How do we come to know mathematical truths? While it would be unreasonable to impose a causal requirement on such knowledge, we still want an informative answer to the question (cf. $\mathrm{\$}\mathrm{\$}1.5$$\mathrm{\$}\mathrm{\$}1.5$$$1.5\$ \$ 1.5 and 7.1).
我们如何认识数学真理？虽然对这类知识强加因果性要求并不合理，但我们仍希望对此问题获得富有启发性的解答（参见 $\mathrm{\$}\mathrm{\$}1.5$$\mathrm{\$}\mathrm{\$}1.5$$$1.5\$ \$ 1.5 及 7.1 节）。

One answer is that mathematical knowledge is broadly empirical (cf. chap. 6). But this answer struggles to do justice to mathematics as it is actually practiced. The most sophisticated form of empiricism about mathematics is Quine’s, which compares the epistemological status of mathematics to that of theoretical physics. The comparison is problematic, however. Elementary mathematics appears to enjoy a far stronger and more direct form of evidence than the often tenuous and indirect evidence enjoyed by theoretical physics.
一种观点认为数学知识大体上是经验性的（参见第 6 章）。但这种解释难以公正地反映数学的实际实践。关于数学最复杂的经验主义理论是奎因提出的，他将数学的认识论地位与理论物理学相类比。然而这种类比存在问题。基础数学所获得的证据形式，似乎远比理论物理学通常依赖的脆弱间接证据更为坚实直接。

We shall now take a closer look at some accounts of mathematical knowledge that do not assimilate it so strongly to empirical knowledge. One option came up in our discussions of Kant, Hilbert, and Brouwer, namely that some form of mathematical intuition provides evidence for certain mathematical truths. This idea will be our main focus in the present chapter.
现在我们将更详细地考察一些不将数学知识强烈同化于经验知识的解释。在讨论康德、希尔伯特和布劳威尔时曾提到一种观点，即某种形式的数学直觉为某些数学真理提供了证据。这一观点将是本章的主要关注点。

There are other options too, to be discussed in later chapters. A brief overview may be useful. According to Frege and his followers, logic-or at least broadly conceptual considerationsprovide a source of evidence in mathematics (cf. chap. 9). Another important issue in the epistemology of mathematics is that of extrapolation. This issue plays an important role in our set-theoretic reasoning, where some very natural ideas about collecting objects are extrapolated from their most secure home in applications to finite domains and applied to huge infinite domains (cf. chap. 10). Last, Russell and Gödel seek
还有其他选择将在后续章节讨论。简要概述或许有所帮助。根据弗雷格及其追随者的观点，逻辑——或至少是广义的概念性考量——为数学提供了证据来源（参见第 9 章）。数学认识论中另一个重要问题是外推法。这一问题在我们的集合论推理中扮演重要角色，其中关于对象收集的一些非常自然的想法从其最安全的有限领域应用中被外推，并应用于庞大的无限领域（参见第 10 章）。最后，罗素和哥德尔试图...
inspiration from the natural sciences, where a theoretical hypothesis can be supported by its capacity to systematize, explain, and predict more elementary observations. Both thinkers argue that analogous considerations can serve as evidence in mathematics, including for its higher reaches (cf. chap. 12).
灵感源自自然科学，其中理论假设能够通过其系统化、解释和预测更基础观察的能力得到支持。两位思想家都认为，类似的考量可以作为数学中的证据，包括其更高深的领域（参见第 12 章）。

When discussing the various possible sources of evidence in mathematics, two points should be kept in mind. First, the availability of one form of evidence need not preclude the availability of other forms. As we shall see, Gödel was a pluralist about mathematical evidence, defending a role for mathematical intuition, conceptual analysis, and indirect explanatory evidence. Second, evidence may come in degrees. While elementary mathematics may enjoy a particularly strong and direct form of evidence, only more attenuated forms of evidence may be available for parts of higher mathematics. For example, Parsons argues that there is such a thing as intuitive evidence-but that it does not take us beyond parts of finitary mathematics. In short, we should take seriously the possibility of a conception of mathematical evidence that is both pluralist and gradualist.
在讨论数学中各种可能的证据来源时，有两点需要牢记。首先，一种证据形式的存在并不排斥其他形式的存在。正如我们将看到的，哥德尔在数学证据上持多元主义立场，主张数学直觉、概念分析和间接解释性证据各自发挥作用。其次，证据可能有程度之分。初等数学可能享有特别强烈和直接的证据形式，而高等数学的某些部分可能只有更为间接的证据形式。例如，帕森斯认为存在直觉证据这种东西——但它无法带我们超越有限数学的部分领域。简而言之，我们应当认真考虑一种既多元又渐进的数学证据观的可能性。

“Intuition” is multiply ambiguous. The term is often used simply in the sense of an immediate or pretheoretic opinion. Philosophers sometimes appeal to intuitions in this sense. Such appeals are more widespread and less controversial in linguistics, however, where we have immediate reactions concerning the grammaticality of sentences. Consider, for example, the following well-known pair (where, as usual, ungrammaticality is marked by ${}^{(\ast )}$${}^{(\ast )}$^((**)){ }^{(*)} ):
“直觉”一词具有多重含义。该术语常被简单地用来指代一种即时或前理论的观点。哲学家有时会以这种意义援引直觉。然而，在语言学中，这种对直觉的引用更为普遍且争议较少，因为我们对句子的语法性有着直接的反应。例如，考虑以下著名的一对句子（按照惯例，不合语法的部分用 ${}^{(\ast )}$${}^{(\ast )}$^((**)){ }^{(*)} 标记）：
(4) I wish/hope that John will leave.
(4) 我希望/但愿约翰会离开。
(5) I wish/*hope John to leave.
(5) 我希望/*但愿约翰离开。

Another notion of intuition is that of an immediate rational insight. This notion figures prominently in the rationalists, for instance in Descartes’s appeal to “the natural light of reason,” which can be trusted and serve as a source of knowledge. We shall not have anything to say about these two notions of intuition.
另一种直觉概念是指直接的理性洞察。这一概念在理性主义者中尤为突出，例如笛卡尔诉诸“理性的自然之光”，认为其可被信赖并作为知识的来源。我们对这两种直觉概念将不作讨论。

We shall instead focus on a notion of intuition employed by Kant and other thinkers whom he inspired. In particular, Hilbert argues that we have intuition of what we are now calling Hilbert strokes (cf. $\mathrm{\S }4.4$$\mathrm{\S }4.4$§4.4\S 4.4 ). Since these strokes are understood as types, not tokens, they are abstract and thus cannot strictly speaking be perceived. Hilbert nevertheless contends that we have a broadly perceptual mode of access to types, provided by a form of mathematical intuition. We shall discuss some recent attempts to defend this idea.
我们将转而关注康德及其启发的一批思想家所采用的直觉概念。特别是，希尔伯特提出我们拥有对所谓“希尔伯特笔划”（参见 $\mathrm{\S }4.4$$\mathrm{\S }4.4$§4.4\S 4.4 ）的直觉。由于这些笔划被理解为类型而非标记，它们是抽象的，因而严格来说无法被感知。然而希尔伯特主张，我们通过一种数学直觉的形式，获得了对类型的广义感知访问方式。我们将探讨近期为这一观点辩护的若干尝试。

The idea of intuitive evidence for mathematics has encountered substantial skepticism in recent philosophy.
关于数学直觉证据的观点在当代哲学中遭遇了强烈的质疑。

One reason is that such evidence may seem irrelevant. True, mathematicians used to appeal to intuitive evidence. For example, the intermediate value theorem used to be regarded as intuitively obvious. Surely, any continuous function whose value begins below some number but ends above it must at some point have this number as its value (cf. $\mathrm{\$}2.1$$\mathrm{\$}2.1$$2.1\$ 2.1 ). But Bolzano showed that a better proof is possible, replacing the appeal to intuition with a rigorous analytical argument. This development continued throughout the nineteenth century with the rigorization of analysis. Appeals to intuition gradually gave way to analytical and eventually set-theoretic arguments.
其中一个原因是这类证据可能显得无关紧要。诚然，数学家过去常依赖直观证据。例如，介值定理曾被视为直观上显而易见——任何连续函数若其值从某数下方开始、最终升至该数上方，必然在某一点取到该数值（参见 $\mathrm{\$}2.1$$\mathrm{\$}2.1$$2.1\$ 2.1 ）。但波尔查诺证明了存在更优的证明方法，用严谨的分析论证替代了对直觉的依赖。这一发展贯穿整个十九世纪分析的严格化进程，直观诉求逐渐让位于分析性论证，并最终演变为集合论论证。

Ultimately, however, this development only pushes the epistemological question back to set theory. Might intuition play a role there? Gödel famously thought so:
然而归根结底，这种发展只是将认识论问题推给了集合论。直觉是否可能在此发挥作用？哥德尔对此持有著名论断：

But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves on us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception. (1964, pp. 483-84)
尽管远离感官经验，我们对集合论对象确实拥有某种类似感知的能力，这体现为公理会迫使我们承认其真实性。我认为没有理由对这种感知——即数学直觉——的信任度应低于感官知觉。（1964 年，第 483-84 页）

However, do we really have something like perceptual access to the vast infinite sets postulated by modern set theory? And do
然而，我们真的对现代集合论所假设的那些庞大无限集合拥有某种感知上的接触吗？而且，
the set-theoretic axioms really “force themselves on us”? Many thinkers have dismissed such claims as simply incredible. So a second reason for skepticism about mathematical intuition is that this alleged capacity seems mysterious.
集合论的公理真的“强加于我们”吗？许多思想家认为这种主张简直难以置信。因此，对数学直觉持怀疑态度的第二个原因是，这种所谓的能力似乎神秘莫测。

It cannot be denied that some appeals to intuition are merely “just so” stories, devoid of explanatory power. Russell’s talk of “acquaintance” with universals provides an example. At one stage, Russell took mathematical propositions to be concerned with abstract universals; for example, “the proposition ‘two and two are four’ […] states a relation between the universal ‘two’ and the universal ‘four’” (1912, p. 103). This raises the question of how we acquire knowledge of such universals and their relations to one another. Russell answers that we have “knowledge by acquaintance” not only of “sensible qualities” but also of “relations of space and time, similarity, and certain abstract logical universals” (p. 109). He concludes that “all our knowledge of truths depends on our intuitive knowledge.” But Russell offers no account of how this acquaintance works. Unlike sensible qualities, which he takes to be “exemplified in sense-data,” the abstract logical universals are not given in sensation. How, then, do we apprehend them? Is this the work of some special mental faculty? Instead of giving proper answers, Russell simply postulates a form of acquaintance with universals with no real explanation.
不可否认，某些诉诸直觉的说法不过是缺乏解释力的“凭空捏造”。罗素关于与共相“亲知”的论述便是一例。在某一阶段，罗素认为数学命题涉及抽象共相；例如，“命题‘二加二等于四’……陈述了共相‘二’与共相‘四’之间的关系”（1912 年，第 103 页）。这引出了我们如何获得关于此类共相及其相互关系的知识的问题。罗素回答道，我们不仅对“可感性质”具有“亲知的知识”，还对“空间与时间的关系、相似性以及某些抽象逻辑共相”具有此类知识（第 109 页）。他总结道：“我们所有关于真理的知识都依赖于我们的直觉知识。”但罗素并未说明这种亲知如何运作。与他认为“在感觉材料中得到例示”的可感性质不同，抽象逻辑共相并非通过感官给予。那么我们如何把握它们？这是某种特殊心理官能的作用吗？ 罗素并未给出合理的解答，而是简单地假设了一种对共相的直观认识，却未提供真正的解释。

In sum, to be convincing, an account of mathematical intuition must explain why such intuition is neither irrelevant nor mysterious.
总之，要使数学直觉的论述令人信服，必须阐明为何这种直觉既非无关紧要也不显得神秘。

Penelope Maddy has argued that we can perceive impure sets, such as the set of twelve eggs in a carton. ${}^1$${}^1$^(1){ }^{1} Her argument has obvious potential relevance to the epistemology of set theory.
佩内洛普·麦迪提出，我们可以感知不纯的集合，比如一盒鸡蛋中的十二枚鸡蛋组成的集合。 ${}^1$${}^1$^(1){ }^{1} 她的论点显然对集合论的认识论具有潜在的相关性。

And she takes great care to ensure that the form of perception she describes is scientifically respectable. The central idea is that we have an ability to represent many objects considered together as a form of unity. This unity (or set) is located where its constituents (or elements) are, thus ensuring that sets of concrete objects are themselves concrete. Maddy even advances some hypotheses about the neural mechanisms underlying our perception of such concrete sets.
她极其谨慎地确保所描述的感知形式在科学上是可信的。核心观点是我们具备一种能力，能将多个对象共同表征为一种统一体形式。这种统一体（或集合）位于其构成成分（或元素）所在之处，从而确保由具体对象组成的集合本身也是具体的。梅迪甚至提出了一些关于我们感知此类具体集合背后神经机制的假说。

As noted, Gödel expressed high hopes concerning a form of perception of sets. Maddy’s account of set perception has far more limited scope. Since the objects to be considered as a unity need to be given in perception, there can on her account be no perception of pure sets, which involve no perceptible objects. It is not even clear how perception of sets of sets of concrete objects would work. (What would it be to perceive the powerset of the mentioned set of eggs?) Nor does Maddy claim that we have perceptual access to infinite sets, even when these are impure (and thus concrete).
如前所述，哥德尔对集合的某种感知形式寄予厚望。而梅迪关于集合感知的解释范围则局限得多。由于需要被统一考虑的对象必须是在感知中呈现的，因此她的理论中不存在对纯集合（即不涉及可感知对象的集合）的感知。甚至对于由具体对象集合的集合（例如鸡蛋集合的幂集）如何被感知也不甚明了。梅迪也并未宣称我们能通过感知接触无限集合，即便这些集合是非纯的（因而是具体的）。

In contrast to Maddy’s naturalistic approach, Charles Parsons defends a broadly Kantian conception of mathematical intuition. Such intuition, he argues, presents us with types of perceptible tokens in a way that is “strongly analogous” to how ordinary concrete objects are given in perception; he therefore calls such intuition quasi-perceptual (Parsons, 1980, p. 162). If correct, Parsons’ claim has great significance for the epistemology of finitary mathematics. As Hilbert realized, finitary mathematics can be understood as concerned with syntactic types, such as Hilbert strokes, and their basic syntactic properties (cf. §4.4). An intuitive mode of access to such objects would therefore provide a particularly strong and direct form of evidence for this part of elementary mathematics-in stark contrast to the more attenuated forms of evidence available for set theory, to which finitary mathematics could otherwise be reduced.
与 Maddy 的自然主义方法不同，Charles Parsons 捍卫了一种广义的康德式数学直觉观。他认为，这种直觉以"强烈类比"于普通具体对象在感知中被给予的方式，向我们呈现可感知符号的类型；因此他将这种直觉称为准感知性的（Parsons, 1980, p. 162）。若该主张成立，Parsons 的论点对有限数学的认识论具有重大意义。正如 Hilbert 所认识到的，有限数学可被理解为关注句法类型（如 Hilbert strokes）及其基本句法特性（参见§4.4）。对此类对象的直觉访问方式，将为初等数学的这一部分提供特别强大而直接的证据形式——与集合论（有限数学可被归约至该理论）所能获得的更为间接的证据形式形成鲜明对比。

Notice that Parsons goes beyond Maddy by defending a form of intuition of objects that are not concrete. As we shall see, however, it is important for his account that the syntactic types that we intuit be at least “quasi-concrete”-in the sense that they have canonical instantiations in spacetime.
值得注意的是，帕森斯比麦迪更进一步，他捍卫了一种对非具体对象的直觉形式。然而，我们将看到，在他的解释中，关键之处在于我们所直觉的句法类型至少是“准具体的”——即它们在时空中具有典范实例化。

Why believe that we have intuition of types, not just perception of corresponding tokens? Parsons’s argument seeks to establish that types play an important role in our sensory experience. Our identification of a type is often firmer and more explicit than the identification of the corresponding token. Consider, for example, episodes of perceiving the word ‘dog’. Very often, we do not notice the accent with which the token was pronounced, the font with which it was written, or any other property that would distinguish one token of the word from another. What registers most firmly and explicitly in our minds is only a phonetic or orthographic type. In cases such as these, our experience seems to be directed at a type, not a token.
为何相信我们拥有对类型的直觉，而不仅仅是对相应标记的感知？帕森斯的论证试图确立类型在我们感官经验中扮演的重要角色。我们对一个类型的识别往往比对相应标记的识别更为牢固和明确。以感知单词“狗”的多个实例为例，我们通常不会注意到发音时的口音、书写时的字体或其他能区分该单词不同标记的属性。在我们头脑中留下最牢固、最明确印象的仅是一个语音或拼写类型。在此类情况下，我们的经验似乎指向的是类型，而非标记。

This is a plausible description of how things seem to the subject. One might nevertheless worry that this form of intuition would be mysterious. Would it not require an entirely new mode of access to reality in addition to our familiar and fairly well understood senses? Parsons’s response tries to “dispel the widespread impression that mathematical intuition is a ‘special’ faculty, which perhaps comes into play only in doing pure mathematics” (1980, pp. 154-55). Borrowing an idea from Husserl, he proposes that intuition is always “founded” on ordinary sensations or imaginings. That is, we perceive or imagine something particular, which is an instance of the more abstract object of our intuition. By anchoring intuition in this way to capacities that we indisputably have, there is no need for any mysterious “special faculty” of intuition.
这是对主体感知方式的合理描述。然而人们或许会担忧，这种直觉形式显得神秘莫测——难道除了我们熟悉且相对易于理解的感官之外，还需要一种全新的现实认知方式吗？帕森斯的回应试图"消除数学直觉是'特殊'能力的普遍印象，这种能力或许仅在进行纯数学时才发挥作用"（1980 年，第 154-55 页）。借鉴胡塞尔的观点，他提出直觉始终"奠基于"普通感觉或想象之上。也就是说，我们通过感知或想象某个具体事物，来直觉更为抽象的对象。通过将直觉如此锚定于我们确凿拥有的能力之上，便无需任何神秘的"特殊直觉能力"了。

One difference between mathematical intuition and ordinary perception is thus that the former is “founded.” This difference has an important consequence concerning the possible objects of mathematical intuition. For the required “founding” to be possible, the objects of intuition must be quasiconcrete. It is this that allows the intuition to be “founded” on perceptions or imaginings of corresponding tokens. So by requiring that mathematical intuition be “founded,” Parsons also limits its scope. We cannot have intuition of abstract objects that do not have tokens in space and time, such as numbers or pure sets, which are not quasi-concrete but purely abstract.
数学直觉与普通感知的一个区别在于前者是“有根基的”。这一区别对数学直觉的可能对象产生了重要影响。为了使所需的“根基”成为可能，直觉的对象必须是准具体的。正是这一点使得直觉能够“建立”在对相应标记的感知或想象之上。因此，通过要求数学直觉“有根基”，帕森斯也限制了其范围。我们无法对没有时空标记的抽象对象（如数字或纯集合）产生直觉，因为它们不是准具体的，而是纯粹抽象的。

Another difference is that mathematical intuition relies more heavily on conceptualization than ordinary perception does. As Parsons observes, " $[w]$$[w]$[w][w] hat is intuited depends on the concept brought to the situation by the subject" (1980, p. 162). For example, one cannot intuit linguistic types without having at least an implicit grasp of the difference between types and tokens and of how the relevant types are individuated. Mathematical intuition therefore requires more sophistication and training than ordinary perception. This observation goes some way toward explaining why our capacity for mathematical intuition is so much less obvious to us than our capacity for ordinary perception. Without the training needed to acquire the relevant concepts, we cannot enjoy the benefits of such intuition.
另一个区别在于，数学直觉比普通感知更依赖于概念化。正如帕森斯所言："被直觉到的内容取决于主体带入情境的概念"（1980 年，第 162 页）。例如，若不具备对类型与标记之间差异的基本理解，以及相关类型如何被个体化的认知，就无法直觉语言类型。因此，数学直觉需要比普通感知更高的专业素养和训练。这一观察在一定程度上解释了为何我们的数学直觉能力远不如普通感知能力那样显而易见。若缺乏获取相关概念所需的训练，我们就无法享有这种直觉带来的益处。

Where Parsons draws on Kant and Hilbert, Dagfinn Føllesdal looks to Husserl in an attempt to develop a phenomenological conception of mathematical intuition. ${}^2$${}^2$^(2){ }^{2} Føllesdal first asks us to consider Jastrow’s famous duck-rabbit drawing, which can equally well be seen as a duck and a rabbit, depending on what we focus on. The drawing thus shows that “we can experience a multitude of different objects when we are in a given sensory situation: a duck, a rabbit, but also an ear of a rabbit, an eye, or even the color or the front side of an object” (Føllesdal, 1995, p. 429). Our perception is an active process, where we choose what to focus on and are actively involved in interpreting the sensory information that we receive. This active character of our perception suggests that we can equally well choose to focus on an object’s more abstract features, such as its shape. When we stand in front of a tree with a nice triangular shape, for example, we can choose to concentrate on its shape. Doing so results in an intuition of triangularity.
帕森斯借鉴了康德和希尔伯特的思想，而达格芬·弗勒斯达尔则转向胡塞尔，试图发展一种关于数学直觉的现象学概念。 ${}^2$${}^2$^(2){ }^{2} 弗勒斯达尔首先让我们思考贾斯特罗著名的鸭兔图，根据我们关注的重点不同，这幅图既可以看作鸭子也可以看作兔子。这幅图因此表明"在特定的感官情境中，我们可以体验到多种不同的对象：鸭子、兔子，也可以是兔子的耳朵、眼睛，甚至物体的颜色或正面"（弗勒斯达尔，1995 年，第 429 页）。我们的感知是一个主动的过程，我们选择关注什么，并积极参与解释所接收的感官信息。这种感知的主动性表明，我们同样可以选择关注物体更抽象的特征，比如它的形状。例如，当我们站在一棵形状优美的三角形树前时，可以选择专注于它的形状。这样做会产生对三角形性的直觉。

Føllesdal’s account of intuition has much in common with Parsons’s. Both emphasize that such intuition is “founded” on acts of perceiving or imagining; both accounts are closely connected with the distinction between types and tokens; and both acknowledge the reliance of intuition on concepts and concep-
Føllesdal 对直觉的解释与 Parsons 的观点有许多共同之处。两者都强调这种直觉“基于”感知或想象的行为；两种解释都与类型和标记之间的区别密切相关；并且两者都承认直觉对概念和概念化的依赖。

tualization. But Føllesdal places more emphasis than Parsons does on our own contribution to the structuring of what we perceive or intuit. This difference emerges most clearly when we consider concepts that are more abstract than that of triangularity. Consider the concept of topological genus, which can be thought of as the number of holes that an object has. An example will help to convey the idea. A cup with a handle is superficially far more similar to a glass than to a donut: both the cup and the glass are shaped so as to serve as drinking vessels. But let us use our imagination to explore how an object can be transformed into another in a continuous manner, that is, by compressing, stretching, and twisting, but with no tearing. Since the cup shares with the donut the property of having a single hole, we can imagine the former being continuously transformed into the latter. ${}^3$${}^3$^(3){ }^{3} By contrast, the glass cannot be continuously transformed to the cup, since a hole would at some point have to be torn in the glass to make a handle. With some training, we can in this way become perceptually aware of a highly abstract property that the cup shares with the donut but not the glass, despite the cup’s far greater superficial similarity to the glass than to the donut.
但弗勒斯达尔比帕森斯更强调我们自身对所感知或直觉事物的结构化的贡献。当我们考虑比三角形更抽象的概念时，这种差异最为明显。以拓扑属的概念为例，它可以理解为一个物体拥有的孔洞数量。举例来说，带把手的杯子在表面上与玻璃杯的相似度远高于与甜甜圈的相似度：杯子和玻璃杯的形状都设计为饮用容器。但让我们运用想象力探索一个物体如何通过连续变形（即压缩、拉伸和扭曲，但不撕裂）转变为另一个物体。由于杯子和甜甜圈都具有一个孔洞的特性，我们可以想象前者被连续变形为后者。相比之下，玻璃杯无法连续变形为带把手的杯子，因为在制作把手的过程中，必须在玻璃杯上撕裂出一个孔洞。 经过一定训练后，我们能够以这种方式在感知层面意识到杯子与甜甜圈共有的高度抽象属性，而这种属性在杯子与玻璃杯之间并不存在，尽管杯子在表面特征上与玻璃杯的相似度远高于与甜甜圈的相似度。

Let us take stock. We have found that a strong case can be made for the existence of a broadly perceptual source of evidence about mathematical objects that are either concrete (as Maddy thinks is the case with impure sets) or quasi-concrete (as Parsons’s types). If defensible, this source of evidence will play an important role in the epistemology of elementary mathematics. However, we have found reasons to doubt that this broadly perceptual evidence will provide much support for what Hilbert calls “infinitary mathematics.” In order to find evidence for higher mathematics, it appears we must look elsewhere. Two options were mentioned in $\mathrm{\S }8.1$$\mathrm{\S }8.1$§8.1\S 8.1. We may analyze our mathematical concepts, such as that of set. Or we may seek indirect evidence for mathematical axioms via their ability to systematize and explain more elementary observations. These options will be explored in Chapters 10 and 12, respectively.
让我们总结一下。我们发现，对于数学对象（无论是马迪认为的具体不纯集合，还是帕森斯所称的准具体类型）存在一种广义感知性证据来源的观点，可以提出强有力的论证。若这一观点成立，该证据来源将在基础数学认识论中发挥重要作用。然而，我们也找到了质疑这种广义感知证据能否为希尔伯特所谓"无穷数学"提供充分支持的理由。要寻找高等数学的证据，似乎必须另辟蹊径。此前 $\mathrm{\S }8.1$$\mathrm{\S }8.1$§8.1\S 8.1 中提到了两种途径：我们可以分析集合等数学概念本身，或者通过数学公理对基础观察的系统化解释能力来获取间接证据。第十章和第十二章将分别探讨这两种方案。

The accounts of mathematical intuition proposed by Parsons and Føllesdal seem quite plausible from the point of view of the perceiving subject. But there is a residual worry. Do the accounts succeed in explaining our access to abstract objects in a way that meets the integration challenge (cf. §1.5)?
帕森斯和弗勒斯达尔提出的数学直觉解释从感知主体的角度来看相当合理。但仍有一个挥之不去的疑虑：这些解释是否真能以符合整合挑战（参见§1.5）的方式，阐明我们对于抽象对象的认知途径？

Let us grant that we can perceive a ball to be round and some marbles to be five in number. This means that properties and simple structural features can be present in our perception. But it does not establish that we can perceive abstract objects. In the mentioned examples, the proper objects of perception are the ball and the marbles, respectively. Roundness and “fiveness” are merely predicated of these objects and are not themselves the proper objects of perception. The idea of mathematical intuition promises the existence of quasi-perceptual acts in which such properties and structural features figure as the proper objects of perception and thus also as the subjects of other predications. An example would be an intuition that roundness is distinct from being cubical.
姑且承认我们能感知一个球是圆的，也能感知某些弹子有五颗。这意味着属性和简单结构特征可以呈现在我们的感知中。但这并不能证明我们能感知抽象对象本身。在上述例子中，感知的恰当对象分别是球和弹子，而圆润性和“五数性”只是这些对象的谓词，并非感知本身的恰当对象。数学直觉的概念承诺了存在一种类感知行为，其中这些属性和结构特征作为感知的恰当对象显现，并因此成为其他谓述的主词。例如，直觉到圆润性与立方体性质截然不同便属此类。

The required shift from predicables to proper objects of predication is an instance of what philosophers call reification; that is, coming to regard an item as an object. Now, wherever there are objects, it must make sense to ask questions about identity and distinctness: is this object identical with that? (As Quine famously put it, there can be “no entity without identity.”) This marks an important shift. Questions about the identity and distinctness of properties were not obligatory prior to their reification, when they were merely put to predicative use. For example, we can perceive that the ball is round without taking a stand on whether roundness is identical with some other property which we perceive another object to have (say being a good but not perfect approximation of a sphere). This neutrality is no longer possible when roundness is recognized as an object.
从可述谓项转向作为述谓固有对象的必要转变，是哲学家所称的“物化”实例，即开始将某物视为对象。一旦存在对象，关于同一性与差异性的提问便必然成立：此对象是否与彼对象相同？（正如奎因那句名言所言，“无同一性则无实体。”）这标志着一个重要转变。在属性仅被用作述谓功能、尚未被物化之前，关于属性同一性与差异性的问题并非强制性的。例如，我们可以感知球体是圆的，却无需判定“圆性”是否等同于我们从另一物体感知到的其他属性（比如作为球体良好但不完美近似的属性）。然而当“圆性”被确认为对象时，这种中立性便不复存在。

What does a question about the identity and distinctness of properties turn on? How does the world have to be in order to make the relevant identity statement true? Criteria of identity
关于属性同一性与差异性的问题取决于什么？世界需以何种状态才能使相关同一性陈述为真？同一性标准
are supposed to systematize our answers to such questions. Cardinality properties, such as “fourness,” provide a nice example. Is the cardinality of these things identical with the cardinality of those? It is plausible to take the answer to be ‘yes’ just in case the former things can be one-to-one correlated with the latter. That is, Hume’s Principle is plausibly taken to provide a criterion of identity for cardinality properties (cf. §2.4).
这些问题旨在系统化我们对这类问题的回答。基数性质（如“四性”）提供了一个很好的例子。这些事物的基数是否等同于那些事物的基数？一个合理的回答是“是”，仅当前者可以与后者建立一一对应关系。也就是说，休谟原则（Hume’s Principle）被认为为基数性质提供了一个同一性标准（参见§2.4）。

We are now in a position to connect the above discussion with another topic that figures prominently in this book, namely abstraction (cf. $\mathrm{\$}2.4$$\mathrm{\$}2.4$$2.4\$ 2.4 ). We began by noting that we perceive and imagine objects to have various properties and stand in various relations. We then asked what is required for such properties to be regarded as objects in their own right. It is widely believed that we need a criterion of identity to specify when two property instantiations count as instantiations of the same property. But to specify when two objects count as instantiating the same property is nothing other than to provide an equivalence relation on objects; that is, a relation that is reflexive, symmetric, and transitive. The discussion in this chapter shows that our handle on this equivalence relation can be implicit and perceptual, rather than explicit and conceptual. In cases where we rely on perception to determine whether two objects stand in the equivalence relation, it seems reasonable to talk about a quasi-perceptual mode of access to the ensuing properties. But in other cases, perception plays little or no role. In the next chapter, we shall therefore undertake a general examination of how equivalence relationsserving as criteria of identity-can underlie our apprehension of abstract objects.
我们现在可以将上述讨论与本书中另一个重要主题——抽象（参见 $\mathrm{\$}2.4$$\mathrm{\$}2.4$$2.4\$ 2.4 ）——联系起来。我们首先注意到，我们感知并想象物体具有各种属性并处于各种关系中。接着我们探讨了这些属性若要被视为独立对象需要满足什么条件。学界普遍认为，需要借助同一性标准来判定两个属性实例何时属于同一属性。而判定两个对象何时实例化同一属性，本质上就是在对象间建立等价关系——即满足自反性、对称性和传递性的关系。本章论述表明，我们把握这种等价关系的方式可以是隐含的、感知性的，而非显性的、概念化的。当我们依赖感知来判断两个对象是否具有等价关系时，采用"准感知模式"来理解由此产生的属性似乎是合理的。但在其他情况下，感知几乎不起作用或完全无关。 因此在下一章中，我们将对等价关系如何作为同一性标准——支撑我们对抽象对象的理解——进行总体考察。

The approaches to mathematical intuition discussed in this chapter are clearly presented in Maddy (1990, chap. 2); Gödel (1964); Parsons (1980); and Føllesdal (1995). Parsons (2008, chap. 5) is a more complete exposition of Parsons’ approach. Tieszen (1989) provides a fuller development of a Husserlian approach.
本章讨论的数学直觉方法在以下文献中有清晰阐述：Maddy (1990, 第 2 章)；Gödel (1964)；Parsons (1980)；以及 Føllesdal (1995)。Parsons (2008, 第 5 章)是对 Parsons 方法更完整的阐述。Tieszen (1989)则对胡塞尔式进路进行了更全面的发展。