理查德·戴德金Richard Dedekind

理查德·戴德金 的父亲是不伦瑞克 Collegium Carolinum 的一位教授。他的母亲是一位也在 Collegium Carolinum 工作的教授的女儿。戴德金 是四个孩子中最小的，从未结婚。他成年后的大部分时间都与他的一个姐妹一起生活，这位姐妹也一直未婚。

他从七岁起在不伦瑞克上学，在这个阶段数学并不是他的主要兴趣。这所学校 Martino-Catharineum 是一所好学校，戴德金 学习科学，尤其是物理学和化学。然而，物理学在 戴德金 看来变得不那么令人满意，因为他认为其逻辑结构不精确，于是他的注意力转向了数学。

Collegium Carolinum 是一所介于中学和大学之间的教育机构，他于 1848 年 16 岁时进入该校。在那里，他通过学习微分和积分、解析几何以及分析的基础，对基础数学有了很好的理解。他于 1850 年春季进入哥廷根大学，在数学方面已有扎实的基础。

当时哥廷根并不是一个学习数学的令人满意的地方，它还没有成为不久之后变成的那个活跃的研究中心。数学由 M A 莫里茨·亚伯拉罕·斯特恩 和 G Ulrich 主持。卡尔·弗里德里希·高斯 也教授数学课程，但大多在初等水平。物理系由 利斯廷 和 威廉·韦伯 主持。这两个系联合发起了一个讨论班，戴德金 从一开始就参加了。在那里他学习了 数论，这是他学过的最深的内容。他的其他课程涵盖微分和积分等内容，而他对这些已经有了很好的理解。第一门真正让 戴德金 产生热情的课程，相当出人意料，是 威廉·韦伯 讲授的一门实验物理课程。更可能的是，激励 戴德金 的是 威廉·韦伯，而不是这门课程的主题。

1850年秋季学期，戴德金听了卡尔·弗里德里希·高斯讲授的第一门课程。这是一门关于最小二乘法和[1]的课程：-

……五十年后，戴德金仍记得这些讲座是他听过的最优美的讲座，他写道，自己怀着不断增长的兴趣跟随卡尔·弗里德里希·高斯学习，并且无法忘记这段经历。

你可以在THIS LINK看到戴德金对这门课程的记述。

戴德金在卡尔·弗里德里希·高斯的指导下用了四个学期完成博士工作，并提交了一篇关于欧拉积分理论的学位论文。他于1852年在哥廷根获得博士学位，并且是卡尔·弗里德里希·高斯的最后一位学生。然而，他在高等数学方面训练不足，并充分意识到自己数学教育中的缺陷。

此时柏林是讲授最新数学发展的地方，但戴德金在哥廷根无法学到这类内容。此时波恩哈德·黎曼也在哥廷根，他也发现数学教育面向的是打算成为中学教师的学生，而不是那些具有最高才能、将从事研究事业的学生。因此，戴德金在获得博士学位后的两年里学习了最新数学发展，并为他的教授资格论文（Habilitation）而努力。

1854年，波恩哈德·黎曼和戴德金在几周内先后获得了任教资格。戴德金随后取得了大学教师资格，并开始在哥廷根授课，讲授probability和几何学课程。

卡尔·弗里德里希·高斯于1855年去世，约翰·彼得·古斯塔夫·勒热纳·狄利克雷被任命填补哥廷根的空缺讲席。这对戴德金来说是极其重要的事件，他发现与约翰·彼得·古斯塔夫·勒热纳·狄利克雷合作极为有益。他听了约翰·彼得·古斯塔夫·勒热纳·狄利克雷关于数论、potential theory、定积分以及偏微分方程的课程。戴德金和约翰·彼得·古斯塔夫·勒热纳·狄利克雷很快成为密友，这种关系在许多方面造就了戴德金，两人之间的讨论使他的数学兴趣焕发出新的活力。当时在哥廷根求学的保罗·巴赫曼[12]：-

……晚年回忆说，他只是见过戴德金，因为戴德金总是和约翰·彼得·古斯塔夫·勒热纳·狄利克雷一起来去，并且完全被约翰·彼得·古斯塔夫·勒热纳·狄利克雷的光芒所掩盖。

戴德金在1856年7月的一封信中写道[4]：-

对我最有帮助的是几乎每天与约翰·彼得·古斯塔夫·勒热纳·狄利克雷的交往，我第一次开始跟他真正地学习；他总是对我极为和蔼，并且直截了当地告诉我需要填补哪些空白，同时给我指导和手段去做。我已经为他给我的无数事情而感谢他，毫无疑问还会有更多。

戴德金此时肯定仍然像学生一样通过听课继续学习数学，例如波恩哈德·黎曼关于abelian functions和椭圆函数的课程。大约在这段时间，戴德金研究了埃瓦里斯特·伽罗瓦的工作，并且他是第一个讲授伽罗瓦理论的人，当时他在此期间于哥廷根教授这门课程。

在哥廷根期间，戴德金申请了苏黎世理工学院的J L 约瑟夫·路德维希·拉比讲席。约翰·彼得·古斯塔夫·勒热纳·狄利克雷支持他的申请，写道戴德金是“一位杰出的教育家”。1858年春天，负责任命的瑞士委员来到哥廷根，戴德金很快被选中担任该职位。戴德金被任命到苏黎世理工学院，并于1858年秋季开始在那里授课。

事实上，正是在他思考如何教授微分和积分——这是他第一次教授这个主题——的时候，戴德金分割的想法出现在他脑海中。他叙述说这个想法是在1858年11月24日产生的。他的想法是每个实数$r$将有理的数分成两个子集，即大于$r$的数和小于$r$的数。戴德金的绝妙想法是用有理数的这种分割来表示实数。

戴德金和波恩哈德·黎曼于1859年9月一同前往柏林，当时正值波恩哈德·黎曼当选柏林科学院之际。在柏林，戴德金会见了卡尔·魏尔斯特拉斯、恩斯特·爱德华·库默尔、卡尔·威廉·博尔夏特和利奥波德·克罗内克。

布伦斯威克的卡罗琳学院在19世纪60年代已升格为布伦斯威克理工学院，戴德金于1862年被任命到该理工学院。通过这次任命，他回到了家乡，甚至回到了他旧日的教育机构，他的父亲曾在那里担任高级管理人员多年。戴德金在那里度过了余生，于1894年4月1日退休。他一生都在布伦斯威克担任教授[1]：-

……与他的兄弟和姐妹密切相伴，无视一切改变或获得更高活动领域的机会。他所生活的那个小而熟悉的世界完全满足了他的要求：在其中，他的亲属完全取代了他自己的妻子和儿女，并且他在那里找到了足够的闲暇和自由来进行基础数学研究的科学工作。他并不感到有压力要在外部世界产生更显著的影响：这种对自身的确认是不必要的。

退休后，戴德金继续偶尔授课，并在漫长的退休生活中保持健康。戴德金经历的唯一一段健康不佳的时期是在他被任命到布伦斯威克理工学院10年后，当时他患了一场重病，就在他父亲去世后不久。然而他完全康复了，并且正如我们提到的，一直保持健康。

戴德金 对数学做出了许多极其重要的贡献，他的工作将数学的风格改变为我们今天所熟悉的样子。一项引人注目的工作是他用 戴德金 分割重新定义了 无理数 数，正如我们上面提到的，这最早在 1858 年就出现在他的脑海中。他在 1872 年的 Stetigkeit und Irrationale Zahlen Ⓣ（连续性与无理数）中发表了这一成果。在其中他写道：-

现在，在每一种情况下，当存在一个不由任何有理数产生的分割 $(A_{1}, A_{2})$ 时，我们就创造一个新的无理数 a，我们认为它由这个分割完全定义；我们将说这个数 a 对应于这个分割，或者说它产生了这个分割。

除了他对数之本性的分析之外，他在数学归纳法方面的工作，包括有限集与无限集的定义，以及他在数论方面的工作，特别是在 代数数域 中的工作，都具有重大重要性。

戴德金 喜欢在瑞士、奥地利蒂罗尔或德国南部的黑森林度假。在 1872 年的一次这样的假期中，他在 Gersau 村遇到了 格奥尔格·康托尔，他们讨论了无限的概念。[31] 1874 年，在美丽的因特拉肯城逗留期间，两人讨论了集合论。戴德金 对 格奥尔格·康托尔 的理论做出了重要贡献，尽管当 格奥尔格·康托尔 发表它们时这些贡献未被承认。[31] 戴德金 同情 格奥尔格·康托尔 的集合论，这可以从 Was sind und was sollen die Zahlen Ⓣ（数是什么以及它们将是什么）（1888）中关于确定一个给定元素是否属于一个给定集合的这段话看出：-

这种确定以何种方式产生，或者我们是否知道一种判定它的方法，在接下来的内容中无关紧要。将要发展的一般规律根本不依赖于这一点。

在这段话中，戴德金 是在反驳 利奥波德·克罗内克 对无限的反对意见，因此，他同意 格奥尔格·康托尔 的观点。

戴德金对数学的其他显著贡献是他编辑了约翰·彼得·古斯塔夫·勒热纳·狄利克雷、卡尔·弗里德里希·高斯和波恩哈德·黎曼的全集。事实上，戴德金对约翰·彼得·古斯塔夫·勒热纳·狄利克雷工作的研究导致了他自己对代数数域的研究，以及他对理想的引入。戴德金编辑了约翰·彼得·古斯塔夫·勒热纳·狄利克雷关于数论的讲义，并于1863年以Vorlesungen über Zahlentheorie Ⓣ（数论讲义）为名出版。在[12]中指出：-

尽管这本书无疑是基于约翰·彼得·古斯塔夫·勒热纳·狄利克雷的讲义，并且尽管戴德金一生都称这本书为约翰·彼得·古斯塔夫·勒热纳·狄利克雷的，但书本身完全是由戴德金撰写的，大部分是在约翰·彼得·古斯塔夫·勒热纳·狄利克雷去世后。

正是在1879年和1894年出版的Vorlesungen über Zahlentheorie Ⓣ（数论讲义）第三版和第四版中，戴德金撰写了补篇，在其中他引入了对环论至关重要的理想概念。戴德金在代数数域的整数环中阐述了他的理论。一般术语“环”并未出现，它是由大卫·希尔伯特后来引入的。

戴德金在1882年与海因里希·马丁·韦伯合作发表的一篇论文中，将他的理想理论应用于波恩哈德·黎曼曲面理论。这产生了强有力的结果，例如波恩哈德·黎曼-Roch定理的纯代数证明。

戴德金的工作很快被接受，部分是因为他阐述思想的清晰性，部分是因为海因里希·马丁·韦伯在柯尼斯堡大学就这些主题向大卫·希尔伯特授课。戴德金的理想概念被大卫·希尔伯特采纳并扩展，后来又由埃米·诺特进一步发展。这导致整数唯一分解为primes的幂被推广到其他环中的理想。

1879年，戴德金出版了Über die Theorie der ganzen algebraischen Zahlen Ⓣ（论代数整数理论），这本书再次对数学基础产生了重大影响。在戴德金 [1]的书中：-

……提出了数和完全归纳法的逻辑理论，提出了他对算术本质的主要构想，并在空间连续性问题中处理了实数完全系统在几何中的作用。除其他内容外，他通过使用映射的概念，为集合的无限性或有限性提供了一个独立于数概念的定義，并处理了对序数理论如此重要的递归定义。

戴德金的卓越不仅在于他所研究的定理和概念，而且由于他能够如此清晰地表述和表达自己的思想，他引入了一种新的数学风格，从此对数学家产生了重大影响。正如12在[12]中所写：-

戴德金的遗产……不仅包括重要的定理、例子和概念，还包括一种整体的数学风格，这种风格激励了每一代后来者。

由于戴德金的杰出工作，他获得了许多荣誉，尽管他对自己能力和成就始终极为谦逊。他于1862年当选哥廷根科学院会士，1880年当选柏林科学院会士，还当选为罗马科学院、利奥波德-卡罗莱纳自然 curiosorum 科学院院士，并于1900年当选巴黎Académie des Sciences会士。克里斯蒂安尼亚（奥斯陆）、苏黎世和不伦瑞克大学授予他荣誉博士学位。

Richard Dedekind's father was a professor at the Collegium Carolinum in Brunswick. His mother was the daughter of a professor who also worked at the Collegium Carolinum. Richard was the youngest of four children and never married. He was to live with one of his sisters, who also remained unmarried, for most of his adult life.

He attended school in Brunswick from the age of seven and at this stage mathematics was not his main interest. The school, Martino-Catharineum, was a good one and Dedekind studied science, in particular physics and chemistry. However, physics became less than satisfactory to Dedekind with what he considered an imprecise logical structure and his attention turned towards mathematics.

The Collegium Carolinum was an educational institution between a high school and a university and he entered it in 1848 at the age of 16. There he was to receive a good understanding of basic mathematics studying differential and integral calculus, analytic geometry and the foundations of analysis. He entered the University of Göttingen in the spring of 1850 with a solid grounding in mathematics.

Göttingen was a rather disappointing place to study mathematics at this time, and it had not yet become the vigorous research centre that it turned into soon afterwards. Mathematics was directed by M A Stern and G Ulrich. Gauss also taught courses in mathematics, but mostly at an elementary level. The physics department was directed by Listing and Wilhelm Weber. The two departments combined to initiate a seminar which Dedekind joined from its beginning. There he learnt number theory which was the most advanced material he studied. His other courses covered material such as the differential and integral calculus, of which he already had a good understanding. The first course to really make Dedekind enthusiastic was, rather surprisingly, a course on experimental physics taught by Weber. More likely it was Weber who inspired Dedekind rather than the topic of the course.

In the autumn term of 1850, Dedekind attended his first course given by Gauss. It was a course on least squares and [1]:-

... fifty years later Dedekind remembered the lectures as the most beautiful he had ever heard, writing that he had followed Gauss with constantly increasing interest and that he could not forget the experience.

You can see Dedekind's account of this course at THIS LINK.

Dedekind did his doctoral work in four semesters under Gauss's supervision and submitted a thesis on the theory of Eulerian integrals. He received his doctorate from Göttingen in 1852 and he was to be the last pupil of Gauss. However he was not well trained in advanced mathematics and fully realised the deficiencies in his mathematical education.

At this time Berlin was the place where courses were given on the latest mathematical developments but Dedekind had not been able to learn such material at Göttingen. By this time Riemann was also at Göttingen and he too found that the mathematical education was aimed at students who were intending to become secondary school teachers, not those with the very top abilities who would go on to research careers. Dedekind therefore spent the two years following the award of his doctorate learning the latest mathematical developments and working for his habilitation.

In 1854 both Riemann and Dedekind were awarded their habilitation degrees within a few weeks of each other. Dedekind was then qualified as a university teacher and he began teaching at Göttingen giving courses on probability and geometry.

Gauss died in 1855 and Dirichlet was appointed to fill the vacant chair at Göttingen. This was an extremely important event for Dedekind who found working with Dirichlet extremely profitable. He attended courses by Dirichlet on the theory of numbers, on potential theory, on definite integrals, and on partial differential equations. Dedekind and Dirichlet soon became close friends and the relationship was in many ways the making of Dedekind, whose mathematical interests took a new lease of life with the discussions between the two. Bachmann, who was a student in Göttingen at this time [12]:-

... recalled in later years that he only knew Dedekind by sight because Dedekind always arrived and left with Dirichlet and was completely eclipsed by him.

Dedekind wrote in a letter in July 1856 [4]:-

What is most useful to me is the almost daily association with Dirichlet, with whom I am for the first time beginning to learn properly; he is always completely amiable towards me, and he tells me without beating about the bush what gaps I need to fill and at the same time he gives me the instructions and the means to do it. I thank him already for infinitely many things, and no doubt there will be many more.

Dedekind certainly still continued to learn mathematics at this time as a student would by attending courses, such as those by Riemann on abelian functions and elliptic functions. Around this time Dedekind studied the work of Galois and he was the first to lecture on Galois theory when he taught a course on the topic at Göttingen during this period.

While at Göttingen, Dedekind applied for J L Raabe's chair at the Polytechnikum in Zürich. Dirichlet supported his application writing that Dedekind was 'an exceptional pedagogue'. In the spring of 1858 the Swiss councillor who made appointments came to Göttingen and Dedekind was quickly chosen for the post. Dedekind was appointed to the Polytechnikum in Zürich and began teaching there in the autumn of 1858.

In fact it was while he was thinking how to teach differential and integral calculus, the first time that he had taught the topic, that the idea of a Dedekind cut came to him. He recounts that the idea came to him on 24 November 1858. His idea was that every real number $r$ divides the rational numbers into two subsets, namely those greater than $r$ and those less than $r$. Dedekind's brilliant idea was to represent the real numbers by such divisions of the rationals.

Dedekind and Riemann travelled together to Berlin in September 1859 on the occasion of Riemann's election to the Berlin Academy of Sciences. In Berlin, Dedekind met Weierstrass, Kummer, Borchardt and Kronecker.

The Collegium Carolinum in Brunswick had been upgraded to the Brunswick Polytechnikum by the 1860s, and Dedekind was appointed to the Polytechnikum in 1862. With this appointment he returned to his home town and even to his old educational establishment where his father had been one of the senior administrators for many years. Dedekind remained there for the rest of his life, retiring on 1 April 1894. He lived his life as a professor in Brunswick [1]:-

... in close association with his brother and sister, ignoring all possibilities of change or attainment of a higher sphere of activity. The small, familiar world in which he lived completely satisfied his demands: in it his relatives completely replaced a wife and children of his own and there he found sufficient leisure and freedom for scientific work in basic mathematical research. He did not feel pressed to have a more marked effect in the outside world: such confirmation of himself was unnecessary.

After he retired, Dedekind continued to teach the occasional course and remained in good health in his long retirement. The only spell of bad health which Dedekind had experienced was 10 years after he was appointed to the Brunswick Polytechnikum when he had a serious illness, shortly after the death of his father. However he completely recovered and, as we mentioned, remained in good health.

Dedekind made a number of highly significant contributions to mathematics and his work would change the style of mathematics into what is familiar to us today. One remarkable piece of work was his redefinition of irrational numbers in terms of Dedekind cuts which, as we mentioned above, first came to him as early as 1858. He published this in Stetigkeit und Irrationale Zahlen Ⓣ in 1872. In it he wrote:-

Now, in each case when there is a cut $(A_{1}, A_{2})$ which is not produced by any rational number, then we create a new, irrational number a, which we regard as completely defined by this cut; we will say that this number a corresponds to this cut, or that it produces this cut.

As well as his analysis of the nature of number, his work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is of major importance.

Dedekind loved to take his holidays in Switzerland, the Austrian Tyrol or the Black Forest in southern Germany. On one such holiday in 1872 he met Cantor in the village of Gersau and they discusssed ideas of infinity. [31] In 1874 while staying in the beautiful city of Interlaken, the two discussed set theory. Dedekind made significant contributions to Cantor's theory though they were unacknowledged when Cantor published them. [31] Dedekind was sympathetic to Cantor's set theory as is illustrated by this quote from Was sind und was sollen die Zahlen Ⓣ (1888) regarding determining whether a given element belongs to a given set :-

In what way the determination comes about, or whether we know a way to decide it, is a matter of no consequence in what follows. The general laws that are to be developed do not depend on this at all.

In this quote Dedekind is arguing against Kronecker's objections to the infinite and, therefore, is agreeing with Cantor's views.

Among Dedekind's other notable contributions to mathematics were his editions of the collected works of Peter Dirichlet, Carl Gauss, and Georg Riemann. Dedekind's study of Dirichlet's work did, in fact, lead to his own study of algebraic number fields, as well as to his introduction of ideals. Dedekind edited Dirichlet's lectures on number theory and published these as Vorlesungen über Zahlentheorie Ⓣ in 1863. It is noted in [12] that:-

Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death.

It was in the third and fourth editions of Vorlesungen über Zahlentheorie Ⓣ, published in 1879 and 1894, that Dedekind wrote supplements in which he introduced the notion of an ideal which is fundamental to ring theory. Dedekind formulated his theory in the ring of integers of an algebraic number field. The general term 'ring' does not appear, it was introduced later by Hilbert.

Dedekind, in a joint paper with Heinrich Weber published in 1882, applies his theory of ideals to the theory of Riemann surfaces. This gave powerful results such as a purely algebraic proof of the Riemann-Roch theorem.

Dedekind's work was quickly accepted, partly because of the clarity with which he presented his ideas and partly since Heinrich Weber lectured to Hilbert on these topics at the University of Königsberg. Dedekind's notion of ideal was taken up and extended by Hilbert and then later by Emmy Noether. This led to the unique factorisation of integers into powers of primes to be generalised to ideals in other rings.

In 1879 Dedekind published Über die Theorie der ganzen algebraischen Zahlen Ⓣ which was again to have a large influence on the foundations of mathematics. In the book Dedekind [1]:-

... presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space. Among other things, he provides a definition independent of the concept of number for the infiniteness or finiteness of a set by using the concept of mapping and treating the recursive definition, which is so important to the theory of ordinal numbers.

Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics that been a major influence on mathematicians ever since. As Edwards writes in [12]:-

Dedekind's legacy ... consisted not only of important theorems, examples, and concepts, but a whole style of mathematics that has been an inspiration to each succeeding generation.

Many honours were given to Dedekind for his outstanding work, although he always remained extraordinarily modest regarding his own abilities and achievements. He was elected to the Göttingen Academy (1862), the Berlin Academy (1880), the Academy of Rome, the Leopoldino-Carolina Naturae Curiosorum Academia, and the Académie des Sciences in Paris (1900). Honorary doctorates were awarded to him by the universities of Kristiania (Oslo), Zürich and Brunswick.