他强烈怀疑每一个初等算术问题都可以从皮亚诺算术的公理中得出结论。当然，在某些情况下，用算术解决算术问题是实际上不可能的。数学的历史表明，通过高等数学的“迂回”有时可以导致一个算术命题的证明，这个证明比任何纯粹的算术证明都要简短，并且能提供更多的洞察。

Hilbert realized, albeit somewhat dimly, that some of his convictions can actually be considered to be mathematical conjectures. For a proof in a formal system of higher mathematics or of elementary arithmetic is a finite combinatorial object which can, modulo coding, be considered to be a natural number. But in the 1920s the details of coding proofs as natural numbers were not yet completely understood.
希尔伯特意识到，尽管有些模糊，他的一些信念实际上可以被视为数学猜想。因为高等数学的形式系统或初等算术中的一个证明是一个有限的组合对象，可以通过编码被视作一个自然数。但在 20 世纪 20 年代，将证明编码为自然数的细节尚未完全被理解。

On the formalist view, a minimal requirement of formal systems of higher mathematics is that they are at least consistent. Otherwise every statement of elementary arithmetic can be proved in them. Hilbert also saw (again, dimly) that the consistency of a system of higher mathematics entails that this system is at least partially arithmetically sound. So Hilbert and his students set out to prove statements such as the consistency of the standard postulates of mathematical analysis. Of course such statements would have to be proved in a ‘safe’ part of mathematics, such as elementary arithmetic. Otherwise the proof does not increase our conviction in the consistency of mathematical analysis. And, fortunately, it seemed possible in principle to do this, for in the final analysis consistency statements are, again modulo coding, arithmetical statements. So, to be precise, Hilbert and his students set out to prove the consistency of, e.g., the axioms of mathematical analysis in classical Peano arithmetic. This project was known as Hilbert’s program (Zach 2006). It turned out to be more difficult than they had expected. In fact, they did not even succeed in proving the consistency of the axioms of Peano Arithmetic in Peano Arithmetic.
在形式主义观点中，高等数学形式系统的一个最低要求是它们至少是一致的。否则，每一个初等算术的命题都可以在其中被证明。希尔伯特也看到了（再次，有些模糊）一个高等数学系统的一致性意味着这个系统在算术上至少是部分可靠的。因此，希尔伯特和他的学生们着手证明诸如数学分析的标准公理的一致性这样的命题。当然，这样的命题必须在一个“安全”的数学领域，如初等算术中被证明。否则，这个证明并不能增加我们对数学分析一致性的信念。而且，幸运的是，从原则上讲，似乎有可能做到这一点，因为最终一致性陈述是，再次通过编码，算术陈述。因此，确切地说，希尔伯特和他的学生们着手在经典皮亚诺算术中证明，例如，数学分析的公理的一致性。这个项目被称为希尔伯特计划（Zach 2006）。事实证明，这比他们预期的要困难得多。实际上，他们甚至没有成功地在皮亚诺算术中证明皮亚诺算术公理的一致性。

Then Kurt Gödel proved that there exist arithmetical statements that are undecidable in Peano Arithmetic (Gödel 1931). This has become known as his Gödel’s first incompleteness theorem. This did not bode well for Hilbert’s program, but it left open the possibility that the consistency of higher mathematics is not one of these undecidable statements. Unfortunately, Gödel then quickly realized that, unless (God forbid!) Peano Arithmetic is inconsistent, the consistency of Peano Arithmetic is independent of Peano Arithmetic. This is Gödel’s second incompleteness theorem. Gödel’s incompleteness theorems turn out to be generally applicable to all sufficiently strong but consistent recursively axiomatizable theories. Together, they entail that Hilbert’s program fails. It turns out that higher mathematics cannot be interpreted in a purely instrumental way. Higher mathematics can prove arithmetical sentences, such as consistency statements, that are beyond the reach of Peano Arithmetic.
然后，库尔特·哥德尔证明了在皮亚诺算术中存在不可判定的算术命题（Gödel 1931）。这被称为他的哥德尔第一不完全性定理。这对希尔伯特计划来说并不是一个好兆头，但它仍然留下了高等数学的一致性不是这些不可判定命题之一的可能性。不幸的是，哥德尔随后很快意识到，除非（天佑吾等！）皮亚诺算术是不一致的，否则皮亚诺算术的一致性与皮亚诺算术无关。这是哥德尔的第二不完全性定理。哥德尔的不完全性定理结果普遍适用于所有足够强大但一致的递归公理化理论。它们共同导致了希尔伯特计划的失败。事实证明，高等数学不能以纯粹的工具主义方式被解释。高等数学可以证明像一致性陈述这样的算术句子，这些句子超出了皮亚诺算术的范围。

All this does not spell the end of formalism. Even in the face of the incompleteness theorems, it is coherent to maintain that mathematics is the science of formal systems.
所有这些并没有宣告形式主义的终结。即使面对不完全性定理，坚持认为数学是形式系统的科学仍然是合理的。

One version of this view was proposed by Curry (Curry 1958). On this view, mathematics consists of a collection of formal systems which have no interpretation or subject matter. (Curry here makes an exception for metamathematics.) Relative to a formal system, one can say that a statement is true if and only if it is derivable in the system. But on a fundamental level, all mathematical systems are on a par. There can be at most pragmatical reasons for preferring one system over another. Inconsistent systems can prove all statements and therefore are pretty useless. So when a system is found to be inconsistent, it must be modified. It is simply a lesson from Gödel’s incompleteness theorems that a sufficiently strong consistent system cannot prove its own consistency.
这种观点的一个版本是由卡里提出的（Curry 1958）。在这种观点下，数学由一系列没有解释或主题的形式系统组成。（卡里在这里为元数学做了一个例外。）相对于一个形式系统，可以说一个命题是真的当且仅当它可以在系统中被推导出来。但在根本层面上，所有数学系统都是平等的。最多只能出于实用主义的原因而偏好一个系统而不是另一个系统。不一致的系统可以证明所有命题，因此相当无用。因此，当一个系统被发现是不一致的时候，它必须被修改。这只是哥德尔不完全性定理的一个教训，即一个足够强大且一致的系统不能证明它自身的一致性。

There is a canonical objection against Curry’s formalist position. Mathematicians do not in fact treat all apparently consistent formal systems as being on a par. Most of them are unwilling to admit that the preference of arithmetical systems in which the arithmetical sentence expressing the consistency of Peano Arithmetic are derivable over those in which its negation is derivable, for instance, can ultimately be explained in purely pragmatical terms. Many mathematicians want to maintain that the perceived correctness (incorrectness) of certain formal systems must ultimately be explained by the fact that they correctly (incorrectly) describe certain subject matters.
对于卡里的形式主义立场有一个经典的反对意见。事实上，数学家们并不把所有表面上一致的形式系统都视为平等的。他们中的大多数人不愿承认，例如，对那些可以推导出表达皮亚诺算术一致性的算术命题的算术系统的偏好，而不是那些可以推导出其否定的系统，最终可以用纯粹的实用主义术语来解释。许多数学家希望坚持认为，某些形式系统的正确性（或不正确性）必须最终用它们正确地（或不正确地）描述某些主题的事实来解释。

Detlefsen has emphasized that the incompleteness theorems do not preclude that the consistency of parts of higher mathematics that are in practice used for solving arithmetical problems that mathematicians are interested in can be arithmetically established (Detlefsen 1986). In this sense, something can perhaps be rescued from the flames even if Hilbert’s instrumentalist stance towards all of higher mathematics is ultimately untenable.
德特勒芬强调，不完全性定理并不排除在实践中用于解决数学家感兴趣的算术问题的高等数学的部分的一致性可以用算术来确定（Detlefsen 1986）。在这个意义上，即使希尔伯特对所有高等数学的工具主义立场最终是不可维持的，也许还可以从火焰中拯救出一些东西。

Another attempt to salvage a part of Hilbert’s program was made by Isaacson (Isaacson 1987). He defends the view that in some sense, Peano Arithmetic may be complete after all (Isaacson 1987). He argues that true sentences undecidable in Peano Arithmetic can only be proved by means of higher-order concepts. For instance, the consistency of Peano Arithmetic can be proved by induction up to a transfinite ordinal number (Gentzen 1938). But the notion of an ordinal number is a set-theoretic, and hence non-arithmetical, concept. If the only ways of proving the consistency of arithmetic make essential use of notions which arguably belong to higher-order mathematics, then the consistency of arithmetic, even though it can be expressed in the language of Peano Arithmetic, is a non-arithmetical problem. And generalizing from this, one can wonder whether Hilbert’s conjecture that every problem of arithmetic can be decided from the axioms of Peano Arithmetic might not still be true.
另一次试图挽救希尔伯特计划的一部分是由艾萨克森做出的（Isaacson 1987）。他捍卫这样一种观点，即在某种意义上，皮亚诺算术最终可能是完整的（Isaacson 1987）。他认为，只能用高阶概念来证明皮亚诺算术中不可判定的真命题。例如，可以通过归纳到一个超限序数来证明皮亚诺算术的一致性（Gentzen 1938）。但是序数的概念是集合论的，因此是非算术的。如果证明算术一致性的唯一方法都必然使用属于高阶数学的概念，那么算术的一致性，尽管可以用皮亚诺算术的语言表达，却是一个非算术问题。由此推广，我们可以思考希尔伯特的猜想，即每一个算术问题都可以从皮亚诺算术的公理中得出结论，或许仍然是正确的。

2.4 Predicativism

预设主义

As was mentioned earlier, predicativism is not ordinarily described as one of the schools. But it is only for contingent reasons that before the advent of the second world war predicativism did not rise to the level of prominence of the other schools.
如前文所述，预设主义通常不被视为其中一个学派。但只是由于偶然的原因，在第二次世界大战之前，预设主义没有达到其他学派的显著程度。

The origin of predicativism lies in the work of Russell. On a cue of Poincaré, he arrived at the following diagnosis of the Russell paradox. The argument of the Russell paradox defines the collection $C$ of all mathematical entities that satisfy $\neg x\in x$. The argument then proceeds by asking whether $C$ itself meets this condition, and derives a contradiction.
预设主义的起源可以追溯到罗素的工作。在庞加莱的提示下，他得出了对罗素悖论的如下诊断。罗素悖论的论证定义了满足 $\neg x\in x$ 的所有数学实体的集合 $C$。然后，论证继续询问 $C$ 本身是否满足这个条件，并得出了一个矛盾。

The Poincaré-Russell diagnosis of this argument states that this definition does not pick out a collection at all: it is impossible to define a collection S by a condition that implicitly refers to S itself. This is called the vicious circle principle. Definitions that violate the vicious circle principle are called impredicative. A sound definition of a collection only refers to entities that exist independently from the defined collection. Such definitions are called predicative. As Gödel later pointed out, a platonist would find this line of reasoning unconvincing. If mathematical collections exist independently of the act of defining, then it is not immediately clear why there could not be collections that can only be defined impredicatively (Gödel 1944).
庞加莱 - 罗素对这一论证的诊断指出，这个定义根本就没有挑选出一个集合：不可能通过一个隐含地指涉其自身的条件来定义一个集合 S。这被称为恶性循环原则。违反恶性循环原则的定义被称为非预设的。一个集合的合理定义只指涉那些独立于被定义集合而存在的实体。这样的定义被称为预设的。正如哥德尔后来指出的，一个柏拉图主义者会觉得这种推理不具说服力。如果数学集合的存在独立于定义行为，那么并不清楚为什么不能存在只能通过非预设方式定义的集合（Gödel 1944）。

All this led Russell to develop the simple and the ramified theory of types, in which syntactical restrictions were built in that make impredicative definitions ill-formed. In simple type theory, the free variables in defining formulas range over entities to which the collection to be defined do not belong. In ramified type theory, it is required in addition that the range of the bound variables in defining formulas do not include the collection to be defined. It was pointed out in section 2.1 that Russell’s type theory cannot be seen as a reduction of mathematics to logic. But even aside from that, it was observed early on that especially in ramified type theory it is too cumbersome to formalize ordinary mathematical arguments.
所有这些促使罗素发展了简单类型论和分支类型论，在其中构建了语法限制，使非预设定义成为不合逻辑的。在简单类型论中，定义公式中的自由变量的范围是那些被定义的集合不属于的实体。在分支类型论中，还要求定义公式中约束变量的范围不包括被定义的集合。正如在第 2.1 节中指出的，罗素的类型论不能被视为将数学归结为逻辑。但即使撇开这一点不谈，早在一开始人们就观察到，特别是在分支类型论中，形式化普通的数学论证太繁琐了。

When Russell turned to other areas of analytical philosophy, Hermann Weyl took up the predicativist cause (Weyl 1918). Like Poincaré, Weyl did not share Russell’s desire to reduce mathematics to logic. And right from the start he saw that it would be in practice impossible to work in a ramified type theory. Weyl developed a philosophical stance that is in a sense intermediate between intuitionism and platonism. He took the collection of natural numbers as unproblematically given. But the concept of an arbitrary subset of the natural numbers was not taken to be immediately given in mathematical intuition. Only those subsets which are determined by arithmetical (i.e., first-order) predicates are taken to be predicatively acceptable.
当罗素转向分析哲学的其他领域时，赫尔曼·外尔承担起了预设主义的事业（Weyl 1918）。和庞加莱一样，外尔并不认同罗素将数学归结为逻辑的愿望。而且从一开始他就看到了在实践中在分支类型论中工作是不可能的。外尔发展了一种在某种意义上介于直觉主义和柏拉图主义之间的哲学立场。他把自然数的集合看作是毫无问题地给定的。但是任意子集的概念并没有在数学直觉中被立即给出。只有那些由算术的（即一阶的）谓词确定的子集才被认为是预设可接受的。

On the one hand, it emerged that many of the standard definitions in mathematical analysis are impredicative. For instance, the minimal closure of an operation on a set is ordinarily defined as the intersection of all sets that are closed under applications of the operation. But the minimal closure itself is one of the sets that are closed under applications of the operation. Thus, the definition is impredicative. In this way, attention gradually shifted away from concern about the set-theoretical paradoxes to the role of impredicativity in mainstream mathematics. On the other hand, Weyl showed that it is often possible to bypass impredicative notions. It even emerged that most of mainstream nineteenth century mathematical analysis can be vindicated on a predicative basis (Feferman 1988).
一方面，逐渐发现数学分析中的许多标准定义都是非预设的。例如，一个集合上一个运算的最小闭包通常被定义为所有在该运算下封闭的集合的交集。但是，最小闭包本身也是在该运算下封闭的集合之一。因此，这个定义是非预设的。通过这种方式，人们的关注逐渐从集合论悖论转移到主流数学中非预设性的角色。另一方面，外尔表明，通常可以绕过非预设的概念。甚至发现，19 世纪主流数学分析的大部分可以在预设的基础上得到证明（Feferman 1988）。

In the 1920s, History intervened. Weyl was won over to Brouwer’s more radical intuitionistic project. In the meantime, mathematicians became convinced that the highly impredicative transfinite set theory developed by Cantor and Zermelo was less acutely threatened by Russell’s paradox than previously suspected. These factors caused predicativism to lapse into a dormant state for several decades.
在 20 世纪 20 年代，历史介入了。外尔被布劳威尔更激进的直觉主义项目所吸引。与此同时，数学家们相信，康托和策梅洛发展的高度非预设的超限集合论并没有像以前怀疑的那样受到罗素悖论的严重威胁。这些因素导致预设主义在数十年间陷入休眠状态。

Building on work in generalized recursion theory, Solomon Feferman extended the predicativist project in the 1960s (Feferman 2005). He realized that Weyl’s strategy could be iterated into the transfinite. Also those sets of numbers that can be defined by using quantification over the sets that Weyl regarded as predicatively justified, should be counted as predicatively acceptable, and so on. This process can be propagated along an ordinal path. This ordinal path stretches as far into the transfinite as the predicative ordinals reach, where an ordinal is predicative if it measures the length of a provable well-ordering of the natural numbers. This calibration of the strength of predicative mathematics, which is due to Feferman and (independently) Schütte, is nowadays fairly generally accepted. Feferman then investigated how much of standard mathematical analysis can be carried out within a predicativist framework. The research of Feferman and others (most notably Harvey Friedman) shows that most of twentieth century analysis is acceptable from a predicativist point of view. But it is also clear that not all of contemporary mathematics that is generally accepted by the mathematical community is acceptable from a predicativist standpoint: transfinite set theory is a case in point.
基于广义递归理论的工作，索洛蒙·费弗曼在 20 世纪 60 年代扩展了预设主义项目（Feferman 2005）。他意识到外尔的策略可以迭代到超限。此外，那些可以通过对外尔认为是预设合理的集合进行量化的集合，也应该被视为预设可接受的，依此类推。这个过程可以沿着一个序数路径传播。这个序数路径延伸到超限的程度，取决于预设序数能达到的范围，其中序数是预设的，如果它测量了自然数的一个可证明的良好排序的长度。这种对预设数学强度的校准，归功于费弗曼和（独立地）舒特，如今已被相当广泛地接受。费弗曼随后研究了在预设主义框架内可以进行多少标准数学分析。费弗曼和其他人（尤其是哈维·弗里德曼）的研究表明，20 世纪的大部分分析从预设主义的角度来看是可以接受的。但也很清楚，并非所有被数学界普遍接受的当代数学都从预设主义的角度来看是可以接受的：超限集合论就是一个例子。

3. Platonism

柏拉图主义

In the years before the second world war it became clear that weighty objections had been raised against each of the three anti-platonist programs in the philosophy of mathematics. Predicativism was perhaps an exception, but it was at the time a program without defenders. Thus room was created for a renewed interest in the prospects of platonistic views about the nature of mathematics. On the platonistic conception, the subject matter of mathematics consists of abstract entities.
在第二次世界大战之前，人们已经对数学哲学中的三种反柏拉图主义项目提出了有力的反对意见。预设主义或许是一个例外，但在当时这是一个没有支持者的项目。因此，人们重新对柏拉图主义关于数学本质的观点产生了兴趣。在柏拉图主义的观点中，数学的主题是抽象实体。

3.1 Gödel’s Platonism

哥德尔的柏拉图主义

Gödel was a platonist with respect to mathematical objects and with respect to mathematical concepts (Gödel 1944; Gödel 1964). But his platonistic view was more sophisticated than that of the mathematician in the street.
哥德尔在数学对象和数学概念方面都是柏拉图主义者（Gödel 1944；Gödel 1964）。但他的柏拉图主义观点比普通数学家的更为复杂。

Gödel held that there is a strong parallelism between plausible theories of mathematical objects and concepts on the one hand, and plausible theories of physical objects and properties on the other hand. Like physical objects and properties, mathematical objects and concepts are not constructed by humans. Like physical objects and properties, mathematical objects and concepts are not reducible to mental entities. Mathematical objects and concepts are as objective as physical objects and properties. Mathematical objects and concepts are, like physical objects and properties, postulated in order to obtain a good satisfactory theory of our experience. Indeed, in a way that is analogous to our perceptual relation to physical objects and properties, through mathematical intuition we stand in a quasi-perceptual relation with mathematical objects and concepts. Our perception of physical objects and concepts is fallible and can be corrected. In the same way, mathematical intuition is not fool-proof — as the history of Frege’s Basic Law V shows— but it can be trained and improved. Unlike physical objects and properties, mathematical objects do not exist in space and time, and mathematical concepts are not instantiated in space or time.
哥德尔认为，数学对象和概念的合理理论与物理对象和属性的合理理论之间存在着强烈的平行关系。就像物理对象和属性一样，数学对象和概念不是由人类构造的。就像物理对象和属性一样，数学对象和概念不能被还原为心理实体。数学对象和概念与物理对象和属性一样是客观的。数学对象和概念，就像物理对象和属性一样，是为了获得一个关于我们经验的良好满意的理论而被假定的。实际上，与我们对物理对象和属性的感知关系类似，通过数学直觉，我们与数学对象和概念处于一种准感知关系。我们对物理对象和概念的感知是可错的，并且可以被纠正。同样地，数学直觉也不是万无一失的——正如弗雷格的基本法则 V 的历史所显示的那样——但它可以被训练和改进。与物理对象和属性不同，数学对象不存在于空间和时间中，数学概念也不在空间和时间中被实例化。

Our mathematical intuition provides intrinsic evidence for mathematical principles. Virtually all of our mathematical knowledge can be deduced from the axioms of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). In Gödel’s view, we have compelling intrinsic evidence for the truth of these axioms. But he also worried that mathematical intuition might not be strong enough to provide compelling evidence for axioms that significantly exceed the strength of ZFC.
我们的数学直觉为数学原理提供了内在证据。我们几乎所有的数学知识都可以从策梅洛 - 弗兰克尔集合论与选择公理（ZFC）的公理中推导出来。在哥德尔看来，我们有令人信服的内在证据证明这些公理的真实性。但他也担心，数学直觉可能不够强大，无法为那些显著超出 ZFC 强度的公理提供令人信服的证据。

Aside from intrinsic evidence, it is in Gödel’s view also possible to obtain extrinsic evidence for mathematical principles. If mathematical principles are successful, then, even if we are unable to obtain intuitive evidence for them, they may be regarded as probably true. Gödel says that:
除了内在证据之外，在哥德尔看来，还可以为数学原理获得外在证据。如果数学原理是成功的，那么即使我们无法获得对它们的直觉证据，它们也可以被视为可能是真的。哥德尔说：

… success here means fruitfulness in consequences, particularly in ‘verifiable’ consequences, i.e. consequences verifiable without the new axiom, whose proof with the help of the new axiom, however, are considerably simpler and easier to discover, and which make it possible to contract into one proof many different proofs […] There might exist axioms so abundant in their verifiable consequences, shedding so much light on a whole field, yielding such powerful methods for solving problems […] that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory. (Gödel 1947, p. 477)
…这里的成功意味着结果的丰富性，特别是在‘可验证’的结果方面，即那些可以在没有新公理的情况下验证的结果，然而借助新公理，它们的证明却大大简化，更容易被发现，并且可以将许多不同的证明合并为一个证明……可能存在一些公理，它们的可验证结果如此丰富，为整个领域带来如此多的光明，提供了如此强大的解决问题的方法……不管它们是否内在必要，都必须至少像任何已确立的物理理论一样被接受。（哥德尔 1947，第 477 页）

This inspired Gödel to search for new axioms which can be extrinsically motivated and which can decide questions such as the continuum hypothesis which are highly independent of ZFC (cf. section 5.1).
这激发了哥德尔去寻找新的公理，这些公理可以从外部得到激励，并且能够决定像连续统假设这样高度独立于 ZFC 的问题（参见第 5.1 节）。

Gödel shared Hilbert’s conviction that all mathematical questions have definite answers. But platonism in the philosophy of mathematics should not be taken to be ipso facto committed to holding that all set-theoretical propositions have determinate truth values. There are versions of platonism that maintain, for instance, that all theorems of ZFC are made true by determinate set-theoretical facts, but that there are no set-theoretical facts that make certain statements that are highly independent of ZFC truth-determinate. It seems that the famous set theorist Paul Cohen held some such view (Cohen 1971).
哥德尔与希尔伯特一样坚信所有数学问题都有确定的答案。但在数学哲学中，柏拉图主义不应被当作必然承诺所有集合论命题都有确定的真值。有些柏拉图主义版本认为，例如，ZFC 的所有定理都是由确定的集合论事实所证实的，但并没有集合论事实使某些高度独立于 ZFC 的陈述具有确定的真值。著名的集合论学家保罗·科恩似乎持有某种类似的观点（Cohen 1971）。

3.2 Naturalism and Indispensability

自然主义与不可或缺性

Quine formulated a methodological critique of traditional philosophy. He suggested a different philosophical methodology instead, which has become known as naturalism (Quine 1969). According to naturalism, our best theories are our best scientific theories. If we want to obtain the best available answer to philosophical questions such as What do we know? and Which kinds of entities exist?, we should not appeal to traditional epistemological and metaphysical theories. We should also refrain from embarking on a fundamental epistemological or metaphysical inquiry starting from first principles. Rather, we should consult and analyze our best scientific theories. They contain, albeit often implicitly, our currently best account of what exists, what we know, and how we know it. 奎因对传统哲学提出了方法论上的批判。他提出了另一种哲学方法论，即所谓的自然主义（Quine 1969）。根据自然主义，我们最好的理论就是我们最好的科学理论。如果我们想要获得关于哲学问题如*我们知道了什么？和存在哪些种类的实体？*的最佳答案，我们不应诉诸传统的认识论和形而上学理论。我们也应避免从第一原理出发进行基本的认识论或形而上学探究。相反，我们应该参考和分析我们最好的科学理论。尽管这些理论常常是隐含的，但它们包含了我们目前对存在什么、我们知道什么以及我们如何知道的最好解释。

Putnam applied Quine’s naturalistic stance to mathematical ontology (Putnam 1972). At least since Galilei, our best theories from the natural sciences are mathematically expressed. Newton’s theory of gravitation, for instance, relies heavily on the classical theory of the real numbers. Thus an ontological commitment to mathematical entities seems inherent to our best scientific theories. This line of reasoning can be strengthened by appealing to the Quinean thesis of confirmational holism. Empirical evidence does not bestow its confirmatory power on any one individual hypothesis. Rather, experience globally confirms the theory in which the individual hypothesis is embedded. Since mathematical theories are part and parcel of scientific theories, they too are confirmed by experience. Thus, we have empirical confirmation for mathematical theories. Even more appears true. It seems that mathematics is indispensable to our best scientific theories: it is not at all obvious how we could express them without using mathematical vocabulary. Hence the naturalist stance commands us to accept mathematical entities as part of our philosophical ontology. This line of argumentation is called an indispensability argument (Colyvan 2001). 普特南将奎因的自然主义立场应用于数学本体论（Putnam 1972）。至少自伽利略以来，我们自然科学中最好的理论都是用数学表达的。例如，牛顿的引力理论严重依赖于实数的经典理论。因此，对数学实体的本体论承诺似乎是我们最好的科学理论所固有的。通过引用奎因的确认整体论，可以加强这一推理。经验证据并不赋予任何一个单独假设以确认的力量。相反，经验全局地确认了包含该单独假设的理论。由于数学理论是科学理论的一部分，它们也通过经验得到确认。因此，我们对数学理论有经验上的确认。似乎还有更真实的情况。数学似乎对我们最好的科学理论是不可或缺的：我们根本不清楚如何在不使用数学词汇的情况下表达它们。因此，自然主义立场要求我们将数学实体作为我们哲学本体论的一部分。这种论证被称为不可或缺性论证（Colyvan 2001）。

If we take the mathematics that is involved in our best scientific theories at face value, then we appear to be committed to a form of platonism. But it is a more modest form of platonism than Gödel’s platonism. For it appears that the natural sciences can get by with (roughly) function spaces on the real numbers. The higher regions of transfinite set theory appear to be largely irrelevant to even our most advanced theories in the natural sciences. Nevertheless, Quine thought (at some point) that the sets that are postulated by ZFC are acceptable from a naturalistic point of view; they can be regarded as a generous rounding off of the mathematics that is involved in our scientific theories. Quine’s judgement on this matter is not universally accepted. Feferman, for instance, argues that all the mathematical theories that are essentially used in our currently best scientific theories are predicatively reducible (Feferman 2005). Maddy even argues that naturalism in the philosophy of mathematics is perfectly compatible with a non-realist view about sets (Maddy 2007, part IV). 如果我们对涉及在我们最好的科学理论中的数学采取表面价值，那么我们似乎就承诺了一种柏拉图主义的形式。但这是一种比哥德尔的柏拉图主义更为温和的形式。因为似乎自然科学可以用（大致）实数上的函数空间来解决。超限集合论的更高区域似乎与我们自然科学中甚至最先进的理论都几乎无关。然而，奎因（在某个时候）认为，ZFC 所假设的集合从自然主义的角度来看是可以接受的；它们可以被视为我们科学理论中所涉及的数学的慷慨概括。奎因在这个问题上的判断并不是普遍接受的。例如，费弗曼认为，在我们目前最好的科学理论中本质上使用的数学理论都是预设可约的（Feferman 2005）。马迪甚至认为，数学哲学中的自然主义与集合的非实在论观点是完全兼容的（Maddy 2007，第四部分）。

In Quine’s philosophy, the natural sciences are the ultimate arbiters concerning mathematical existence and mathematical truth. This has led Charles Parsons to object that this picture makes the obviousness of elementary mathematics somewhat mysterious (Parsons 1980). For instance, the question whether every natural number has a successor ultimately depends, in Quine’s view, on our best empirical theories; however, somehow this fact appears more immediate than that. In a kindred spirit, Maddy notes that mathematicians do not take themselves to be in any way restricted in their activity by the natural sciences. Indeed, one might wonder whether mathematics should not be regarded as a science in its own right, and whether the ontological commitments of mathematics should not be judged rather on the basis of the rational methods that are implicit in mathematical practice. 在奎因的哲学中，自然科学是关于数学存在和数学真理的最终裁决者。这使得查尔斯·帕森斯反对这种观点，认为它使初等数学的明显性变得有些神秘（Parsons 1980）。例如，在奎因看来，每个自然数是否都有后继数的问题最终取决于我们最好的经验理论；然而，不知何故，这一事实似乎比这更直接。出于类似的精神，马迪指出，数学家们并不认为他们的活动在任何方面受到自然科学的限制。实际上，人们可能会怀疑，数学是否不应该被视为一门独立的科学，以及数学的本体论承诺是否不应该更多地根据数学实践中隐含的理性方法来判断。

Motivated by these considerations, Maddy set out to inquire into the standards of existence implicit in mathematical practice, and into the implicit ontological commitments of mathematics that follow from these standards (Maddy 1990). She focussed on set theory, and on the methodological considerations that are brought to bear by the mathematical community on the question which large cardinal axioms can be taken to be true. Thus her view is closer to that of Gödel than to that of Quine. In more recent work, she isolates two maxims that seem to be guiding set theorists when contemplating the acceptability of new set theoretic principles: unify and maximize (Maddy 1997). The maxim “unify” is an instigation for set theory to provide a single system in which all mathematical objects and structures of mathematics can be instantiated or modelled. The maxim “maximize” means that set theory should adopt set theoretic principles that are as powerful and mathematically fruitful as possible. 受这些考虑的激励，马迪开始研究数学实践中隐含的存在标准，以及由此产生的数学的隐含本体论承诺（Maddy 1990）。她专注于集合论，以及数学界在考虑哪些大基数公理可以被认为是真时所采用的方法论考虑。因此，她的观点更接近哥德尔而不是奎因。在她最近的研究中，她提出了两个似乎在指导集合论者考虑新的集合论原理的可接受性时的准则：“统一”和“最大化”（Maddy 1997）。准则“统一”是促使集合论提供一个单一的系统，在这个系统中，所有的数学对象和数学结构都可以被实例化或被建模。准则“最大化”意味着集合论应该采用尽可能强大且数学上有成果的集合论原理。

3.3 Deflating Platonism

柏拉图主义的消解

Bernays observed that when a mathematician is at work she “naively” treats the objects she is dealing with in a platonistic way. Every working mathematician, he says, is a platonist (Bernays 1935). But when the mathematician is caught off duty by a philosopher who quizzes her about her ontological commitments, she is apt to shuffle her feet and withdraw to a vaguely non-platonistic position. This has been taken by some to indicate that there is something wrong with philosophical questions about the nature of mathematical objects and of mathematical knowledge. 伯奈斯观察到，当一位数学家在工作时，她会“天真地”以一种柏拉图主义的方式对待她所处理的对象。他说，每一位工作的数学家都是柏拉图主义者（Bernays 1935）。但是，当一位数学家被一位哲学家拦住并询问她关于她的本体论承诺时，她往往会感到不安并退缩到一种模糊的非柏拉图主义立场。一些人认为这表明关于数学对象和数学知识的本质的哲学问题是出了什么问题。

Carnap introduced a distinction between questions that are internal to a framework and questions that are external to a framework (Carnap 1950). It has been argued that Carnap’s distinction in some guise survives the demise of the logical empiricist framework in which it was first articulated (Burgess 2004b). Tait has attempted to work out in detail how the resulting distinction can be applied to mathematics (Tait 2005). This has resulted in what might be regarded as a deflationary versions of platonism. 卡纳普引入了一种框架内的问题和框架外的问题之间的区别（Carnap 1950）。有人认为，卡纳普的这种区别以某种形式在逻辑实证主义框架中首次被表述后，已经幸存下来（Burgess 2004b）。泰特试图详细阐述这种区别如何应用于数学（Tait 2005）。这导致了可能被视为柏拉图主义的消解版本。

According to Tait, questions of existence of mathematical entities can only be sensibly asked and reasonably answered from within (axiomatic) mathematical frameworks. If one is working in number theory, for instance, then one can ask whether there are prime numbers that have a given property. Such questions are then to be decided on purely mathematical grounds. Philosophers have a tendency to step outside the framework of mathematics and ask “from the outside” whether mathematical objects really exist and whether mathematical propositions are really true. In this question they are asking for supra-mathematical or metaphysical grounds for mathematical truth and existence claims. Tait argues that it is hard to see how any sense can be made of such external questions. He attempts to deflate them, and bring them back to where they belong: to mathematical practice itself. Of course not everyone agrees with Tait on this point. Linsky and Zalta have developed a systematic way of answering precisely the sort of external questions that Tait approaches with disdain (Linsky & Zalta 1995). 根据泰特，关于数学实体的存在问题只能在（公理化的）数学框架内合理地提出并合理地回答。例如，如果一个人在数论中工作，那么他可以问是否存在具有某种属性的素数。然后这些问题应该在纯粹的数学基础上来决定。哲学家们倾向于走出数学的框架，并从“外部”问数学对象是否真正存在，数学命题是否真正为真。在这个问题中，他们要求对数学真理和存在主张提供超数学的或形而上学的根据。泰特认为，很难看出这种外部问题能有什么意义。他试图消解这些问题，并把它们带回它们所属的地方：数学实践本身。当然，并不是每个人都同意泰特的这个观点。林斯基和扎尔塔已经发展了一种系统的方法来回答泰特不屑一顾的正是这种外部问题（Linsky & Zalta 1995）。

It comes as no surprise that Tait has little use for Gödelian appeals to mathematical intuition in the philosophy of mathematics, or for the philosophical thesis that mathematical objects exist “outside space and time”. More generally, Tait believes that mathematics is not in need of a philosophical foundation; he wants to let mathematics speak for itself. In this sense, his position is reminiscent of the (in some sense Wittgensteinian) natural ontological attitude that is advocated by Arthur Fine in the realism debate in the philosophy of science. 毫不奇怪，泰特对哥德尔在数学哲学中对数学直觉的呼吁，或者对数学对象存在于“空间和时间之外”的哲学论题并不感兴趣。更一般地说，泰特认为数学不需要哲学基础；他想让数学为自己说话。在这个意义上，他的立场让人想起亚瑟·芬在科学哲学的实在论辩论中所倡导的（某种意义上是维特根斯坦式的）自然本体论态度。

3.4 Benacerraf’s Epistemological Problem

贝纳塞拉夫的认识论问题

Benacerraf formulated an epistemological problem for a variety of platonistic positions in the philosophy of science (Benacerraf 1973). The argument is specifically directed against accounts of mathematical intuition such as that of Gödel. Benacerraf’s argument starts from the premise that our best theory of knowledge is the causal theory of knowledge. It is then noted that according to platonism, abstract objects are not spatially or temporally localized, whereas flesh and blood mathematicians are spatially and temporally localized. Our best epistemological theory then tells us that knowledge of mathematical entities should result from causal interaction with these entities. But it is difficult to imagine how this could be the case. 贝纳塞拉夫为科学哲学中的一系列柏拉图主义立场提出了一个认识论问题（Benacerraf 1973）。这个论证特别针对像哥德尔那样的数学直觉理论。贝纳塞拉夫的论证从我们的最佳知识理论是因果知识理论这一前提开始。然后指出，根据柏拉图主义，抽象对象在空间和时间上没有定位，而有血有肉的数学家在空间和时间上是有定位的。我们最好的认识论理论告诉我们，对数学实体的知识应该来自于与这些实体的因果互动。但很难想象这怎么可能。

Today few epistemologists hold that the causal theory of knowledge is our best theory of knowledge. But it turns out that Benacerraf’s problem is remarkably robust under variation of epistemological theory. For instance, let us assume for the sake of argument that reliabilism is our best theory of knowledge. Then the problem becomes to explain how we succeed in obtaining reliable beliefs about mathematical entities. 如今，很少有认识论者认为因果知识理论是我们最好的知识理论。但事实证明，贝纳塞拉夫的问题在认识论理论的变化下非常稳健。例如，为了论证的需要，让我们假设可靠主义是我们最好的知识理论。那么问题就变成了如何解释我们如何成功地获得关于数学实体的可靠信念。

Hodes has formulated a semantical variant of Benacerraf’s epistemological problem (Hodes 1984). According to our currently best semantic theory, causal-historical connections between humans and the world of concreta enable our words to refer to physical entities and properties. According to platonism, mathematics refers to abstract entities. The platonist therefore owes us a plausible account of how we (physically embodied humans) are able to refer to them. On the face of it, it appears that the causal theory of reference will be unable to supply us with the required account of the ‘microstructure of reference’ of mathematical discourse. 霍德斯提出了贝纳塞拉夫认识论问题的一个语义变体（Hodes 1984）。根据我们目前最好的语义理论，人类与具体世界之间的因果 - 历史联系使我们的词汇能够指代物理实体和属性。根据柏拉图主义，数学指代抽象实体。因此，柏拉图主义者需要向我们提供一个合理的解释，说明我们（具有物理身体的人类）如何能够指代它们。乍一看，似乎因果指代理论无法为我们提供数学话语的“指代微观结构”所需的解释。

3.5 Plenitudinous Platonism

充裕的柏拉图主义

A version of platonism has been developed which is intended to provide a solution to Benacerraf’s epistemological problem (Linsky & Zalta 1995; Balaguer 1998). This position is known as plenitudinous platonism. The central thesis of this theory is that every logically consistent mathematical theory necessarily refers to an abstract entity. Whether the mathematician who formulated the theory knows that it refers or does not know this, is largely immaterial. By entertaining a consistent mathematical theory, a mathematician automatically acquires knowledge about the subject matter of the theory. So, on this view, there is no epistemological problem to solve anymore. 已经发展了一种柏拉图主义版本，旨在解决贝纳塞拉夫的认识论问题（Linsky & Zalta 1995；Balaguer 1998）。这种立场被称为充裕的柏拉图主义。这个理论的核心论点是，每一个逻辑上一致的数学理论必然指代一个抽象实体。提出这个理论的数学家是否知道它指代什么，或者不知道，这在很大程度上是无关紧要的。通过思考一个一致的数学理论，数学家自动获得了关于该理论主题的知识。因此，在这种观点下，已经没有认识论问题需要解决了。

In Balaguer’s version, plenitudinous platonism postulates a multiplicity of mathematical universes, each corresponding to a consistent mathematical theory. Thus, in particular a question such as the continuum problem (cf. section 5.1) does not receive a unique answer: in some set-theoretical universes the continuum hypothesis holds, in others it fails to hold. However, not everyone agrees that this picture can be maintained. Martin has developed an argument to show that multiple universes can always to a large extent be “accumulated” into a single universe (Martin 2001).
在巴拉格的版本中，充裕的柏拉图主义假设了多个数学宇宙，每个都对应于一个一致的数学理论。因此，特别是像连续统问题（参见第 5.1 节）这样的问题并没有一个唯一的答案：在一些集合论宇宙中，连续统假设成立，在其他宇宙中则不成立。然而，并非每个人都同意这种观点可以维持。马丁已经发展了一个论证来表明，多个宇宙总是可以在很大程度上被“累积”到一个单一的宇宙中（Martin 2001）。

In Linsky and Zalta’s version of plenitudinous platonism, the mathematical entity that is postulated by a consistent mathematical theory has exactly the mathematical properties which are attributed to it by the theory. The abstract entity corresponding to ZFC, for instance, is partial in the sense that it neither makes the continuum hypothesis true nor false. The reason is that ZFC neither entails the continuum hypothesis nor its negation. This does not entail that all ways of consistently extending ZFC are on a par. Some ways may be fruitful and powerful, others less so. But the view does deny that certain consistent ways of extending ZFC are preferable because they consist of true principles, whereas others contain false principles.
在林斯基和扎尔塔的充裕柏拉图主义版本中，一个一致的数学理论所假设的数学实体具有该理论赋予它的所有数学属性。例如，对应于 ZFC 的抽象实体是部分的，因为它既不使连续统假设为真，也不使其为假。原因是 ZFC 既不蕴含连续统假设，也不蕴含其否定。这并不意味着所有一致扩展 ZFC 的方式都是等同的。有些方式可能是富有成果和强大的，而其他方式则不然。但这种观点确实否认某些一致扩展 ZFC 的方式是可取的，因为它们由真实的原理组成，而其他方式则包含虚假的原理。

4. Structuralism and Nominalism

结构主义和唯名论

Benacerraf’s work motivated philosophers to develop both structuralist and nominalist theories in the philosophy of mathematics (Reck & Price 2000). And since the late 1980s, combinations of structuralism and nominalism have also been developed. 贝纳塞拉夫的工作激励哲学家在数学哲学中发展结构主义和唯名论理论（Reck & Price 2000）。自 20 世纪 80 年代末以来，结构主义和唯名论的结合也得到了发展。

4.1 What Numbers Could Not Be

数字不可能是什么

As if saddling platonism with one difficult problem were not enough (section 3.4), Benacerraf formulated a challenge for set-theoretic platonism (Benacerraf 1965). The challenge takes the following form.
仿佛给柏拉图主义加上一个难题还不够（第 3.4 节），本纳塞尔提出了对集合论柏拉图主义的挑战（本纳塞尔 1965）。挑战的形式如下。

There exist infinitely many ways of identifying the natural numbers with pure sets. Let us restrict, without essential loss of generality, our discussion to two such ways:
存在无限多种方法可以将自然数与纯集合识别。让我们在不失一般性的情况下，将讨论限制在两种这样的方法上：

Ⅰ : 0 = ∅ 1 = { ∅ } 2 = { { ∅ } } 3 = { { { ∅ } } } ⋮ Ⅱ : 0 = ∅ 1 = { ∅ } 2 = { ∅ , { ∅ } } 3 = { ∅ , { ∅ } , { ∅ , { ∅ } } } ⋮ \begin{align*} \mathrm Ⅰ:\\ 0 & = \varnothing \\ 1 & = \{\varnothing\} \\ 2 & = \{\{\varnothing\}\} \\ 3 & = \{\{\{\varnothing\}\}\} \\ & \vdots\\ \mathrm Ⅱ:\\ 0 & = \varnothing \\ 1 & = \{\varnothing\} \\ 2 & = \{\varnothing, \{\varnothing\}\} \\ 3 & = \{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\} \\ & \vdots \end{align*} Ⅰ:0123Ⅱ:0123​=∅={∅}={{∅}}={{{∅}}}⋮=∅={∅}={∅,{∅}}={∅,{∅},{∅,{∅}}}⋮​

The simple question that Benacerraf asks is:
本纳塞尔提出的简单问题是：

Which of these consists solely of true identity statements: Ⅰ or Ⅱ?
这些中哪些完全由真实的身份陈述组成：Ⅰ 还是 Ⅱ？

It seems very difficult to answer this question. It is not hard to see how a successor function and addition and multiplication operations can be defined on the number-candidates of I and on the number-candidates of II so that all the arithmetical statements that we take to be true come out true. Indeed, if this is done in the natural way, then we arrive at isomorphic structures (in the set-theoretic sense of the word), and isomorphic structures make the same sentences true (they are elementarily equivalent). It is only when we ask extra-arithmetical questions, such as ‘$1\in 3$?’ that the two accounts of the natural numbers yield diverging answers. So it is impossible that both accounts are correct. According to story I, 3 = { { { ∅ } } } 3=\{\{\{\emptyset\}\}\} 3={{{∅}}}, whereas according to story II, 3 = { ∅ , { ∅ } , { ∅ , { ∅ } } } 3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\} 3={∅,{∅},{∅,{∅}}}. If both accounts were correct, then the transitivity of identity would yield a purely set theoretic falsehood.
回答这个问题似乎非常困难。不难看出，可以在 I 的数候选者和 II 的数候选者上定义后继函数以及加法和乘法运算，使得我们所认为的算术陈述都为真。实际上，如果以自然的方式进行定义，那么我们得到的是同构 的结构（在集合论的意义上），同构结构使相同的句子为真（它们是初等等价的）。只有当我们问一些额外的算术问题，比如“$1\in 3$?”时，这两种关于自然数的解释才会得出不同的答案。因此，这两种解释不可能都是正确的。根据 I 的说法， 3 = { { { ∅ } } } 3=\{\{\{\emptyset\}\}\} 3={{{∅}}}，而根据 II 的说法， 3 = { ∅ , { ∅ } , { ∅ , { ∅ } } } 3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\} 3={∅,{∅},{∅,{∅}}}。如果这两种解释都是正确的，那么根据同一性的传递性，就会得出一个纯粹的集合论假命题。

Summing up, we arrive at the following situation. On the one hand, there appear to be no reasons why one account is superior to the other. On the other hand, the accounts cannot both be correct. This predicament is sometimes called labelled Benacerraf’s identification problem.
总结一下，我们得到了如下情况。一方面，似乎没有任何理由认为一种解释优于另一种。另一方面，这两种解释不可能都是正确的。这种困境有时被称为贝纳塞拉夫的识别问题。

The proper conclusion to draw from this conundrum appears to be that neither account I nor account II is correct. Since similar considerations would emerge from comparing other reasonable-looking attempts to reduce natural numbers to sets, it appears that natural numbers are not sets after all. It is clear, moreover, that a similar argument can be formulated for the rational numbers, the real numbers… Benacerraf concludes that they, too, are not sets at all.
从这个难题中得出的正确结论似乎是，I 和 II 这两种解释都不正确。此外，通过比较其他将自然数归约为集合的合理尝试，也会出现类似的考虑，因此自然数似乎根本不是集合。此外，很明显，可以为有理数、实数……制定类似的论证。贝纳塞拉夫得出结论，它们也不是集合。

It is not at all clear whether Gödel, for instance, is committed to reducing the natural numbers to pure sets. A platonist can uphold the claim that the natural numbers can be embedded into the set-theoretic universe while maintaining that the embedding should not be seen as an ontological reduction. Indeed, on Linsky and Zalta’s plenitudinous platonist account, the natural numbers have no properties beyond those that are attributed to them by our theory of the natural numbers (Peano Arithmetic). But then it seems that platonists would have to take a similar line with respect to the rational numbers, the complex numbers, …. Whereas maintaining that the natural numbers are sui generis admittedly has some appeal, it is perhaps less natural to maintain that the complex numbers, for instance, are also sui generis. And, anyway, even if the natural numbers, the complex numbers, … are in some sense not reducible to anything else, one may wonder if there may not be another way to elucidate their nature.
并不清楚哥德尔是否致力于将自然数归约为纯集合。一个柏拉图主义者可以坚持认为自然数可以嵌入到集合论宇宙中，同时认为这种嵌入不应被视为一种本体论上的归约。实际上，在林斯基和扎尔塔的充裕柏拉图主义理论中，自然数除了在我们的自然数理论（皮亚诺算术）中赋予它们的属性之外，没有任何属性。但随后似乎柏拉图主义者必须对有理数、复数……采取类似的立场。尽管承认自然数是独特的有一定吸引力，但认为复数也是独特的或许不太自然。而且，无论如何，即使自然数、复数……在某种意义上不能归约为其他东西，人们可能会怀疑是否还有其他方式来阐明它们的本质。

4.2 Ante Rem Structuralism

先验结构主义

Shapiro draws a useful distinction between algebraic and non-algebraic mathematical theories (Shapiro 1997). Roughly, non-algebraic theories are theories which appear at first sight to be about a unique model: the intended model of the theory. We have seen examples of such theories: arithmetic, mathematical analysis… Algebraic theories, in contrast, do not carry a prima facie claim to be about a unique model. Examples are group theory, topology, graph theory…
沙皮罗在代数的 和非代数的 数学理论之间做出了一个有用的区分（Shapiro 1997）。大致来说，非代数理论似乎是关于一个独特模型的理论：理论的预期模型。我们已经看到了这类理论的例子：算术、数学分析……相比之下，代数理论并没有表面上声称是关于一个独特模型的。例如群论、拓扑学、图论……

Benacerraf’s challenge can be mounted for the objects that non-algebraic theories appear to describe. But his challenge does not apply to algebraic theories. Algebraic theories are not interested in mathematical objects per se; they are interested in structural aspects of mathematical objects. This led Benacerraf to speculate whether the same could not be true also of non-algebraic theories. Perhaps the lesson to be drawn from Benacerraf’s identification problem is that even arithmetic does not describe specific mathematical objects, but instead only describes structural relations?
贝纳塞拉夫的挑战可以针对非代数理论所描述的对象。但他的挑战并不适用于代数理论。代数理论并不关心数学对象本身；它们感兴趣的是数学对象的结构方面。这促使贝纳塞拉夫推测，非代数理论是否也可能如此。也许从贝纳塞拉夫的识别问题中得出的教训是，甚至算术也不描述具体的数学对象，而是只描述结构关系？

Shapiro and Resnik hold that all mathematical theories, even non-algebraic ones, describe structures. This position is known as structuralism (Shapiro 1997; Resnik 1997). Structures consists of places that stand in structural relations to each other. Thus, derivatively, mathematical theories describe places or positions in structures. But they do not describe objects. The number three, for instance, will on this view not be an object but a place in the structure of the natural numbers.
沙皮罗和雷斯尼克认为，所有的数学理论，即使是非代数的，也描述结构。这种立场被称为结构主义（Shapiro 1997；Resnik 1997）。结构由处于结构关系中的位置组成。因此，从派生意义上说，数学理论描述结构中的位置或位置。但它们不描述对象。例如，数字三将不是对象，而是自然数结构中的一个位置。

Systems are instantiations of structures. The systems that instantiate the structure that is described by a non-algebraic theory are isomorphic with each other, and thus, for the purposes of the theory, equally good. The systems I and II that were described in section 4.1 can be seen as instantiations of the natural number structure. { { { ∅ } } } \{\{\{\emptyset\}\}\} {{{∅}}} and { ∅ , { ∅ } , { ∅ , { ∅ } } } \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\} {∅,{∅},{∅,{∅}}} are equally suitable for playing the role of the number three. But neither are the number three. For the number three is an open place in the natural number structure, and this open place does not have any internal structure. Systems typically contain structural properties over and above those that are relevant for the structures that they are taken to instantiate.
系统是结构的实例化。实例化非代数理论所描述的结构的系统彼此是同构的，因此，对于理论的目的来说，它们是同等良好的。在第 4.1 节中描述的系统 I 和 II 可以被视为自然数结构的实例化。 { { { ∅ } } } \{\{\{\emptyset\}\}\} {{{∅}}} 和 { ∅ , { ∅ } , { ∅ , { ∅ } } } \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\} {∅,{∅},{∅,{∅}}} 同样适合扮演数字三的角色。但它们都不是数字三。因为数字三在自然数结构中是一个开放的位置，而这个位置没有任何内部结构。系统通常包含超出它们所实例化的结构相关的结构属性。

Sensible identity questions are those that can be asked from within a structure. They are those questions that can be answered on the basis of structural aspects of the structure. Identity questions that go beyond a structure do not make sense. One can pose the question whether $3\in 4$, but not cogently: this question involves a category mistake. The question mixes two different structures: $\in$ is a set-theoretical notion, whereas 3 and 4 are places in the structure of the natural numbers. This seems to constitute a satisfactory answer to Benacerraf’s challenge.
合理的同一性问题是在结构内部可以提出的问题。这些问题是可以基于结构的结构方面来回答的问题。超出结构的同一性问题是毫无意义的。人们可以提出 $3\in 4$ 的问题，但这是不合理的：这个问题涉及了一个范畴错误。这个问题混合了两种不同的结构：$\in$ 是一个集合论的概念，而 3 和 4 是自然数结构中的位置。这似乎是对贝纳塞拉夫挑战的一个满意的答案。

In Shapiro’s view, structures are not ontologically dependent on the existence of systems that instantiate them. Even if there were no infinite systems to be found in Nature, the structure of the natural numbers would exist. Thus structures as Shapiro understands them are abstract, platonic entities. Shapiro’s brand of structuralism is often labeled ante rem structuralism.
在沙皮罗看来，结构并不依赖于实例化它们的系统的存在。即使在自然界中找不到任何无限的系统，自然数的结构也会存在。因此，沙皮罗所理解的结构是抽象的、柏拉图式的实体。沙皮罗的结构主义通常被称为先验 结构主义。

In textbooks on set theory we also find a notion of structure. Roughly, the set theoretic definition says that a structure is an ordered $n+1$-tuple consisting of a set, a number of relations on this set, and a number of distinguished elements of this set. But this cannot be the notion of structure that structuralism in the philosophy of mathematics has in mind. For the set theoretic notion of structure presupposes the concept of set, which, according to structuralism, should itself be explained in structural terms. Or, to put the point differently, a set-theoretical structure is merely a system that instantiates a structure that is ontologically prior to it.
在集合论的教科书中，我们也发现了一个结构的概念。大致上，集合论的定义说，一个结构是一个有序的$n+1$ 元组，由一个集合、这个集合上的若干关系以及这个集合的若干特殊元素组成。但这不能是数学哲学中的结构主义所指的结构概念。因为集合论的结构概念预设了集合的概念，而根据结构主义，集合的概念本身应该用结构术语来解释。或者，换一种说法，一个集合论的结构只是一个实例化一个本体论上优先于它的结构的系统。

Nonetheless, the motivation for extending ante rem structuralism even to the most encompassing mathematical discipline (set theory) is not entirely evident (Burgess 2015). Recall that the main motivation for arriving at a structuralist understanding of a mathematical discipline lies in Benacerraf’s identification problem. For set theory, it seems hard to mount an identification challenge: sets are not usually defined in terms of more primitive concepts.
然而，将先验 结构主义扩展到最广泛的数学学科（集合论）的动机并不完全明显（Burgess 2015）。回想一下，对一个数学学科进行结构主义理解的主要动机在于贝纳塞拉夫的识别问题。对于集合论来说，似乎很难提出一个识别挑战：集合通常不是用更原始的概念来定义的。

It appears that ante rem structuralism describes the notion of a structure in a somewhat circular manner. A structure is described as places that stand in relation to each other, but a place cannot be described independently of the structure to which it belongs. Yet this is not necessarily a problem. For the ante rem structuralist, the notion of structure is a primitive concept, which cannot be defined in other more basic terms. At best, we can construct an axiomatic theory of mathematical structures.
似乎先验 结构主义以一种有点循环的方式描述了结构的概念。结构被描述为相互关联的位置，但位置不能独立于它所属的结构来描述。然而，这不一定是问题。对于先验 结构主义者来说，结构的概念是一个原始概念，不能用其他更基本的术语来定义。最多，我们可以构建一个数学结构的公理理论。

But Benacerraf’s epistemological problem still appears to be urgent. Structures and places in structures may not be objects, but they are abstract. So it is natural to wonder how we succeed in obtaining knowledge of them. This problem has been taken by certain philosophers as a reason for developing a nominalist theory of mathematics and then to reconcile this theory with basic tenets of structuralism.
但贝纳塞拉夫的认识论问题似乎仍然紧迫。结构和结构中的位置可能不是对象，但它们是抽象的。因此，很自然地会想知道我们是如何获得对它们的知识的。一些哲学家将这个问题作为发展数学唯名论理论并将其与结构主义的基本信条相协调的理由。

4.3 Mathematics Without Abstract Entities

无抽象实体的数学

Goodman and Quine tried early on to bite the bullet: they embarked on a project to reformulate theories from natural science without making use of abstract entities (Goodman & Quine 1947). The nominalistic reconstruction of scientific theories proved to be a difficult task. Quine, for one, abandoned it after this initial attempt. In the past decades many theories have been proposed that purport to give a nominalistic reconstruction of mathematics. (Burgess & Rosen 1997) contains a good critical discussion of such views.
古德曼和奎因很早就尝试直面问题：他们着手开展一个项目，试图在不使用抽象实体的情况下重新表述自然科学的理论（Goodman & Quine 1947）。对科学理论进行唯名论的重构被证明是一项艰巨的任务。奎因在最初的尝试之后就放弃了这一方向。在过去的几十年中，有许多理论被提出，声称要对数学进行唯名论的重构。（Burgess & Rosen 1997）对这类观点进行了很好的批判性讨论。

In a nominalist reconstruction of mathematics, concrete entities will have to play the role that abstract entities play in platonistic accounts of mathematics, and concrete relations (such as the part-whole relation) have to be used to simulate mathematical relations between mathematical objects. But here problems arise. First, already Hilbert observed that, given the discretization of nature in quantum mechanics, the natural sciences may in the end claim that there are only finitely many concrete entities (Hilbert 1925). Yet it seems that we would need infinitely many of them to play the role of the natural numbers — never mind the real numbers. Where does the nominalist find the required collection of concrete entities? Secondly, even if the existence of infinitely many concrete objects is assumed, it is not clear that even elementary mathematical theories such as Primitive Recursive Arithmetic can be “simulated” by means of nominalistic relations (Niebergall 2000).
在数学的唯名论重构中，具体实体将不得不扮演在柏拉图主义数学理论中抽象实体所扮演的角色，并且需要使用具体关系（如整体与部分的关系）来模拟数学对象之间的数学关系。但这里出现了问题。首先，希尔伯特已经观察到，鉴于量子力学中自然的离散化，自然科学最终可能会声称只存在有限数量的具体实体（Hilbert 1925）。然而，似乎我们需要无限多的具体实体来扮演自然数的角色——更不用说实数了。唯名论者在哪里能找到所需的具体实体集合呢？其次，即使假设存在无限多的具体对象，也不清楚即使是原始递归算术这样的初等数学理论是否能通过唯名论关系来“模拟”（Niebergall 2000）。

Field made an earnest attempt to carry out a nominalistic reconstruction of Newtonian mechanics (Field 1980). The basic idea is this. Field wanted to use concrete surrogates of the real numbers and functions on them. He adopted a realist stance toward the spatial continuum, and took regions of space to be as physically real as chairs and tables. And he took regions of space to be concrete (after all, they are spatially located). If we also count the very disconnected ones, then there are as many regions of Newtonian space as there are subsets of the real numbers. And then there are enough concrete entities to play the role of the natural numbers, the real numbers, and functions on the real numbers. And the theory of the real numbers and functions on them is all that is needed to formulate Newtonian mechanics. Of course it would be even more interesting to have a nominalistic reconstruction of a truly contemporary scientific theory such as Quantum Mechanics. But given that the project can be carried out for Newtonian mechanics, some degree of initial optimism seems justified.
菲尔德认真地尝试对牛顿力学进行唯名论的重构（Field 1980）。基本的想法是这样的。菲尔德想要使用实数及其函数的具体替代品。他采取了对空间连续体的实在论立场，并认为空间区域与椅子和桌子一样在物理上是真实的。而且他认为空间区域是具体的（毕竟，它们在空间上有位置）。如果我们把那些非常不连续的区域也计算在内，那么牛顿空间的区域数量与实数子集的数量一样多。然后就有足够具体实体来扮演自然数、实数以及实数函数的角色。而实数及其函数的理论正是表述牛顿力学所需要的。当然，对量子力学这样的真正当代科学理论进行唯名论的重构会更有趣。但鉴于这个项目可以用于牛顿力学，一定程度的初步乐观似乎是合理的。

This project clearly has its limitations. It may be possible nominalistically to interpret theories of function spaces on the real numbers, say. But it seems far-fetched to think that along Fieldian lines a nominalistic interpretation of set theory can be found. Nevertheless, if it is successful within its confines, then Field’s program has really achieved something. For it would mean that, to some extent at least, mathematical entities appear to be dispensable after all. He would thereby have taken an important step towards undermining the indispensability argument for Quinean modest platonism in mathematics, for, to some extent, mathematical entities appear to be dispensable after all.
这个项目显然有其局限性。也许可以唯名论地解释实数上的函数空间理论。但沿着菲尔德的路线找到集合论的唯名论解释似乎不太可能。然而，如果在其范围内取得成功，那么菲尔德的计划确实取得了某种成就。因为这意味着，至少在某种程度上，数学实体似乎是多余的。他因此迈出了重要的一步，削弱了奎因式温和柏拉图主义在数学中的不可或缺性论证，因为数学实体在某种程度上似乎是多余的。

Field’s strategy only has a chance of working if Hilbert’s fear that in a very fundamental sense our best scientific theories may entail that there are only finitely many concrete entities, is ill-founded. If one sympathizes with Hilbert’s concern but does not believe in the existence of abstract entities, then one might bite the bullet and claim that there are only finitely many mathematical entities, thus contradicting the basic principles of elementary arithmetic. This leads to a position that has been called ultra-finitism (Essenin-Volpin 1961).
菲尔德的策略只有在希尔伯特的担忧是错误的情况下才有可能奏效，希尔伯特担心我们最好的科学理论在非常根本的意义上可能意味着只有有限数量的具体实体。如果有人同情希尔伯特的担忧，但不相信抽象实体的存在，那么他可能会直面问题，声称只有有限数量的数学实体，从而与初等算术的基本原理相矛盾。这就导致了一种被称为超有限主义的立场（Essenin-Volpin 1961）。

On most accounts, ultra-finitism leads, like intuitionism, to revisionism in mathematics. For it would seem that one would then have to say that there is a largest natural number, for instance. From the outside, a theory postulating only a finite mathematical universe appears proof-theoretically weak, and therefore very likely to be consistent. But Woodin has developed an argument that purports to show that from the ultra-finitist perspective, there are no grounds for asserting that the ultra-finitist theory is likely to be consistent (Woodin 2011).
在大多数说法中，超有限主义像直觉主义一样导致数学中的修正主义。因为似乎人们会说存在最大的自然数。从外部来看，一个只假设有限数学宇宙的理论在证明理论上显得很弱，因此很可能是一致的。但伍丁已经发展了一个论证，声称从超有限主义的角度来看，没有理由认为超有限主义理论很可能是一致的（Woodin 2011）。

Regardless of this argument (the details of which are not discussed here), many already find the assertion that there is a largest number hard to swallow. But Lavine has articulated a sophisticated form of set-theoretical ultra-finitism which is mathematically non-revisionist (Lavine 1994). He has developed a detailed account of how the principles of ZFC can be taken to be principles that describe determinately finite sets, if these are taken to include indefinitely large ones.
不管这个论证（这里不讨论其细节），许多人已经觉得存在最大数字的说法很难接受。但是拉文已经阐述了一种复杂的集合论超有限主义形式，这种形式在数学上是非修正主义的（Lavine 1994）。他详细说明了如果将这些集合包括无限大的集合，那么 ZFC 的公理可以被看作是描述确定有限集合的公理。

4.4 In Rebus structuralism

事物中结构主义

Field’s physicalist interpretation of arithmetic and analysis not only undermines the Quine-Putnam indispensability argument. It also partially provides an answer to Benacerraf’s epistemological challenge. Admittedly it is not a simple task to give an account of how humans obtain knowledge of spacetime regions. But at least according to many (but not all) philosophers spacetime regions are physically real. So we are no longer required to explicate how flesh and blood mathematicians stand in contact with non-physical entities. But Benacerraf’s identification problem remains. One may wonder why one spacetime point or region rather than another plays the role of the number $\pi$, for instance.
菲尔德对算术和分析的物理主义解释不仅削弱了奎因 - 普特南的不可或缺性论证。它还部分地回答了贝纳塞拉夫的认识论挑战。承认解释人类如何获得对时空区域的知识并非易事。但至少根据许多（而非全部）哲学家的说法，时空区域在物理上是真实的。因此，我们不再需要解释有血有肉的数学家如何与非物理实体接触。但贝纳塞拉夫的识别问题仍然存在。人们可能会好奇，为什么是某个时空点或区域，而不是另一个，在扮演数字$\pi$ 的角色。

In response to the identification problem, it seems attractive to combine a structuralist approach with Field’s nominalism. This leads to versions of nominalist structuralism, which can be outlined as follows. Let us focus on mathematical analysis. The nominalist structuralist denies that any concrete physical system is the unique intended interpretation of analysis. All concrete physical systems that satisfy the basic principles of Real Analysis (RA) would do equally well. So the content of a sentence $\varphi$ of the language of analysis is (roughly) given by:
Every concrete system $S$ that makes RA true, also makes $\varphi$ true.
为了应对识别问题，将结构主义方法与菲尔德的唯名论结合起来似乎很有吸引力。这导致了唯名论结构主义的版本，可以概述如下。让我们专注于数学分析。唯名论结构主义者否认任何具体的物理系统是分析的唯一预期解释。所有满足实分析（RA）基本原则的具体物理系统同样适用。因此，分析语言中的一个句子$\varphi$ 的内容（大致）是：

Every concrete system $S$ that makes RA true, also makes $\varphi$ true.
使 RA 为真的每一个具体系统 $S$，也使 $\varphi$ 为真。

This entails that, as with ante rem structuralism, only structural aspects are relevant to the truth or falsehood of mathematical statements. But unlike ante rem structuralism, no abstract structure is postulated above and beyond concrete systems.
根据这一点，与先验 结构主义一样，只有结构方面的内容与数学陈述的真实性或虚假性有关。但与先验 结构主义不同的是，这里并没有假设超出具体系统之外的任何抽象结构。

According to in rebus structuralism, no abstract structures exist over and above the systems that instantiate them; structures exist only in the systems that instantiate them. For this reason nominalist in rebus structuralism is sometimes described as “structuralism without structures”. Nominalist structuralism is a form of in rebus structuralism. But in rebus structuralism is not exhausted by nominalist structuralism. Even the version of platonism that takes mathematics to be about structures in the set-theoretic sense of the word can be viewed as a form of in rebus structuralism.
根据事物中 结构主义，不存在超出实例化它们的系统之外的抽象结构；结构只存在于实例化它们的系统之中。因此，唯名论事物中 结构主义有时被描述为“没有结构的结构主义”。唯名论结构主义是事物中 结构主义的一种形式。但事物中 结构主义并不完全等同于唯名论结构主义。即使是将数学视为关于集合论意义上的结构的柏拉图主义版本，也可以被视为事物中 结构主义的一种形式。

In mathematical discourse, non-algebraic structures (such as ‘the’ natural numbers) and mathematical objects (such as ‘the’ number 1) are referred to by definite descriptions. This strongly suggests that mathematical symbols (N, 1) have a unique reference rather than a ‘distributed’ one as in rebus structuralism would have it. But in rebus structuralists argue that such mathematical symbols function as dedicated variables in much the same way as in ‘Tommy needs his letters from home’, a world war II slogan, the name ‘Tommy’ is chosen to stand for some arbitrary concrete soldier, and re-used on many occasions without changing its reference (Pettigrew 2008). 在数学话语中，非代数结构（如“自然数”）和数学对象（如“数字 1”）通过特定描述来指代。这强有力地表明，数学符号（N，1）有唯一的指代，而不是像事物中 结构主义所说的“分散的”指代。但是事物中 结构主义者认为，这样的数学符号就像在“Tommy 需要来自家乡的信”，这个二战时期的口号中，“Tommy”这个名字被选来代表某个任意的具体士兵，并且在许多场合反复使用而不改变其指代（Pettigrew 2008）。

If Hilbert’s worry is wellfounded in the sense that there are no concrete physical systems that make the postulates of mathematical analysis true, then the above nominalist structuralist rendering of the content of a sentence $\varphi$ of the language of analysis gets the truth conditions of such sentences wrong. For then for every universally quantified sentence $\varphi$, its paraphrase will come out vacuously true. So an existential assumption to the effect that there exist concrete physical systems that can serve as a model for RA is needed to back up the above analysis of the content of mathematical statements. Perhaps something like Field’s construction fits the bill. 如果希尔伯特的担忧是合理的，即不存在使数学分析的公理为真的具体物理系统，那么上述唯名论结构主义对分析语言中的句子$\varphi$ 的内容的解释就会使这类句子的真值条件出错。因为那时对于每一个全称量词句子$\varphi$，它的释义将空洞地为真。因此，需要一个存在性的假设，即存在可以作为 RA 模型的具体物理系统，来支持上述对数学陈述内容的分析。也许菲尔德的构造就是这样的。

Putnam noticed early on that if the above explication of the content of mathematical sentences is modified somewhat, a substantially weaker background assumption is sufficient to obtain the correct truth conditions (Putnam 1967). Putnam proposed the following modal rendering of the content of a sentence $\varphi$ of the language of analysis:
普特南很早就注意到，如果对上述数学句子的内容的解释进行一些修改，那么一个明显更弱的背景假设就足以得到正确的真值条件（Putnam 1967）。普特南提出了以下对分析语言中的句子$\varphi$ 的内容的模态解释：

Necessarily, every concrete system $S$ that makes RA true, also makes $\varphi$ true.
必然地，使 RA 为真的每一个具体系统$S$，也使$\varphi$ 为真。

This is a stronger statement than the nonmodal rendering that was presented earlier. But it seems equally plausible. And an advantage of this rendering is that the following modal existential background assumption is sufficient to make the truth conditions of mathematical statements come out right:
这比之前提出的非模态解释是一个更强的表述。但它似乎同样合理。而且这种解释的一个优点是，以下模态存在性背景假设足以使数学陈述的真值条件正确：

It is possible that there exists a concrete physical system that can serve as a model for RA.
可能存在一个可以作为 RA 模型的具体物理系统。

(‘It is possible that’ here means ‘It is or might have been the case that’.) Now Hilbert’s concern seems adequately addressed. For on Putnam’s account, the truth of mathematical sentences no longer depends on physical assumptions about the actual world.
（这里的“可能”意味着“是或可能是这样的情况。”）现在，希尔伯特的担忧似乎得到了充分的解决。因为在普特南的解释中，数学句子的真值不再依赖于对现实世界的物理假设。

It is admittedly not easy to give a satisfying account of how we know that this modal existential assumption is fulfilled. But it may be hoped that the task is less daunting than the task of explaining how we succeed in knowing facts about abstract entities. And it should not be forgotten that the structuralist aspect of this (modal) nominalist position keeps Benacerraf’s identification challenge at bay.
承认很难给出一个令人满意的解释，说明我们是如何知道这个模态存在性假设得到了满足。但或许可以希望，这个任务比解释我们是如何成功地了解关于抽象实体的事实的任务要不那么艰巨。而且不应忘记，这种（模态）唯名论立场的结构主义方面仍然使贝纳塞拉夫的识别挑战无法得逞。

Putnam’s strategy also has its limitations. Chihara sought to apply Putnam’s strategy not only to arithmetic and analysis but also to set theory (Chihara 1973). Then a crude version of the relevant modal existential assumption becomes:
普特南的策略也有其局限性。齐哈拉试图将普特南的策略不仅应用于算术和分析，还应用于集合论（Chihara 1973）。那么，相关的模态存在性假设的一个粗糙版本就变成了：

It is possible that there exist concrete physical systems that can serve as a model for ZFC.
可能存在可以作为 ZFC 模型的具体物理系统。

Parsons has noted that when possible worlds are needed which contain collections of physical entities that have large transfinite cardinalities or perhaps are even too large to have a cardinal number, it becomes hard to see these as possible concrete or physical systems (Parsons 1990a). We seem to have no reason to believe that there could be physical worlds that contain highly transfinitely many entities.
帕森斯指出，当需要包含具有大超限基数或甚至太大而无法有基数的物理实体集合的可能世界时，很难将这些视为可能的具体或物理系统（Parsons 1990a）。我们似乎没有理由相信可能存在包含高度超限多实体的物理世界。

4.5 Fictionalism

虚构主义

According to the previous proposals, the statements of ordinary mathematics are true when suitably, i.e., nominalistically, interpreted. The nominalistic account of mathematics that will now be discussed holds that all existential mathematical statements are false simply because there are no mathematical entities.
根据前面的提议，普通数学的陈述在适当地，即唯名论地，解释时是真的。现在将要讨论的唯名论数学解释认为，所有存在性的数学陈述都是假的，仅仅是因为不存在数学实体。

For the same reason all universal mathematical statements will be trivially true. Fictionalism holds that mathematical theories are like fiction stories such as fairy tales and novels. Mathematical theories describe fictional entities, in the same way that literary fiction describes fictional characters. This position was first articulated in the introductory chapter of (Field 1989), and has in recent years been gaining in popularity.
同样的原因，所有全称的数学陈述将平凡地为真。虚构主义认为数学理论就像童话和小说这样的虚构故事。数学理论描述虚构实体，就像文学虚构描述虚构人物一样。这种立场最初在（Field 1989）的引言章节中被阐述，并且在近年来越来越受欢迎。

This crude description of the fictionalist position immediately opens up the question what sort of entities fictional entities are. This appears to be a deep metaphysical ontological problem. One way to avoid this question altogether is to deny that there exist fictional entities. Mathematical theories should be viewed as invitations to participate in games of pretence, in which we act as if certain mathematical entities exist. Pretence or make-believe operators shield their propositional objects from existential exportation (Leng 2010).
这种对虚构主义立场的简单描述立即引发了关于虚构实体是什么样的实体的问题。这似乎是一个深奥的形而上学本体论问题。完全回避这个问题的一个方法是否认存在虚构实体。数学理论应该被视为邀请人们参与假装游戏，在这些游戏中，我们表现得好像某些数学实体存在一样。假装或虚构算子保护它们的命题对象免受存在性导出（Leng 2010）。

Anyway, as said above, on the fictionalist view, a mathematical theory isn’t literally true. Nonetheless, mathematics is used to get truths across. So we must subtract something from what is literally said when we assert a physical theory that involves mathematics, if we want to get at the truth. But this requires a theory of how this subtraction of content works. Such a theory has been developed in (Yablo, 2014).
无论如何，正如前面所说，在虚构主义的观点中，数学理论并不是字面意义上的真。然而，数学被用来传达真理。因此，如果我们想要得到真理，当我们断言一个涉及数学的物理理论时，我们必须从字面上所说的中减去一些东西。但这需要一个关于这种内容减法是如何工作的理论。这样的理论已经在（Yablo, 2014）中得到了发展。

If the fictionalist thesis is correct, then one demand that must be imposed on mathematical theories is surely consistency. Yet Field adds to this a second requirement: mathematics must be conservative over natural science. This means, roughly, that whenever a statement of an empirical theory can be derived using mathematics, it can in principle also be derived without using any mathematical theories. If this were not the case, then an indispensability argument could be played out against fictionalism. Whether mathematics is in fact conservative over physics, for instance, is currently a matter of controversy. Shapiro has formulated an incompleteness argument that intends to refute Field’s claim (Shapiro 1983).
如果虚构主义的观点是正确的，那么必须对数学理论提出的一个要求肯定是它们必须是一致的。然而，菲尔德又加上了第二个要求：数学必须对自然科学是保守的。这意味着，大致上，每当一个经验理论的陈述可以用数学推导出来时，原则上也可以不用任何数学理论来推导。如果不是这样的话，那么就可以对虚构主义提出一个不可或缺性论证。例如，数学是否实际上对物理学是保守的，目前还是一个有争议的问题。夏皮罗已经提出了一个不完全性论证，旨在驳斥菲尔德的说法（Shapiro 1983）。

If there are indeed no mathematical (fictional) entities, as one form of fictionalism has it, then Benacerraf’s epistemological problem does not arise. Fictionalism then shares this advantage over most forms of platonism with nominalistic reconstructions of mathematics. But the appeal to pretence operators entails that the logical form of mathematical sentences then differs somewhat from their surface form. If there are fictional objects, then the surface form of mathematical sentences can be taken to coincide with their logical form. But if they exist as abstract entities, then Benacerraf’s epistemological problem reappears.
如果确实不存在数学（虚构的）实体，就像一种虚构主义所说的那样，那么贝纳塞拉夫的认识论问题就不会出现。虚构主义在这一点上与柏拉图主义的大多数形式相比，与数学的唯名论重构共享这一优势。但对虚构算子的诉求意味着数学句子的逻辑形式与它们的表面形式有所不同。如果存在虚构对象，那么数学句子的表面形式可以被认为与其逻辑形式一致。但如果它们作为抽象实体存在，那么贝纳塞拉夫的认识论问题又会重新出现。

Whether Benacerraf’s identification problem is solved is not completely clear. In general, fictionalism is a non-reductionist account. Whether an entity in one mathematical theory is identical with an entity that occurs in another theory is usually left indeterminate by mathematical “stories”. Yet Burgess has rightly emphasized that mathematics differs from literary fiction in the fact that fictional characters are usually confined to one work of fiction, whereas the same mathematical entities turn up in diverse mathematical theories (Burgess 2004). After all, entities with the same name (such as $\pi$) turn up in different theories. Perhaps the fictionalist can maintain that when mathematicians develop a new theory in which an “old” mathematical entity occurs, the entity in question is made more precise. More determinate properties are ascribed to it than before, and this is all right as long as overall consistency is maintained.
贝纳塞拉夫的识别问题是否得到解决还不完全清楚。一般来说，虚构主义是一种非还原论的解释。一个数学理论中的实体是否与另一个理论中出现的实体相同，通常由数学“故事”来决定。然而，伯吉斯正确地强调，数学与文学虚构不同，虚构人物通常局限于一部虚构作品，而相同的数学实体会出现在不同的数学理论中（Burgess 2004）。毕竟，具有相同名称（如$\pi$）的实体会在不同的理论中出现。也许虚构主义者可以坚持认为，当数学家发展一个新的理论，其中出现了一个“旧的”数学实体时，该实体被进一步精确化。比以前赋予了更明确的属性，只要保持整体一致性，这都是可以接受的。

The canonical objection to formalism seems also applicable to fictionalism. The fictionalists should find some explanation of the fact that extending a mathematical theory in one way, is often considered preferable over continuing it in a another way that is incompatible with the first. There is often at least an appearance that there is a right way to extend a mathematical theory.
对形式主义的经典反对意见似乎也适用于虚构主义。虚构主义者应该找到一种解释，即以一种方式扩展一个数学理论，通常被认为比以另一种与第一种不兼容的方式继续它更可取。通常至少有一种感觉，即有一种正确的方式来扩展一个数学理论。

The canonical objection to formalism seems also applicable to fictionalism. The fictionalists should find some explanation of the fact that extending a mathematical theory in one way, is often considered preferable over continuing it in another way that is incompatible with the first. There is often at least an appearance that there is a right way to extend a mathematical theory. 对形式主义的经典反对意见似乎也适用于虚构主义。虚构主义者应该找到一种解释，即以一种方式扩展一个数学理论，通常被认为比以另一种与第一种不兼容的方式继续它更可取。通常至少有一种感觉，即有一种正确的方式来扩展一个数学理论。

5. Special Topics

专题

In recent years, subdisciplines of the philosophy of mathematics have started to arise. They evolve in a way that is not completely determined by the “big debates” about the nature of mathematics. In this section, we look at a few of these disciplines.
在近年来，数学哲学的子学科开始出现。它们的发展方式并不完全由关于数学本质的“大辩论”决定。在本节中，我们将关注其中的一些学科。

5.1 Foundations and Set Theory

基础与集合论

Many regard set theory as in some sense the foundation of mathematics. It seems that just about any piece of mathematics can be carried out in set theory, even though it is sometimes an awkward setting for doing so. In recent years, the philosophy of set theory is emerging as a philosophical discipline of its own. This is not to say that in specific debates in the philosophy of set theory it cannot make an enormous difference whether one approaches it from a formalistic point of view or from a platonistic point of view, for instance.
许多认为集合论在某种意义上是数学的基础。似乎几乎任何数学都可以在集合论中进行，尽管这有时是一个笨拙的环境。近年来，集合论哲学作为一个独立的哲学学科正在兴起。这并不是说，在集合论哲学的具体辩论中，从形式主义观点还是柏拉图主义观点来探讨并没有巨大差异。

The thesis that set theory is most suitable for serving as the foundations of mathematics is by no means uncontroversial. Over the past decades, category theory has presented itself as a rival for this role. Category theory is a mathematical theory that was developed in the middle of the twentieth century. Unlike in set theory, in category theory mathematical objects are only defined up to isomorphism. This means that Benacerraf’s identification problem cannot be raised for category theoretical concepts and ‘objects’. At the same time, (roughly) everything that can be done in set theory can be done in category theory (but not always in a natural manner), and vice versa (again not always in a natural manner). This means that for a structuralist perspective, category theory is an attractive candidate for providing the foundations of mathematics (McLarty 2004).
认为集合论最适合充当数学基础的论点绝非毫无争议。在过去的几十年中，范畴论已成为这一角色的竞争者。范畴论是 20 世纪中期发展起来的数学理论。与集合论不同，在范畴论中，数学对象仅仅定义到同构为止。这意味着贝纳塞拉夫的识别问题无法针对范畴论的概念和“对象”提出。同时，（大致上）在集合论中可以做的事情在范畴论中也可以做到（但并不总是以自然的方式），反之亦然（同样不总是以自然的方式）。这意味着从结构主义的角度来看，范畴论是为数学提供基础的有吸引力的候选者（McLarty 2004）。

One question that has been important from the beginning of set theory concerns the difference between sets and proper classes. (This question has a natural counterpart for category theory: the difference between small and large categories.) Cantor’s diagonal argument forces us to recognize that the set-theoretical universe as a whole cannot be regarded as a set. Cantor’s Theorem shows that the power set (i.e., the set of all subsets) of any given set has a larger cardinality than the given set itself. Now suppose that the set-theoretical universe forms a set: the set of all sets. Then the power set of the set of all sets would have to be a subset of the set of all sets. This would contradict the fact that the power set of the set of all sets would have a larger cardinality than the set of all sets. So we must conclude that the set-theoretical universe cannot form a set.
从集合论的开始，一个重要问题涉及集合与真类之间的区别。（这个问题在范畴论中有自然的对应物：小范畴与大范畴之间的区别。）康托的对角线论证迫使我们认识到，集合论宇宙整体不能被视为一个集合。康托定理表明，任何给定集合的幂集（即所有子集的集合）的基数大于给定集合本身的基数。现在假设集合论宇宙形成一个集合：所有集合的集合。那么所有集合的集合的幂集必须是所有集合的集合的一个子集。这将与所有集合的集合的幂集的基数大于所有集合的集合的基数这一事实相矛盾。因此，我们必须得出结论，集合论宇宙不能形成一个集合。

Cantor called pluralities that are too large to be considered as a set inconsistent multiplicities (Cantor 1932). Today, Cantor’s inconsistent multiplicities are called proper classes. Some philosophers of mathematics hold that proper classes still constitute unities, and hence can be seen as a sort of collection. They are, in a Cantorian spirit, just collections that are too large to be sets. Nevertheless, there are problems with this view. Just as there can be no set of all sets, there can for diagonalization reasons also not be a proper class of all proper classes. So the proper class view seems compelled to recognize in addition a realm of super-proper classes, and so on. For this reason, Zermelo claimed that proper classes simply do not exist. This position is less strange than it looks at first sight. On close inspection, one sees that in ZFC one never needs to quantify over entities that are too large to be sets (although there exist systems of set theory that do quantify over proper classes). On this view, the set-theoretical universe is potentially infinite in an absolute sense of the word. It never exists as a completed whole, but is forever growing, and hence forever unfinished (Zermelo 1930). This way of speaking indicates that in our attempts to understand this notion of potential infinity, we are drawn to temporal metaphors. It is not surprising that these temporal metaphors cause some philosophers of mathematics acute discomfort. For this reason, contemporary philosophers of mathematics who are sympathetic to Zermelo’s potentialist interpretation of the set theoretic universe, tend to regard the modality involved in this interpretation as a non-temporal one: the nature of this modality is hotly debated (Linnebo 2013, Studd 2019).
康托将那些太大而不能被视为集合的多重性称为不一致的多重性（Cantor 1932）。如今，康托的不一致多重性被称为真类。一些数学哲学家认为真类仍然构成统一体，因此可以被视为一种集合。它们只是太大而不能成为集合的集合，具有康托精神。然而，这种观点存在问题。正如不能有所有集合的集合一样，由于对角化原因，也不能有所有真类的真类。因此，真类观点似乎不得不承认还有超真类的领域，依此类推。因此，策梅洛声称真类根本不存在。这一立场并不像乍看起来那么奇怪。仔细审查后会发现，在 ZFC 中，从不需要对那些太大而不能成为集合的实体进行量化（尽管存在一些集合论系统会对真类进行量化）。在这种观点下，集合论宇宙在绝对意义上是潜在无限的。它从未作为一个完整的整体存在，而是永远在增长，因此永远未完成（Zermelo 1930）。这种说法表明，在我们试图理解这种潜在无限的概念时，我们倾向于使用时间隐喻。这些时间隐喻给一些数学哲学家带来了极大的不适，这并不奇怪。因此，当代对策梅洛集合论宇宙的潜能论解释持同情态度的数学哲学家，倾向于将这种解释中涉及的模态视为非时间性的：这种模态的性质是激烈争论的话题（Linnebo 2013, Studd 2019）。

A second subject in the philosophy of set theory concerns the justification of the accepted basic principles of mathematics, i.e., the axioms of ZFC. An important historical case study is the process by which the Axiom of Choice came to be accepted by the mathematical community in the early decades of the twentieth century (Moore 1982). The importance of this case study is largely due to the fact that an open and explicit discussion of its acceptability was held in the mathematical community. In this discussion, general reasons for accepting or refusing to accept a principle as a basic axiom came to the surface. On the systematic side, two conceptions of the notion of set have been elaborated which aim to justify all axioms of ZFC in one fell swoop. On the one hand, there is the iterative conception of sets, which describes how the set-theoretical universe can be thought of as generated from the empty set by means of the power set operation (Boolos 1971, Linnebo 2013). On the other hand, there is the limitation of size conception of sets, which states that every collection which is not too big to be a set, is a set (Hallett 1984). The iterative conception motivates some axioms of ZFC very well (the power set axiom, for instance), but fares less well with respect to other axioms, such as the replacement axiom (Potter 2004, Part IV). The limitation of size conception motivates other axioms better (such as the restricted comprehension axiom). It seems fair to say that there is no uniform conception that clearly justifies all axioms of ZFC.
集合论哲学的第二个主题涉及对数学中被接受的基本原则，即 ZFC 的公理的合理性证明。一个重要的历史案例研究是选择公理在 20 世纪早期几十年被数学界接受的过程（Moore 1982）。这个案例研究的重要性主要在于数学界对它的可接受性进行了公开和明确的讨论。在这场讨论中，接受或拒绝接受一个原则作为基本公理的一般理由浮出水面。从系统方面来看，已经详细阐述了两种关于集合概念的观念，旨在一举证明 ZFC 的所有公理。一方面，有集合的迭代观念，它描述了如何通过幂集运算从空集生成集合论宇宙（Boolos 1971, Linnebo 2013）。另一方面，有集合的大小限制观念，它声称每一个不是太大而不能成为集合的集合，就是一个集合（Hallett 1984）。迭代观念很好地激励了 ZFC 的一些公理（例如幂集公理），但在其他公理（如替换公理）方面表现不佳（Potter 2004, 第四部分）。大小限制观念更好地激励了其他公理（如受限概括公理）。可以说，没有一个统一的观念能够清楚地证明 ZFC 的所有公理。

The motivation of putative axioms that go beyond ZFC constitutes a third concern of the philosophy of set theory (Maddy 1988; Martin 1998). One such class of principles is constituted by the large cardinal axioms. Nowadays, large cardinal hypotheses are really taken to mean some kind of embedding properties between the set theoretic universe and inner models of set theory (Kanamori 2009). Most of the time, large cardinal principles entail the existence of sets that are larger than any sets which can be guaranteed by ZFC to exist.
大于 ZFC 的公理动机构成了集合论哲学的第三个关注点（Maddy 1988；Martin 1998）。其中一类原则是由大基数公理构成的。如今，大基数假设实际上被理解为集合论宇宙与集合论内模型之间的一种嵌入性质（Kanamori 2009）。大多数情况下，大基数原则意味着存在比 ZFC 所能保证存在的任何集合都大的集合。

The weaker of the large cardinal principles are supported by intrinsic evidence (see section 3.1). They follow from what are called reflection principles. These are principles that state that the set theoretic universe as a whole is so rich that it is very similar to some set-sized initial segment of it. The stronger of the large cardinal principles hitherto only enjoy extrinsic support. Many researchers are skeptical about the possibility that reflection principles, for instance, can be found that support them (Koellner 2009); others, however, disagree (Welch & Horsten 2016).
大基数原则中较弱的那些得到了内在证据的支持（见第 3.1 节）。它们遵循所谓的反射原则。这些原则表明，集合论宇宙作为一个整体是如此丰富，以至于它与它的一个集合大小的初始片段非常相似。到目前为止，较强的大基数原则只得到了外在支持。许多研究者对反射原则能否支持它们持怀疑态度（Koellner 2009）；然而，其他人不同意（Welch & Horsten 2016）。

Gödel hoped that on the basis of such large cardinal axioms, the most important open question of set theory could eventually be settled. This is the continuum problem. The continuum hypothesis was proposed by Cantor in the late nineteenth century. It states that there are no sets $S$ which are too large for there to be a one-to-one correspondence between $S$ and the natural numbers, but too small for there to exist a one-to-one correspondence between $S$ and the real numbers. Despite strenuous efforts, all attempts to settle the continuum problem failed. Gödel came to suspect that the continuum hypothesis is independent of the accepted principles of set theory (ZFC). Around 1940, he managed to show that the continuum hypothesis is consistent with ZFC. A few decades later, Paul Cohen proved that the negation of the continuum hypothesis is also consistent with ZFC. Thus Gödel’s conjecture of the independence of the continuum hypothesis was eventually confirmed.
哥德尔希望基于这样的大基数公理，最终能够解决集合论中最重要的未决问题。这就是连续统问题。连续统假设是由康托在 19 世纪末提出的。它声称不存在集合$S$，使得$S$ 与自然数之间无法建立一一对应关系，但与实数之间可以建立一一对应关系。尽管经过了艰苦的努力，所有解决连续统问题的尝试都失败了。哥德尔开始怀疑连续统假设独立于集合论的公认原则（ZFC）。大约在 1940 年，他设法证明了连续统假设与 ZFC 一致。几十年后，保罗·科恩证明了连续统假设的否定也与 ZFC 一致。因此，哥德尔关于连续统假设独立性的猜想最终得到了证实。

But Gödel’s hope that large cardinal axioms could solve the continuum problem turned out to be unfounded. The continuum hypothesis is independent of ZFC even in the context of large cardinal axioms. Nevertheless, large cardinal principles have manage to settle restricted versions of the continuum hypothesis (in the affirmative). The existence of so-called Woodin cardinals ensures that sets definable in analysis are either countable or the size of the continuum. Thus the definable continuum problem is settled.
但哥德尔希望大基数公理能够解决连续统问题的希望最终被证明是不成立的。即使在大基数公理的背景下，连续统假设也独立于 ZFC。然而，大基数原则已经设法解决了连续统假设的受限版本（肯定）。所谓 Woodin 基数的存在确保了在分析中可定义的集合要么是可数的，要么是连续统的大小。因此，可定义的连续统问题得到了解决。

In recent years, attempts have been focused on finding principles of a different kind which might be justifiable and which might yet decide the continuum hypothesis (Woodin 2001a, Woodin 2001b). One of the more general philosophical questions that have emerged from this research is the following: which conditions have to be satisfied in order for a principle to be a putative basic axiom of mathematics?
近年来，研究的重点是寻找不同种类的原则，这些原则可能是合理的，并且可能最终决定连续统假设（Woodin 2001a, Woodin 2001b）。从这项研究中出现的一个更一般的哲学问题是：一个原则要满足哪些条件才能成为数学的潜在基本公理？

Some of the researchers who seek to decide the continuum hypothesis think that it is true; others think that it is false. But there are also many set theorists and philosophers of mathematics who believe that the continuum hypothesis not just undecidable in ZFC but absolutely undecidable, i.e. that it is neither provable (in the informal sense of the word) nor disprovable (in the informal sense of the word) because it is neither true nor false. If the mathematical universe is a set theoretic multiverse, for instance, then there are equally models that make the continuum hypothesis true and equally good models that make it false, and there is no more to be said (Hamkins, 2015).
一些试图决定连续统假设的研究者认为它是真的；另一些人则认为它是假的。但也有许多集合论者和数学哲学家相信，连续统假设不仅在 ZFC 中不可判定，而且是绝对不可判定的，即它既不能被证明（在非正式的意义上），也不能被反驳（在非正式的意义上），因为它既不是真的，也不是假的。例如，如果数学宇宙是一个集合论多宇宙，那么就有同样多的模型使连续统假设为真，也有同样多的模型使它为假，而且没有什么更多的可说的了（Hamkins, 2015）。

5.2 Categoricity and Pluralism

范畴性与多元性

In the second half of the nineteenth century Dedekind proved that the basic axioms of arithmetic have, up to isomorphism, exactly one model, and that the same holds for the basic axioms of Real Analysis. If a theory has, up to isomorphism, exactly one model, then it is said to be categorical. So modulo isomorphisms, arithmetic and analysis each have exactly one intended model. Half a century later Zermelo proved that the principles of set theory are “almost” categorical or quasi-categorical: for any two models $M_1$ and $M_2$ of the principles of set theory, either $M_1$ is isomorphic to $M_2$, or $M_1$ is isomorphic to a strongly inaccessible rank of $M_2$, or $M_2$ is isomorphic to a strongly inaccessible rank of $M_1$ (Zermelo 1930). In recent years, attempts have been made to develop arguments to the effect that Zermelo’s conclusion can be strengthened to a full categoricity assertion (McGee 1997; Martin 2001), but we will not discuss these arguments here.
在十九世纪下半叶，戴德金证明了算术的基本公理在同构的意义上恰好有一个模型，实分析的基本公理也是如此。如果一个理论在同构的意义上恰好有一个模型，那么它被称为范畴的（categorical）。因此，算术和分析在同构的意义上各自恰好有一个预期的模型。半个世纪后，策梅洛证明了集合论的原则是“几乎”范畴的，或者说是准范畴的（quasi-categorical）：对于集合论原则的任意两个模型 $M_1$ 和 $M_2$，要么 $M_1$ 与 $M_2$ 同构，要么 $M_1$ 与 $M_2$ 的某个强不可达层级同构，要么 $M_2$ 与 $M_1$ 的某个强不可达层级同构（策梅洛，1930）。近年来，有人试图发展论证，认为策梅洛的结论可以被加强为一个完全的范畴性断言（麦基，1997；马丁，2001），但这里我们不讨论这些论证。

At the same time, the Löwenheim-Skolem theorem says that every first-order formal theory that has at least one model with an infinite domain, must have models with domains of all infinite cardinalities. Since the principles of arithmetic, analysis and set theory had better possess at least one infinite model, the Löwenheim-Skolem theorem appears to apply to them. Is this not in tension with Dedekind’s categoricity theorems?
与此同时，洛文海姆 - 斯科伦定理指出，每一个具有一阶形式化的且至少有一个无限域的模型的理论，必然存在所有无限基数的模型。由于算术、分析和集合论的原则最好至少有一个无限模型，洛文海姆 - 斯科伦定理似乎适用于它们。这与戴德金的范畴性定理不是存在矛盾吗？

The solution of this conundrum lies in the fact that Dedekind did not even implicitly work with first-order formalizations of the basic principles of arithmetic and analysis. Instead, he informally worked with second-order formalizations.
这个悖论的解决方法在于戴德金甚至没有隐含地使用算术和分析的基本原则的一阶形式化。相反，他非正式地使用了二阶形式化。

Let us focus on arithmetic to see what this amounts to. The basic postulates of arithmetic contain the induction axiom. In first-order formalizations of arithmetic, this is formulated as a scheme: for each first-order arithmetical formula of the language of arithmetic with one free variable, one instance of the induction principle is included in the formalization of arithmetic. Elementary cardinality considerations reveal that there are infinitely many properties of natural numbers that are not expressed by a first-order formula. But intuitively, it seems that the induction principle holds for all properties of natural numbers. So in a first-order language, the full force of the principle of mathematical induction cannot be expressed. For this reason, a number of philosophers of mathematics insist that the postulates of arithmetic should be formulated in a second-order language (Shapiro 1991). Second-order languages contain not just first-order quantifiers that range over elements of the domain, but also second-order quantifiers that range over properties (or subsets) of the domain. In full second-order logic, it is insisted that these second-order quantifiers range over all subsets of the domain. If the principles of arithmetic are formulated in a second-order language, then Dedekind’s argument goes through and we have a categorical theory. For similar reasons, we also obtain a categorical theory if we formulate the basic principles of real analysis in a second-order language, and the second-order formulation of set theory turns out to be quasi-categorical.
让我们专注于算术来看看这意味着什么。算术的基本公理包含归纳公理。在算术的一阶形式化中，这是作为一个方案来表述的：对于算术语言中每一个带有一个自由变量的一阶算术公式，归纳原则的一个实例被包含在算术的形式化中。基本的基数考虑表明，有无限多自然数的性质不能用一阶公式来表达。但直觉上，似乎归纳原则适用于所有自然数的性质。因此，在一阶语言中，数学归纳原理的全部力量无法被表达。因此，许多数学哲学家坚持认为算术的公理应该在一个二阶语言中表述（夏皮罗，1991）。二阶语言不仅包含在一阶语言中对定义域元素进行量化的量词，还包含对定义域的性质（或子集）进行量化的二阶量词。在完全的二阶逻辑中，坚持这些二阶量词是对定义域的所有子集进行量化。如果算术的原则在一个二阶语言中表述，那么戴德金的论证就成立了，我们得到了一个范畴的理论。出于类似的原因，如果我们用二阶语言表述实分析的基本原则，我们也得到了一个范畴的理论，而集合论的二阶表述结果是准范畴的。

Ante rem structuralism, as well as the modal nominalist structuralist interpretation of mathematics, could benefit from a second-order formulation. If the ante rem structuralist wants to insists that the natural number structure is fixed up to isomorphism by the Peano axioms, then she will want to formulate the Peano axioms in second-order logic. And the modal nominalist structuralist will want to insist that the relevant concrete systems for arithmetic are those that make the second-order Peano axioms true (Hellman 1989). Similarly for real analysis and set theory. Thus the appeal to second-order logic appears as the final step in the structuralist project of isolating the intended models of mathematics.
Ante rem 结构主义以及数学的模态唯名论结构主义解释都可以从二阶表述中获益。如果ante rem 结构主义者想要坚持自然数结构是由皮亚诺公理固定到同构的，那么她就会想要用二阶逻辑来表述皮亚诺公理。而模态唯名论结构主义者会坚持认为，对于算术来说，相关的具体系统是那些使二阶皮亚诺公理为真的系统（赫尔曼，1989）。实分析和集合论也是如此。因此，对二阶逻辑的诉求似乎是在结构主义项目中孤立数学的预期模型的最后一步。

Yet appeal to second-order logic in the philosophy of mathematics is by no means uncontroversial. A first objection is that the ontological commitment of second-order logic is higher than the ontological commitment of first-order logic. After all, use of second-order logic seems to commit us to the existence of abstract objects: classes. In response to this problem, Boolos has articulated an interpretation of second-order logic which avoids this commitment to abstract entities (Boolos 1985). His interpretation spells out the truth clauses for the second-order quantifiers in terms of plural expressions, without invoking classes. For instance, an second-order expression of the form $\exists xF(x)$ is interpreted as: “there are some (first-order objects) $x$ such that they have the property $F$”. This interpretation is called the plural interpretation of second-order logic. It is controversial whether there is a real difference between the mathematical use of pluralities and of sets (Linnebo 2003). Nevertheless it is clear that an appeal to the plural interpretation of second-order logic will be tempting for nominalist versions of structuralism.
然而，在数学哲学中对二阶逻辑的诉求绝非没有争议。第一个反对意见是，二阶逻辑的本体论承诺高于一阶逻辑的本体论承诺。毕竟，使用二阶逻辑似乎使我们承诺了抽象对象的存在：类。针对这个问题，布卢斯阐述了一种二阶逻辑的解释，避免了对抽象实体的这种承诺（布卢斯，1985）。他的解释用复数表达式来说明二阶量词的真实性条款，而不涉及类。例如，形式为 $\exists xF(x)$ 的二阶表达式被解释为：“有一些（一阶对象）$x$，它们具有性质 $F$”。这种解释被称为二阶逻辑的复数解释。数学中使用复数与使用集合之间是否存在真正的区别是有争议的（林内博，2003）。然而，很明显，对二阶逻辑的复数解释的诉求对结构主义的唯名论版本来说是很有吸引力的。

A second objection against second-order logic can be traced back to Quine (Quine 1970). This objection states that the interpretation of full second-order logic is connected with set-theoretical questions. This is already indicated by the fact that most regimentations of second-order logic adopt a version of the axiom of choice as one of its axioms. But more worrisome is the fact that second-order logic is inextricably intertwined with deep problems in set theory, such as the continuum hypothesis. For theories such as arithmetic that intend to describe an infinite collection of objects, even a matter as elementary as the question of the cardinality of the range of the second-order quantifiers, is equivalent to the continuum problem. Also, it turns out that there exists a sentence which is a second-order logical truth if and only if the continuum hypothesis holds (Boolos 1975). We have seen that the continuum problem is independent of the currently accepted principles of set theory. And many researchers believe it to be absolutely truth-valueless. If this is so, then there is an inherent indeterminacy in the very notion of second-order infinite model. And many contemporary philosophers of mathematics take the latter not to have a determinate truth value. Thus, it is argued, the very notion of an (infinite) model of full second-order logic is inherently indeterminate.
对二阶逻辑的第二个反对意见可以追溯到奎因（奎因，1970）。这个反对意见指出，完全二阶逻辑的解释与集合论问题有关。这已经从大多数二阶逻辑的规范化采用选择公理的某个版本作为其公理之一这一事实中得到了表明。但更令人担忧的是，二阶逻辑与集合论中的深刻问题，如连续统假设，是不可分割地交织在一起的。对于打算描述无限对象集合的理论，如算术，甚至像二阶量词范围的基数这样基本的问题，也等同于连续统问题。此外，结果表明，存在一个句子，当且仅当连续统假设成立时，它才是二阶逻辑真理（布卢斯，1975）。我们已经看到，连续统问题是独立于目前接受的集合论原则的。许多研究者认为它是绝对无真值的。如果事实如此，那么二阶无限模型的概念本身就存在固有的不确定性。许多当代数学哲学家认为后者没有确定的真值。因此，有人认为，完全二阶逻辑的（无限）模型的概念本身是固有不确定的。

If one does not want to appeal to full second-order logic, then there are other ways to ensure categoricity of mathematical theories. One idea would be to make use of quantifiers which are somehow intermediate between first-order and second-order quantifiers. For instance, one might treat “there are finitely many $x$” as a primitive quantifier. This will allow one to, for instance, construct a categorical axiomatization of arithmetic.
如果不想求助于完全的二阶逻辑，那么还有其他方法可以确保数学理论的范畴性。一个想法是使用某种介于一阶和二阶量词之间的量词。例如，可以把“有有限多个 $x$”当作一个原始量词。这将允许人们，例如，构建一个算术的范畴公理化。

But ensuring categoricity of mathematical theories does not require introducing stronger quantifiers. Another option would be to take the informal concept of algorithmic computability as a primitive notion (Halbach & Horsten 2005; Horsten 2012). A theorem of Tennenbaum states that all first-order models of Peano Arithmetic in which addition and multiplication are computable functions, are isomorphic to each other. Now our operations of addition and multiplication are computable: otherwise we could never have learned these operations. This, then, is another way in which we may be able to isolate the intended models of our principles of arithmetic. Against this account, however, it may be pointed out that it seems that the categoricity of intended models for real analysis, for instance, cannot be ensured in this manner. For computation on models of the principles of real analysis, we do not have a theorem that plays the role of Tennenbaum’s theorem.
但确保数学理论的范畴性并不需要引入更强的量词。另一个选择是把非正式的算法可计算性概念当作一个原始概念（哈尔巴赫和霍斯特，2005；霍斯特，2012）。腾纳鲍姆定理指出，所有皮亚诺算术的一阶模型，其中加法和乘法是可计算函数，彼此都是同构的。现在我们的加法和乘法运算是可计算的：否则我们永远无法学会这些运算。因此，这是另一种我们可以用以确定我们的算术原则的预期模型的方法。然而，反对这种观点的人可能会指出，似乎实分析的预期模型的范畴性无法以这种方式得到保证。对于实分析原则的模型的计算，我们没有一个起腾纳鲍姆定理作用的定理。

If one accepts a certain open-endedness of the collection of arithmetical predicates, then a categoricity theorem of sorts for arithmetic can be obtained without overstepping the bounds of first-order logic and without appealing to an informal concept of computability. Suppose that there are two mathematicians, A and B, who both assert the first-order Peano-axioms in their own idiolect. Suppose furthermore that A and B regard the collection of predicates for which mathematical induction is permissible as open-ended, and are both willing to accept the other’s induction scheme as true. Then A and B have the wherewithal to convince themselves that both idiolects describe isomorphic structures (Parsons 1990b). Such arguments are called internal categoricity arguments. They are widely debated in contempory philosophy of mathematics: see for instance (Button & Walsh 2019).
如果接受算术谓词集合的某种开放性，那么可以在不超出一阶逻辑范围且不求助于非正式的可计算性概念的情况下，获得一种算术的范畴性定理。假设有两位数学家 A 和 B，他们都用自己的方言断言一阶皮亚诺公理。进一步假设 A 和 B 都认为数学归纳法所允许的谓词集合是开放的，并且都愿意接受对方的归纳方案为真。那么 A 和 B 有能力使自己相信，两种方言所描述的结构是同构的（帕森斯，1990b）。这种论证被称为内部范畴性论证。它们在当代数学哲学中被广泛讨论：例如（巴顿和沃尔什，2019）。

Many of those who are sceptical of the philosophical use of categoricity argments in the philosophy of mathematics take all of our consistent mathematical theories to have many structurally different models, and take all or many of those models to be on a par with one another. As we saw in the previous sub-section, the set theoretic multiverse view is a case in point, and so is set theoretic potentialism. But one can go further, and defend the thesis that any consistent mathematical theory describes a free-standing mathematical universe, and that no such theory is more true than any other (Linsky & Zalta 1995, Bueno 2011).
许多对数学哲学中范畴性论证的哲学使用持怀疑态度的人认为，我们所有一致的数学理论都有许多结构不同的模型，并且认为所有或许多这些模型都是平等的。正如我们在上一个小节中看到的，集合论的多宇宙观就是一个例子，集合论的潜能主义也是如此。但人们还可以走得更远，捍卫这样一个论点：任何一致的数学理论都描述了一个独立的数学宇宙，而且没有任何这样的理论比其他理论更真实（林斯基和扎尔塔，1995；布埃诺，2011）。

These theories belong to a family of views that is called mathematical pluralism, which is an increasingly prominent theme in the philosophy of mathematics. Historically, this constellation of views has roots in the work of Hilbert and of Carnap. In a debate with Frege, Hilbert insisted that consistency suffices for a mathematical theory to have a subject matter (Resnik 1974); Carnap argued that choice between alternative large-scale theories (frameworks) is ultimately never more than a pragmatic matter (Carnap 1950).
这些理论属于一个被称为数学多元主义的理论家族，这是数学哲学中越来越突出的主题。从历史上看，这一理论群源于希尔伯特和卡尔纳普的工作。在与弗雷格的辩论中，希尔伯特坚持认为一致性足以使一个数学理论有研究对象（雷斯尼克，1974）；卡尔纳普认为，在选择不同的大规模理论（框架）之间，最终不过是一个实用问题（卡尔纳普，1950）。

As is everywhere the case in philosophy, there is disagreement here: for a critique of the doctrine that mathematical truth is an irrevocably use-relative notion, see (Koellner 2009b), and for a retort, see (Warren 2015). Some react to mathematical pluralism by taking it one step further still, and argue that also all inconsistent mathematical theories should be regarded as true (in a relativised sense). Moreover, some mathematical theories that are trivial in the sense of being inconsistent, are commonly taken to be just as valuable as many venerable consistent ones: “Historically, there are three [to the author’s knowledge] mathematical theories which had a profound impact on mathematics and logic, and were found to be trivial. There are Cantor’s naive set theory, Frege’s formal theory of logic and the first version of Church’s formal theory of mathematical logic. All three had profound repercussions on subsequent mathematics” (Friend 2013, p. 294).
正如在哲学的任何地方一样，这里也存在分歧：对数学真理是一种不可撤销的使用相对概念的批判，可参见（科勒纳，2009b），而反驳意见可参见（沃伦，2015）。一些人对数学多元主义的反应是更进一步，认为所有不一致的数学理论也应被视为真（在相对化的意义上）。此外，一些在意义上不一致的、因而被认为是平凡的数学理论，通常被认为和许多著名的、一致的理论一样有价值：“历史上，据作者所知，有三种数学理论对数学和逻辑产生了深远影响，但被发现是平凡的。它们是康托尔的朴素集合论、弗雷格的形式逻辑理论和丘奇的数理逻辑形式理论的第一个版本。这三种理论都对后来的数学产生了深远的影响”（弗兰德，2013，第294页）。

5.3 Computation

计算

Until fairly recently, the subject of computation did not receive much attention in the philosophy of mathematics. This may be due in part to the fact that in Hilbert-style axiomatizations of number theory, computation is reduced to proof in Peano Arithmetic. But this situation has changed in recent years. It seems that along with the increased importance of computation in mathematical practice, philosophical reflections on the notion of computation will occupy a more prominent place in the philosophy of mathematics in the years to come.
直到不久之前，计算这一主题在数学哲学中并没有受到太多关注。这可能部分是由于在希尔伯特风格的数论公理化中，计算被简化为皮亚诺算术中的证明。但近年来这种情况发生了变化。随着计算在数学实践中重要性的增加，对计算概念的哲学思考在未来几年可能会在数学哲学中占据更重要的位置。

Church’s Thesis occupies a central place in computability theory. It says that every algorithmically computable function on the natural numbers can be computed by a Turing machine.
丘奇论题在可计算性理论中占据核心地位。它表明，自然数上每一个算法可计算的函数都可以通过图灵机计算。

As a principle, Church’s Thesis has a somewhat curious status. It appears to be a basic principle. On the one hand, the principle is almost universally held to be true. On the other hand, it is hard to see how it can be mathematically proved. The reason is that its antecedent contains an informal notion (algorithmic computability) whereas its consequent contains a purely mathematical notion (Turing machine computability). Mathematical proofs can only connect purely mathematical notions—or so it seems. The received view was that our evidence for Church’s Thesis is quasi-empirical. Attempts to find convincing counterexamples to Church’s Thesis have come to naught. Independently, various proposals have been made to mathematically capture the algorithmically computable functions on the natural numbers. Instead of Turing machine computability, the notions of general recursiveness, Herbrand-Gödel computability, lambda-definability… have been proposed. But these mathematical notions all turn out to be equivalent. Thus, to use Gödelian terminology, we have accumulated extrinsic evidence for the truth of Church’s Thesis.
作为一个原则，丘奇论题具有一种有点奇特的地位。它似乎是一个基本原则。一方面，该原则几乎被普遍认为是正确的。另一方面，很难看出它如何能够被数学地证明。原因是它的前件包含一个非正式的概念（算法可计算性），而它的后件包含一个纯粹的数学概念（图灵机可计算性）。数学证明似乎只能连接纯粹的数学概念。普遍的观点是，我们对丘奇论题的证据是准经验的。试图找到令人信服的丘奇论题的反例的尝试都失败了。独立地，已经提出了各种建议，试图数学地捕捉自然数上的算法可计算函数。除了图灵机可计算性之外，还提出了通用递归性、赫布兰德 - 哥德尔可计算性、λ - 可定义性……等概念。但这些数学概念最终都证明是等价的。因此，用哥德尔的术语来说，我们已经为丘奇论题的真实性积累了外在证据。

Kreisel pointed out long ago that even if a thesis cannot be formally proved, it may still be possible to obtain intrinsic evidence for it from a rigorous but informal analysis of intuitive notions (Kreisel 1967). Kreisel calls these exercises in informal rigour. Detailed scholarship by Sieg revealed that the seminal article (Turing 1936) constitutes an exquisite example of just this sort of analysis of the intuitive concept of algorithmic computability (Sieg 1994).
克雷塞尔很久以前就指出，即使一个论题不能被形式地证明，也仍然可能通过严格但非正式的直观概念分析来获得内在证据（Kreisel 1967）。克雷塞尔称这些练习为非形式的严格性。齐格的详细研究揭示了开创性的文章（图灵 1936）正是这种对算法可计算性的直观概念进行分析的绝佳例子（齐格 1994）。

Currently, the most active subjects of investigation in the domain of foundations and philosophy of computation appear to be the following. First, energy has been invested in developing theories of algorithmic computation on structures other than the natural numbers. In particular, efforts have been made to obtain analogues of Church’s Thesis for algorithmic computation on various structures. In this context, substantial progress has been made in recent decades in developing a theory of effective computation on the real numbers (Pour-El 1999). Second, attempts have been made to explicate notions of computability other than algorithmic computability by humans. One area of particular interest here is the area of quantum computation (Deutsch et al. 2000).
目前，在计算的基础和哲学领域中最活跃的研究主题似乎是以下这些。首先，人们投入精力发展非自然数结构上的算法计算理论。特别是，人们努力获得各种结构上的算法计算的丘奇论题的类似物。在这方面，近几十年来在发展实数上的有效计算理论方面取得了实质性进展（Pour-El 1999）。其次，人们试图阐明人类算法可计算性之外的可计算性概念。这里特别感兴趣的一个领域是量子计算（Deutsch et al. 2000）。

5.4 Mathematical Proof

数学证明

We know much about the concepts of formal proof and formal provability, their connection with algorithmic computability, and the principles by which these concepts are governed. We know, for instance, that the proofs of a formal system are computably enumerable, and that provability in a sound (strong enough) formal system is subject to Gödel’s incompleteness theorems. But a mathematical proof as you find it in a mathematical journal is not a formal proof in the sense of the logicians: it is a (rigorous) informal proof (Myhill 1960, Detlefsen 1992, Antonutti 2010).
我们对形式证明和形式可证明性的概念、它们与算法可计算性的联系以及支配这些概念的原则了解很多。例如，我们知道一个形式系统的证明是可计算枚举的，以及在一个可靠（足够强）的形式系统中的可证明性受到哥德尔不完全性定理的制约。但在数学期刊中找到的数学证明并不是逻辑学家意义上的形式证明：它是一个（严格的）非形式证明（Myhill 1960, Detlefsen 1992, Antonutti 2010）。

First, whereas the collection of sentences provable in a formal system is always computably enumerable, we know much less about the extension of the concept of informal provability. Lucas (Lucas 1961), and later Penrose (Penrose 1989, 1994), have argued that informal mathematical provability outstrips provability in any given formal system. But their arguments are widely regarded as unpersuasive. Benacerraf has argued against Lucas and Penrose that it cannot be excluded that there is a formal system $T$ such that in fact mathematical provability extensionally coincides with provability in $T$, even though we cannot know that it does (Benacerraf 1967). Others have argued that the concept of informal mathematical provability is not even clear enough for the question whether its extension is computably enumerable to have a definite answer (Horsten & Welch 2016).
首先，尽管一个形式系统中可证明的句子集合总是可计算枚举的，但我们对非形式可证明性概念的外延了解甚少。卢卡斯（Lucas 1961）以及后来的彭罗斯（Penrose 1989, 1994）认为，非形式的数学可证明性超越了任何给定形式系统中的可证明性。但他们的论证被广泛认为是不具说服力的。贝纳塞拉夫反驳卢卡斯和彭罗斯，认为不能排除存在一个形式系统 $T$，使得实际上数学可证明性与在$T$ 中的可证明性外延上一致，尽管我们无法知道这一点（贝纳塞拉夫 1967）。还有人认为，非形式数学可证明性的概念甚至不清楚到足以对它的外延是否可计算枚举这一问题有一个明确答案（霍尔斯特 & 韦尔奇 2016）。

Second, there is no agreement about what the standard is for an argument to qualify as a mathematical proof. According to what may be called the received view, a mathematical argument for a statement $p$ constitutes an informal mathematical proof if the argument allows a competent mathematician to transform it into a formal deduction of $p$ from generally accepted mathematical axioms (Avigad 2021). An informal mathematical proof can then be taken to be a derivation-indicator for $p$ (Azzouni 2004). But the received view of the standard of mathematical proof has come under attack in recent years. It has been argued, for instance, that the interpolations of reasons in an informal mathematical proof until a logically correct and non-elliptical first-order derivation is reached, can be an infinite process (Rav 1999, p.14-15). Others are mounting a defence of the received view, so that there is a lively debate about these issues at the moment (Tatton-Brown forthcoming, Di Toffoli 2021).
其次，对于什么样的论证才能算作数学证明这一标准，人们并没有达成一致。按照可以被称为普遍接受的观点，如果一个论证能够让一个有能力的数学家将其转化为从普遍接受的数学公理出发对$p$ 的形式演绎，那么这个论证就可以算作对陈述$p$ 的非形式数学证明（Avigad 2021）。那么，非形式数学证明就可以被看作是$p$ 的一个推导指示器（Azzouni 2004）。然而，近年来，普遍接受的数学证明标准观点已经受到了攻击。例如，有人认为，在非形式数学证明中插入理由，直到达到逻辑正确且非省略的一阶推导，可能是一个无限的过程（Rav 1999, 第 14-15 页）。其他人则在为普遍接受的观点进行辩护，因此目前关于这些问题有一个热烈的辩论（Tatton-Brown 即将发表，Di Toffoli 2021）。

The past decades have witnessed the first occurrences of mathematical proofs in which computers appear to play an essential role. The four-colour theorem is one example. It says that for every map, only four colours are needed to colour countries in such a way that no two countries that have a common border receive the same color. This theorem was proved in 1976 (Appel et al. 1977). But the proof distinguishes many cases which were verified by a computer. These computer verifications are too long to be double-checked by humans. The proof of the four colour theorem gave rise to a debate about the question to what extent computer-assisted proofs count as proofs in the true sense of the word.
在过去的几十年里，首次出现了计算机似乎发挥了重要作用的数学证明。四色定理就是一个例子。它表明，对于每一张地图，只需要四种颜色就可以给国家上色，使得没有两个有共同边界的国家得到相同的颜色。这个定理在 1976 年得到了证明（阿佩尔等人 1977）。但这个证明区分了许多由计算机验证的情况。这些计算机验证太长了，无法由人类进行双重检查。四色定理的证明引发了一场关于计算机辅助证明在多大程度上算是真正意义上的证明的辩论。

The received view has it that mathematical proofs yield a priori knowledge. Yet when we rely on a computer to generate part of a proof, we appear to rely on the proper functioning of computer hardware and on the correctness of a computer program. These appear to be empirical factors. Thus one is tempted to conclude that computer proofs yield quasi-empirical knowledge (Tymoczko 1979). In other words, through the advent of computer proofs the notion of proof has lost its purely a priori character. Burge, in contrast, held the view that because the empirical factors on which we rely when we accept computer proofs do not appear as premises in the argument, computer proofs can yield a priori knowledge after all (Burge 1998). (Burge later retracted this claim: see (Burge 2013, p.31).)
普遍接受的观点认为，数学证明产生了先验知识。然而，当我们依赖计算机来生成证明的一部分时，我们似乎依赖于计算机硬件的正常运行和计算机程序的正确性。这些似乎是经验因素。因此，人们倾向于得出结论，计算机证明产生了准经验知识（Tymoczko 1979）。换句话说，由于计算机证明的出现，证明的概念已经失去了其纯粹的先验性。相比之下，伯奇认为，由于我们在接受计算机证明时所依赖的经验因素并没有作为论证的前提出现，计算机证明最终可以产生先验知识（伯奇 1998）。（伯奇后来收回了这一说法：见（伯奇 2013，第 31 页）。）

6. The Future

未来

In the twentieth century, research in the philosophy of mathematics revolved mostly around the nature of mathematical objects, the fundamental laws that govern them, and how we acquire mathematical knowledge about them. These are foundational concerns that are intimately connected with traditional metaphysical and epistemological questions.
在二十世纪，数学哲学的研究主要围绕数学对象的本质、支配它们的基本规律以及我们如何获得关于它们的数学知识。这些都是与传统形而上学和认识论问题密切相关的基础性问题。

In the second half of the twentieth century, research in the philosophy of science to a significant extent moved away from foundational concerns. Instead, philosophical questions relating to the growth of scientific knowledge and of scientific understanding became more central. As early as the 1970s, there were voices that argued that a similar shift of attention should take place in the philosophy of mathematics. Lakatos initiated the philosophical investigation of the evolution of mathematical concepts (Lakatos 1976). He argued that the content of a mathematical concept evolves in roughly the following way. A mathematician formulates a deep conjecture, but is unable to prove it. Then counterexamples against the conjecture are found. In response, the definition of one or more central concepts in the conjecture is changed in such a way that the counterexamples are at least eliminated. Still the thus revised conjecture cannot be proved, and gradually new counterexamples appear. The procedure of revising the definition of one or more central concepts is applied again and again, until a proof of the conjecture is found. Lakatos calls this procedure concept stretching.
在二十世纪下半叶，科学哲学的研究在很大程度上从基础性问题转向。相反，与科学知识和科学理解的增长相关的哲学问题变得更加核心。早在 20 世纪 70 年代，就有人主张数学哲学也应该发生类似的注意力转移。拉卡托斯开始了对数学概念演变的哲学研究（Lakatos 1976）。他认为，数学概念的内容大致以如下方式演变。数学家提出了一个深刻的猜想，但却无法证明它。然后发现了针对该猜想的反例。作为回应，猜想中一个或多个核心概念的定义被改变，以使反例至少被消除。然而，经过修订的猜想仍然无法被证明，新的反例逐渐出现。修订一个或多个核心概念定义的程序被反复应用，直到找到该猜想的证明。拉卡托斯将这一程序称为概念拉伸。

For some decades, the view that the philosophy of mathematics should take a historical and sociological turn remained restricted to a somewhat marginal school of thought in the philosophy of mathematics. However, in recent years the opposition between this new movement of mathematical practice on the one hand, and “mainstream” philosophy of mathematics on the other hand, is softening. Philosophical questions relating to mathematical practice, the evolution of mathematical theories, and mathematical explanation and understanding have become more prominent, and have been related to more traditional themes from the philosophy of mathematics (Mancosu 2008). This trend will doubtlessly continue in the years to come.
在随后的几十年里，认为数学哲学应该转向历史和社会学的观点一直局限于数学哲学中一个相对边缘的思想流派。然而，近年来，这种新的数学实践运动与“主流”数学哲学之间的对立正在缓和。与数学实践、数学理论的演变以及数学解释和理解相关的哲学问题变得更加突出，并且与数学哲学中的更传统主题联系在一起（Mancosu 2008）。这一趋势无疑将在未来几年继续。

For an example, let us briefly return to the subject of computer proofs (see section 5.3). The source of the discomfort that mathematicians experience when confronted with computer proofs appears to be the following. A “good” mathematical proof should do more than to convince us that a certain statement is true. It should also explain why the statement in question holds. And this is done by referring to deep relations between deep mathematical concepts that often link different mathematical domains (Manders 1989). Until now, computer proofs typically only employ fairly low-level mathematical concepts. They are notoriously weak at developing deep concepts on their own, and have difficulties with linking concepts in from different mathematical fields. All this leads us to a philosophical question which is just now beginning to receive the attention that it deserves: what is mathematical understanding?
以计算机证明（见第 5.3 节）为例。当数学家面对计算机证明时所感受到的不适似乎源于以下原因。一个“好的”数学证明应该不仅仅让我们相信某个陈述是真的，它还应该解释为什么该陈述成立。这是通过引用常常联系不同数学领域的深刻数学概念之间的深刻关系来完成的（Manders 1989）。到目前为止，计算机证明通常只使用相当低层次的数学概念。它们在自行发展深刻概念方面臭名昭著地薄弱，并且在引入不同数学领域的概念时存在困难。所有这些都引导我们提出一个哲学问题，这个问题现在才刚刚开始得到它应得的关注：什么是数学理解？

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Other Internet Resources

• Mathematics, Foundations of, by Detlefsen, M., Routledge Encyclopedia of Philosophy
• Category Archives: Philosophy of Mathematics, Internet Encyclopedia of Philosophy
• The Infinite, entry by Bradley Dowden, Internet Encyclopedia of Philosophy
• FOM – Foundations Of Mathematics, e-mail discussion forum

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另一篇

• 数学哲学 | 穿越古代智慧与现代争论的旅程-优快云博客
  https://blog.youkuaiyun.com/u013669912/article/details/154950903

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via:

• The Relationship Between Philosophy and Mathematics
  https://sapienthink.com/articles/the-relationship-between-philosophy-and-mathematics

• Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)
  https://plato.stanford.edu/entries/philosophy-mathematics/