亚伯拉罕·鲁滨逊Abraham Robinson

我们一开始就应该注意到，事实上亚伯拉罕·鲁滨逊的姓氏是Robinsohn而不是鲁滨逊，但在本文中我们将始终使用鲁滨逊，即他在1940年之后使用的版本。鲁滨逊的父亲也叫鲁滨逊，他的母亲是Hedwig Lotte。他是家中的第二个孩子，他的哥哥Saul鲁滨逊也走上了杰出的职业道路；他成为比较教育方面的专家。老鲁滨逊也是一位极具天赋的人。在学习化学之后，他成为一位重要的作家和哲学家，但小马克斯·亚伯拉罕从未见过他的父亲，因为父亲在小亚伯拉罕出生前不久就去世了。

这是一个犹太家庭，尽管老鲁滨逊是一位犹太复国主义者，但他从未去过巴勒斯坦，但他在去世前刚刚接受了耶路撒冷希伯来国家图书馆馆长的职位。海德维希鲁滨逊是一名教师，她在德国抚养两个儿子直到1933年，当时亚伯拉罕十四岁。尽管对这些年亚伯拉罕的情况知之甚少，但他拥有的一些笔记本保存了下来[18]：-

……其中包含诗歌和戏剧，表明他是一个敏感、善于观察且怀有写作抱负的孩子。

显然，这个家庭一直对耶路撒冷心向往之，但1933年德国推行的反犹立法非常清楚地表明，是时候离开了。1933年1月30日希特勒上台，1933年4月7日的《公务员法》提供了将犹太教师从学校和大学中清除的手段，当然也提供了将犹太裔人士从其他职位上清除的手段。所有非雅利安血统的公务员（只要有一位祖父母信奉犹太教，即被视为非雅利安人）都应被令退休。Hedwig、亚伯拉罕和Saul 鲁滨逊通过在巴勒斯坦开始新生活，避开了1933年起犹太人在德国将会遇到的问题。

鲁滨逊完成了学业，并于1935年开始在耶路撒冷希伯来大学跟随亚伯拉罕·弗兰克尔和雅各布·列维茨基学习数学。鲁滨逊是一名才华横溢的学生，1939年毕业后，他获得奖学金前往巴黎索邦大学学习。仅仅学习了几个月后，当德国人入侵法国时，他被迫逃离。在乘坐从波尔多出发撤离难民的最后几艘小船之一抵达英格兰后，他将自己的姓氏从Robinsohn改掉。

在希伯来大学读本科时，鲁滨逊就对代数和数理逻辑都感兴趣。然而，一到英国，他就加入了自由法国空军，而且因为他是数学家，1941年被派往法恩伯勒的皇家航空研究院，成为科学军官。他迅速成为空气动力学专家，在第二次世界大战余下的时间里，他研究三角翼和超音速流。1945年，他仍以科学军官的身份被派往德国，随后于1946年被任命为克兰菲尔德航空学院的资深讲师。Young在[20]中写道：-

在克兰菲尔德，他对机翼空气动力学理论——包括亚音速流和超音速流——的兴趣不断拓宽，并变得日益全面。

此时，鲁滨逊已是空气动力学领域的世界领先权威，但他仍继续保持着对数理逻辑的兴趣。1946年，他获得耶路撒冷希伯来大学的硕士学位，此后他开始在伦敦大学进行研究，并于1949年获得伦敦大学博士学位，其博士论文是在模型论和代数系统的元数学方面的开创性工作。

他于1951年前往多伦多大学，担任应用数学讲席，但于1957年前往耶路撒冷，接任希伯来大学亚伯拉罕·弗兰克尔的讲席。他在那里担任数学系主任，直到1962年接受加利福尼亚大学洛杉矶分校数学与哲学教授职位。1967年他再次调动，但仍留在美国，前往耶鲁大学任数学教授。除了1971年在耶鲁改任斯特林数学教授外，他一直留在那里，直至去世。1973年，他被诊断患有胰腺癌，于当年11月接受手术，但几个月后去世。1975年出版了一部Model theory and algebra文集，作为对鲁滨逊的纪念致敬。编者的前言写道：-

鲁滨逊的突然致命疾病给全世界许多人带来了巨大震惊。因为鲁滨逊不仅仅是一位杰出的数学家。他还是一个人们很快就会非常喜欢的人。1973年11月至1974年4月那些迅速而悲伤的月份，对耶鲁的人来说，带有一种不真实感。在任何人能够理解正在发生的事情之前，他就已经离去了。我们寻求一种方式来表达我们的敬意和个人失落感。这个卷册是我们所知道的最好方式。

鲁滨逊是数学中截然不同领域的领先专家。文章[20]列出了他撰写的130篇论文和九本书。让我们首先考察他对应用数学的贡献。他只有一本书涉及应用数学，但这可能会让那些只把鲁滨逊视为数学逻辑学家的数学家感到惊讶，因为他几乎一半的论文都是关于应用数学的，特别是空气动力学。那本应用数学书是Wing theory，与J A Laurmann合著，于1956年出版。詹姆斯·莱特希尔在评论这项工作时写道：-

这是一部关于翼型和机翼空气动力学数学理论的令人钦佩的汇编。几乎所有重要结果都被提及，尽管对于更困难的主题只能简要参考文献。……对于具有数学头脑的学生来说，它应该是对机翼空气动力学的宝贵介绍，同时也是该领域及相关领域所有工作者参考的可靠依据。

然而，鲁滨逊最为人所知的是他在数学逻辑方面的工作。他1949年从希伯来大学获得的博士学位论文是The metamathematics of algebraic systems，这成为他1951年出版的第一本书。他于1956年出版了Complete theories，该书旨在研究模型完备性和界限变换的性质。他将这两个概念应用于某些数学结构的基本理论。鲁滨逊对模型论的贡献是在多伦多大学期间发展起来的。他将许多贡献和论文编织成一部专著Introduction to model theory and to the metamathematics of algebra，于1963年出版。Engeler写道：-

这是……首次尝试对模型论这一新主题进行连贯阐述。作品的主体部分由作者对该主题主要贡献的改写版本组成，这些版本被整理成流畅且极其可读的顺序。……其结果是从任何单一作者那里所能期望的尽可能完整的综述。

鲁滨逊最著名的发明是非标准分析，他在1961年引入了它。Kochen在[20]中写道：-

我想强调，非标准分析并不是鲁滨逊突然转向的一个旁支方向。相反，它是把他早先应用于代数上的同一观点系统地应用于分析的研究。

Fenyo解释了该理论背后的思想：-

[鲁滨逊的]理论基于这样一个元数学事实：实数系统是不完全的。因此，存在实数域的扩张，它们具有实数系统在用某个给定关系集以低阶谓词演算表述时所具有的全部性质。不完全理论的真扩张通常被称为非标准模型。实数系统的非标准模型具有这样的特征：它是一个非阿基米德全序域，其中包含实数系统的一个副本。

1966年，鲁滨逊出版了他的著名教科书Non-standard analysis。格奥尔格·克里泽尔写道：-

这本书在哥特弗里德·威廉·莱布尼茨去世仅250年后问世，它给出了一套严格而有效的无穷小理论，正如哥特弗里德·威廉·莱布尼茨所希望的那样，这些无穷小遵循与普通数相同的规律。

我们以对鲁滨逊性格的一些评论来结束这篇传记。在[20]中给出了这样的评价：-

他具有真正伟人的谦逊与善良，他对人感兴趣，容易喜欢上别人，从不居高临下。他深切关注人类文化的大多数形式和创造力，而在所有这些方面，他都能以逻辑、洞察力和知识的那种迷人结合与人交谈，这正是他数学工作的特点。

Macintyre在[18]中写道：-

鲁滨逊是一位绅士，始终彬彬有礼，热情不知疲倦。他对自己获得的许多荣誉怀有适度的喜悦。他因愿意倾听以及建议的真诚而深受尊敬。

最后，让我们引用鲁滨逊在1974年9月15日耶鲁大学追悼会上Korner的悼词：-

当人们考虑到他兴趣的丰富、深刻与多样，以及他的思想中纯数学、应用数学、逻辑和哲学的持续相互作用时，就会不断想起哥特弗里德·威廉·莱布尼茨，他对他怀有一种天然的亲近感，并对他怀有最深切的钦佩。莱布尼茨式的观念中，他几乎看不出有什么价值的一个是哥特弗里德·威廉·莱布尼茨的“最佳原则”，根据这一原则，世界是一切可能世界中最好的世界。我记得他不止一次以温和讽刺的方式问我，我能否对这个原则作出任何合理的解释。今天，我想提供一个部分答案：一个鲁滨逊能够生活和思考的世界，一个他的妻子和朋友们能够珍视对他的记忆的世界，一个只要逻辑、数学和哲学对人类仍然重要，他毕生的工作就会被铭记的世界，不可能是一个完全糟糕的世界。

We should note at the outset that in fact Abraham Robinson's family name was Robinsohn rather than Robinson, but we shall refer throughout this article to Robinson, the version which he used after 1940. Abraham Robinson's father was also named Abraham Robinson and his mother was Hedwig Lotte. He was the second child of the family, his older brother Saul Robinson also went on to have an outstanding career; he became an expert on comparative education. Abraham Robinson senior was also a highly talented man. After studying chemistry he became an important writer and philosopher but Abraham junior never knew his father for he died shortly before Abraham junior was born.

It was a Jewish family and although Abraham Robinson senior was a Zionist he had never been to Palestine but he had accepted the position of head of the Hebrew National Library in Jerusalem just before he died. Hedwig Robinson was a teacher and she brought up her two sons in Germany until 1933 when Abraham was fourteen years old. Although little is known of Abraham during these years, some notebooks which he owned have survived [18]:-

... containing poems and plays, suggesting a sensitive observant child with an ambition to write.

Clearly the family had always been attracted to Jerusalem but the anti-Jewish legislation introduced into Germany in 1933 indicated very clearly that it was time to leave. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the schools and universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Hedwig, Abraham and Saul Robinson avoided the problems that Jews would have in Germany from 1933 by starting a new life in Palestine.

There Robinson completed his schooling and, in 1935, began studying mathematics under Fraenkel and Levitzki at the Hebrew University of Jerusalem. Robinson was a brilliant student and, after graduating in 1939, he was awarded a scholarship to allow him to study at the Sorbonne in Paris. After only a few months of study he was forced to flee when the Germans invaded France. After reaching England on one of the last small boats from Bordeaux to evacuate refugees, he changed his name from Robinsohn.

As an undergraduate at the Hebrew University Robinson has been interested in both algebra and mathematical logic. However, once in England he enlisted in the Free French Air Force and, because he was a mathematician, he was sent in 1941 to the Royal Aircraft Establishment at Farnborough where he became a Scientific Officer. Rapidly he became an expert in aerodynamics and for the rest of World War II he worked on delta wings and supersonic flow. He was sent to Germany in 1945, still in his role as Scientific Officer, and then in 1946 he was appointed as a senior lecturer at the College of Aeronautics at Cranfield. Young writes in [20]:-

At Cranfield his interests in the aerodynamic theory of wings, both in subsonic and supersonic flow, broadened and became increasingly comprehensive.

By now Robinson was a world leading authority in aerodynamics yet he continued with his interest in mathematical logic. In 1946 he was awarded a Master's Degree from the Hebrew University in Jerusalem and, following this, he began research at London University receiving a Ph.D. from London in 1949 for pioneering work in model theory and the metamathematics of algebraic systems.

He went to the University of Toronto in 1951 to take up a chair of applied mathematics but left for Jerusalem in 1957 to fill Fraenkel's chair at the Hebrew University. He was Chairman of the Mathematics Department there until 1962 when he accepted the professorship of Mathematics and Philosophy at the University of California, Los Angeles. In 1967 he moved again, but remaining in the United States he went to Yale University as Professor of Mathematics. Other than changing his chair to the Sterling Professor of Mathematics at Yale in 1971 he remained there until his death. He was diagnosed as having cancer of the pancreas in 1973, underwent an operation in November of that year, but died a few months later. A collection of papers Model theory and algebra was published in 1975 as a memorial tribute to Robinson. The editors' foreword states:-

The sudden fatal illness of Abraham Robinson came as a great shock to many people around the world. For Robinson was more than an excellent mathematician. He was also a person whom one came very quickly to like very much. Those swift sad months of November 1973-April 1974 were for those at Yale tinged with a sense of unreality. He was gone before anyone could come to grips with what was happening. We sought a way of expressing our respect and our sense of personal loss. This volume was the best way we knew.

Robinson was a leading expert in remarkably different areas of mathematics. The article [20] lists 130 papers and nine books which he wrote. Let us examine first his contributions to applied mathematics. Only one of his books deals with applied mathematics but it may surprise mathematicians who think of Robinson only as a mathematical logician to realise that almost half his papers are on applied mathematics, particularly on aerodynamics. The one applied mathematics book is Wing theory written jointly with J A Laurmann and published in 1956. Lighthill, reviewing the work, wrote:-

This is an admirable compendium of the mathematical theories of the aerodynamics of aerofoils and wings. Almost all the important results are referred to, even though there can be only a brief reference to literature in connection with the more difficult topics. ... It should be an invaluable introduction to wing aerodynamics for mathematically-minded students, as well as a solid stand-by for purposes of reference for all workers in this and allied fields.

Robinson is best known, however, for his work on mathematical logic. His doctorate from the Hebrew University in 1949 was The metamathematics of algebraic systems and this became his first book published in 1951. He published Complete theories in 1956 which was written to study the properties of model-completeness and bounding transform. He applied these two concepts to the elementary theories of certain mathematical structures. Robinson's contributions to model theory were developed during his time at the University of Toronto. He weaved his many contributions and papers into a treatise Introduction to model theory and to the metamathematics of algebra published in 1963. Engeler wrote:-

This is ... the first attempt to write a connected exposition of the new subject of model theory. The main body of the work consists of rewritten versions of the author's main contributions to the subject, which are brought into a smooth and eminently readable sequence. ... there results as complete a survey as can be expected at this time from any single author.

Robinson's most famous invention was non-standard analysis which he introduced in 1961. Kochen writes in [20]:-

I want to emphasise that non-standard analysis was not a sudden tangential direction in which Robinson moved. Rather, it was the systematic application of the same viewpoint which he earlier applied to algebra to the study of analysis.

Fenyo has explained the ideas behind the theory:-

[Robinson's] theory is based on the metamathematical fact that the system of real numbers is incomplete. Thus, there exist extensions of the field of real numbers that possess all the properties of the system of real numbers that are formulated in the lower predicate calculus in terms of some given set of relations. Proper extensions of noncomplete theories are often referred to as non-standard models. A non-standard model for the system of real numbers has the feature of being a non-Archimedean totally ordered field which contains a copy of the real number system.

In 1966 Robinson published his famous text Non-standard analysis. Kreisel wrote:-

This book, which appeared just 250 years after Leibniz's death, presents a rigorous and efficient theory of infinitesimals obeying, as Leibniz wanted, the same laws as the ordinary numbers.

We end this biography by giving some comments on Robinson's personality. In [20] this appreciation is given:-

He had the humility and the kindness of the truly great, he was interested in people and he found it easy to like them and he patronised no-one. He was deeply concerned with most forms of human culture and creativity, and on all he could converse with the fascinating combination of logic, insight and knowledge that characterised his mathematics.

Macintyre, in [18], writes:-

Robinson was a gentleman, unfailingly courteous, with inexhaustible enthusiasm. He took modest pleasure in his many honours. He was much respected for his willingness to listen, and for the sincerity of his advice.

Finally let us quote from Korner's tribute to Robinson during the memorial service at Yale University on 15 September 1974:-

When one considers the wealth, profundity and diversity of his interests and the continuous interplay in his thinking of pure mathematics, applied mathematics, logic and philosophy one is constantly reminded of Leibniz to whom he felt a natural affinity and for whom he had the deepest admiration. The one Leibnizian idea in which he could see little merit was Leibniz's 'principe de meilleur' according to which the world is the best of all possible worlds. I remember his asking me more than once in his gently ironic way whether I could make any sense of this principle. Today I should like to offer a partial answer: It cannot be a wholly bad worlds in which an Abraham Robinson could live and think; in which his wife and friends are able to cherish his memory; and in which his life's work will be remembered as long as logic, mathematics and philosophy matter to mankind.