维基百科：数学基础(zslcn周生烈编译摘注评)

译者注：本人所译文章(以及其中本人的所注、所编和所评，用绿色正体示出，仅供参考，阅读时可以略去)，首先是出于自身研究工作的需要；同时也兼顾 作为同行们和学友们 的非正式参考。文中诸多错误和谬误，恳望读者审查、指正。 不难发现,数学术语的译名,常常比较艰涩难读(但不应是晦涩难懂)，想来是为了避免 与容易产生常义二义性的习常词汇相混淆，以保证数学术语涵义的唯一性和确切性。译者把这一条 作为自己译作的信条之一；出于类似的考虑，在本人译作的译文中，亦常尝试着,采用插入空格、短逗号(正常逗号只用于 独立句但不是完整句 的场合)、增加虚词等‘不规范’的辅助方式，来尽量避免 译意的模糊性和二义性，提高译文的可读性。还应指出，译者将译作中 第一次明确出现的、译者‘杜撰’的数学术语的译名(后加原文名),以及原文中相应部分,用阴影加以强调。愿读者不吝赐教。（在本段落中即有部分体现。请见带阴影 的部分。） 为了避免术语译义上的混乱，本人译作中认为需要杜撰的重要术语,後附术语原文，必要时更附上已经存在的汉译术语，并一直保持。 周生烈 数学哲学 实数分析 群论 投影几何 布尔代数和逻辑 皮亚诺算术 基础性危机 悖论 http://en.wikipedia.org/wiki/Foundations_of_mathematics This page was last modified on 22 January 2014 at 02:54 　　 Foundations of mathematics 数学基础 　　From Wikipedia, the free encyclopedia 取自维基百科，开放的百科全书 　　 This article needs additional citations for verification. Please helpimprove this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (January 2014) 为了验证本条目，需要附加引证。请通过增加引证可靠资源 帮助改善本条目。没有资源的材料，可以质疑和删除。（2014年1月） For the book by Hilbert and Bernays, see Grundlagen der Mathematik.对于希尔伯特和伯奈斯的著作，见数学基础(德文)。 　　 　　Foundations of mathematics is the study of the basic mathematical concepts (number, geometrical figure, set, function...) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms...) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges. 数学基础研究基本数学概念（数、几何形状、集合、函数...），及其 如何构成更复杂结构和概念的 层次结构，特别是一类 也被称为元数学概念的 基础性重要结构，用它们来形成数学语言（公式、理论、以及它们的 用来表意公式、定义、证明、算法...的模型），着眼於哲学方面 以及数学的统一性。对数学基础的探索 是数学哲学的一个中心论题；数学对象的抽象性质 向哲学提出了特殊的挑战。 　　 　　The foundations of mathematics as a whole does not aim to contain the foundations of every mathematical topic. Generally, the foundations of a field of study refers to a more-or-less systematic analysis of its most basic or fundamental concepts, its conceptual unity and its natural ordering or hierarchy of concepts, which may help to connect it with the rest of human knowledge. The development, emergence and clarification of the foundations can come late in the history of a field, and may not be viewed by everyone as its most interesting part. 数学基础作为一个整体 并不瞄準於包含每个数学论题的基础。一般说来 建立一个研究领域 指的是一种系统分析，或多或少地建立 其最基本的或基础的概念、其概念的一致性、以及其概念的本性顺序或层次结构；这可以有助于 将其与其它人类知识 联系起来。但是在一个领域的历史上，基础的开发、出现、和滤清往往来得晚些，以致无法让每一个人 都能观察到 其最感兴趣部分的来龙去脉。 　　Mathematics always played a special role in scientific thought, serving since ancient times as a model of truth and rigor for rational inquiry, and giving tools or even a foundation for other sciences (especially physics). Mathematics' many developments towards higher abstractions in the 19th century brought new challenges and paradoxes, urging for a deeper and more systematic examination of the nature and criteria of mathematical truth, as well as a unification of the diverse branches of mathematics into a coherent whole. 在科学思维中 数学总是起着特殊的作用；其自古以来 一直是作为 理性探讨真理性和严谨性的一种范型，并作为 其他科学(特别是物理学)的工具,甚至是基础。在19世纪中，数学的 趋于更高抽象的 许多开发,带来了新的挑战和悖论，迫切需要对数学真理的本性和准则，进行更深入、更系统的考察，以及将各个不同的数学分支 统一成一个连贯的整体。 　　 　　The systematic search for the foundations of mathematics started at the end of the 19th century and formed a new mathematical discipline called mathematical logic, with strong links to theoretical computer science. It went through a series of crises with paradoxical results, until the discoveries stabilized during the 20th century as a large and coherent body of mathematical knowledge with several aspects or components (set theory, model theory, proof theory...), whose detailed properties and possible variants are still an active research field. Its high level of technical sophistication inspired many philosophers to conjecture that it can serve as a model or pattern for the foundations of other sciences. 系统地探索数学基础 始于19世纪末，并形成了一个 与理论计算机科学 有紧密联系的、称之为数理逻辑 的新数学学科。它历经了 种种相悖结论 的一系列危机，直到 在20世纪期间 发掘出 作为具有多个方位或组成部分（集合论，模型论，证明论·····）的 一个庞大的、条理分明的 数学知识体系 而稳定下来。研究其详尽的属性和可能的变体，仍然是一个活跃的研究领域。它的深邃的技术内涵，激励了许多哲学家去揣测，它可能作为一种 成为其他科学的基础 的模型或模式。 　　 Contents 1 Historical context 历史背景 1.1 Ancient Greek mathematics 古希腊的数学 1.2 Platonism as a traditional philosophy of mathematics 柏拉图主义作为一种传统的数学哲学 1.3 Middle Ages and Renaissance 中世纪和文艺复兴时期 1.4 19th century 19世纪 1.4.1 Real Analysis 实数分析 1.4.2 Group theory 群论 1.4.3 Non-Euclidean Geometries 非欧几何 1.4.4 Projective geometry 投影几何 1.4.5 Boolean algebra and logic 布尔代数和逻辑 1.4.6 Peano Arithmetic 皮亚诺算术 2 Foundational crisis 基础性危机 2.1 Philosophical views 哲学观点 2.1.1 Formalism 形式主义 2.1.2 Intuitionism 直觉主义 2.1.3 Logicism 逻辑主义 2.1.4 Set-theoretical Platonism 集合论的柏拉图主义 2.1.5 Indispensability argument for realism 对现实主义的不可或缺的论证 2.1.6 Rough-and-ready realism 粗线条的现实主义 2.1.7 Philosophical consequences of the Completeness Theorem 完备性定理的哲学推论 2.2 More paradoxes 更多的悖论 3 Partial resolution of the crisis 危机的部分解决 4 See also 参见 5 Notes 注解 6 References 参考文献 7 External links 外部链接 　　 ·1.Historical context[edit] 历史背景 See also: History of logic and History of mathematics.另请参阅：逻辑史和数学史。 　　 　　1.1 Ancient Greek mathematics[edit] 古希腊的数学 　　 　　While the practice of mathematics previously developed in other civilizations, the special interests for its theoretical and foundational aspects really started with Ancient Greeks. Early Greek philosophers disputed as to which is more basic, arithmetic or geometry. Zeno of Elea (490 BC – ca. 430 BC) produced four paradoxes that seem to show that change is impossible. 虽然早在其他文明时代 就已有数学的实践，但对于其理论和基础方面的特殊兴趣 实际上是从古希腊人开始的。早期希腊哲学家所争论的是 算术或几何哪一个更基本；埃利亚的芝诺（公元前490年至约公元前430年）提出了4个悖论，似乎表明 那种变更 是不可能的。 　　 　　The Pythagorean school of mathematics originally insisted that only natural and rational numbers exist. The discovery of the irrationality of √2, the ratio of the diagonal of a square to its side (around 5th century BC), was a shock to them which they only reluctantly accepted. The discrepancy between rationals and reals was finally resolved by Eudoxus of Cnidus, a student of Plato, who reduced the comparison of irrational ratios to comparisons of multiples (rational ratios), thus anticipating Richard Dedekind's definition of real numbers. 数学的毕达哥拉斯学派 最初坚持认为 只存在自然数和有理数。√2，即正方形的对角线与其边之比，其非有理性的发现(约公元前5世纪),是对他们的一个冲击，他们只是勉强接受。有理数和实数之间的冲突 是由克尼得岛的欧多克斯,柏拉图的一个学生,最终解决的；他将无理比率的比较，简化为倍乘(有理比率)的比较，从而预见到(可从 用垂线对正直角三角形进行不断分割，形成一系列(无穷)镶套的正直角三角形；根据 三角形两边之和大于第三边,排序 镶套正直角三角形各弦，而导出) 实数的理查德·戴德金定义。 　　 　　In the Posterior Analytics, Aristotle (384 BC – 322 BC) laid down the axiomatic method, to organize a field of knowledge logically by means of primitive concepts, axioms, postulates, definitions, and theorems, taking a majority of his examples from arithmetic and geometry. This method reached its high point with Euclid's Elements(300 BC), a monumental treatise on geometry structured with very high standards of rigor: each proposition is justified by a demonstration in the form of chains of syllogisms (though they do not always conform strictly to Aristotelian templates). Aristotle's syllogistic logic, together with the Axiomatic Method exemplified by Euclid's Elements, are universally recognized as towering scientific achievements of ancient Greece. 在(工具论之)后分析篇中，亚里士多德（公元前384 -公元前322年）提出了公理化方法；他从算术和几何的主要范例中，将原始概念、公理、公设、定义、和定理，从逻辑上组织成 一个知识领域。这一方法 在欧几里德的几何原本著作中(公元前300年）达到了高峰。几何原本是一本关于几何的里程碑式著作，它以十分严谨的标准写成；每个命题 都是通过一个 用三段论链接形式的论证 来合理化（虽然它们并不总是严格地遵守亚里士多德的模式）。亚里士多德的三段论逻辑 加上公理化方法，通过欧几里德 几何原本的实例化，被公认为是古希腊的顶尖科学成就。 　　 　　1.2 Platonism as a traditional philosophy of mathematics[edit] 作为一种传统数学哲学的柏拉图主义 　　 　　Starting from the end of the 19th century, a Platonist view of mathematics became common among practicing mathematicians. 从十九世纪末开始，一种柏拉图学派的数学观点，曾普遍存在于实用数学家之中。 　　The concepts or, as Platonists would have it, the objects of mathematics are abstract and remote from everyday perceptual experience: geometrical figures are conceived as idealities to be distinguished from effective drawings and shapes of objects, and numbers are not confused with the counting of concrete objects. Their existence and nature present special philosophical challenges: How do mathematical objects differ from their concrete representation? Are they located in their representation, or in our minds, or somewhere else? How can we know them? 柏拉图学派具有这样的概念，即 数学的客体是抽象的，远离日常的感性经验：几何图形理想化，以区别于客体的实际图样和形状；数字不与具体客体的计算相混淆。它们的存在和本性 出现了特殊的哲学挑战：如何做到 数学客体不同于具体表现？它们是位于其表现形式中，或者是在我们的头脑中，还是别的什么地方？我们怎样才能知道它们呢？ 　　 　　The ancient Greek philosophers took such questions very seriously. Indeed, many of their general philosophical discussions were carried on with extensive reference to geometry and arithmetic. Plato (424/423 BC – 348/347 BC) insisted that mathematical objects, like other platonic Ideas (forms or essences), must be perfectly abstract and have a separate, non-material kind of existence, in a world of mathematical objects independent of humans. He believed that the truths about these objects also exists independently of the human mind, but is discovered by humans. In the Meno Plato’s teacher Socrates asserts that it is possible to come to know this truth by a process akin to memory retrieval. 古希腊哲学家非常严肃地对待这些问题。事实上，他们之间 许多通常的哲学讨论 就是广泛引用几何和算术 来进行的。 柏拉图（公元前423/424年-公元前34/348年）坚持认为，数学客体 像其他柏拉图理念（形式或本质）一样，必须完善地抽象，且在一个独立于人类的 数学客体世界中，具有一种独立的、非物质类别的存在。他认为，关于这些客体的真实性，也独立于人类的脑海而存在，但被人类发现了。在梅诺 柏拉图的老师苏格拉底 声称，通过一种类似于记忆提取的过程，有可能发现这种真实性。 　　 　　Above the gateway to Plato's academy appeared a famous inscription: "Let no one who is ignorant of geometry enter here". 在通向网关柏拉图学院的必经之途上，呈现有一句著名的题词：“不让任何一个对几何无知的人 进入这里”。 　　In this way Plato indicated his high opinion of geometry. He regarded geometry as ``the first essential in the training of philosophers", because of its abstract character. 通过这种方式 柏拉图表明了他对几何的高度评价。由于几何的抽象特征，他把几何尊为“培训哲学家中的第一要素”。 　　 　　This philosophy of Platonist mathematical realism, is shared by many mathematicians. It can be argued that Platonism somehow comes as a necessary assumption underlying any mathematical work.[1] 这种柏拉图式数学现实主义哲学，是许多数学家都共有的。 这可以表明 柏拉图主义 在某种程度上 是一种 构建任何数学工作 的必要前提。 [1] 　　 　　In this view, the laws of nature and the laws of mathematics have a similar status, and the effectiveness ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation. 这种观点认为，自然规律和数学法则 也有类似的情况，有效性不再是不合理的。 不是我们的公理，而是数学客体的十分现实的世界 形成了这一基础。 　　 　　Aristotle dissected and rejected this view in his Metaphysics. These questions provide much fuel for philosophical analysis and debate. 亚里士多德 以他的形而上学立场 解剖并拒绝这一观点。这些问题 为哲学分析和辩论 注入了大量的‘燃料’。 　　 1.3 Middle Ages and Renaissance[edit] 中世纪和文艺复兴时期 [编辑] 　　 　　For over 2,000 years, Euclid’s Elements stood as a perfectly solid foundation for mathematics, as its methodology of rational exploration guided mathematicians, philosophers, and scientists well into the 19th century. 几何原本 在过去的2000多年里，一直作为一种用于数学的完美坚实基础；其合理探索的方法论，引导数学家、哲学家、和科学家,顺当地进入19世纪。 　　 　　The Middle Ages saw a dispute over the ontological status of the universals (platonic Ideas): Realism asserted their existence independently of perception; conceptualism asserted their existence within the mind only;nominalism, denied either, only seeing universals as names of collections of individual objects (following older speculations that they are words, "logos"). 在中世纪，出现过一种 关于一般概念（柏拉图理念）的本体论状态的争议：现实主义断言，它们的存在 独立于感知；观念主义宣称，它们只存在于头脑中；唯名主义则对这两种断言都拒绝，将一般概念 仅仅看作为 独个客体集合的名称（它们是词‘logos（标识）’顺着原意推测）。 　　 　　René Descartes published La Géométrie (1637) aimed to reduce geometry to algebra by means of coordinate systems, giving algebra a more foundational role (while the Greeks embedded arithmetic into geometry by identifying whole numbers with evenly spaced points on a line). It became famous after 1649 and paved the way to infinitesimal calculus. 勒内·笛卡尔发表了几何学(1637)，旨在通过坐标系 将几何简化到代数，予代数以更基础的角色（而希腊人 则通过确认 将全部数 均匀分布在一条线上的点，把算术嵌入至几何）。这一著作铺平了通向无限小运算的道路，并在1649年以后名声大增。 　　 　　Isaac Newton (1642 – 1727) in England and Leibniz (1646 – 1716) in Germany independently developed theinfinitesimal calculus based on heuristic methods greatly efficient, but direly lacking rigorous justifications. Leibniz even went on to explicitly describe infinitesimals as actual infinitely small numbers (close to zero). Leibniz also worked on formal logic but most of his writings on it remained unpublished until 1903. 艾萨克·牛顿（1642-1727）在英格兰，莱布尼茨（1646-1716）在德国，各自独立开发了 基于探试法 的无限小运算，十分有效 但极缺乏严格的理据。莱布尼茨更进而阐明 无穷小是作为实际无限小的数（接近于零）。莱布尼茨还参与了形式逻辑的研究；但直到1903年，他的大部分有关著作仍未发表。 　　 　　The Christian philosopher George Berkeley (1685–1753), in his campaign against the religious implications of Newtonian mechanics, wrote a pamphlet on the lack of rational justifications of infinitesimal calculus:[2] “They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?” 基督教徒哲学家乔治·伯克利(1685–1753)，在他攻击牛顿力学宗教含义的运动中，写了一本 关于无限小运算缺乏合理理据的小册子：[2]“他们既不是有限的数量，也不是数量无限小，还 什么也不是。那么 我们不可以称它们为 数量已经死亡 的鬼吗？” 　　 　　Then mathematics developed very rapidly and successfully in physical applications, but with little attention to logical foundations. 自此之后，在物理应用中 数学取得了十分迅速和成功的发展，却很少关注逻辑基础。 　　 1.4 19th century[edit] 19世纪[编辑] 　　 　　In the 19th century, mathematics became increasingly abstract. Concerns about logical gaps and inconsistencies in different fields led to the development of axiomatic systems. 在19世纪，数学变得越来越抽象。出于对不同的领域中逻辑漏洞和不一致性的关注，导致公理系统的开发。 　　 　　1.4.1 Real Analysis[edit] 实分析 [编辑] 　　 　　Cauchy (1789 – 1857) started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors. In his 1821 work Cours d'Analyse he defines infinitely small quantities in terms of decreasing sequences that converge to 0, which he then used to define continuity. But he did not formalize his notion of convergence. 柯西（1789-1857）排斥为早期作者所使用的 通用代数的探索式原理，开始用一种严谨方法 来设计无限小演算定理的公式化和证明。他在1821年的著作分析教程中，依据 收敛趋于0的递减序列，定义了无穷小量，然后 以此来定义连续性。但他没有形式化他的收敛的概念。 　　 　　The modern (ε, δ)-definition of limit and continuous functions was first developed by Bolzano in 1817, but remained relatively unknown. It gives a rigorous foundation of infinitesimal calculus based on the set of real numbers, arguably resolving the Zeno paradoxes and Berkeley's arguments. 近代的(ε, δ)-极限定义和连续函数 是由 博尔扎诺于1817年首先开发的，但这相对不为人知。他基于实数集合，赋予无限小演算 一个严格的基础，可以认为 解决了芝诺悖论 及伯克利的争辩。 　　 　　Mathematicians such as Karl Weierstrass (1815 – 1897) discovered pathological functions such as continuous, nowhere-differentiable functions. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis, to axiomatize analysis using properties of the natural numbers. In 1858, Dedekind proposed a definition of the real numbers as cuts of rational numbers. This reduction of real numbers and continuous functions in terms of rational numbers and thus of natural numbers, was later integrated by Cantor in his set theory, and axiomatized in terms of second order arithmetic by Hilbert and Bernays. 数学家们 如卡尔·维尔斯特拉斯(1815 – 1897) 发现了 诸如连续的,无处可微函数之类的 异态函数。以前的 一个函数作为一种 用于计算规则 或平滑曲线 的观念，已不再恰当。维尔斯特拉斯开始提倡分析算术化，采用自然数的属性 来将分析公理化。 1858年，戴德金提出了一种 将实数作为分隔有理数 的定义。这种 依据有理数乃至自然数 来简化实数和连续函数，后来被康托尔综合到他的集合论中，并由希尔伯特和伯内斯 依据二阶算术 公理化。 　　 　　1.4.2 Group theory[edit] 群论 [编辑] 　　 　　For the first time, the limits of mathematics were explored. Niels Henrik Abel (1802 – 1829), a Norwegian, andÉvariste Galois, (1811 – 1832) a Frenchman, investigated the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four (Abel–Ruffini theorem). With these concepts, Pierre Wantzel (1837) proved that straightedge and compass alone cannot trisect an arbitrary angle nor double a cube, nor construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks. 最初，探讨的是数学的局限性。尼尔斯·亨里克·阿贝尔(1802 – 1829),一位挪威人，和埃瓦里斯特·伽罗瓦(1811 – 1832),一位法国人，研究了各种多项式方程的解，并证明了 对于大于四次的方程 不存在一般代数解(阿贝尔-鲁菲尼定理)。根据这些概念，Pierre Wantzel（1837皮埃尔）证明，只用直尺和圆规 不能三等分任意角，不能加倍立方体，也不能构建一个 面积等于一已知圆 的正方形。而在此前，数学家一直徒劳地企图解决 自古希腊时代提出的 所有这些问题。 　　 　　Abel and Galois's works opened the way for the developments of group theory (which would later be used to studysymmetry in physics and other fields), and abstract algebra. Concepts of vector spaces emerged from the conception of barycentric coordinates by Möbius in 1827, to the modern definition of vector spaces and linear maps by Peano in 1888. Geometry was no more limited to 3 dimensions. These concepts did not generalize numbers but combined notions of functions and sets which were not yet formalized, breaking away from familiar mathematical objects. 阿贝尔和伽罗华的工作 开辟了 开发群论（这后来被用来 在物理学和其他领域中 研究对称性）和抽象代数 的道路；从1827年 莫比乌斯构想重心坐标，到1888年 皮亚诺定义了 向量空间和线性映射的现代定义，向量空间的概念出现了；几何也没有更多受限于3维。这些概念并没有将数一般化，但是 组合了函数和集合的概念,那时还尚未形式化，从此告别了人们熟悉的数学客体。 　　1.4.3 Non-Euclidean Geometries[edit] 非欧几何 [编辑] 　　 　　After many failed attempts to derive the parallel postulate from other axioms, the study of the still hypotheticalhyperbolic geometry by Johann Heinrich Lambert (1728 – 1777) led him to introduce the hyperbolic functions and compute the area of a hyperbolic triangle (where the sum of angles is less than 180°). Then the Russian mathematician Nikolai Lobachevsky (1792–1856) established in 1826 (and published in 1829) the coherence of this geometry (thus the independence of the parallel postulate), in parallel with the Hungarian mathematician János Bolyai (1802–60) in 1832, and with Gauss. Later in the 19th century, the German mathematician Bernhard Riemanndeveloped Elliptic geometry, another non-Euclidean geometry where no parallel can be found and the sum of angles in a triangle is more than 180°. It was proved consistent by defining point to mean a pair of antipodal points on a fixed sphere and line to mean a great circle on the sphere. At that time, the main method for proving the consistency of a set of axioms was to provide a model for it. 约翰·海因里希·兰伯特（1728-1777）从(几何的)其他公理 导出平行公设的企图，经过多次失败之后，研究了 仍然是假设性的双曲几何，导致他引入双曲函数 来计算一个双曲三角形的面积（其中 三角形三角总和 小于180°）。之后 俄罗斯数学家 尼古拉·罗巴切夫斯基(1792–1856)于1826年（发表于1829年）、与此同时，还有匈牙利数学家亚诺什·波尔约(1802–1860)于1832年、以及高斯，确定了该几何的一致性（从而独立于平行公设）。19世纪后期，德国数学家伯恩哈德·黎曼开发的椭圆几何--又一种非欧几何--找不到平行线，其三角形内角之和大于180°；通过定义 点 意味着 一固定球体上的一对对极点，线 意味着 该球体上的一大圆弧，来证明其一致性。那时 为了证明一组公理的一致性，其主要方法是 为一致性提供一个模型。 　　 　　1.4.4 Projective geometry[edit] 射影几何 [编辑] 　　 　　One of the traps in a deductive system is circular reasoning, a problem that seemed to befall projective geometryuntil it was resolved by Karl von Staudt. As explained by Laptev & Rosenfeld (1996): 在一个演绎系统中 陷阱之一是循环论证；这个问题似乎也降临到射影几何，直到这个问题 为卡尔·冯·施陶特所解决。正如拉普捷夫海和罗森菲尔德（1996）所解释的那样： In the mid-nineteenth century there was an acrimonious controversy between the proponents of synthetic and analytic methods in projective geometry, the two sides accusing each other of mixing projective and metric concepts. Indeed the basic concept that is applied in the synthetic presentation of projective geometry, thecross-ratio of four points of a line, was introduced through consideration of the lengths of intervals. 十九世纪中叶 在射影几何中 有一场 发生在综合方法支持者和分析方法支持者之间 的激烈争论，双方彼此指责对方 混淆了投影和度量的概念。的确，应用于投影几何综合表述 的基本概念,一线上4点的交比，是通过考察区间的长度而引入的。 　　The purely geometric approach of von Staudt was based on the complete quadrilateral to express the relation of projective harmonic conjugates. Then he created a means of expressing the familiar numeric properties with his Algebra of Throws. English language versions of this process of deducing the properties of a field can be found in either the book by Oswald Veblen and John Young, Projective Geometry (1938), or more recently in John Stillwell'sFour Pillars of Geometry (2005). Stillwell writes on page 120 冯·施陶特的纯几何方法 是基于完全四线形 来表达投射调和共轭的关系。 然后，他采用他的投掷代数 创建了一种 表达熟知的数字属性 的工具。这一演绎一域属性的过程 的英文版本，可以从 奥斯瓦尔德·凡勃伦和约翰·扬的著作射影几何 （1938），或最近的约翰·史迪威著作几何的四大支柱（2005年）中 任一本书中找到。史迪威在第120页上写道： 　　...projective geometry is simpler than algebra in a certain sense, because we use only five geometric axioms to derive the nine field axioms. ...射影几何比较于代数 在某种意义上 更简单 ，因为我们只使用五个几何公理 就推导出九个域公理。 　　The algebra of throws is commonly seen as a feature of cross-ratios since students ordinarily rely upon numberswithout worry about their basis. However, cross-ratio calculations use metric features of geometry, features not admitted by purists. For instance, in 1961 Coxeter wrote Introduction to Geometry without mention of cross-ratio. 投掷代数一般被视为交比的一个特点，因为学生们通常依赖于数字 并不担心自身的基础。 然而，交比计算使用的 几何度量特征，并不为纯粹主义者所承认。 例如 考克斯特在1961年所著的几何简介中 就没有提及交比。 　　 　　1.4.5 Boolean algebra and logic[edit] 布尔代数和逻辑 [编辑] 　　 　　Attempts of formal treatment of mathematics had started with Leibniz and Lambert (1728 – 1777), and continued with works by algebraists such as George Peacock (1791 – 1858). Systematic mathematical treatments of logic came with the British mathematician George Boole (1847) who devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1 and logical combinations (conjunction, disjunction, implication and negation) are operations similar to the addition and multiplication of integers. Also De Morgan publishes his laws (1847). Logic becomes a branch of mathematics. Boolean algebra is the starting point of mathematical logic and has important applications in computer science. 数学的 形式处理的尝试 始于莱布尼茨和兰伯特（1728-1777），并为代数学家们 诸如乔治·皮科克(1791 – 1858) 继承和发展。逻辑的系统数学处理 则是伴随着英国数学家乔治·布尔（1847）而来。布尔发明了一种代数，很快就演变成 现在称谓的布尔代数，其中只有数字0和1 以及逻辑组合（合取、析取、蕴涵、和否取），是类似于整数加和乘的运算。德·摩根也发表了他的运算法则（1847）。逻辑成了数学的一个分支。布尔代数是数理逻辑的出发点，在计算机科学中具有重要应用。 　　 　　Charles Sanders Peirce built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. 在布尔工作的基础上 查尔斯·桑德斯·皮尔士开发了一个用于关系和数量的逻辑系统；从1870年至1885年，他发表的几篇关于这方面的论文。 　　 　　The German mathematician Gottlob Frege (1848–1925) presented an independent development of logic with quantifiers in his Begriffsschrift (formula language) published in 1879, a work generally considered as marking a turning point in the history of logic. He exposed deficiencies in Aristotle's Logic, and pointed out the 3 expected properties of a mathematical theory 德国数学家弗雷格(1848–1925)，在他的 出版于1879年的 概念文字（公式语言）中，提出一种 具有数量的逻辑 的独立开发。这项工作 在逻辑史中 通常被认为 标志着一个转折点。他揭示了亚里士多德逻辑中的缺陷，并指出了 一种数学理论的3个预期属性： 　　Consistency: impossibility to prove contradictory statements 一致性：证明矛盾陈述的不可能性 　　Completeness: any statement is either provable or refutable (i.e. its negation is provable). 完备性：任何陈述不是可证的，就是可驳的（即它的否取是可证明的）。 　　Decidability: there is a decision procedure to test any statement in the theory.可判定性：存在一种决策步骤，来测试该理论中的任何陈述。 　　 　　He then showed in Grundgesetze der Arithmetik (Basic Laws of Arithmetic) how arithmetic could be formalised in his new logic. 之后 他在Grundgesetze DER Arithmetik（算术的基本规律）中证示了 算术如何可以用他的新逻辑的概念 形式化。 　　 　　Frege's work was popularized by Bertrand Russell near the turn of the century. But Frege's two-dimensional notation had no success. Popular notations were (x) for universal and (∃x) for existential quantifiers, coming from Giuseppe Peano and William Ernest Johnson until the ∀ symbol was introduced by Gentzen in 1935 and became canonical in the 1960s. 弗雷格的工作 是由罗素 在接近世纪之交 推广的。然而 弗雷格的二维符号 没有成功。流行的符号是 用于全称命题的（x）、和用于存在量词的（∃x）--它们来自杰赛普·皮亚诺和威廉·欧内斯特·约翰逊、以及 直到1935年根岑引入的 符号∀；它们在20世纪60年代成为经典。 　　 　　From 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century. 从1890年到1905年间，恩斯特·施罗德发表Vorlesungen über die Algebra der (Vorlesungen超级模代数逻辑学)三卷本。这项工作 总结和扩展了 布尔、德·摩根、和皮尔斯的工作；这是一本关于符号逻辑的综合参考，因为它基于19世纪末年代的认识。 　　 　　1.4.6 Peano Arithmetic[edit] 皮亚诺算术 　　 　　The formalization of arithmetic (the theory of natural numbers) as an axiomatic theory, started with Peirce in 1881, and continued with Richard Dedekind and Giuseppe Peano in 1888. This was still a second-order axiomatization (expressing induction in terms of arbitrary subsets, thus with an implicit use of set theory) as concerns for expressing theories in first-order logic were not yet understood. In Dedekind's work, this approach appears as completely characterizing natural numbers and providing recursive definitions of addition and multiplication from the successor function and mathematical induction. 作为一种公理化理论，算术（自然数理论）的形式化，始于1881年皮尔斯，并于1888年 为理查德·戴德和杰赛普·皮亚诺所继续。这仍然是一个二阶公理化（依据任意子集表达归纳，于是 采取一种隐含使用集合论 的方法）；因为对于用一阶逻辑表达理论 尚不能理解。在戴德金的工作中，这种做法 似乎完整地表征了自然数，并根据后继函数和数学归纳法 提供了加法和乘法的递归定义。 　　 ·2. Foundational crisis[edit] 基础性危机 　　 　　The foundational crisis of mathematics (in German wikipedia article: Grundlagenkrise der Mathematik) was the early 20th century's term for the search for proper foundations of mathematics. 数学基础的危机（在德文维基百科的文章：Grundlagenkrise DER Mathematik中）是20世纪初的术语，用来探索数学的恰当基础。 　　 　　Several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, as the assumption that mathematics had any foundation that could be consistently stated within mathematics itself was heavily challenged by the discovery of various paradoxes (such as Russell's paradox). 　　在20世纪中，数学哲学的几个学派，由于认定 数学已经具备了 数学自身就能说明的一致性，而受到 发现各种悖论(如罗素悖论)的沉重挑战，一个跟着一个进入了困境。 　　 　　The name "paradox" should not be confused with contradiction. A contradiction in a formal theory is a formal proof of an absurdity inside the theory (such as 2 + 2 = 5), showing that this theory is inconsistent and must be rejected. But a paradox may either refer to a surprising but true result in a given formal theory, or to an informal argument leading to a contradiction, so that a candidate theory where a formalization of the argument might be attempted must disallow at least one of its steps; in this case the problem is to find a satisfying theory without contradiction. Both meanings may apply if the formalized version of the argument forms the proof of a surprising truth. For instance, Russell's paradox may be expressed as "there is no set of all sets" (except in some marginal axiomatic set theories). 名称‘悖论’ 不应该与矛盾相混淆。在一种形式理论中，如果存在着 某种谬论（如2 +2 = 5）却能在该理论中 得到形式证明，这就是一个矛盾；这表明 该理论是不一致的，必须被拒绝。而一个悖论 或者指的是 在一种所给的形式理论中 是一种惊异 却为真的结果；或者是 一种导致矛盾的 非形式论据；这样，要使待选理论的论据形式化，就必须禁止其中至少一个步骤；在这种情况下，问题归结为 去寻找一种没有矛盾的满意理论。如果论据的形式化版本 形成了一个惊异事实的证明，这两种涵义都可适用。例如，罗素悖论可以表达为“不存在所有集合的集合”（除了在某些边缘化的公理化集合理论中）。 　　 　　Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a logical system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols (van Dalen, 2008). The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time. 关于恰当处理数学基础的思路，有着各种流派，他们彼此激烈反对对方。领先的流派是形式主义者方法，其中大卫·希尔伯特是最重要的倡导者，最终归结为著名的希尔伯特规划；规划设想 将数学建立在 一个逻辑系统的很小基础上，而这个基础 通过元数学的有穷手段 证明是健壮的。这一流派的主要对手 是直观主义者流派，由 L.E.J.布劳威尔带头，坚决摒弃形式主义，认为是一个毫无意义的符号游戏（范·达伦，2008）。争斗很激烈。1920年 希尔伯特成功地摆布了布劳威尔：希尔伯特认为布劳威尔是数学的一个威胁，于是从数学年鉴编辑部赶走了布劳威尔。数学年鉴 是当时一本主要的数学杂志。 　　 　　2.1 Philosophical views[edit] 哲学观点 　　Main article: Philosophy of mathematics 主条目：数学哲学 　　 　　At the beginning of the 20th century, 3 schools of philosophy of mathematics were opposing each other: Formalism, Intuitionism and Logicism. 20世纪初，数学哲学的三个流派是彼此对立的：形式主义、直觉主义、和逻辑主义。 （这几种‘对立’的哲学观点，似乎可以统一起来：柏拉图主义 对客观世界和客体的观点，以及将客体和客观世界 抽象出 人们能理解的 其属性共性 的认识，是本质；自觉主义肯定了 人类认识客观事物的特殊途径和抽象方法（思维），不能离开人类思维而独立存在，是宇宙中‘人类’这个客体的特有属性；逻辑主义强调的逻辑 是人类思维进行抽象的基础；形式主义强调的形式 是逻辑在数学方向的延伸。） 　　 　　2.1.1 Formalism 形式主义[edit] 　　Main article: Formalism (mathematics) 主条目：形式主义（数学） 　　 　　It has been claimed that formalists, such as David Hilbert (1862–1943), hold that mathematics is only a language and a series of games. Indeed he used the words "formula game" in his 1927 response to L. E. J. Brouwer's criticisms: 形式主义者 如大卫·希尔伯特（1862-1943），一直持有这样的主张，即 认为数学只是一种语言和一系列博弈。事实上 在1927年 针对L.E.J.布劳威尔的批评，他使用字眼“公式博弈”来回应： 　　 　　"And to what has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it in such a way that, at the same time, the interconnections between the individual propositions and facts become clear......The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed. These rules form a closed system that can be discovered and definitively stated."[3] “为什么有了公式博弈就可能获得成功？这种公式博弈，使我们能够 以统一的方式 表达数学科学的整个思想内容，并以这样一种途径 即 同时将独立命题和事实 互联起来进行开发,这样一种思路 变得清晰起来......被布劳威尔如此反对的公式博弈，除了它的数学价值，更有一个重要的普遍哲学意义。对于这种公式博弈 是按照某种明确的规则来完成的，其中表达了我们思维的技巧。这些规则 构成了一个 可以揭示并明确说明的 封闭系统。”[ 3 ] 　　 　　Thus Hilbert is insisting that mathematics is not an arbitrary game with arbitrary rules; rather it must agree with how our thinking, and then our speaking and writing, proceeds.[3] 因此 希尔伯特坚持认为，数学不是一个 采用任意规则的任意游戏，而是必须与 我们如何思考 乃至我们说和写 的进程 相一致。[ 3 ] 　　 　　"We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise."[4] “在这里 我们并非说 是任何意义上的随意性。数学并不像游戏那样 其任务可通过任意约定的规则 来确定，相反 这是一个 具有内部必要性的 概念系统，它只能如此 否则就决不。” [ 4 ] 　　 　　The foundational philosophy of formalism, as exemplified by David Hilbert, is a response to the paradoxes of set theory, and is based on formal logic. Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is represented by the fact that the statement can be derived from the axioms of set theory using the rules of formal logic. 形式主义的基本哲学思想 如大卫·希尔伯特所例证的 是一种 对集合论的悖论 基于形式逻辑的回应。实际上 几乎所有的数学定理 今天都可归结为 集合论的定理。以这样的视角观察，判定一个数学主题的真实性，就可 通过使用形式逻辑 根据集合论公理 导出该主题，来确认。 　　 　　Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why do "true" mathematical statements (e.g., the laws of arithmetic) appear to be true, and so on. Hermann Weyl would ask these very questions of Hilbert: 仅仅只使用形式主义 还不能解释几个疑问：为什么我们应当使用的公理 是我们强调的那些 而不是某些其他的，为什么我们应当使用的逻辑规则 是我们强调的那些 而不是某些其他的，为什么强调 “真”数学语句（例如，算术法则）看起来是真实的，诸如此类。赫尔曼·外尔问及了这些很有疑问的希尔伯特的疑题： 　　"What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? Consistency is indeed a necessary but not a sufficient condition. For the time being we probably cannot answer this question…"[5] “至于‘真实性’或客观性 可以归结到 我们世界的这种理性结构，这远远超出了我们所要考虑的，是一个深奥的哲学问题。这与下面的进一步问题 紧密联系在一起：是什么促使我们 非要采取 由希尔伯特开发的 特有的公理系统 作为一种基础？一致性确实是一个必要条件，但不是充分条件。暂且我们或许不能回答这个疑问......“ [ 5 ] 　　 　　In some cases these questions may be sufficiently answered through the study of formal theories, in disciplines such as reverse mathematics and computational complexity theory. As noted by Weyl, formal logical systems also run the risk of inconsistency; in Peano arithmetic, this arguably has already been settled with several proofs ofconsistency, but there is debate over whether or not they are sufficiently finitary to be meaningful. Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own consistency. What Hilbert wanted to do was prove a logical system S was consistent, based on principles P that only made up a small part of S. But Gödel proved that the principles P could not even prove P to be consistent, let alone S! 在某些情况下 这些疑题 在诸如逆向数学和计算复杂性理论等学科中 可以通过形式理论的研究 作出充分的回答。正如魏尔所指出的，形式逻辑系统 也冒着不一致的风险；在皮亚诺算术中，这可以说 通过若干一致性证明 而已经解决，但在 它们足够有穷方面 是否有意义 存在争论。哥德尔第二不完备性定理 确立了 算术逻辑系统 从来不能包含它们自身一致性的有效证明。至于希尔伯特打算要做的 是证明 基于原理P的一个逻辑系统S 是一致的，而P仅由S的一小部分组成。然而哥德尔已经证明，原理P甚至不能证明P是一致的，更何况S！ 　　 　　2.1.2 Intuitionism 直觉主义[edit] 　　Main article: Intuitionism 主条目：直觉主义 　　 　　Intuitionists, such as L. E. J. Brouwer (1882–1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them. 直觉主义，如L.E.J.布劳威尔（1882-1966），认为数学是一种人类心智的创造。数字 就像童话人物 仅仅是精神实体，如果从未有过任何人的头脑去思考它们，也就不存在了。 　　 　　The foundational philosophy of intuitionism or constructivism, as exemplified in the extreme by Brouwer and more coherently by Stephen Kleene, requires proofs to be "constructive" in nature – the existence of an object must be demonstrated rather than inferred from a demonstration of the impossibility of its non-existence. For example, as a consequence of this the form of proof known as reductio ad absurdum is suspect. 直觉主义或建构主义的基本哲学思想，由布劳威尔极端化，更为斯蒂芬·克莱尼条理化；正如所例证的那样，需要证明 是按本性‘构建’的--一个客体的存在 必须被论证，而不是 从论证其不存在的不可能性 来推断。由此推论，称之为归谬法的证明形式，是受怀疑的。 　　 　　Some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter remains controversial. 在数学哲学中的某些现代理论 否认存在原来意义上的基础。有些理论 往往把重点放在数学实践上，旨在描述和分析 作为一个社会群体的数学家们的 实际工作。其他的 则试图建立一种数学的认知科学，当其应用于现实世界时，其专注于 将人类的认知 作为数学可靠性的起点；这些理论会建议 只在人类的思维中 寻找基础，而不在任何客观的外部结构中寻找。这件事仍存在着争议。 　　 　　2.1.3 Logicism 逻辑主义[edit] 　　Main article: Logicism 主要文章：逻辑主义 　　 　　Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russelland Alfred North Whitehead championed this theory fathered by Gottlob Frege. 在数学哲学中 逻辑主义是思维学派之一，把理论推向 数学是逻辑的延伸，因而部分或全部数学 就可简化到逻辑。伯特兰·罗素和艾尔弗雷德·诺斯·怀特海倡导这一 由弗雷格创建的理论。 　　 　　2.1.4 Set-theoretical Platonism 集合论的柏拉图主义[edit] 　　 　　Many researchers in axiomatic set theory have subscribed to what is known as set-theoretical Platonism, exemplified by mathematician Kurt Gödel. 许多公理集合论的研究者 都认同 由数学家库尔特·哥德尔例证了的 集合论的柏拉图主义。 　　 　　Several set theorists followed this approach and actively searched for possible axioms that may be considered as true for heuristic reasons and that would decide the continuum hypothesis. Many large cardinal axioms were studied but the continuum hypothesis remained independent from them. Other types of axioms were considered, but none of them has as yet reached consensus as a solution to the continuum problem. 若干集合理论家 遵循这一方法；并积极寻找这样一类可能的公理，即 其对于探索式推理 可以认为为真，以及 能判定连续统假设的公理。他们曾对许多大基数公理 进行了研究，但对连续统假设的公理 的研究 独立进行。其他类型的公理 也作了考虑，然而 没有一个公理 可以成为他们 对连续统问题解案 的共识。 　　2.1.5 Indispensability argument for realism 对于现实主义不可或缺的论证[edit] 　　 　　This argument by Willard Quine and Hilary Putnam says (in Putnam's shorter words),这一由威拉德·奎因和希拉里·普特南提出的论据 (用普特南更短的话)说， 　　quantification over mathematical entities is indispensable for science...; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question. 遍及数学实体的量化 对于科学是不可缺少的 ......，因此，我们应该接受这样的量化，但这使我们承受 接受有疑问数学实体的存在。 　　 　　However Putnam was not a Platonist. 然而 普特南不是柏拉图主义者。 　　2.1.6 Rough-and-ready realism 粗线条的现实主义[edit] 　　 　　Few mathematicians are typically concerned on a daily, working basis over logicism, formalism or any other philosophical position. Instead, their primary concern is that the mathematical enterprise as a whole always remains productive. Typically, they see this as insured by remaining open-minded, practical and busy; as potentially threatened by becoming overly-ideological, fanatically reductionistic or lazy. Such a view was expressed by the Physics Nobel Prize laureate Richard Feynman 通常 很少有数学家 天天关心 基于什么哲学立场-逻辑主义、形式主义、还是任何其他的, 在进行工作。相反，他们主要关注的是，数学事业 作为一个整体 始终保持卓有成效。通常 他们认为 这是通过保持思想开放、务实、和勤奋 来保证的，而潜在威胁 则是由于变得过于意识形态、狂热地还原论、或懒散。诺贝尔物理学奖获得者理查德·费曼表达了这种观点。 　　People say to me, “Are you looking for the ultimate laws of physics?” No, I’m not… If it turns out there is a simple ultimate law which explains everything, so be it — that would be very nice to discover. If it turns out it’s like an onion with millions of layers… then that’s the way it is. But either way there’s Nature and she’s going to come out the way She is. So therefore when we go to investigate we shouldn’t predecide what it is we’re looking for only to find out more about it. Now you ask: “Why do you try to find out more about it?” If you began your investigation to get an answer to some deep philosophical question, you may be wrong. It may be that you can’t get an answer to that particular question just by finding out more about the character of Nature. But that’s not my interest in science; my interest in science is to simply find out about the world and the more I find out the better it is, I like to find out…[6] 人们对我说，“您是否在寻找物理学的终极规律？”不，我不是......如果事实证明存在一种 能解释一切的、一种简单的终极法则，顺其自然-那是非常美好的发现。如果事实证明 这像一个无数层的洋葱.....于是 事情本来就是这样的。但无论怎样 ‘本性’总是存在的，她总是要出现的。所以，当我们进行研究时，我们不应该预先判定 这是什么，我们只期待发现更多的其所有关。现在，你要问：“为什么你不尝试找出得再多一点呢？”如果你在为 对一些深层次的哲学疑题 求取一个答案 而开始你的研究，你可能是错误的。很可能 对于那个具体疑题 正是由于发现更多关于本性的特征 而不能回答。而这 并非我在科学上的兴趣所在，我对科学的兴趣 仅仅是发现世界，发现得越多越好，我酷爱发现... [ 6 ] 　　Philosophers, incidentally, say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong[7] 顺带提一下， 哲学家说了大量 关于什么是 对于科学所绝对必要 的话，迄今为止 如人们所见 总是 相当幼稚，而且或许是错的[ 7 ] 　　 　　and also Steven Weinberg[8] 史蒂芬·温伯格[ 8 ]也是这种观点 　　The insights of philosophers have occasionally benefited physicists, but generally in a negative fashion—by protecting them from the preconceptions of other philosophers.(...) without some guidance from our preconceptions one could do nothing at all. It is just that philosophical principles have not generally provided us with the right preconceptions. 哲学家的见解 偶尔会有益于物理学家，但通常却以一种消极的方式-以保护他们免受其他哲学家的偏见。（...）对我们的观念没有某种指导，人们可以什么都没有做。恰恰是 哲学原理通常并没有为我们提供了正确的观念。 　　 　　Physicists do of course carry around with them a working philosophy. For most of us, it is a rough-and-ready realism, a belief in the objective reality of the ingredients of our scientific theories. But this has been learned through the experience of scientific research and rarely from the teachings of philosophers. (...) we should not expect [the philosophy of science] to provide today's scientists with any useful guidance about how to go about their work or about what they are likely to find. (...) 当然，物理学家们的工作理念总是不离身影地伴随着他们。对于我们中的大多数而言，这差不离是一种现实主义，是组成我们科学理论的 客观现实中的信念。但是，这是通过科学研究认识到的，极少来自哲学家的教诲。（...）我们不应指望[科学哲学] 来为今天的科学家提供 关于如何去进行他们的工作 或者他们可能会发现什么 的任何有用的指导。（...） 　　After a few years' infatuation with philosophy as an undergraduate I became disenchanted. The insights of the philosophers I studied seemed murky and inconsequential compared with the dazzling successes of physics and mathematics. From time to time since then I have tried to read current work on the philosophy of science. Some of it I found to be written in a jargon so impenetrable that I can only think that it aimed at impressing those who confound obscurity with profundity. (...) But only rarely did it seem to me to have anything to do with the work of science as I knew it. (...) 作为一个大学生 我迷恋哲学几年之后，变得不再抱有幻想。我学的哲学家的见解 与物理和数学的耀眼成就相比 显得昏暗和无足轻重。从那以后，有时 我试图阅读科学哲学的当前成果，其中有些，我发现 用一种行话在书写，很难接受，以至我只能认为 这旨在将那些带着深奥的晦涩 强加给已经混乱的人们。（...）却只有 似乎罕见对我科学工作中的任何事情有什么帮助，如我已经了解的。（...） 　　I am not alone in this; I know of no one who has participated actively in the advance of physics in the postwar period whose research has been significantly helped by the work of philosophers. I raised in the previous chapter the problem of what Wigner calls the "unreasonable effectiveness" of mathematics; here I want to take up another equally puzzling phenomenon, the unreasonable ineffectiveness of philosophy. 这 不仅我一个，我知道 没有一个 战后时期 积极参与了物理学进展 的人，其研究 是由于哲学家工作的极大帮助。我在前面的章节中提到过 维格纳称之为数学的“不合理有效性”问题；在这里 我要称 另外一个同样的费解现象 为：哲学的不合理无效。 　　Even where philosophical doctrines have in the past been useful to scientists, they have generally lingered on too long, becoming of more harm than ever they were of use. 即使在过去 哲学信条曾有益于科学家，它们通常也因为遗留太久，而成为 比以往使用它们的任何时候 造成更大的伤害。 　　 　　He believed that any undecidability in mathematics, such as the continuum hypothesis, could be potentially resolved despite the incompleteness theorem, by finding suitable further axioms to add to set theory. 他认为，数学中的任何不可判定性 如连续统假设，可以撇开不完备性定理，通过寻找适当的更深层公理，添加到集合论中，从而有能力得以解决。 　　 　　2.1.7 Philosophical consequences of the Completeness Theorem 完备性定理的哲学推论[edit] 　　 　　The Completeness theorem establishes an equivalence in first-order logic, between the formal provability of a formula, and its truth in all possible models. Precisely, for any consistent first-order theory it gives an "explicit construction" of a model described by the theory; and this model will be countable if the language of the theory is countable. However this "explicit construction" is not algorithmic. It is based on an iterative process of completion of the theory, where each step of the iteration consists in adding a formula to the axioms if it keeps the theory consistent; but this consistency question is only semi-decidable (an algorithm is available to find any contradiction but if there is none this consistency fact can remain unprovable). 完备性定理，在所有可能模型中 一个公式的形式可证性与其真实性之间，建立起一种 一阶逻辑中的 等价关系。严格地说，对于任何一致的一阶理论，它为一个 该理论所描述的模型，给出一种‘清晰的架构’；如果理论的语言是可数的，该模型也是可数的。然而 这种‘清晰的架构’不是算法。它是基于 理论完成的 迭代过程，其中迭代的每一步 是由加入一个公式 给公理 来组成，如果它保持理论是一致的；但这个一致性疑题 只是半可判定的（一种算法 可用于寻找任何矛盾，但如果什么也没有 这种一致性事实 就能保持不可证明）。 　　 　　This can be seen as a giving a sort of justification to the Platonist view that the objects of our mathematical theories are real. More precisely, it shows that the mere assumption of the existence of the set of natural numbers as a totality (an actual infinity) suffices to imply the existence of a model (a world of objects) of any consistent theory. However several difficulties remain: 这可以被看作是 对柏拉图主义者观点的 一种合理解释，即 我们数学理论的客体 是真实的。更确切地说，它表明 仅仅假设 自然数集合作为一个整体（一个实际的无穷大）存在 就足以意味着 存在一个 任何一致理论的模型（一个 客体的世界）。然而仍然存在几个困难： For any consistent theory this usually does not give just one world of objects, but an infinity of possible worlds that the theory might equally describe, with a possible diversity of truths between them. 对于任何一致的理论，这通常并非只给出 一种客体世界，而是该理论可以同样描述的 无穷多的可能世界，其真实性是多种多样的。 In the case of set theory, none of the models obtained by this construction resemble the intended model, as they are countable while set theory intends to describe uncountable infinities. Similar remarks can be made in many other cases. For example, with theories that include arithmetic, such constructions generally give models that include non-standard numbers, unless the construction method was specifically designed to avoid them. 在集合论的情况下，通过这种架构 获得不了 任何类似的预期模型，这是由于 它们是可数的，而集合论旨在描述不可数的无穷。类似的情况 也可能在其他许多场合下出现。例如，对于包含算术的理论，这样的架构 通常给出 包含非标准数的模型，除非 架构方法被设计成 刻意避免它们。 As it gives models to all consistent theories without distinction, it gives no reason to accept or reject any axiom as long as the theory remains consistent, but regards all consistent axiomatic theories as referring to equally existing worlds. It gives no indication on which axiomatic system should be preferred as a foundation of mathematics. 因为它给出的模型 对所有一致的理论 都没有区别，只要理论保持一致 就没有理由 接受或拒绝任何公理，而 对所有一致的公理化理论 等同地看作为 已有的世界。它没有指明 哪一个公理化系统更应当作为数学基础。 As claims of consistency are usually unprovable, they remain a matter of belief or non-rigorous kinds of justifications. Hence the existence of models as given by the completeness