阿克塞尔·图厄Axel Thue

阿克塞尔·图厄是Niels Thue（1825-1895）的儿子，Niels Thue是一名船长，后来成为船主，母亲是Nicoline Cathinka Eger（1837-1866）。图厄在奥斯陆峡湾畔的滕斯贝格长大，并在那里上了中学。起初他被物理学吸引，但通过一件相当奇怪的事情对数学产生了兴趣。他看到一则广告，宣传Elling Holst的一本书，题为“论摆对几何的意义”，他很自然地认为这是一本关于摆的物理学书籍，便去函索要一本。当书到达时，他发现广告中有印刷错误，书名是“论让-维克托·彭赛列对几何的意义”。这本书由数学家Elling Bolt Holst（1849-1915）撰写，于1878年出版，是一本获奖的几何教科书。当图厄发现这是一本数学书而不是他预期的物理学书时，他还是读了它。事实上，这本书记录了让-维克托·彭赛列对现代几何的影响，图厄被其中描述的数学迷住了。他于1880年在滕斯贝格参加了中学考试，同年晚些时候成为奥斯陆Aars and Voss学校的学生。

图厄于1880年成为该校的学生，Elling Holst是他的数学老师。Holst在1876年被任命为奥斯陆Aars and Voss学校的教师之前，曾师从索菲斯·李和菲利克斯·克莱因。物理学家Kristian Birkeland也在Aars and Voss学校受教于Holst，他在Holst去世时写道：——

在我年轻时，没有其他人像Elling那样深深地触动过我，我感激认识他。他是如此特别的人。

显然，Holst对图厄也有同样的意义。1883年，他准备在文理中学（Gymnasium）参加期末考试。他在日记中记录了这些考试前发生的重要事件[6]：-

那是在考试期间。我读了又读，以获得尽可能高的分数。就在考试前，Holst给我看了丹麦数学期刊上的一些问题。我幸运地解决了这些问题，并将解答寄给了哥本哈根。有一天，Holst进来递给我一份丹麦数学期刊，令我高兴的是，我在其中发现了一篇关于我的一个解答的描述。当天晚些时候，我们在学校聚会上聚集时，Holst在班上传阅了这份期刊，我得到了令人满意的评价。前一天，我在Asker拜访了Holst。我在那里度过了一个愉快的夜晚。我们喝了一杯潘趣酒，谈论了索菲斯·李的球面几何、链环系统以及其他愉快的事情。几天后，我前往Tönsberg度假。

随后，他进入奥斯陆大学数学与自然科学学院，于1889年毕业。在本科期间，他在Holst创办的数学研讨会上做了十五次讲座。其中一次讲座由索菲斯·李参加，现在Holst和索菲斯·李都开始建议这位年轻人申请旅行奖学金。1889年初，索菲斯·李写信给Holst支持图厄[6]：-

我已建议图厄今年冬天申请旅行奖学金。图厄在一系列原创研究中表明，他具备成为杰出数学家所需的许多素质。……1886年我离开奥斯陆时，我就知道图厄的工作。我很高兴地从他后来的工作中看到，他的知识有了相当大的增长，并且他为自己设定了更高的目标。然而，我仍然有一个明确的印象，即他的数学知识并不配得上他的天赋和热情。正是因为这个原因，我认为他绝对有必要去……因此，我恳请您尽一切努力立即为图厄获得旅行奖学金。

Holst于1889年2月写了一封信支持图厄的申请[6]：-

图厄，目前学习科学，自他在Aars and Voss中学科学方向的第一年起，就一直是我的数学学生。……在他的整个学校生涯中，他越来越证实了我对他的印象：他的数学天赋超过了我以前作为学校或大学教师所遇到的任何人。我不会否认，拥有这样的学生既是一种难得的快乐，也是一份不小的责任。近年来，他的独立研究涉及数论和变分法，这是两门最困难、且（就挪威而言）最不熟悉的数学学科。

获得奖学金后，图厄于1891年前往莱比锡，师从索菲斯·李学习。然而，维戈·布朗写道[1]：-

……他的作品并未显示出索菲斯·李的影响，可能是因为图厄无法遵循任何其他人的思路。

我们从图厄写给Holst的一封信中，确实得到了他在莱比锡期间情况的很好说明。这告诉我们，图厄在莱比锡时病得很重，这必定意味着他没有获得本可以获得的那么多收获。见THIS LINK。

他还在柏林待了一段时间，在那里他听了Georg Hermann Hettner（1854-1914）、赫尔曼·冯·亥姆霍兹、拉扎勒斯·福克斯和利奥波德·克罗内克的讲座。同样，我们在图厄1891年6月写给Holst的一封信中，对他对柏林和在那里讲课的数学家的看法有了很好的描述。见THIS LINK。

回到奥斯陆后，图厄从1891年到1894年持有数学奖学金。1894年7月6日，他与Lucie Collett Lund结婚，她是演员Anders Jacob Lund（1824-1896）和Marie Collett（1855-1891）的女儿。Lucie生于1873年8月4日，比图厄小十岁。图厄和Lucie 图厄在随后的几年里有了七个孩子。奖学金结束后，图厄被任命到特隆赫姆技术学院，从1894年到1903年在那里工作。尽管这是一所优秀的学院，被认为是该国培养工程师最好的学院，但图厄在那里并不快乐。他继续给Holst写信，在1902年8月28日的一封信中，他写道（例如见[6]）：-

尽管如此，在这种破坏性的孤立中，没有人对我的材料感兴趣，我产生的工作比之前更多也更好。因此，我发展了一种理论，根据该理论，除其他外，我在e和π方面得出了与夏尔·埃尔米特和费迪南德·冯·林德曼相同的结果。

在这封信中，他还告诉Holst，他正在申请奥斯陆大学空缺的应用数学教授职位。他的申请成功了，并于1903年被任命为奥斯陆大学应用数学教授。他一直担任这个职位，直到1922年去世。在被任命后的几年里，他忙于撰写力学讲义，几乎没有时间进行研究。然而，从1906年到1916年这段时间是图厄做出最佳工作的时期，尽管这并不是在轻松的环境中完成的。他的健康状况不佳，经常遭受胸痛的折磨。他的大家庭（我们上面提到他有七个孩子）很难靠教授的薪水维持，他还在奥斯陆的军事学校兼职教学，努力给他们更好的生活。然而，鉴于他的健康问题，额外的工作是他最不应该做的。他的身体健康问题似乎对他产生了另一个影响，那就是他产生了对独处的极大恐惧。维戈·布朗写道[6]：-

在1913年奥斯陆的斯堪的纳维亚数学大会上，他在一个晚会结束时请求我陪他去他兄弟在奥斯陆的家，他要在那里过夜，因为太晚了无法回到Nordstrand。当我们到达他兄弟家时，那里没有人，因为他兄弟去拜访别人了。我在那里坐到凌晨2点，因为我能理解图厄害怕独自在房子里的恐惧。

现在让我们看看图厄的一些数学贡献。1909年，他发表了一篇重要的论文，发表在奥古斯都·利奥波德·克雷勒的期刊上，关于代数数，表明例如$y^{3} - 2x^{2} = 1$不能被无限多对整数满足。他的工作由卡尔·西格尔在1920年扩展，并由克劳斯·罗特在1958年再次扩展。

埃德蒙·朗道在1922年将图厄的工作描述为：-

……我所知道的基本数论中最重要的发现。

图厄的定理指出：-

如果 f (x, y) 是一个整系数齐次多项式，在有理数上不可约且次数大于 2，c 是一个非零整数，那么 f (x, y) = c 只有有限个整数解。

他对 丢番图方程 理论的贡献在 [3] 中讨论。事实上，图厄 写了 35 篇关于数论的论文，主要关于丢番图方程理论，这些论文收录在 [2] 中。

图厄 的另一项著名贡献是他的四篇论文 Über unendliche Zeichenreihen Ⓣ（论无限字符行）（1906）、Die Lösung eines Spezialfalles eines generellen logischen Problems Ⓣ（一个一般逻辑问题特殊情况的解）（1910）、Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen Ⓣ（论某些字符串相等部分的相互位置）（1912）和 Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln Ⓣ（关于按给定规则改变字符串的问题）（1914）。这些关于词和语言的论文提出了关于有限表示半群的词问题的重要工作。当 图厄 写这些论文时，它们并未被视为重要，但后来的发展表明，他在考虑这些后来引起极大兴趣的困难问题方面远远超前于他的时代。他的许多结果被那些不知道 图厄 贡献的数学家重新发现。例如，他考虑了在有限字母表上寻找一个不包含两个相邻相同块的无限序列的问题。今天，这个性质被称为“无平方性”。这些论文在 [4]、[9] 和 [10] 中有详细讨论。在 [10] 中，Magnus Steinby 和 Wolfgang Thomas 写道：-

图厄 的许多思想在理论计算机科学中具有影响力。特别是，图厄 系统、半 图厄 系统以及他在词的组合学方面的工作广为人知。……他 1910 年的论文……包含许多关于树、项重写和词问题的概念和思想，这些概念和思想惊人地现代，后来在数学、逻辑学和计算机科学中发挥了重要作用。

James Power 在 [9] 中写道：-

图厄 发表了四篇直接与词和语言理论相关的论文：两篇关于无限字符串中的模式，分别于 1906 年和 1912 年；两篇关于更一般的变换问题，分别于 1910 年和 1914 年。1906 年和 1912 年的论文都被 Jean Berstel [4] 翻译并广泛讨论，并且除其他贡献外，以提出 图厄-莫尔斯 序列而闻名。图厄 1910 年的论文处理树之间的变换，因此是他 1914 年论文的更直接前身。它已被 Steinby 和 Thomas [10] 讨论。图厄 1914 年的论文……主要因证明了一个早期不可判定问题的例子而闻名，波斯特 在《图厄 问题的递归不可解性》（1947）中显著引用了它。

如果这项工作对一位应用数学教授来说显得有些奇怪，那么引用图厄的一些话将阐明他在应用问题上的立场。他在奥斯陆任教授期间写的一篇文章中写道：-

为了逻辑科学的发展，重要的是为困难问题的思辨处理找到广阔的领域，而不考虑最终的应用。

图厄关于应用数学的另一段引文（例如见[1]）是：-

越远离有用性或实际应用，就越重要。

然而，他确实对应用数学做出了贡献。例如，他的论文The Principle of Virtual Velocity是[1]：-

……一个在文献中没有类似表述的原创性陈述。

然而，他基于传递力的杆件的力学工作，大部分在他去世时仍未完成。应当提及图厄工作的一个特点。他似乎对他人的数学兴趣不大，他的论文以参考文献极少而著称。例如，上面提到的他1906年和1910年的论文完全没有参考文献，而1912年的论文只引用了他自己1906年的论文，1914年的论文只引用了他自己1910年和1912年的论文。这既可以被视为一个弱点，也可以被视为一个优点，因为它使他能够产生极具原创性的材料。

图厄于1894年当选为挪威科学与文学院院士，1895年当选为特隆赫姆的挪威皇家科学学会院士，以此获得荣誉。他从1916年到1922年担任Acta Mathematica的编辑。随着健康状况恶化，大学于1920年给他休假。到1921年，他已成为奥斯陆Ulleval医院的病人，尽管他有自己的房间，仍然能够从事数学工作。事实上，他生命的最后一段时间都在研究皮埃尔·德·费马最后定理，去世时留下了400页在研究期间写的手稿。他被安葬在Nordstrand公墓。他的妻子于1962年4月22日去世，几乎正好在他丈夫去世40年后。她也被安葬在Nordstrand公墓的同一座墓中。

让我们以Carl Stürmer [14]的一段引文来结束这篇传记：-

随着图厄教授于今年3月7日去世，Academy of Science失去了一位最杰出的成员，一位数学天才，他将非凡的原创性天赋与罕见的洞察力和逻辑感结合在一起。

图厄去世后其他纪念演讲的摘录见THIS LINK。

Axel Thue was the son of Niels Thue (1825-1895), a master mariner and later a ship owner, and Nicoline Cathinka Eger (1837-1866). Axel was brought up in Tönsberg, a town on the Oslofjord, and attended middle school there. Initially he was attracted to physics but he became interested in mathematics through a rather strange event. He saw an advertisement for a book by Elling Holst entitled "On pendulum's meaning of geometry" and, quite naturally, believing it to be a physics book about the pendulum he sent off for a copy. When the book arrived he discovered that there had been a misprint in the advertisement and the title of the book was "On Poncelet's meaning of geometry". The book, written by the mathematician Elling Bolt Holst (1849-1915), was published in 1878 and was a prize winning geometry text. When Thue discovered that it was a mathematics book rather than the physics book he had expected, nevertheless he read it. In fact, the book documents the influence on modern geometry of Jean-Victor Poncelet, and Thue became fascinated by the mathematics it described. He took the middle school examinations in Tönsberg in 1880 and, later that year, became a pupil at the Aars and Voss's school in Oslo.

When Thue became a pupil at the school in 1880, Elling Holst was his mathematics teacher. Holst had studied under Sophus Lie and Felix Klein before being appointed as a teacher at the Aars and Voss's school in Oslo in 1876. The physicist Kristian Birkeland, who was also taught by Holst at Aars and Voss's school, wrote on Holst's death:-

No other man has ever touched me as deeply as Elling when I was young and I am thankful to have known him. He was such a special person.

It is clear that Holst meant the same to Thue. In 1883 he prepared to take his final examinations at the gymnasium. He recorded in his diary important events that took place in the run up to these examinations [6]:-

It was during the examinations. I read and read to get as high a mark as possible. Just before the examination Holst showed me some problems in the Danish mathematical periodical. I was fortunate enough to solve these, and I sent the solutions in to Copenhagen. One day Holst came in and handed me a copy of the Danish mathematical periodical, where, to my pleasure I found a description of one of my solutions. Holst circulated this copy in the class when later in the day we were gathered at a school party, where I was given a gratifying testimonial. The day before I had visited Holst in Asker. I had an agreeable evening there. We had a glass of punch and talked about Lie's geometry of spheres and about link systems and other pleasant things. A few days afterwards I left for Tönsberg on vacation.

He then enrolled in the Faculty of Mathematics and Natural Sciences of the University of Oslo graduating in 1889. While he was an undergraduate he gave fifteen lectures in the mathematical seminar which had been founded by Holst. One of these lectures was attended by Sophus Lie and now both Holst and Lie began to advise the young man on applying for a travelling scholarship. Early in 1889, Sophus Lie wrote a letter to Holst supporting Thue [6]:-

I have advised Axel Thue to apply for a travelling scholarship this winter. Thue has shown in a series of original investigations that he has many of the qualifications needed to become an outstanding mathematician. ... I knew of Thue's work when I left Oslo in 1886. I have been pleased to see from his later work that there has been a considerable increase in his knowledge and that he has set himself higher aims. Yet I still have a definite impression that his mathematical knowledge does not do justice to his gifts and his enthusiasm. It is for that reason that I consider it to be absolutely necessary for him to go ... I therefore beg you to do all you can to obtain a travelling scholarship for Thue immediately.

Holst wrote a letter supporting Thue's application in February 1889 [6]:-

Axel Thue, currently studying sciences, has been a pupil of mine in mathematics since his first year in the science line at Aars and Voss's gymnasium. ... Throughout his school career he provided increasing confirmation of the impression I had received of him as one whose mathematical gifts exceeded those of anyone I had previously encountered as a school or university teacher. I will not deny that it was both a rare pleasure and no little responsibility to have such a pupil. Of recent years his independent studies have been concerned with the theory of numbers and the calculus of variations, two of the most difficult, and (as far as Norway is concerned) least familiar mathematical disciplines.

After the award of the scholarship, Thue went to Leipzig in 1891 to study under Lie. However, Viggo Brun writes that [1]:-

... his works do not reveal Lie's influence, probably because of Thue's inability to follow anyone else's line of thought.

We do have a very good indication of his time in Leipzig from a letter he wrote to Holst. This tells us that Thue was seriously ill while in Leipzig and it must have meant that he did not profit as much as he might have done. See THIS LINK.

He also spent a while in Berlin where he attended lectures by Georg Hermann Hettner (1854-1914), Helmholtz, Fuchs and Kronecker. Again we have a nice description of Thue's opinion of Berlin and the mathematicians who lectured there in a letter he wrote to Holst in June 1891. See THIS LINK.

Back in Olso, Thue held a scholarship in mathematics from 1891 to 1894. On 6 July 1894 he married Lucie Collett Lund, the daughter of the actor Anders Jacob Lund (1824-1896) and Marie Collett (1855-1891). Lucie had been born on 4 August 1873 and was ten years younger than Thue. Axel and Lucie Thue had seven children over the following years. After his scholarship ended, Thue was appointed to Trondheim Technical College where he worked from 1894 until 1903. Although this was an excellent college, considered the best in the country to train engineers, Thue was not happy there. He continued to write to Holst and, in a letter dated 28 August 1902, he writes (see for example [6]):-

All the same, in this destructive isolation, where nobody has any interest in my material, I have produced more and better work than previously. Thus I have developed a theory according to which inter alia I come to the same results as Hermite and Lindemann concerning e and π.

Also in this letter he tells Holst that he is applying for the vacant applied mathematics professorship at Oslo University. His application was successful and he was appointed as professor of applied mathematics at Oslo University in 1903. He held this post until his death in 1922. In the years immediately following his appointment he was busy writing mechanics lectures and had little time for research. However, the period from 1906 to 1916 was when Thue did his best work, although it was not accomplished in easy circumstances. His health was poor and he suffered from periods of chest pains. His large family (we mentioned above that he had seven children) was difficult to support with a professor's salary and took on extra work teaching at the military school in Oslo in an effort to give them a better life. However, given his health problems, extra work was the last thing he should have been doing. His physical heath problems seem to have had another effect on him, namely that he developed a great fear of being alone. Viggo Brun writes [6]:-

At the Scandinavian Mathematical Congress in Oslo in 1913, he asked me, at the end of an evening party, to accompany him to his brother's house in Oslo, where he was to spend the night, as he couldn't get home to Nordstrand so late. When we reached his brother's, there was nobody there, as his brother was visiting somebody. I sat there until 2 a.m., as I could appreciate Thue's fear of being alone in the house.

Let us now look at some of Thue's mathematical contributions. In 1909 he produced an important paper, published in Crelle's Journal, on algebraic numbers showing that, for example, $y^{3} - 2x^{2} = 1$ cannot be satisfied by infinitely many pairs of integers. His work was extended by Carl Ludwig Siegel in 1920 and again by Klaus Roth in 1958.

Edmund Landau, in 1922, described Thue's work as:-

... the most important discovery in elementary number theory that I know.

Thue's Theorem states that:-

If f (x, y) is a homogeneous polynomial with integer coefficients, irreducible in the rationals and of degree > 2 and c is a non-zero integer then f (x, y) = c has only a finite number of integer solutions.

His contributions to the theory of Diophantine equations are discussed in [3]. In fact Thue wrote 35 papers on number theory, mostly on the theory of Diophantine equations, and these are reproduced in [2].

Another famous contribution made by Thue was his four papers Über unendliche Zeichenreihen Ⓣ (1906), Die Lösung eines Spezialfalles eines generellen logischen Problems Ⓣ (1910), Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen Ⓣ (1912) and Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln Ⓣ (1914). These papers on words and languages present important work on the word problem for finitely presented semigroups. These papers were not seen as important when Thue wrote them but subsequent developments have shown him to be well ahead of his time in considering difficult problems that have since attracted a great deal of interest. Many of his results were rediscovered by mathematicians who were unaware of Thue's contributions. For example, he considered the problem of finding an infinite sequence over a finite alphabet that does not contain two adjacent identical blocks. Today this property has been called 'square freeness'. These papers are discussed in detail in [4], [9] and [10]. In [10] Magnus Steinby and Wolfgang Thomas write:-

Many of Axel Thue's ideas have been influential in theoretical computer science. In particular, Thue systems, semi-Thue systems and his work on the combinatorics of words are well-known. ... his 1910 paper ... contains many notions and ideas about trees, term rewriting and word problems which are surprisingly modern and have later come to play important roles in mathematics, logic, and computer science.

James Power writes in [9]:-

Axel Thue published four papers directly relating to the theory of words and languages: two on patterns in infinite strings in 1906 and 1912 and two on the more general problem of transformations in 1910 and 1914. Both the 1906 and 1912 papers have been translated and discussed extensively by Jean Berstel [4], and are known, among other contributions, for their presentation of the Thue-Morse sequence. Thue's 1910 paper deals with transformations between trees, and is thus a more direct predecessor of his 1914 paper. It been discussed by Steinby and Thomas [10]. Thue's 1914 paper ... is mainly famous for proving an early example of an undecidable problem, cited prominently by Emil Post in 'Recursive unsolvability of a problem of Thue' (1947).

If this work seems a little strange for a professor of applied mathematics then some quotes from Thue will clarify where he stood on the issue of applications. He wrote in one of his articles written while professor in Olso:-

For the development of the logical sciences it will be important to find wide fields for the speculative treatment of difficult problems, without regard to eventual applications.

Another quote from Thue on applied mathematics (see for example [1]) is:-

The further removed from usefulness or practical application, the more important.

He did, however, make contributions to applied mathematics. For example his paper The Principle of Virtual Velocity is [1]:-

... an original statement that has no parallel in the literature.

However, much of his work in mechanics, based on rods which transmitted forces, remained unfinished at his death. One characteristic of Thue's work should be mentioned. He seemed little interested in other people's mathematics and his papers are remarkable for the few references that they contain. For example, his 1906 and 1910 papers mentioned above contain no references at all while the 1912 paper only references his own 1906 paper, and the 1914 paper only references his own 1910 and 1912 papers. This might be considered both a weakness and a strength in allowing him to produce highly original material.

Thue was honoured by being elected to the Norwegian Academy of Science and Letters in 1894 and to the Royal Norwegian Society of Sciences in Trondheim 1895. He served as an editor of Acta Mathematica from 1916 to 1922. As his health deteriorated, the university gave him leave of absence in 1920. By 1921 he was a patient in Ulleval hospital in Oslo although he had his own room and was still able to work at his mathematics. In fact he spent this final period of his life working on Fermat's last Theorem and when he died he left 400 pages of manuscript which he had written during his investigations. He was buried in the Nordstrand cemetery. His wife died on 22 April 1962, almost exactly 40 years after her husband. She is buried in the same tomb in the Nordstrand cemetery.

Let us end this biography with a quote from Carl Stürmer [14]:-

With the death of Professor Thue on 7 March this year, the Academy of Science has lost one of its most illustrious members, a mathematical genius who united the gift of extraordinary originality with a rare perspicacity and sense of logic.

Extracts from other commemorative speeches following Thue's death are given at THIS LINK.