伯特兰·罗素Bertrand Russell

伯特兰·罗素出版了大量关于逻辑学、知识论以及许多其他主题的书籍。他是20世纪最重要的逻辑学家之一。

罗素的数学贡献

在漫长而多样的职业生涯中，罗素对数学基础、当代形式逻辑的发展以及分析哲学做出了开创性贡献。他与数学相关的贡献包括发现罗素悖论、为逻辑主义辩护（即认为数学在某种重要意义上可归约为形式逻辑的观点）、引入类型论，以及完善和普及一阶谓词演算。与库尔特·弗雷德里希·哥德尔一起，他通常被认为是20世纪两位最重要的逻辑学家之一。

罗素于1901年5月发现以他命名的悖论，当时他正在撰写他的Principles of Mathematics（1903年）。该悖论产生于所有不是自身成员的集合所构成的集合。这样的集合，如果存在，将是自身的成员当且仅当它不是自身的成员。该悖论的重要性随之而来，因为在经典逻辑中，所有句子都由矛盾推出。在许多数学家（包括大卫·希尔伯特和勒伊岑·布劳威尔）看来，因此一旦发现显然支撑全部数学的逻辑是矛盾的，就没有任何证明可以信赖。这促成了本世纪早期在逻辑学、集合论以及数学哲学和基础方面的大量工作。

罗素的悖论源于朴素集合论所谓的无限制概括（或抽象）公理。该公理最初由格奥尔格·康托尔引入，它指出任何包含$x$作为自由变量的谓词表达式$P(x)$将确定一个集合，其成员恰好是那些满足$P(x)$的对象。该公理赋予了一种直觉以形式，即任何连贯的条件都可以用来确定一个集合（或类）。因此，大多数解决罗素悖论的尝试都集中在限制或放弃该公理的各种方式上。

罗素本人对悖论的回应伴随着他的类型论的引入。他的基本思想是，通过将所有句子排列成一个层级（从最低层次的关于个体的句子开始，次低层次的关于个体集合的句子，再次低层次的关于个体集合的集合的句子，等等），可以避免指涉麻烦的集合（例如所有不是自身成员的集合的集合）。利用儒勒·昂利·庞加莱也采用的恶性循环原则，连同他所谓的类的“无类”理论，罗素随后能够解释为什么无限制概括公理失败：命题函数，例如函数“x是一个集合”，不应应用于自身，因为自我应用会涉及恶性循环。根据这种观点，可以推出，只有当给定条件（或谓词）所适用的对象全都处于同一层次或具有相同的“类型”时，才可能指涉满足该条件的对象的汇集。

尽管罗素于1903年在Principles中首次引入类型论，但他的类型论在其1908年的文章Mathematical Logic as Based on the Theory of Types以及他与阿尔弗雷德·诺思·怀特海合著的巨著Principia Mathematica（1910、1912、1913）中得到了成熟的表述。因此，就其细节而言，该理论有两种版本：“简单理论”和“分支理论”。该理论的两种版本后来都受到了攻击。对一些人来说，它们太弱，因为它们未能解决所有已知的悖论。对另一些人来说，它们太强，因为它们不允许许多数学定义，这些定义虽然一致，却违反了恶性循环原则。罗素对第二种反对意见的回应是在分支理论中引入可化归性公理。尽管该公理成功地缩小了恶性循环原则的适用范围，但许多人声称它太ad hoc，无法在哲学上得到辩护。

在同一时期同样重要的是罗素对逻辑主义的辩护，即数学在某种重要意义上可化归为逻辑的理论。罗素的逻辑主义首先在其Principles中得到辩护，后来在Principia Mathematica中更详细地得到辩护，它由两个主要论点组成。第一个论点是，所有数学真理都可以翻译为逻辑真理，或者换句话说，数学的词汇构成逻辑词汇的真子集。第二个论点是，所有数学证明都可以重新表述为逻辑证明，或者换句话说，数学定理构成逻辑定理的真子集。

与戈特洛布·弗雷格一样，罗素为逻辑主义辩护的基本思想是，数可以被等同于类的类，数论陈述可以用量词和同一性来解释。因此，数1将被等同于所有单元类的类，数2将被等同于所有二元类的类，等等。诸如“有两本书”这样的陈述将被重新表述为“有一本书$x$，有一本书$y$，并且$x$与$y$不同一”。由此推出，数论运算可以用集合论运算如交、并等来解释。在Principia Mathematica中，阿尔弗雷德·诺思·怀特海和罗素能够提供集合论、有限与超限算术以及初等测度论中许多主要定理的详细推导。关于几何学的第四卷曾有计划，但从未完成。

在很大程度上，正如罗素想用逻辑来澄清数学基础中的问题一样，他也想用逻辑来澄清哲学中的问题。作为“分析哲学”的创始人之一，罗素因其使用一阶逻辑来表明广泛的指称短语如何可以用谓词和量化变量重新表述的工作而被铭记。因此，他也因强调逻辑形式对于解决许多相关哲学问题的重要性而被铭记。在这里，如同在数学中一样，罗素的希望是，通过应用逻辑机制和洞见，人们将能够解决否则难以处理的困难。

罗素的生平与公共影响

罗素是约翰·史考特·罗素勋爵的孙子，后者曾在维多利亚女王治下两次出任首相。母亲（1874年）和父亲（1876年）相继去世后，罗素和弟弟便去与祖父母同住。（尽管罗素的父亲曾将罗素及其弟弟的监护权授予两位无神论者，罗素的祖父母却没费多少力气就推翻了他的遗嘱。）祖父（1878年）去世后，罗素由祖母罗素夫人抚养。罗素先在家中接受教育，后入剑桥大学三一学院，在数学与道德科学两个领域均获得一等学位。

尽管罗素于1908年当选皇家学会，他在三一学院的职业生涯似乎在1916年走到了尽头，当时他因反战活动被定罪并处以罚款。由于这一判决，他被学院解职。（解职的详情见戈弗雷·哈罗德·哈代所著Bertrand Russell and Trinity（1942年）。）两年后，罗素第二次被定罪。这一次他在狱中度过了六个月。正是在狱中，他写下了广受好评的Introduction to Mathematical Philosophy（1919年）。直到1944年他才重返三一学院。罗素结过四次婚，并因众多风流韵事而声名狼藉，他还曾在1907年、1922年和1923年竞选议员，均未成功。20世纪20年代末至30年代初，他与第二任妻子一起开办并经营了一所实验学校。1931年弟弟去世后，他成为第三代罗素伯爵。

20世纪30年代末在美国任教期间，罗素获得了纽约城市学院的一个教职。但在大量公众抗议以及1940年一项司法裁决之后，这一聘任被撤销，裁决称他在道德上不适合在该学院任教。九年后，他被授予功绩勋章。1950年，他获得诺贝尔文学奖。

20世纪50年代和60年代，由于持续不断的反战与反核抗议，罗素在大量理想主义青年中成为一种鼓舞。1955年，他与阿尔伯特·爱因斯坦共同发表了罗素-阿尔伯特·爱因斯坦宣言，呼吁削减核武器。1957年，他是第一次帕格沃什会议的主要组织者，该会议汇集了关注核武器扩散的科学家。1958年，他成为核裁军运动的创始主席，并于1961年再次入狱，这一次与反核抗议有关。经上诉，他两个月的刑期被减为在监狱医院服刑一周。他一直是引人注目的公众人物，直到九年后去世，享年97岁。

Bertrand Russell published a large number of books on logic, the theory of knowledge, and many other topics. He is one of the most important logicians of the 20th Century.

Russell's Mathematical Contributions

Over a long and varied career, Bertrand Russell made ground-breaking contributions to the foundations of mathematics and to the development of contemporary formal logic, as well as to analytic philosophy. His contributions relating to mathematics include his discovery of Russell's paradox, his defence of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his introduction of the theory of types, and his refining and popularizing of the first-order predicate calculus. Along with Kurt Gödel, he is usually credited with being one of the two most important logicians of the twentieth century.

Russell discovered the paradox which bears his name in May 1901, while working on his Principles of Mathematics (1903). The paradox arose in connection with the set of all sets which are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it is not a member of itself. The significance of the paradox follows since, in classical logic, all sentences are entailed by a contradiction. In the eyes of many mathematicians (including David Hilbert and Luitzen Brouwer) it therefore appeared that no proof could be trusted once it was discovered that the logic apparently underlying all of mathematics was contradictory. A large amount of work throughout the early part of this century in logic, set theory, and the philosophy and foundations of mathematics was thus prompted.

Russell's paradox arises as a result of naive set theory's so-called unrestricted comprehension (or abstraction) axiom. Originally introduced by Georg Cantor, the axiom states that any predicate expression, $P(x)$, which contains $x$ as a free variable, will determine a set whose members are exactly those objects which satisfy $P(x)$. The axiom gives form to the intuition that any coherent condition may be used to determine a set (or class). Most attempts at resolving Russell's paradox have therefore concentrated on various ways of restricting or abandoning this axiom.

Russell's own response to the paradox came with the introduction of his theory of types. His basic idea was that reference to troublesome sets (such as the set of all sets which are not members of themselves) could be avoided by arranging all sentences into a hierarchy (beginning with sentences about individuals at the lowest level, sentences about sets of individuals at the next lowest level, sentences about sets of sets of individuals at the next lowest level, etc.). Using the vicious circle principle also adopted by Henri Poincaré, together with his so-called "no class" theory of classes, Russell was then able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function "x is a set", should not be applied to themselves since self-application would involve a vicious circle. On this view, it follows that it is possible to refer to a collection of objects for which a given condition (or predicate) holds only if they are all at the same level or of the same "type".

Although first introduced by Russell in 1903 in the Principles, his theory of types finds its mature expression in his 1908 article Mathematical Logic as Based on the Theory of Types and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913). Thus, in its details, the theory admits of two versions, the "simple theory" and the "ramified theory". Both versions of the theory later came under attack. For some, they were too weak since they failed to resolve all of the known paradoxes. For others, they were too strong since they disallowed many mathematical definitions which, although consistent, violated the vicious circle principle. Russell's response to the second of these objections was to introduce, within the ramified theory, the axiom of reducibility. Although the axiom successfully lessened the vicious circle principle's scope of application, many claimed that it was simply too ad hoc to be justified philosophically.

Of equal significance during this same period was Russell's defence of logicism, the theory that mathematics was in some important sense reducible to logic. First defended in his Principles, and later in more detail in Principia Mathematica, Russell's logicism consisted of two main theses. The first is that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of that of logic. The second is that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of those of logic.

Like Gottlob Frege, Russell's basic idea for defending logicism was that numbers may be identified with classes of classes and that number-theoretic statements may be explained in terms of quantifiers and identity. Thus the number 1 would be identified with the class of all unit classes, the number 2 with the class of all two-membered classes, and so on. Statements such as "there are two books" would be recast as "there is a book, $x$, and there is a book, $y$, and $x$ is not identical to $y$". It followed that number-theoretic operations could be explained in terms of set-theoretic operations such as intersection, union, and the like. In Principia Mathematica, Whitehead and Russell were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. A fourth volume on geometry was planned but never completed.

In much the same way that Russell wanted to use logic to clarify issues in the foundations of mathematics, he also wanted to use logic to clarify issues in philosophy. As one of the founders of "analytic philosophy", Russell is remembered for his work using first-order logic to show how a broad range of denoting phrases could be recast in terms of predicates and quantified variables. Thus, he is also remembered for his emphasis upon the importance of logical form for the resolution of many related philosophical problems. Here, as in mathematics, it was Russell's hope that by applying logical machinery and insights one would be able to resolve otherwise intractable difficulties.

Russell's Life and Public Influence

Russell was born the grandson of Lord John Russell, who had twice served as Prime Minister under Queen Victoria. Following the death of his mother (in 1874) and of his father (in 1876), Russell and his brother went to live with their grandparents. (Although Russell's father had granted custody of Russell and his brother to two atheists, Russell's grandparents had little difficulty in getting his will overturned.) Following the death of his grandfather (in 1878), Russell was raised by his grandmother, Lady Russell. Educated at first privately, and later at Trinity College, Cambridge, Russell obtained first class degrees both in mathematics and in the moral sciences.

Although elected to the Royal Society in 1908, Russell's career at Trinity appeared to come to an end in 1916 when he was convicted and fined for anti-war activities. He was dismissed from the College as a result of the conviction. (The details of the dismissal are recounted in Bertrand Russell and Trinity (1942) by G H Hardy.) Two years later Russell was convicted a second time. This time he spent six months in prison. It was while in prison that he wrote his well-received Introduction to Mathematical Philosophy (1919). He did not return to Trinity until 1944. Married four times and notorious for his many affairs, Russell also ran unsuccessfully for Parliament, in 1907, 1922, and 1923. Together with his second wife, he opened and ran an experimental school during the late 1920s and early 1930s. He became the third Earl Russell upon the death of his brother in 1931.

While teaching in the United States in the late 1930s, Russell was offered a teaching appointment at City College, New York. The appointment was revoked following a large number of public protests and a judicial decision, in 1940, which stated that he was morally unfit to teach at the College. Nine years later he was awarded the Order of Merit. He received the Nobel Prize for Literature in 1950.

During the 1950s and 1960s, Russell became something of an inspiration to large numbers of idealistic youth as a result of his continued anti-war and anti-nuclear protests. Together with Albert Einstein, he released the Russell-Einstein Manifesto in 1955, calling for the curtailment of nuclear weapons. In 1957, he was a prime organizer of the first Pugwash Conference, which brought together scientists concerned about the proliferation of nuclear weapons. He became the founding president of the Campaign for Nuclear Disarmament in 1958 and was once again imprisoned, this time in connection with anti-nuclear protests, in 1961. Upon appeal, his two-month prison sentence was reduced to one week in the prison hospital. He remained a prominent public figure until his death nine years later at the age of 97.