保罗·贝尔奈斯Paul Bernays

贝尔奈斯的父亲是Julius Bernays（1862-1916），一位商人，他的母亲是Sarah Brecher（1867-1953）。Julius Bernays（被称为Jules）拥有瑞士国籍，并于1887年与Sarah Brecher结婚。事实上，Julius和Sarah关系相当近，因为两人都是贝尔奈斯的曾祖父保罗·贝尔奈斯的后裔。Gregory Moore在[1]中考察了贝尔奈斯的祖先：-

贝尔奈斯出身于一个杰出的德国犹太学者和商人家庭。他的曾祖父贝尔奈斯 ben Jacob Bernays（1792-1849）是汉堡的首席拉比，以严格的东正教观点和现代教育理念而闻名。他的祖父Louis Bernays是一位商人，在帮助创建苏黎世犹太社区之前曾广泛游历，而他的叔祖父Jacob Bernays则是波恩大学的编外讲师。

贝尔奈斯（1830-？），其母为Sara Lea Behrend（1804-1858），已婚，但其妻姓名不详。Jacob Bernays（1824-1881），也是贝尔奈斯的儿子，是一位哲学作家，后来成为波恩大学的编外教授和首席图书馆员。

贝尔奈斯是他父母五个孩子中的老大，有一个弟弟Adolphe（1890-1957）和三个妹妹。在伦敦短暂停留期间，贝尔奈斯出生，之后他的家庭搬到了巴黎。我们在此说明，贝尔奈斯在晚年所写的简历中称自己信仰犹太教，是瑞士公民。从巴黎，全家搬到了柏林，贝尔奈斯于1895年至1907年间就读于Köllnisches 文理中学（Gymnasium）。晚年时，贝尔奈斯常谈起这些年他度过的快乐童年。在文理中学时，他对音乐有浓厚的兴趣，成为了一名非常熟练的钢琴家，随后将他在这一领域的才华转向了作曲。稍后，他又爱上了古代语言和数学。随着学校生活接近尾声，他不得不在音乐和数学之间做出艰难的决定。恩斯特·斯派克写道[18]：-

他在学校似乎相当快乐，是一个天赋异禀、适应良好的孩子，在文学和音乐方面都接受当时的主流文化价值观。确实，他的音乐才华首先引起了注意；他尝试作曲，但从未对自己所取得的成就完全满意，于是他决定投身科学事业。

贝尔奈斯决定学习工程学，并于1907年进入夏洛滕堡技术高等学校开始学习。然而，尽管父母希望他将数学才能用于实际，贝尔奈斯在一个学期（1907年夏季学期）后决定，他必须从工程学转向纯数学。他在柏林大学开始了纯数学的学习，师从伊赛·舒尔、埃德蒙·朗道、费迪南德·格奥尔格·弗罗贝尼乌斯和弗里德里希·赫曼·肖特基学习数学。他还跟随马克斯·普朗克学习物理，并跟随Alois Riehl、Carl Stumpf和Ernst Cassirer学习哲学。从1910年到1912年，他在哥廷根学习，聆听了大卫·希尔伯特、埃德蒙·朗道（他与贝尔奈斯同时搬到哥廷根）、赫尔曼·外尔和菲利克斯·克莱因的数学讲座，Woldemar Voight和马克斯·玻恩的物理讲座，并跟随Leonard Nelson（1882-1927）学习哲学。事实上，他深受Nelson哲学学派的影响，他的第一篇出版物就属于该领域，即Das Moralprinzip bei Sidgwick und bei Kant Ⓣ（西季威克和康德中的道德箴言）（1910年）。正是在哥廷根，他于1912年春季获得博士学位，与埃德蒙·朗道合作研究解析数论和二元二次型。他的学位论文题目是Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht-quadratischen Diskriminante Ⓣ（论用非二次判别式的本原二元二次型表示正整数）。他于1912年底向苏黎世大学提交了关于模椭圆函数的教授资格论文（Habilitation）学位论文，题为Zur elementaren Theorie der Landauschen Funktion $\phi(a)$ Ⓣ（论Landau函数φ的初等理论）。

在苏黎世大学，贝尔奈斯被任命为privatdocent和恩斯特·策梅洛的助手。他在那里工作到1917年，但这对他来说并不是一个多产的时期。他又发表了两篇论文，即Über den transzendentalen Idealismus Ⓣ（论先验唯心主义）和Über die Bedenklichkeiten der neueren Relativitätstheorie Ⓣ（对较新的相对论理论的质疑），均于1913年发表，但这些是他在哥廷根与Nelson合作的成果。他后来写道（见[18]）：-

第一次世界大战开始时，我正在撰写对Alfred Kastil关于Fries哲学的批评的答复。这篇答复没有发表——等到有机会发表时，我已经不再完全同意其中的所有内容。

事实上，他于1918年提交了题为Beiträge zur axiomatischen Behandlung des Logikkalküls Ⓣ（对逻辑演算公理化处理的贡献）的论文，但未能发表。他在写道时解释了这一点：-

诚然，这篇论文具有明确的数学性质，但受数学逻辑启发的探究并未被十分严肃地对待——它们被视为娱乐数学中有趣的一半。我自己也有这种倾向，因此没有急于及时发表它。它直到很久以后才出现，严格说来并不十分完整，只有某些部分。因此，我在论文中的许多内容并未在数学逻辑发展的描述中得到相应的记录。

1916年，恩斯特·策梅洛离开了苏黎世，部分出于健康原因，部分因为与大学管理层的争执。贝尔奈斯在恩斯特·策梅洛离开后接替了他的讲座课程。在许多方面，苏黎世对贝尔奈斯来说是一段美好时光，因为他与乔治·波利亚成为朋友，多次拜访阿尔伯特·爱因斯坦，并在赫尔曼·外尔家频繁享受社交场合。1917年，大卫·希尔伯特访问苏黎世讲学，并邀请贝尔奈斯担任他在哥廷根的助手。Constance Reid在[5]中写道：-

1917年春天，在访问苏黎世时，大卫·希尔伯特安排阿道夫·赫维兹周围圈子中的两位年轻数学家陪他散步。其中一位是赫尔曼·外尔的朋友乔治·波利亚。另一位是一位矜持、害羞且有些神经质的男子，名叫贝尔奈斯。令乔治·波利亚和贝尔奈斯惊讶的是，在登上苏黎世山山顶的散步中，谈话的主题不是数学而是哲学。他们两人都没有专攻那个领域。然而，贝尔奈斯曾学习过一些哲学，并且在哥廷根的学生时代与Leonard Nelson关系密切。事实上，他的第一篇出版物就发表在Nelson的哲学期刊上。现在，尽管他沉默寡言，贝尔奈斯比通常健谈的乔治·波利亚有更多话要说。散步结束时，大卫·希尔伯特邀请贝尔奈斯来哥廷根担任他的助手。贝尔奈斯接受了。

我们应该指出，这段叙述中有一个小错误。正如Reid所写，散步发生在1917年春天大卫·希尔伯特访问苏黎世时，但贝尔奈斯直到1917年9月大卫·希尔伯特返回苏黎世时才收到成为大卫·希尔伯特助手的邀请。在哥廷根，贝尔奈斯致力于大卫·希尔伯特课程“数学原理”的讲义。这些讲义后来由威廉·阿克曼编辑，并出版为Grundzüge der theoretischen Logik Ⓣ（理论逻辑纲要）。贝尔奈斯写了第二篇资格论文，其中他确立了命题逻辑的完备性；这实际上是对伯特兰·罗素和阿尔弗雷德·诺思·怀特海的Principia Mathematica的研究，并使用了恩斯特·施勒德的思想。Constance Reid在[5]中写道关于贝尔奈斯与大卫·希尔伯特的合作：-

然而，这些日子里大卫·希尔伯特的真正合作者是贝尔奈斯。在某些人看来，他甚至是在剥削他的逻辑助手。贝尔奈斯不是年轻学生，而是一个三十五六岁的成熟数学家。作为大卫·希尔伯特的助手，他领取薪水，并且在抵达哥廷根后不久取得任教资格，还从听其讲座的学生那里收取费用。他靠所得可以生活，但肯定无法结婚。……除了准备自己的讲座外，贝尔奈斯还帮助大卫·希尔伯特准备讲座，陪他上课，并常常承担部分课时的教学，指导大卫·希尔伯特的攻读博士学位的学生，研究和消化他们工作所需的文献，并为他们的合著《⟦E1⟧》做了大量写作工作。在贝尔奈斯，大卫·希尔伯特找到了一个对数学基础与他同样感兴趣的人。他毫无顾忌地让助手像自己一样努力工作。……然而，两人有时会就基础问题发生相当激烈的争论。贝尔奈斯将这些争论的情感特质归因于大卫·希尔伯特对数学感受的根本“对立”。……贝尔奈斯并不总是同意大卫·希尔伯特关于他们纲领的看法，但他欣赏这样一个事实：尽管大卫·希尔伯特在争论中充满激情，但当助手采取相反立场时，他从不针对助手个人怀恨在心。工作完成后，大卫·希尔伯特和贝尔奈斯常常争论政治。……音乐常常在逻辑或政治争论之后带来平静。贝尔奈斯热爱音乐，在苏黎世时曾与阿道夫·赫维兹弹过“四手联弹”。

1922年，大卫·希尔伯特推荐贝尔奈斯担任哥廷根大学的编外教授。在推荐信中，大卫·希尔伯特写道（原文为德文，英文翻译见[3]）：-

贝尔奈斯的著作扩展到数学科学最不同的领域：用二元二次型表示正整数（学位论文，由Landau指导），埃德蒙·朗道函数在埃米尔·皮卡定理中的初等理论（苏黎世的任教资格论文），变分法中的阿德里安-马里·勒让德条件，作为遍历系统例子的一维气体，伯特兰·罗素命题演算的公理化处理（哥廷根的任教资格论文，未印刷）。几篇关于哲学主题的论文，发表在《Abhandlungen der Friesschen Schule》上。所有这些科学工作的特点是彻底和可靠。它常常聚焦于一个需要阐明的具体基本问题，并以敏锐和远见进行分析，或者聚焦于一个现有的重要困难，然后以高超的技巧加以克服。贝尔奈斯的知识异常广博和深刻，并延伸到哲学、物理学和生物学领域。他的特点是对科学充满热爱的奉献；此外，他是一个品格可靠、处处受到高度尊敬的高尚之人。在所有关于科学基础的问题上，尤其是数学科学的基础问题，他是最有知识的专家，特别是我的工作人员中最受重视和最成功的成员。

贝尔奈斯被任命为无终身职位的编外教授。他继续听课，尤其是埃米·诺特、巴特尔·伦德特·范德瓦尔登和古斯塔夫·赫格洛茨的课。他发现通过听课比读书更容易学到新思想。每个假期他都回到柏林，与家人住在一起。

1933年纳粹政权颁布针对犹太人的指令时，贝尔奈斯失去了他在哥廷根的职位。1933年4月，他被通知在最终决定作出前停止教学，8月他的助理聘任被终止。9月，他的授课资格被正式撤销。大卫·希尔伯特让他作为私人助理留任了六个月，但不久他就被迫离开德国。他仍是瑞士公民，因此迁往苏黎世并不太困难。在苏黎世，他从1934年起在苏黎世联邦理工学院（ETH：瑞士联邦理工学院）担任临时职位。他在1935-36学年访问了普林斯顿，讲授数理逻辑和集合论课程。1939年，苏黎世联邦理工学院授予他授课资格，但仅限四年。1943年四年期满时获得延长。他从1945年起在苏黎世联邦理工学院获得了一个半职永久职位，但有人批评该学院没有以更体面的方式对待像贝尔奈斯这样的杰出学者。然而贝尔奈斯从未这样看待此事，他对苏黎世联邦理工学院在极大困难时期伸出援手极为感激。他担任这一兼职职位直到1959年退休，并被授予荣休教授称号。

贝尔奈斯最为人所知的是他与大卫·希尔伯特合著的两卷本著作Grundlagen der Mathematik Ⓣ（《数学基础》）（1934-39）。尽管这本书是合著出版物，但两位作者的贡献非常不同，所有文本均由贝尔奈斯撰写，而许多内容则是贝尔奈斯对大卫·希尔伯特常常相当模糊的问题作出解答。这部著作试图从符号逻辑构建数学。Akihiro Kanamori关于这部著名著作写道[8]：-

这部里程碑式的著作经历多次重印，并在三十年后最终出了第二版，它对大卫·希尔伯特学派在一阶逻辑形式化和证明论方面的工作，以及库尔特·弗雷德里希·哥德尔在不完备性及其周边方面的工作，包括第二不完备性定理的第一个完整证明，提供了权威性的阐述。最近对贝尔奈斯作用的重新评价实际上将他置于数学逻辑和大卫·希尔伯特纲领发展的中心。

1899年，大卫·希尔伯特写了Grundlagen der GeometrieⓉ（几何基础），1956年，贝尔奈斯修订了这部关于几何基础的作品。贝尔奈斯受大卫·希尔伯特思想的影响，认为整个数学结构可以统一为一个连贯的整体。为了开始这一过程，有必要设计一套公理，以此为基础建立这样一个完整的理论。因此，他试图将集合论置于公理基础之上，以避免悖论。1937年至1954年间，贝尔奈斯在Journal of Symbolic Logic上发表了一系列文章，试图实现这一目标。他试图修改冯·诺伊曼的公理系统，以纳入恩斯特·策梅洛系统的特征。他提出了依赖选择原则，这是一种选择公理的形式，后来由阿尔弗雷德·塔斯基独立研究。他使用了类似于威廉·阿克曼所使用的数论模型来证明其公理的独立性。1958年，贝尔奈斯出版了Axiomatic Set Theory，在其中汇集了他关于集合论公理化的所有工作。

贝尔奈斯在苏黎世的工作受到他的同事Ferdinand Gonseth（1890-1975）的影响。他写道（见[4]）：-

由于我的思想与Kant、Fries和Nelson的哲学的交锋，我已经接近了Ferdinand Gonseth的观点，因此我加入了他的哲学学派。

让我们通过引用Die Mathematik als ein zugleich Vertrautes und UnbekanntesⓉ（既熟悉又陌生的数学）中的一个例子来说明贝尔奈斯的思想，他在其中论证我们对自然数的概念必定是通过试错过程产生的：-

首先，我们意识到我们有自由从计数过程中达到的一个位置前进到下一个位置。但随后我们采取了一步连接，通过这一步，一个将后继数与每一个数相关联的函数被设定。在这里，一个“无限进展”取代了“不定进展”。但是，这种无限数列的想法能否实现并不是立即显而易见的；其成功实现的知识经验对于培养一种熟悉感，甚至是不言而喻感，作为获得的证据，是至关重要的。

贝尔奈斯关于数学公理基础的工作由库尔特·弗雷德里希·哥德尔进一步推进。

现在让我们引用查尔斯·帕森斯关于贝尔奈斯成就的一些总结性评论[4]：-

贝尔奈斯作为数学哲学作家的优点显而易见：与实际数学，尤其是数理逻辑的接触，加上对哲学家所关注问题的熟悉，以及对哲学立场易于陷入的困难的敏感。大概最后一点对数学家来说是最难做到的。……人们可以辨别出一些已被证明具有持久价值的哲学贡献：他在“论柏拉图主义”中对数学概念的分层，该文中指向一方是有限主义与直觉主义、另一方是“严格有限主义”之间区分的观察……，可能还有对无限集的任意子集概念的“准组合”描述。

正如我们已经指出的，考虑到贝尔奈斯成就的 magnitude，他担任的职位相对次要。有人可能会说，其原因在于反犹的纳粹政策，但即使在纳粹掌权之前，他已经担任大卫·希尔伯特的助手十六年。同样，他因其成就获得的荣誉也不如人们可能期望的那样多。然而，他当选为比利时皇家科学院和挪威科学与文学院的通讯院士。慕尼黑大学于1976年授予他荣誉博士学位。他还当选为国际科学哲学科学院院长，并被授予德国数学逻辑与精确科学基础研究学会荣誉讲席。他是多家期刊的编委，Dialectica、Journal of Symbolic Logic和Archiv für mathematische Logik und GrundlagenforschungⓉ（数学逻辑与基础研究档案）。

最后，我们给出[18]中提供的关于他个性的细节：-

正如他大量的通信、对访客的友好、直到生命最后几年仍接受大会邀请所清楚表明的，他喜欢与他人接触。他极其仁慈，帮助许多作者修改论文——从大卫·希尔伯特到一位有小发现的高中教师。另一方面，他生活在一种超然的光环中。他独特地拒绝评判他人；他从未说过任何人的坏话，有充分理由假定他甚至不会对他人有不好的想法。有一次，有人提到一位几乎被普遍认为是本世纪恶棍之一的政治家，以诱使他作出负面判断，他回答说：“我的情况与他如此不同，轮不到我来评判”。毫无疑问，他处处看到最好的一面，并在看不到任何好处时避免评判的天赋，极大地帮助基础研究摆脱了不同学派被期望相互争斗的局面。

贝尔奈斯从未结婚，但搬到苏黎世后，他与母亲和两个未婚姐妹住在一起。1953年母亲去世后，他继续与姐妹们同住。他直到80多岁仍保持研究活跃。他在短暂患病后死于心脏病，一个姐妹几年前已去世。他的妹妹玛莎比他活得久。

Paul Bernays's father was Julius Bernays (1862-1916), a businessman, and his mother was Sarah Brecher (1867-1953). Julius Bernays (known as Jules) had Swiss nationality and married Sarah Brecher in 1887. In fact Julius and Sarah were quite closely related as both were descended from Paul's great-grandfather Isaac Bernays. Gregory Moore looks at Paul's ancestors in [1]:-

Bernays came from a distinguished German-Jewish family of scholars and businessmen. His great-grandfather, Isaac ben Jacob Bernays (1792-1849), chief rabbi of Hamburg, was known for both strict Orthodox views and modern educational ideas. His grandfather, Louis Bernays, a merchant, travelled widely before helping to found the Jewish community in Zürich, while his great-uncle, Jacob Bernays, was a Privatdozent at the University of Bonn.

Louis Bernays (1830-?), whose mother was Sara Lea Behrend (1804-1858), married but the name of his wife is not known. Jacob Bernays (1824-1881), also the son of Isaac Bernays, was a writer on philosophy who became an extraordinary professor and chief librarian at the University of Bonn.

Paul was the oldest of his parents' five children having a younger brother, Adolphe (1890-1957), and three younger sisters. After a short time in London, during which Paul was born, his family moved to Paris. Let us make clear at this point that Bernays, in a CV that he wrote later in life, says that he was of the Jewish faith and a citizen of Switzerland. From Paris the family moved to Berlin where Paul attended the Köllnisches Gymnasium from 1895 to 1907. In later life Bernays was to speak of the happy childhood he had during these years. At the Gymnasium he had a strong interest in music, became a highly proficient pianist, then put his talent in this area into composing. Slightly later came a love for ancient languages and mathematics. As his school days drew to an end he had to make the difficult decision between music and mathematics. E Specker writes [18]:-

He seems to have been quite happy at school, a gifted, well adapted child accepting the prevailing cultural values in literature as well as in music. It was indeed his musical talent that first attracted attention; he tried his hand at composing, but being never quite satisfied with what he achieved, he decided a scientific career.

Bernays's decision was to take up engineering and he entered the Technische Hoschule in Charlottenburg where he began his studies in 1907. However, despite his parents' wish that he put his mathematical talents to practical use, Bernays decided after one semester (the 1907 summer semester) that he must make the change from engineering to pure mathematics. He began his pure mathematics studies at the University of Berlin where he was taught mathematics by Issai Schur, Edmund Landau, Georg Frobenius and Friedrich Schottky. He also studied physics with Max Planck, and philosophy with Alois Riehl, Carl Stumpf and Ernst Cassirer. From 1910 until 1912 he studied at Göttingen where he attended mathematics lectures by David Hilbert, Edmund Landau (who moved to Göttingen at the same time as Bernays), Hermann Weyl and Felix Klein, physics lectures by Woldemar Voight and Max Born, and he studied philosophy with Leonard Nelson (1882-1927). In fact he was much influenced by Nelson's philosophy school and his first publication was in that area, namely Das Moralprinzip bei Sidgwick und bei Kant Ⓣ (1910). It was at Göttingen that he obtained his doctorate in the spring of 1912, working with Edmund Landau on analytic number theory and binary quadratic forms. The title of his thesis was Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht-quadratischen Diskriminante Ⓣ. He submitted his habilitation thesis on modular elliptic functions entitled Zur elementaren Theorie der Landauschen Funktion $\phi(a)$ Ⓣ to the University of Zürich towards the end of 1912.

At the University of Zürich, Bernays was appointed as a privatdocent and an assistant to Ernst Zermelo. He worked there until 1917 but this was not a productive period for him. He published two further papers, namely Über den transzendentalen Idealismus Ⓣ and Über die Bedenklichkeiten der neueren Relativitätstheorie Ⓣ, both in 1913 but these were consequences of his work with Nelson in Göttingen. He wrote later (see [18]):-

At the beginning of the First World War, I worked on a reply to a critique by Alfred Kastil of the Fries philosophy. This reply was not published - by the time there was an opportunity to have it published I no longer agreed with all of it.

In fact he submitted the paper with title Beiträge zur axiomatischen Behandlung des Logikkalküls Ⓣ in 1918 but it was not published. He explained this when he wrote:-

To be sure, the paper was of definite mathematical character, but investigations inspired by mathematical logic were not taken quite seriously - they were thought of as amusing, half-way part of recreational mathematics. I myself had this tendency, and therefore did not take pains to publish it in time. It has appeared only much later, and strictly speaking not quite complete, only certain parts. Many things I had in the paper have therefore not been recorded accordingly in descriptions of the development of mathematical logic.

In 1916 Zermelo left Zürich, partly for health reasons, partly because of a dispute with the university administration. Bernays took over Zermelo's lecture courses after he left. In many ways Zürich was a good time for Bernays for he became friends with Georg Pólya, made several visits to Albert Einstein and enjoyed frequent social occasions at Hermann Weyl's. In 1917 Hilbert visited Zürich to lecture and offered Bernays a post as his assistant at Göttingen. Constance Reid writes in [5]:-

In the spring of 1917, on a visit to Zürich, Hilbert arranged for two of the young mathematicians in the circle around Hurwitz to accompany him on a walk. One of these was Weyl's friend Polya. The other was a reserved, shy and somewhat nervous man named Paul Bernays. To the surprise of Polya and Bernays the subject of conversation on the walk to the top of the Zürichberg was not mathematics but philosophy. Neither of them had specialised in that field. Bernays, however, had studied some philosophy and, during his student days at Göttingen, had been close to Leonard Nelson. In fact, his first publication had been in Nelson's philosophical journal. Now, in spite of his quietness, Bernays had much more to say than the usually voluble Polya. At the end of the walk Hilbert asked Bernays to come to Göttingen as his assistant. Bernays accepted.

We should point out that there is a small error in this account. The walk took place, as Reid writes, when Hilbert visited Zürich in the spring of 1917 but Bernays only received the offer to become Hilbert's assistant when Hilbert returned to Zürich in September 1917. At Göttingen, Bernays worked on the lecture notes to Hilbert's course 'Prinzipien der Mathematik'. These lecture notes were later edited by Wilhelm Ackermann and published as Grundzüge der theoretischen Logik Ⓣ. Bernays wrote a second habilitation in which he established the completeness of propositional logic; this was in fact a study of Russell and Whitehead's Principia Mathematica, and uses ideas from Schröder. Constance Reid writes about Bernays' collaboration with Hilbert in [5]:-

Hilbert's real collaborator during these days, however, was Bernays. To some people it seemed that he was even exploiting his logic assistant. Bernays was no young student but a man in his middle thirties, a mature mathematician. As Hilbert's assistant, he received a salary and, having habilitated shortly after his arrival in Göttingen, also received fees from the students who attended his lectures. He could live on what he received, but certainly he could not marry. ... In addition to preparing his own lectures, Bernays helped Hilbert prepare his lectures, accompanied him to class and often took over the teaching for part of the hour, supervised Hilbert's students who were working for the doctoral degree, studied and digested the literature necessary for their work, and did a great deal of writing on their joint book, which was entitled 'Grundlagen der Mathematik'. In Bernays, Hilbert had found someone as interested in the foundations of mathematics as he was. He had no compunction about working his assistant as hard as he worked himself. ... The two men sometimes, however, got into rather violent arguments over the subject of foundations. Bernays attributes the emotional quality of these arguments to a fundamental "opposition" in Hilbert's feelings about mathematics. ... Bernays did not always agree with Hilbert about their programme, but he appreciated the fact that, passionate though Hilbert was in their disputation, he never held it against his assistant personally when he took the opposite side. After their work was finished, Hilbert and Bernays often argued about politics. ... Music often brought peace after the arguments, logical or political. Bernays loved music and had played "four hands" with Hurwitz when he was in Zürich.

In 1922 Hilbert recommended Bernays for an extraordinary professorship at Göttingen. In his letter of recommendation Hilbert wrote (given in the original German and in an English translation in [3]):-

Paul Bernays' extend over the most diverse fields of mathematical science: the representation of positive integers by binary quadratic forms (dissertation supervised by landau), elementary theory of Landau's function of Picard's theorem (habilitation thesis in Zürich), Legendre's condition in the calculus of variations, one-dimensional gas as an example of an ergodic system, axiomatic treatment of Russell's propositional calculus (habilitation thesis, Göttingen, not printed). Several papers on philosophical topics, in 'Abhandlungen der Friesschen Schule'. All this scientific work is characterised by thoroughness and reliability. Often it focuses on a specific fundamental question requiring elucidation, which is analysed with astuteness and foresight, or on an existing important difficulty that is then overcome with great skill. Bernays' knowledge is extraordinarily broad and deep, and it extends into the domains of philosophy, physics and biology. He is characterised by a loving dedication to science; beyond that he is a noble-minded person of reliable character, highly esteemed everywhere. In all problems concerning the foundations of science, above all those of mathematical science, he is the most knowledgeable expert, and in particular, the most valued and most successful member of my staff.

Bernays was appointed as an extraordinary professor without tenure. He continued to attend lectures, particularly those of Emmy Noether, Bartel van der Waerden and Gustav Herglotz. He found that he learnt new ideas more easily by listening to lectures than from reading books. Every vacation he returned to Berlin and lived with his family.

When the Nazi regime made its directive against Jews in 1933, Bernays lost his post at Göttingen. He was told to stop teaching in April 1933 pending a final decision, and in August his appointment as an assistant was terminated. His right to teach was formally withdrawn in September. Hilbert kept him on as his private assistant for six months but soon he was forced to leave Germany. He was still a Swiss citizen so a move to Zürich was not too difficult. In Zürich he worked at the Eidgenössische Technische Hochschule (ETH: the Swiss Federal Institute of Technology) in a temporary post from 1934. He visited Princeton in session 1935-36 and gave courses on mathematical logic and set theory. In 1939 the ETH granted him the right to teach, but only for four years. It was extended in 1943 when the four year term was up. He obtained a half-time permanent post at the ETH from 1945 but there has been criticism of the ETH for not treating a distinguished academic like Bernays in a more honourable way. However Bernays never saw it that way and he was extremely grateful to the ETH for coming to his rescue at a time of great difficulty. He held this part-time post until 1959 when he retired and was made professor emeritus.

Today Bernays is best known for his joint two volume work Grundlagen der Mathematik Ⓣ (1934-39) with Hilbert. Although the book was a joint publication, the two authors made very different contributions with all the text being written by Bernays and much of the content being Bernays' working out answers to, often rather vague, questions from Hilbert. The work attempted to build mathematics from symbolic logic. Akihiro Kanamori writes about this famous book [8]:-

Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of first-order logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. Recent re-evaluation of Bernays' role actually places him at the centre of the development of mathematical logic and Hilbert's program.

In 1899 Hilbert had written Grundlagen der Geometrie Ⓣ and, in 1956, Bernays revised this work on the foundations of geometry. Bernays, influenced by Hilbert's thinking, believed that the whole structure of mathematics could be unified as a single coherent entity. In order to start this process it was necessary to devise a set of axioms on which such a complete theory could be based. He therefore attempted to put set theory on an axiomatic basis to avoid the paradoxes. Between 1937 and 1954 Bernays wrote a whole series of articles in the Journal of Symbolic Logic which attempted to achieve this goal. He attempted to modify von Neumann's axiom system to include features from Zermelo's. He formulated the principle of dependent choices, a form of the axiom of choice independently studied by Tarski later. He used number theoretic models similar to those used by Ackermann to show the independence of his axioms. In 1958 Bernays published Axiomatic Set Theory in which he brought together all his work on the axiomatisation of set theory.

Now Bernays' work in Zürich was influenced by his colleague Ferdinand Gonseth (1890-1975). He wrote (see [4]):-

I had come close to the views of Ferdinand Gonseth on the basis of the engagement of my thinking with the philosophy of Kant, Fries, and Nelson, and so I attached myself to his philosophical school.

Let us give an example of Bernays' ideas by quoting from Die Mathematik als ein zugleich Vertrautes und Unbekanntes Ⓣ where he is arguing that our conception of natural numbers had to have come about through a process of trial and error:-

First we are conscious of the freedom we have to advance from one position arrived at in the process of counting to the next one. But then we take the step of a connection, through which a function that associates a successor with each and every number is posited. Here a 'progressus in infinitum' replaces the 'progressus in indefinitum'. But it is not immediately obvious that this idea of the infinite number series can be realised; the intellectual experience of its successful realisation is then essential for developing a feeling of familiarity, even of obviousness, as acquired evidence.

Bernays' work on an axiomatic basis for mathematics was taken further by Kurt Gödel.

Let us now quote some of Charles Parsons' concluding remarks about Bernays' achievements [4]:-

Bernays' virtues as a writer on philosophy of mathematics are evident: contact with actual mathematics, especially mathematical logic, combined with a familiarity with the issues that concern philosophers and sensitivity to the difficulties that philosophical positions are prone to. Probably the last is the most difficult for a mathematician to achieve. ... one can discern some philosophical contributions that have proved enduring: his stratification of mathematical conceptions in "Sur le platonisme," the observation in that paper that points to a distinction between finitism and intuitionism on one side and "strict finitism" on the other ..., possibly the "quasi-combinatorial" description of the conception of arbitrary subset of an infinite set.

As we have already indicated, Bernays held relatively minor posts considering the magnitude of his achievements. One might say that the reason for this was a consequence of the anti-Semitic Nazi policies but, even before the Nazis came to power, he had been Hilbert's assistant for sixteen years. Similarly, he didn't receive as many honours for his achievements as one might expect. However, he was elected a corresponding member of the Royal Belgium Academy of Science and of the Norwegian Academy of Science and Letters. The University of Munich awarded him an honorary doctorate in 1976. He was also elected president of the International Academy of the Philosophy of Science, and made an honorary chair of the German Society for Mathematical Logic and Foundational Research in the Exact Sciences. He was on the editorial board of several journals, Dialectica, the Journal of Symbolic Logic, and the Archiv für mathematische Logik und Grundlagenforschung Ⓣ.

Finally, we give details of his personality as given in [18]:-

As his immense correspondence, his friendliness to visitors, his acceptance of invitations to congresses until the last years of his life, clearly show, he liked the contact with other human beings. He was extremely benevolent, helping many an author with his papers - from Hilbert to a high-school teacher having made some small discovery. On the other hand, he lived in an aura of detachment. He was unique in his refusal to judge other people; he never spoke badly of anybody there is every reason to assume that he did not even think badly of others. When, once, reference was made to a statesman almost universally recognized as one of the villains of this century, in order to induce him to a negative judgment, he replied: "My situation is so different from his, that it is not for me to pass judgment". There is no doubt that his gift of seeing everywhere the best and refraining from judgment where he could not see anything good, helped a great deal to free foundational studies from the situation where different schools are expected to fight one another.

Bernays never married but after he moved to Zürich he lived with his mother and two unmarried sisters. After his mother died in 1953, he continued to live with his sisters. He remained research active well into his 80s. He died of a heart condition after a short illness, one sister having died a few years earlier. He was survived by his sister Martha.