詹姆斯·约瑟夫·西尔维斯特James Joseph Sylvester

詹姆斯·约瑟夫·西尔维斯特的父亲是马克斯·亚伯拉罕 西尔维斯特，他是一名商人。第一个奇怪的事实（细心的读者可能已经注意到了！）是西尔维斯特的父亲姓西尔维斯特，而不是西尔维斯特。事实上，这本传记的主人公在成长过程中一直使用西尔维斯特 西尔维斯特这个名字，直到他开始大学学习前不久，他才决定加上西尔维斯特这个姓氏。有人可能会问，他为什么要加一个额外的名字。原因是他的长兄当时决定移民到美国，而他在获准获得居留权之前必须至少有三个名字。我们还应在此提到，西尔维斯特出生在一个犹太家庭，并在犹太信仰中长大，这导致了他晚年生活中的困难，我们将在下面描述。

西尔维斯特在伦敦上过两所学校，第一所是位于Highgate的一所寄宿学校，他在那里读到1827年，之后他在Islington的一所学校又学习了十八个月。1828年，他十四岁时进入伦敦大学学院，并在该学院接收学生的第一年开始学习。这是一个明智的选择，因为与其他一些英国大学不同，它不隶属于任何教派。它还拥有非常有才华的奥古斯塔斯·德摩根作为其数学教授。然而，西尔维斯特的事情进展并不顺利，因为在他入学仅五个月时，他被指控在学院餐厅用刀威胁一名同学。指控提出后，西尔维斯特的家人立即将他从大学学院撤出。他们意识到，尽管西尔维斯特能够从大学教育中获益，但他还不够成熟，无法适应大学生活，于是家人送他到利物浦的皇家学院继续学业。

1831年7月7日，西尔维斯特作为学生被剑桥大学圣弗瑞兹·约翰学院录取，尽管他的学习因长期患病而被迫在1833-34和1834-35这两年的大部分时间里中断。恢复健康后，他于1837年参加了数学荣誉学位考试。与西尔维斯特同年参加荣誉学位考试的还有两位著名数学家，即Duncan Gregory和乔治·格林。西尔维斯特获得第二名，比其他两人大20岁的乔治·格林获得第四名，Duncan Gregory获得第五名。（获得第一名的数学家William Griffin毕业后没有做出什么重要工作：这在当时荣誉学位考试的“速度测试”中是很常见的结果。）

我们谨慎地说西尔维斯特在1837年参加了荣誉学位考试，而不是说他在那一年毕业，因为他无法毕业。当时，学生必须在毕业前签署英国国教的三十九条信纲，而西尔维斯特作为犹太人，自然拒绝宣誓，因此无法毕业。出于同样的原因，他没有资格获得Smith奖，也没有资格竞争研究员职位。

从1838年起的三年里，西尔维斯特担任伦敦大学自然哲学讲席，这是少数几个不因他的宗教而排斥他的地方之一。他以前的老师奥古斯塔斯·德摩根是他的同事之一。然而西尔维斯特是一名数学家，他希望教授的是数学，而不是物理学。他是一位非常活跃的研究者，到1841年辞去自然哲学讲席时，他已经发表了十五篇关于流体动力学和代数方程的论文。西尔维斯特可能当了三年教授，但由于我们上面指出的原因，他仍然没有学位。他在1841年弥补了这一点，当时都柏林三一学院授予他文学学士和文学硕士学位。事实上，正是允许罗马天主教徒在三一学院获得学位的立法，也意味着犹太人可以在那里毕业。

27岁时，他申请美国夏洛茨维尔弗吉尼亚大学的数学讲席并获任命。他的申请得到了奥古斯塔斯·德摩根、约翰·赫歇尔和查尔斯·巴贝奇的大力支持。例如奥古斯塔斯·德摩根写道：-

在这个国家，没有哪个像他这样年纪的人在作为原创数学家方面比西尔维斯特先生享有更高的声誉，或者更有希望通过他的努力来拓展精确科学。根据我对他所做工作的了解，我可以有把握地说，他是一位能力很强的数学家，熟悉这门科学最现代的形式，并且非常热忱地从事他的探究。

然而，他上任仅几个月后就辞职了。Feuer在[11]中记录了以下版本的事件。一名学生曾在西尔维斯特的一次讲座中读报纸，侮辱了他，西尔维斯特用一把剑杖打了他。那名学生因惊吓而倒下，西尔维斯特（错误地）认为自己杀死了他。他逃到了纽约，他的一个哥哥住在那里。虽然这个故事在某种意义上可能是准确的，但在另一种意义上，它给人对事件的印象相当错误。毫无疑问，弗吉尼亚大学的学生行为非常糟糕，而且他们的行为因饮酒而变得更糟。他们也很可能更多地因为西尔维斯特是外国人而对他作出反应，尽管在西尔维斯特到来之前就有许多辱骂行为的记录。当然，西尔维斯特于1842年2月1日向教职员抱怨了一名特定的一年级学生的行为。此后，西尔维斯特遭受这名学生的辱骂变得更严重。虽然这名学生受到大学当局的训斥，但没有对他作出任何制裁。西尔维斯特于3月19日反对教职员的宽容态度，并在三天后辞职。

在纽约，他开始寻找大学职位。在两次申请哥伦比亚学院和哈佛的职位失败后，他于1843年11月20日登船返回英国。他在纽约期间又遭受了一次挫折。他遇到了一位当地女孩马斯顿小姐，并非常喜欢她。他求婚，但她以他信仰犹太教为由拒绝了他。

回到英国后，西尔维斯特担任精算师，并在衡平法律与人寿保险公司担任秘书。然而，他也给包括弗洛伦丝·南丁格尔在内的学生教授数学。他决定学习法律，并且幸运的是，阿瑟·凯莱也在学习成为律师。两人在伦敦林肯律师学院的法庭上相遇，他们在法庭周围散步时讨论数学，尽管性情迥异，他们成为了终身朋友。

西尔维斯特努力想重新成为职业数学家，他于1854年申请伦敦格雷沙姆学院的几何学讲师职位，但未获任命。另一次失败的申请是伍利奇皇家军事学院的数学讲席，但在成功申请者被任命后几个月内去世后，西尔维斯特成为了伍利奇的数学教授。

西尔维斯特在matrix theory方面做了重要工作，这是他在林肯律师学院法庭期间与阿瑟·凯莱散步时产生兴趣的一个主题。1851年，他发现了三次方程的判别式，并首次将“判别式”这一名称用于二次方程及更高阶的这种表达式。

你可以在THIS LINK看到更多关于判别式的信息。

他在1852年和1853年发表了重要论文，即On the principle of the calculus of forms和On the theory of syzygetic relations and two rational integer functions。特别地，他使用矩阵理论研究高维几何。他还对λ矩阵初等因子理论的创立做出了贡献。

奥古斯塔斯·德摩根是伦敦数学会的第一任主席。西尔维斯特于1866年成为该学会的第二任主席。他是该学会为纪念奥古斯塔斯·德摩根而颁发的金质奖章的第一位获得者。他还在1863年当选为巴黎科学院，并自1839年以来一直是伦敦皇家学会的会士。由于身处军事学院，西尔维斯特被迫在55岁时退休。Parshall写道[21]：-

事实上，在那里最后五年期间，西尔维斯特经常发现自己在教学职责上与军事当局发生争执，并越来越感到自己的数学才能衰退。

1870年退休后，他住在伦敦，大部分时间都在雅典娜神庙俱乐部度过（他于1856年当选为该俱乐部成员）。起初看起来他可能会放弃数学，因为他此时出版了他唯一的书，而那是关于诗歌的。显然，西尔维斯特为这部题为The Laws of Verse的著作感到自豪，因为此后他有时署名“J J 西尔维斯特，The Laws of Verse的作者”。Parshall写道[21]：-

西尔维斯特对诗歌和语言的热爱甚至在他的数学著作中也以显著的方式表现出来。他对法语、德语、意大利语和希腊语的精通常常反映在数学新词中——如“meicatecticizant”和“tamisage”——他因此获得了一定的名声。此外，文学典故、诗歌引用和无拘无束的夸张为他的已发表论文和讲座增添了趣味。

有三年时间西尔维斯特似乎没有做任何数学研究，但后来巴夫尼提·列波维奇·切比雪夫访问了伦敦，两人讨论了可用于画直线的机械连杆。在研究这个主题之后，西尔维斯特在皇家学会作了一场题为On recent discoveries in mechanical conversion of motion的晚间讲座。这场讲座的听众中有一位数学家是阿尔弗雷德·布雷·肯普，他对此主题入了迷。阿尔弗雷德·布雷·肯普和西尔维斯特共同研究连杆并做出了重要发现。

1877年，西尔维斯特接受了约翰威廉·霍普金斯大学的讲席，并于1878年创办了American Journal of Mathematics，这是美国第一本数学期刊。事实上，西尔维斯特在约翰霍普金斯大学度过的七年见证了他数学兴趣的复苏。在他的职业生涯中，他第一次在正规的大学环境中教学和进行研究，而且，他能够以一种在他职业生涯其余时间不可能的方式发挥领导作用。他第一次有了研究学生围绕在他身边，他可以按照自己的兴趣方向指导他们的研究。事实上，在约翰霍普金斯大学的七年里，他指导了九名学生的博士学位。他的研究兴趣重新焕发活力，他让学生参与他在这些年进行的划分理论中的重要思想。他在他创办的American Journal of Mathematics上发表了这项工作。鉴于他在约翰霍普金斯大学如此成功，人们不得不问为什么他在那时离开。我们通过他写给菲利克斯·克莱因的一封私人信件了解到他的原因，菲利克斯·克莱因曾考虑接替他，但发现薪水和条件不足。西尔维斯特当然想回到他的祖国，并且有一些个人原因要回去。然而，他的主要原因是（见[5]）：-

……因为我认为我的数学学识不够广博，我的心智结构的活力也不足以使我跟上不断前进的数学进展浪潮，达到人们应当期望于一个实际上肩负着指导和塑造地球上最有才智的种族之一5500万人的数学教育责任的人的那种程度。

当亨利·约翰·斯蒂芬·史密斯于1883年去世时，西尔维斯特虽然此时已68岁，仍被任命为牛津大学萨维尔几何学讲席。然而西尔维斯特只喜欢讲授自己的研究，这在牛津并不受欢迎，那里的学生只想在考试中取得好成绩。1892年，78岁的西尔维斯特被牛津任命了一位代理教授替代他的职位，此时他已部分失明并患有记忆丧失，他回到伦敦，在雅典娜神殿俱乐部度过了他的最后几年。

Macfarlane[17]用以下方式描述西尔维斯特：

西尔维斯特性情如火，充满激情……西尔维斯特写论文时从不放过脚注、附录、补遗，而且校样上的改动和修正之多，令印刷工几乎无法完成任务。……西尔维斯特符合大众对数学家的印象：沉湎于思考，高高在上，不问世事。……西尔维斯特是个演说家，即便算不上诗人，至少也以自己的诗作自豪。

西尔维斯特在约翰霍普金斯大学的一位学生描述了他在那里的教学：-

……他讲课的内容在很大程度上只能是他自己的研究工作，而且通常是刚出炉的成果。结果，指望他连续而系统地呈现任何已经完成的庞大理论体系，是不可能的。任何未解决的困难、任何被提议的推广，在其他讲课人那里可能一提而过，在他那里却不可避免地成为离题发挥的契机，这种离题必定会耗费数周时间，甚至可能使他永久偏离原来的主题。他在巴尔的摩期间发表的重要论文，几乎全都是这样产生的。我们这些听过他讲课的人，可以说亲眼见证了这些论文的诞生过程。

下面这段引自托马斯·阿彻·赫斯特的话，透露了西尔维斯特性格的某些方面：-

周一收到西尔维斯特的来信后，我去雅典娜神殿俱乐部见他。……而且，他极其友好，希望我们住在一起，邀请我去伍利奇与他同住等等。总之，他有一种古怪的深情。

西尔维斯特把下面这道谜题寄给了Educational Times。这既反映了他的一个爱好，也反映了他对谜题的兴趣：-

我有大量面值仅为5便士和17便士的邮票。用这两种不同面值的邮票组合，我无法凑出的最大面值是多少？

[答案是63便士。你能证明这一点吗！]

James Joseph Sylvester's father was Abraham Joseph who was a merchant. The first strange fact (which the careful reader may already have noticed!) is that Sylvester's father had the surname Joseph, and not Sylvester. Indeed the subject of this biography grew up with the name James Joseph and it was only shortly before he began his university studies that he decided to add the surname Sylvester. One might ask at this point why he added an extra name. The reason was that his eldest brother decided at this time to emigrate to the United States and he was required to have at least three names before he was allowed to gain residence. We should also mention at this point the fact that Sylvester was born into a Jewish family, and brought up in the Jewish faith, which would lead to difficulties later in his life which we describe below.

Sylvester attended two schools in London, the first one being a boarding school in Highgate which he attended up to 1827, after which he undertook a further eighteen months study at a school in Islington. In 1828, at the age of fourteen, he entered University College London, and began his studies in the first year that the College received students. This was a sensible choice since, unlike some other British universities, it was non-sectarian. It also had the very talented De Morgan as its professor of mathematics. However things did not go well for Sylvester for, when only five months into his studies, he was accused of threatening a fellow student with a knife in the College refectory. Sylvester's family withdrew him from University College immediately after the allegations were made. Realising that, although Sylvester was able to profit from university education, he was not yet mature enough for university life, his family sent him to continue his schooling at the Royal Institution in Liverpool.

On 7 July 1831 Sylvester matriculated as a student at St John's College, Cambridge, although his studies were interrupted when he was forced to take most of the two years 1833-34 and 1834-35 out due to a lengthy illness. After regaining his health he took the mathematical tripos examination in 1837. Two other famous mathematicians took the tripos examination in the same year as Sylvester, namely Duncan Gregory and George Green. Sylvester came second, Green who was 20 years older than the other two came fourth, with Duncan Gregory fifth. (The mathematician who came first, William Griffin, did little work of importance after graduating: this was not at all uncommon a result of the 'speed test' which the tripos was at that time.)

We were careful to say that Sylvester took the tripos examination in 1837 rather than saying he graduated in that year, for he was not able to graduate. At this time it was necessary for a student to sign up to the Thirty-Nine Articles of the Church of England before graduating and Sylvester, being Jewish, naturally refused to take the necessary oath so could not graduate. For the same reason he was not eligible for a Smith's prize nor was he eligible to compete for a Fellowship.

For the three years from 1838 Sylvester held the chair of natural philosophy at the University of London, one of the few places which did not bar him because of his religion. His former teacher De Morgan was one of his colleagues. However Sylvester was a mathematician and it was mathematics, not physics, that he wished to teach. He was a very active researcher and by the time he resigned the chair of natural philosophy in 1841 he had published fifteen papers on fluid dynamics and algebraic equations. Sylvester might have been a professor for three years but he still had no degree for the reason we indicated above. He remedied this in 1841 when he was awarded a B.A. and an M.A. by Trinity College, Dublin. In fact it was legislation which allowed Roman Catholics to take degrees at Trinity College which also meant that Jews could graduate there.

At the age of 27 he applied for, and was appointed to, the chair of mathematics in the University of Virginia in Charlottesville in the United States. His application was strongly supported by De Morgan, John Herschel and Charles Babbage. For example De Morgan wrote:-

No person of his years in this country has more reputation than Mr Sylvester as an original mathematician, or bids fairer to extend the exact sciences by his labours. From my own knowledge of what he has done, I can safely say that he is a mathematician of great power, well acquainted with the most modern forms of the science, and very zealous in the persecution of his inquiries.

However he resigned after only a few months in post. The following version of events is recorded by Feuer in [11]. A student who had been reading a newspaper in one of Sylvester's lectures insulted him and Sylvester struck him with a sword stick. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. He fled to New York where one of his elder brothers was living. Although this story is probably accurate in one sense, in another it gives a quite false impression of events. There is no doubt that the students at the University of Virginia were very badly behaved and their behaviour was made worse by drink. It is also likely that they reacted more against Sylvester as a foreigner although there are many records of abusive behaviour before Sylvester arrived. Certainly Sylvester complained to the Faculty on 1 February 1842 about the behaviour of a particular first year student. The abuse suffered by Sylvester from this student got worse after this. Although the student was reprimanded by the university authorities, no sanctions were made against him. Sylvester objected to the lenient attitude of the Faculty on 19 March and resigned three days later.

In New York he began to look for university positions. After two unsuccessful bids for posts at Columbia College and Harvard, he boarded a ship back to England on 20 November 1843. He suffered another upset while in New York. He met a local girl, Miss Marston, and became very fond of her. He proposed marriage but she turned him down on the grounds that he was of the Jewish religion.

On his return to England Sylvester worked as an actuary and was secretary at the Equity Law and Life Assurance Company. However, he also gave mathematics tuition with his pupils including Florence Nightingale. He decided to study law and, by good fortune, Cayley was also studying to become a lawyer. Both met at the courts of Lincoln's Inn in London and they discussed mathematics as they walked around the courts and, although very different in temperament, they became life long friends.

Sylvester tried hard to return to being a professional mathematician and he applied for a lectureship in geometry at Gresham College, London in 1854 but he was not appointed. Another failed application was for the chair in mathematics at the Royal Military Academy at Woolwich, but, after the successful applicant died within a few months of being appointed, Sylvester became professor of mathematics at Woolwich.

Sylvester did important work on matrix theory, a topic in which he became interested during walks with Cayley while they were at the courts of Lincoln's Inn. In 1851 he discovered the discriminant of a cubic equation and first used the name 'discriminant' for such expressions of quadratic equations and those of higher order.

You can see more about the discriminant at THIS LINK.

He published important papers in 1852 and 1853, namely On the principle of the calculus of forms and On the theory of syzygetic relations and two rational integer functions. In particular he used matrix theory to study higher dimensional geometry. He also contributed to the creation of the theory of elementary divisors of lambda matrices.

De Morgan was the first president of the London Mathematical Society. Sylvester became the second president of that Society in 1866. He was the first recipient of the gold medal which the Society awarded to honour De Morgan. He had also been elected to the Paris Academy of Sciences in 1863 and had been a fellow of the Royal Society of London since 1839. Being at a military academy Sylvester was forced to retire at age 55. Parshall writes [21]:-

During his last five years there, in fact, Sylvester frequently found himself at odds with the military authorities over his teaching duties and increasingly sensed the diminishment of his mathematical talents.

After retiring in 1870 he lived in London, spending most days at the Athenaeum Club (he had been elected a member of the Club in 1856). At first it looked as though he might give up mathematics since he published his only book at this time and it was on poetry. Clearly Sylvester was proud of this work, entitled The Laws of Verse, since after this he sometimes signed himself "J J Sylvester, author of The Laws of Verse". Parshall writes [21]:-

Sylvester's love of poetry and language manifested itself in notable ways even in his mathematical writings. His mastery of French, German, Italian, and Greek was often reflected in the mathematical neologisms - like "meicatecticizant" and "tamisage" - for which he gained a certain notoriety. Moreover, literary illusions, poetic quotations, and unfettered hyperbole spiced his published papers and lectures.

For three years Sylvester appears to have done no mathematical research but then Chebyshev visited London and the two discussed mechanical linkages which can be used to draw straight lines. After working on this topic Sylvester lectured on it at an evening lecture entitled On recent discoveries in mechanical conversion of motion which he gave at the Royal Institution. One mathematician in the audience at this lecture was Kempe and he became absorbed by this topic. Kempe and Sylvester worked jointly on linkages and made important discoveries.

In 1877 Sylvester accepted a chair at Johns Hopkins University and he founded in 1878 the American Journal of Mathematics, the first mathematical journal in the United States. In fact the seven years that Sylvester spent at Johns Hopkins saw a resurgence in his mathematical interests. For the first time in his career he was teaching and undertaking research in a proper university environment and, moreover, he was able to give leadership in a way that had not been possible throughout the rest of his career. For the first time he had research students around him whose studies he could supervise in directions which his interests took him. In fact he supervised the doctorates of nine students during his seven years at Johns Hopkins. His research interests reinvigorated, he involved his students in important ideas in the theory of partitions which he undertook during these years. He published this work in the American Journal of Mathematics which he had founded. Given that he was so successful at Johns Hopkins, one has to ask why he left when he did. We know of his reasons through a personal letter he wrote to Klein who considered succeeding him but found the salary and conditions insufficient. Sylvester certainly wanted to return to his native land and had some personal reasons to return. However, his main reason was (see [5]):-

... because I did not consider that my mathematical rerudition was sufficiently extensive nor the vigour of my mental constitution adequate to keep me abreast of the continually advancing tide of mathematical progress to that extent which ought to be expected from one on whom practically rests the responsibility of directing and moulding the mathematical education of 55 million of one of the most intellectual races of men upon the face of the earth.

When Smith died in 1883 Sylvester, although 68 years old at this time, was appointed to the Savilian chair of Geometry at Oxford. However Sylvester only liked to lecture on his own research and this was not well liked at Oxford where students wanted only to do well in examinations. In 1892, at the age of 78, Oxford appointed a deputy professor in his place and Sylvester, by this time partially blind and suffering from loss of memory, returned to London where he spent his last years at the Athenaeum Club.

Macfarlane [17] describes Sylvester in the following way:

Sylvester was fiery and passionate ... Sylvester never wrote a paper without foot-notes, appendices, supplements, and the alterations and corrections in his proofs were such that the printers found their task well-nigh impossible. ... Sylvester satisfied the popular idea of a mathematician as one lost in reflection, and high above mundane affairs. ... Sylvester was an orator, and if not a poet, he at least prided himself on his poetry.

One of Sylvester's students at Johns Hopkins University describes his teaching there:-

... the substance of his lectures had to consist largely of his own work, and, as a rule, of work hot from the forge. The consequence was that a continuous and systematic presentation of any extensive body of doctrine already completed was not to be expected from him. Any unsolved difficulty, any suggested extension, such would have been passed by with a mention by other lecturers, became inevitably with him the occasion of a digression which was sure to consume many weeks, if indeed it did not take him away from the original object permanently. Nearly all of the important memoirs which he published, while in Baltimore, arose in this way. We who attended his lectures may be said to have seen these memoirs in the making.

The following quote, from Thomas Hirst, tells us something about Sylvester's personality:-

On Monday having received a letter from Sylvester I went to see him at the Athenaeum Club. ... He was, moreover, excessively friendly, wished we lived together, asked me to go live with him at Woolwich and so forth. In short he was eccentrically affectionate.

Sylvester sent the following puzzle to the Educational Times. It tells us of one of his hobbies as well as his interest in puzzles:-

I have a large number of stamps to the value of 5d and 17d only. What is the largest denomination which I cannot make up with a combination of these two different values.

[The answer is 63d. Can you prove this!]