莫里兹·贝内迪克特·康托尔Moritz Cantor

莫里兹·贝内迪克特·康托尔的父母是Isaac Benedikt 康托尔（1799-1885）和Nanette Meyer Schnapper。Isaac是来自阿姆斯特丹的商人，是Benedict Cantor和Telly de Vries的儿子。Nanette，人称Nelly，是货币兑换商Meyer Wolf Schnapper和Johanna Israel的女儿。Isaac和Nanette 康托尔于1825年11月14日在阿姆斯特丹结婚。Nanette的兄弟Adolph Schnapper是婚礼的见证人。两个家庭都是犹太人。康托尔在阿姆斯特丹长大，他的早期教育来自他父母雇佣的私人教师。他在曼海姆的文理中学完成了中学教育，并为大学学习做准备。

他于1848年进入海德堡大学，在那里受教于Franz Ferdinand Schweins（1780-1856）和Arthur Arneth（1802-1858）。鉴于康托尔作为数学史家的声誉，我们在此应当指出，Arneth是一位杰出的数学史家，撰写了重要著作History of Pure Mathematics in its relation to the History of the Human Mind（1852）。他是最早尝试将数学史建立在一般哲学原则之上的人之一，而不是仅仅将其视为事实的汇编和过去事件的记录。然而，没有迹象表明康托尔在其职业生涯的这一阶段受到数学史方面的引导。在海德堡度过一年后，康托尔前往哥廷根大学，从1849年到1851年在那里度过。在哥廷根，他跟随卡尔·弗里德里希·高斯学习数学和天文学，跟随威廉·韦伯学习物理学，跟随莫里茨·亚伯拉罕·斯特恩学习数学，后者对数论特别感兴趣。他回到海德堡，于1851年提交了博士学位论文。他在哥廷根Ruprecht-Karls大学的导师是Ferdinand Schweins，他于1851年5月6日因其题为Über ein weniger gebräuchliches Koordinatensystem Ⓣ（论一种不常见的坐标系）的学位论文而被授予博士学位。他没有参加大多数学生在这一阶段参加的、取得文理中学教师资格的国家考试，而是前往柏林，在1852年夏季学期旁听了约翰·彼得·古斯塔夫·勒热纳·狄利克雷和雅各布·施泰纳的课程。1853年4月30日，他被任命为海德堡大学的讲师（Docent），此前他提交了教授资格论文（Habilitation）论文Grundzüge einer Elementararithmetik Ⓣ（初等算术的本质）。他此后一生都在海德堡度过。

康托尔的早期工作并非数学史，但他确实写了一篇关于彼得吕斯·拉米斯、米夏埃尔·施蒂费尔和吉罗拉莫·卡尔达诺的短文，并在波恩的一次科学会议上作了报告。该文考察了印度-阿拉伯数字传入欧洲的情况，于1857年发表。后来康托尔告诉弗洛里安·卡乔里，见[5]，他的论文：

……反响极好，以至于他感到受鼓舞，要继续他的历史研究工作。

然而，他在海德堡受到我们上文提到的Arneth以及文化哲学家Eduard Maximilian Röth的影响。尽管Röth是海德堡的哲学和梵语教授，但他曾在巴黎学习数学、物理和化学，师从弗朗索瓦·阿拉戈、让-巴蒂斯特·毕奥、Pierre-Louis Dulong（1785-1838）和Jean-Baptiste-André Dumas（1800-1884）等人。他很可能鼓励康托尔去巴黎访问，后者在1850年代末确实这样做了。在这次访问期间——康托尔是在那次令人鼓舞的波恩会议后不久进行的——他与米歇尔·沙勒和约瑟·伯特兰变得友好。米歇尔·沙勒是公认的几何史 leading expert，并鼓励康托尔在Comptes Rendus上发表更多历史材料。

从1860年起，康托尔讲授数学史，并成为19世纪末德国领先的数学史家之一。他的第一部重要著作是Mathematische Beiträge zum Kulturleben der Völker Ⓣ（数学对民族文化生活的贡献）（1863），该书与其早期著作一样，集中于印度-阿拉伯数字传入欧洲的问题。1863年不仅标志着康托尔第一部重要著作的问世，也是他被晋升为海德堡编外教授的一年。1868年8月23日，在他的生日那天，他与Telly Gerothwohl（1847-1873）结婚；他们育有一子一女。

康托尔的第二本书是Euclid und sein Jahrhundert（欧几里得及其世纪），他在书中总结了欧几里得、阿基米德和阿波罗尼奥斯的工作。该书于1867年出版。他的下一个重要贡献是Die römischen Agrimensoren und ihre Stellung in der Geschichte der Feldmesskunst Ⓣ（罗马测量员及其在测量史上的地位）。乍看之下，这似乎是一个奇怪的题目，因为人们公认罗马人对数学的发展贡献甚微。然而，康托尔看到，罗马测量员在中世纪将埃及和古希腊的实用几何方法传播到欧洲方面发挥了重要作用。

康托尔最为人所铭记的是四卷本著作Vorlesungen über Geschichte der Mathematik，它追溯了直到1799年的数学史。第一卷出版于1880年，最后一卷出版于1908年。第一卷追溯了直到1200年的数学通史。第二卷追溯了直到1668年的历史。康托尔选择1668年，是因为在这一年，艾萨克·牛顿和哥特弗里德·威廉·莱布尼茨即将开始他们的数学研究。第三卷继续概述直到1758年的历史，同样是因为约瑟夫·拉格朗日的工作在此日期后不久开始，具有重要性。对这三卷的评论在[10]中充满赞誉：

在书评的简短篇幅内，不可能对这部巨著做出公正的评价，即使这部著作的崇高地位和作者公认的权威使这项任务变得多余。康托尔的《数学史讲义》是一个人的作品，他坚定不移地将一生奉献给这一单一任务，三十三年前他就因对这一主题的重要贡献而闻名，如今在他伟大著作第一卷的第二版中，他可以自豪地指出，他的努力在他的科学的历史研究中激起了动力和觉醒的兴趣。他有许多前辈，每一位都在某些分支和某些优点上卓尔不群……然而，可以稳妥地说，在构成康托尔这些讲义的这三卷中，深度、准确性和论述的广博，在任何数学史中从未如此彻底和紧密地结合在一起。

George Gibson在评论[7]中写道：-

几乎无需说明，这部历史著作必将在许多年内保持其作为所涉主题标准著作的地位；在完整性、准确性、编排清晰度方面，它无与伦比，并且在其涵盖的时期内，必将成为一部永久性的参考著作。

完成第三卷后，康托尔意识到，在69岁的年纪，他已无力再完成另一卷，因此在1904年海德堡大会上，他组织了一个由另外九位贡献者组成的团队，共同撰写第四卷。作为主编，康托尔设定了高标准，并坚持保留前三卷的风格和公正性。这第四卷同样在一個极其重要的发展之前戛然而止，因为1799年是卡尔·弗里德里希·高斯的学位论文年份。评论[11]遗憾地表示，康托尔未能亲自撰写这一卷：-

自康托尔的《Vorlesungen》第一卷第二版问世以来，已经过去了十五年，去年又需要出第三版。在完成第三卷之后——这部巨著由此推进到1758年——作者感到，年事已高的要求已足够强烈，迫使他将这部巨著的完成留给他人。当他看到，受他的榜样和热情激励，有能力且渴望在他的指导下继续他伟大工作的人出现时，他一定感到自己没有虚度此生，一个无愧于德国传统的历史学派已在他周围成长起来。我们不幸的是，这位年迈学者的健康状况使他无法给予那种充分的亲自监督，而这种监督对于确保总体处理的一致性至关重要，尤其是当半个世纪的数学工作被分到如此多的人手中时，无论他们个人多么能干。当我们说，第四卷由九个人撰写的不少于九篇专论构成时，就不难发现，尽管处理方式的多样性比预期的要少，但这一卷仍不是康托尔若能继续他那庞大任务时所能达到的样子。尽管如此，他原本打算在这部分工作之后附上一篇关于十八世纪后期数学进展的总体论述。然而，我们失望地发现，缺少了本应是对那个时期思想发展的精湛综述，并深感遗憾这一计划不得不被放弃。这位最伟大的数学史家发现自己不得不满足于一份按时间顺序的索引。即便如此，它仍是本书最有用的特色之一。他的《Überblick》在短短二十几页中提供了大量信息，并附有对本卷各节的充分交叉引用。

康托尔是一位出色的语言学家，弗洛里安·卡乔里在[5]中描述了康托尔告诉他的几件趣事，涉及英国数学家在大会上说外语的情况。在此我们注意到，康托尔受邀在1900年巴黎国际数学家大会上作全会报告。他作了L'historiographie des mathématiquesⓉ（数学史编纂学）的报告，用法语发表。他对该语言的掌握如此之好，以至于有些人认为他是法国人。

在[13]中，数学史家大卫·尤金·史密斯讲述了关于康托尔的一些感人的个人回忆：-

我觉得可以允许我提一下我对康托尔教授的个人印象，这是从大约三十年的交往中得来的。这始于一次到他家的拜访，当时我正计划在海德堡大学他的课程中度过一年。当我表示希望把时间投入到微积分的早期阶段时，我永远不会忘记他那种亲切的态度。他问我我认为最好从哪里开始，我说从约翰内斯·开普勒或博纳文图拉·卡瓦列里开始。他赞同的方式令人愉快，并建议也许从阿基米德开始更好，这给当时三十岁的年轻美国人留下了深刻印象，但现在他会开始得更早一些。环境往往会相当突然地改变一个人的生活，这个计划从未在海德堡大学实现，但这并没有妨碍我时不时地到他家拜访几次。我相信最后一次大约是在1910年。那时他几乎失明了。当我走进他的书房时，他从窗边的椅子上站起来，伸出双手，向我走来。我握住他的手，领他回到椅子上，我们谈论了自我们初次见面以来的这些年。……当我离开时，他站起来，用手指向他的书柜（藏书不多）说：“这些是我的书，但我看不见它们。”然后他陪我走到门口，我说了那句惯常的“Auf Wiedersehen”，心里知道这永远不会实现。

弗洛里安·卡乔里在[5]中描述了1915年会见当时86岁的康托尔。康托尔这样描述自己：-

……一个伐木工，用大斧和有力的砍伐粗略地把木材砍成适当的形状和尺寸，但留给后来的人去修整、打磨和完成。

弗洛里安·卡乔里在[5]中讲述了那次拜访的细节。我们摘录一段：-

几乎完全失明迫使他停止了研究。他说他能看到一个人身体的轮廓，但看不清五官。他可以在家中无需帮助地走动。他的听力仍然很好，头脑相当清楚。我们两人都没有触及战争问题，只是他有一次提到我来自一个名义上中立的国家。当天晚些时候，我遇到了他的儿子和女儿，他们坦率地向我表示希望美国很快会真正保持中立。康托尔询问了一些美国和英国数学家的情况，并谈到了当时在德国正在进行的一些特殊历史研究。……在花园露天吃过晚餐后，笔者被告知康托尔有午饭后躺下的习惯，便离开了。

康托尔与格奥尔格·康托尔没有亲属关系，后者称他为Namensvetter（仅同名的表亲）。康托尔于1877年荣幸地当选为德国利奥波第那科学院会士。他葬在海德堡Bergfriedhof公墓。

Moritz Cantor's parents were Isaac Benedikt Cantor (1799-1885) and Nanette Meyer Schnapper. Isaac, who was a merchant from Amsterdam, was the son of Benedict Cantor and Telly de Vries. Nanette, known as Nelly, was the daughter of the money changer Meyer Wolf Schnapper and Johanna Israel. Isaac and Nanette Cantor were married in Amsterdam on 14 November 1825. Nanette's brother Adolph Schnapper was a witness at the marriage ceremony. Both families were Jewish. Moritz was brought up in Amsterdam and his early education was from private tutors employed by his parents. He attended the Gymnasium in Mannheim to complete his secondary education and prepare for his university studies.

He entered Heidelberg University in 1848 where he was taught by Franz Ferdinand Schweins (1780-1856) and Arthur Arneth (1802-1858). Given Cantor's fame as an historian of mathematics, we should note at this point that Arneth was an excellent historian of mathematics who wrote the important work History of Pure Mathematics in its relation to the History of the Human Mind (1852). He was one of the first to try to base the history of mathematics on general philosophical principles, rather than seeing it as merely compiling facts and recording past events. There is, however, no sign that Cantor was influenced towards the history of mathematics at this stage in his career. After a year at Heidelberg, Cantor went to the University of Göttingen where he spent the years from 1849 to 1851. At Göttingen he was taught mathematics and astronomy by Carl Gauss, physics by Wilhelm Weber and mathematics by Moritz Stern, who was particularly interested in number theory. He returned to Heidelberg where he presented his doctoral thesis in 1851. His advisor at the Ruprecht-Karls University of Göttingen was Ferdinand Schweins and he was awarded a doctorate on 6 May 1851 for his thesis entitled Über ein weniger gebräuchliches Koordinatensystem Ⓣ. He did not take the state examination to qualify as a gymnasium teacher, which most students took at this stage, but he went to Berlin where he spent the summer semester of 1852 attending courses by Lejeune Dirichlet and Jakob Steiner. On 30 April 1853 he was appointed as a docent at the University of Heidelberg having submitted his habilitation thesis Grundzüge einer Elementararithmetik Ⓣ. He was to remain at Heidelberg for the rest of his life.

Cantor's early work was not on the history of mathematics but he did write a short paper on Ramus, Stifel and Cardan which he presented to a scientific meeting in Bonn. It looked at the introduction of Hindu-Arabic numerals into Europe, and was published in 1857. Later Cantor told Cajori, see [5], that his paper was:-

... so well received that he felt encouraged to continue his historical work.

However, he was influenced at Heidelberg by Arneth, whom we mentioned above, and by the cultural philosopher Eduard Maximilian Röth. Although Röth was the professor of philosophy and Sanskrit at Heidelberg, he had studied mathematics, physics and chemistry in Paris with teachers such as François Arago, Jean-Baptiste Biot, Pierre-Louis Dulong (1785-1838) and Jean-Baptiste-André Dumas (1800-1884). He may well have encouraged Cantor to visit Paris which he did in the late 1850s. During this visit, which Cantor made shortly after his encouraging Bonn meeting, he became friendly with Michel Chasles and Joseph Bertrand. Chasles was an acknowledged leading expert on the history of geometry and encouraged Cantor to publish further historical material in Comptes Rendus.

From 1860 Cantor lectured on the history of mathematics and became one of the leading German historians of mathematics at the end of the 19th Century. His first significant work was Mathematische Beiträge zum Kulturleben der Völker Ⓣ (1863) (Mathematical Contributions to the Cultural Life of the People) which, like his earlier work, concentrated on the introduction of Hindu-Arabic numerals into Europe. Not only did 1863 mark the first important work by Cantor, but it also was the year in which he was promoted to extraordinary professor at Heidelberg. On his birthday, 23 August 1868, he married Telly Gerothwohl (1847-1873); they had one son and one daughter.

Cantor's second book was Euclid und sein Jahrhundert (Euclid and his century) in which he summarised the work of Euclid, Archimedes and Apollonius. It was published in 1867. His next major contribution was Die römischen Agrimensoren und ihre Stellung in der Geschichte der Feldmesskunst Ⓣ . This may seem a strange topic at first sight since it is recognised that the Romans added little to the development of mathematics. However, Cantor saw that the Roman surveyors had played an important role in transmitting Egyptian and ancient Greek practical geometric methods to Europe in the Middle Ages.

Cantor is best remembered for the four volume work Vorlesungen über Geschichte der Mathematik which traces the history of mathematics up to 1799. The first volume was published in 1880 and the last volume appeared in 1908. The first volume traces the general history of mathematics up to 1200. The second volume traces the history up to 1668. The year 1668 was chosen by Cantor because in this year Newton and Leibniz were just about to embark on their mathematical researches. The third volume continues the overview of the history up to 1758, again chosen because of the significance of Lagrange's work which began shortly after this date. The review of these three volumes in [10] is full of praise:-

It would be impossible to do justice to this monumental work within the brief limits of a book review, even if the task were not rendered supererogatory by the high standing of the work and the acknowledged authority of its author. Cantor's 'Lectures on the History of Mathematics' are the work of a man who has unswervingly devoted a life-time to this single task, who thirty-three years ago was well known for his important contributions to this subject, and who can now in the second edition of the first volume of his great work point with pride to the impulse and awakened interest which his endeavours have aroused in the historical studies of his science. He has had many predecessors, each of whom has distinguished himself in certain branches and by certain excellences ... Nevertheless, it may safely be said that profundity, accuracy, and extensiveness of treatment have never before in any history of mathematics been so thoroughly and intimately united as in the three volumes constituting these Lectures of Moritz Cantor.

George Gibson, in the review [7], writes:-

It hardly requires to be stated that this history is certain to remain for many years the standard work on the subject with which it deals; in completeness, in accuracy, in clearness of arrangement, it stands unrivalled, and for the period which it covers is bound to be a permanent work of reference.

After completing the third volume Cantor realised that, at the age of 69, he was not up to the task of completing another volume, so at the Congress of 1904 in Heidelberg he organised a team with nine further contributors to collaborate on the fourth volume. As editor-in-chief Cantor set high standards and insisted that the style and impartiality of the first three volumes be retained. This fourth volume again stopped just before a highly significant development since 1799 is the year of Gauss's doctoral thesis. The review [11] regrets that Cantor could not have written this himself:-

It is fifteen years since the second edition of the first volume of Cantor's 'Vorlesungen' made its appearance, and last year a third edition was called for. After the completion of Vol. III, which brought this monumental work up to 1758, the author felt that the time had come when the claims of advancing years were strong enough to force him to leave the completion of his magnum opus to others. He must have felt that he had not lived in vain and that an historical school worthy of German traditions had grown up around him, when, inspired by his example and enthusiasm, men were found able and eager to continue his great work under his direction. We are unfortunate in that the health of the aged savant has prevented him from giving that full personal supervision which is so essential in securing a general unity of treatment when the mathematical labours of half a century are divided among so many hands, however individually competent they may be. And when we say that no less than nine monographs by nine men constitute this fourth volume, it is not surprising to find, that although there is less diversity of treatment than might be expected, yet the volume is not what it would have been had Cantor been able to continue his colossal task. As it is, he had intended to append to this portion of the work a general treatment of the progress of mathematics during the latter part of the eighteenth century. We are, however, disappointed at the absence of what would of course have been a masterly survey of the development of ideas in that period, and greatly regret that the plan had to be abandoned. The greatest mathematical historiographer found himself compelled to be content with a chronological index. Even so, it is one of the most useful features in the book. His 'Überblick' gives a mass of information in a couple of dozen pages, fortified with ample cross references to the sections of this volume.

Cantor was an excellent linguist and Cajori describes in [5] some amusing incidents of English speaking mathematicians speaking foreign languages at Congresses told to him by Cantor. Let us note at this point that Cantor was invited to give a plenary lecture at the International Congress of Mathematicians in Paris in 1900. He gave the lecture L'historiographie des mathématiques Ⓣ which he delivered in French. His command of the language was so good that some believed him to be French.

In [13] the historian of mathematics David Smith relates some touching personal memories of Cantor:-

I feel that I may be allowed to mention my personal impressions of Professor Cantor, gained from an acquaintance of about thirty years. It began with a visit to his home when I was planning to spend a year in his course at Heidelberg. I shall never forget his kindliness of manner when I stated that I wished to devote time to the early stages of the calculus. He asked me where I thought it best to begin and I said with Kepler or Cavalieri. The pleasant way in which he approved, with the suggestion that it might be better to start with Archimedes, impressed the young American of thirty, but now he would begin somewhat earlier still. Circumstances have a way of changing one's life rather suddenly, and the plan was never carried out in Heidelberg University, but it did not interfere with the making of several visits from time to time in his home. The last of these was made about 1910, I believe. He was then nearly blind. As I entered his study he rose from a chair near the window, held out his hands, and advanced towards me. I took them and led him back to his chair and we talked over the years since we had first met. ... When I was leaving he rose, motioned with his hand to his bookcase (not a large collection) and said, "these are my books, but I cannot see them." Then he walked with me to the door and I said the conventional "Auf Wiedersehen", knowing that it would never come.

Cajori describes in [5] meeting Cantor, then 86, in 1915. Cantor described himself as:-

... a hewer of timber who with a big axe and with powerful strokes roughly cut the timber to proper form and dimension, but left it for those who follow him to dress, polish and finish.

Cajori relates details of that visit in [5]. We give an extract:-

Almost complete blindness had compelled him to stop research. He said he could see the general outline of a person's body, but could not make out the features. He could move about in his house without assistance. His hearing was still good and his mind fairly clear. Neither of us touched upon war issues, except that he once referred to me as coming from a nominally neutral country. Later in the day I met his son and daughter, who were free in expressing to me the hope that the United States would soon come to observe real neutrality. Cantor made inquiries about certain American and English mathematicians and spoke of special historical researches then in progress in Germany. ... After a dinner which was served in the garden in the open, the writer, having been told that it was Cantor's habit to lie down after the mid-day meal, departed.

Moritz Cantor was no relation to Georg Cantor who referred to him as his Namensvetter (cousin by name only). Moritz Cantor was honoured with election to the German Academy of Sciences Leopoldina in 1877. He is buried in the Heidelberg Bergfriedhof Cemetery.