利奥波德·克罗内克Leopold Kronecker

利奥波德·克罗内克的父母很富有，他的父亲伊西多尔·克罗内克是一位成功的商人，而他的母亲约翰娜·普劳斯尼策也来自一个富裕家庭。这两个家庭都是犹太人，克罗内克一直信奉这一宗教，直到去世前一年皈依基督教。克罗内克的父母聘请私人教师教他，直到他进入利格尼茨的文理中学，这种辅导为他的教育奠定了非常坚实的基础。

克罗内克在利格尼茨中学由恩斯特·爱德华·库默尔教授数学，正是由于恩斯特·爱德华·库默尔，克罗内克才对数学产生了兴趣。恩斯特·爱德华·库默尔立即认识到克罗内克的数学天赋，并把他带到了远超学校预期的水平，鼓励他进行研究。尽管克罗内克接受的是犹太教育，但他在中学接受了路德宗教教育，这无疑表明他的父母在宗教问题上思想开明。

克罗内克于1841年成为柏林大学的学生，在那里他师从约翰·彼得·古斯塔夫·勒热纳·狄利克雷和雅各布·施泰纳。然而，他并不局限于学习数学，还学习了天文学、气象学和化学等其他科目。他尤其对哲学感兴趣，研究了勒内·笛卡儿、哥特弗里德·威廉·莱布尼茨、康德、斯宾诺莎和黑格尔的哲学著作。1843年夏天，他在波恩大学度过，他去那里是因为对天文学而非数学的兴趣，随后他于1843-44年冬季学期前往布雷斯劳大学。他去布雷斯劳的原因当然是因为对数学的兴趣，因为他想再次跟随他的老学校老师恩斯特·爱德华·库默尔学习，后者于1842年被任命为布雷斯劳的讲席。

克罗内克在布雷斯劳待了一年，然后回到柏林度过1844-45年冬季学期。回到柏林后，他在约翰·彼得·古斯塔夫·勒热纳·狄利克雷的指导下撰写关于代数数论的博士论文。论文On complex units于1845年7月30日提交，他于8月14日参加了必要的口试。约翰·彼得·古斯塔夫·勒热纳·狄利克雷在评论该论文时说，克罗内克在其中展示了：-

……非凡的洞察力、极大的勤勉，以及对高等数学当前状况的精确了解。

许多博士生听到克罗内克在口试中被问及广泛的主题时可能会感到惊讶，这些主题包括应用于天文观测的theory of probability、定积分理论、级数和微分方程，以及希腊语和哲学史。

卡尔·古斯塔夫·雅各布·雅可比有健康问题，这使他离开了柯尼斯堡，他在那里拥有一个讲席，并回到了柏林。费迪南·艾森斯坦的健康状况也很差，大约在这个时候在柏林讲学，克罗内克结识了这两人。克罗内克后来数学兴趣的方向很大程度上与此时卡尔·古斯塔夫·雅各布·雅可比和费迪南·艾森斯坦的影响有关。然而，就在他看起来即将开始学术生涯时，克罗内克离开柏林去处理家庭事务。他帮助管理他母亲兄弟的银行业务，并于1848年娶了这位叔叔的女儿Fanny Prausnitzer。他还管理着一份家族地产，但仍抽出时间继续研究数学，尽管他这样做完全是为了自己的乐趣。

克罗内克不需要从事有薪工作，因为他此时已经是一个富有的人。然而，他对数学的喜爱意味着，当1855年情况发生变化，他不再需要住在利格尼茨郊外的庄园时，他回到了柏林。他不想要大学职位，而是想参与大学的数学生活，并与其他数学家互动进行研究。

1855年，恩斯特·爱德华·库默尔来到柏林，填补约翰·彼得·古斯塔夫·勒热纳·狄利克雷前往哥廷根时出现的空缺。卡尔·威廉·博尔夏特自1848年以来一直在柏林讲学，并在1855年底奥古斯都·利奥波德·克雷勒去世后接任Crelle's Journal的编辑。1856年，卡尔·魏尔斯特拉斯来到柏林，因此在克罗内克回到柏林的一年内，由恩斯特·爱德华·库默尔、卡尔·威廉·博尔夏特、卡尔·魏尔斯特拉斯和克罗内克组成的杰出团队就在柏林就位了。

当然，由于克罗内克没有大学职位，他此时没有讲学，但在研究方面非常活跃，迅速连续发表大量著作。这些著作涉及数论、椭圆函数和代数，但更重要的是，他探索了这些主题之间的相互联系。恩斯特·爱德华·库默尔于1860年提名克罗内克竞选柏林科学院，该提名得到卡尔·威廉·博尔夏特和卡尔·魏尔斯特拉斯的附议。1861年1月23日，克罗内克当选为科学院院士，这带来了一个令人惊讶的好处。

Berlin Academy的成员有权在柏林大学授课。尽管克罗内克并未受雇于该大学或任何其他组织，恩斯特·爱德华·库默尔建议克罗内克行使他在大学授课的权利，他于1862年10月开始这样做。他授课的主题与他的研究密切相关：数论、方程理论、行列式理论以及积分理论。在他的讲座[1]中：-

他试图简化和完善现有理论，并从新的视角呈现它们。

对于最优秀的学生来说，他的讲座要求高但富有启发性。然而，对于普通学生来说，他并不是一位受欢迎的教师[1]：-

克罗内克并未吸引大量学生。只有少数听众能够跟上他思想的飞跃，也只有少数人坚持到学期结束。

柏林对克罗内克很有吸引力，以至于1868年当他被授予哥廷根大学数学讲席时，他拒绝了。他确实接受了荣誉，例如在那一年当选为巴黎科学院会士，并且多年来他与柏林及其他地方的同事保持着良好的关系。为了理解为什么在1870年代关系开始恶化，我们需要更仔细地考察克罗内克的数学贡献。

我们已经指出，克罗内克的主要贡献在于方程理论和高阶代数，他在椭圆函数、代数方程理论以及代数数理论方面做出了重大贡献。然而，他所研究的主题受到限制，因为他相信所有数学都应归结为仅涉及整数和有限步骤的论证。克罗内克因其言论而闻名：-

上帝创造了整数，其余都是人的工作。

克罗内克认为数学只应处理有限数和有限次运算。他是第一个质疑非构造性存在性证明重要性的人。似乎从1870年代初开始，克罗内克就反对使用无理数数、上极限和下极限以及伯纳德·波尔查诺-卡尔·魏尔斯特拉斯定理，因为它们具有非构造性。他的数学哲学的另一个后果是，对克罗内克来说，超越的数不可能存在。

1870年，爱德华·海涅在Crelle's Journal上发表了一篇论文On trigonometric series，但克罗内克曾试图说服爱德华·海涅撤回该论文。1877年，克罗内克再次试图阻止格奥尔格·康托尔的工作在Crelle's Journal上发表，这并非出于对格奥尔格·康托尔的任何个人恩怨（一些格奥尔格·康托尔的传记作者曾提出这种看法），而是因为克罗内克认为格奥尔格·康托尔的论文毫无意义，因为它证明了关于克罗内克认为不存在的数学对象的结果。克罗内克是Crelle's Journal的编辑人员，因此他对该期刊发表的内容有着特别强的影响力。1880年卡尔·威廉·博尔夏特去世后，克罗内克作为编辑接管了Crelle's Journal的控制权，他对哪些论文能够发表的影响力也随之增强。

柏林的数学讨论班由恩斯特·爱德华·库默尔和卡尔·魏尔斯特拉斯于1861年联合创立，当恩斯特·爱德华·库默尔于1883年退休时，克罗内克成为该讨论班的共同主任。这增加了克罗内克在柏林的影响力。克罗内克的国际声誉也传播开来，他于1884年1月31日被选为伦敦皇家学会的外籍会士，这是一种荣誉。他在德国数学界也是一个非常有影响力的人物[1]：-

他通过多次出国旅行以及在柏林家中对外国科学家的款待，与他们建立了其他联系。因此，在填补德国和其他地方的数学教授职位时，他的建议经常被征求；他的推荐可能与他昔日朋友卡尔·魏尔斯特拉斯的推荐同样重要。

克罗内克对数学的看法在19世纪70年代和80年代为他的同事们所熟知，但直到1886年他才将这些看法公之于众。在那一年，他反对理查德·戴德金、格奥尔格·康托尔和爱德华·海涅所使用的无理数理论，给出了他反对的论据：-

……引入各种概念，近来（但最早由爱德华·海涅）经常试图借助这些概念来构想和建立一般的“无理数”。甚至无穷级数的概念，例如按变量的确定幂次递增的级数，在我看来也只有在以下保留条件下才被允许：在每一特殊情形下，基于构造项（或系数）的算术法则，……必须证明某些假设成立，这些假设适用于该级数如同有限表达式一样，因而实际上使得超出有限级数概念的扩展成为不必要。

费迪南德·冯·林德曼在1882年证明了π是超越数，而在1886年的一次演讲中，克罗内克称赞费迪南德·冯·林德曼给出了一个漂亮的证明，但他声称，这个证明什么也没有证明，因为超越数并不存在。因此克罗内克在他的论据和信念上是一致的，但许多数学家为自己辛苦得来的成果感到自豪，觉得克罗内克试图改变数学的进程，并将他们的研究方向从未来发展中抹去。克罗内克在1887年的Über den ZahlbergriffⓉ（《论数的概念》）中解释了他的纲领，该纲领基于只研究可以从整数出发通过有限次运算构造出来的数学对象。

克罗内克性格的另一个特点是他倾向于与那些在数学上与他意见不合的人个人交恶。当然，鉴于他认为只有有限可构造的数学对象才存在，他完全反对格奥尔格·康托尔在集合论方面发展中的思想。不仅理查德·戴德金、爱德华·海涅和格奥尔格·康托尔的数学在这种思维方式下是不可接受的，卡尔·魏尔斯特拉斯也逐渐感到克罗内克试图让下一代数学家相信卡尔·魏尔斯特拉斯在分析方面的工作毫无价值。

在1883年恩斯特·爱德华·库默尔退休之前，克罗内克在柏林没有正式职位，那一年他被任命为讲席教授。但到了1888年，卡尔·魏尔斯特拉斯觉得他无法再在柏林与克罗内克共事，决定去瑞士，但随后意识到克罗内克将处于有利地位来影响其继任者的选择，于是决定留在柏林。

克罗内克身材非常矮小，对自己的身高极其敏感。克罗内克如何反应的一个例子发生在1885年，当时赫尔曼·阿曼杜斯·施瓦茨给他寄去一封问候信，其中有一句话：-

不尊重较小者的人，不配得到较大者。

这里赫尔曼·阿曼杜斯·施瓦茨是在拿矮个子克罗内克和高个子卡尔·魏尔斯特拉斯开玩笑。然而克罗内克没有看出这句话的滑稽之处，此后再也没有与赫尔曼·阿曼杜斯·施瓦茨（他是卡尔·魏尔斯特拉斯的学生，恩斯特·爱德华·库默尔的女婿）有过任何往来。不过其他人表现得更有分寸，例如从1871年起担任柏林大学教授的赫尔曼·冯·亥姆霍兹，就设法与克罗内克保持了良好关系。

Deutsche Mathematiker-Vereinigung于1890年成立，协会的第一次会议于1891年9月在哈雷组织举行。尽管格奥尔格·康托尔和克罗内克之间存在激烈的对立，格奥尔格·康托尔仍邀请克罗内克在第一次会议上致辞，以表示对德国数学界一位资深且最杰出人物的尊重。然而，克罗内克从未在该会议上致辞，因为他的妻子在夏天的一次登山事故中受了重伤，于1891年8月23日去世。克罗内克仅比妻子多活了几个月，于1891年12月去世。

我们不应认为克罗内克的数学观点完全古怪。尽管他那个时代的大多数数学家确实不会同意这些观点，而且今天的大多数数学家也不会同意，但这些观点并未被搁置。克罗内克的思想由儒勒·昂利·庞加莱和勒伊岑·布劳威尔进一步发展，他们特别强调直觉。直觉主义强调数学优先于逻辑，数学对象由数学家在头脑中构造和操作，不可能仅仅通过建立若干公理来定义数学对象的性质。

Leopold Kronecker's parents were well off, his father, Isidor Kronecker, being a successful business man while his mother was Johanna Prausnitzer who also came from a wealthy family. The families were Jewish, the religion that Kronecker kept until a year before his death when he became a convert to Christianity. Kronecker's parents employed private tutors to teach him up to the stage when he entered the Gymnasium at Liegnitz, and this tutoring gave him a very sound foundation to his education.

Kronecker was taught mathematics at Liegnitz Gymnasium by Kummer, and it was due to Kummer that Kronecker became interested in mathematics. Kummer immediately recognised Kronecker's talent for mathematics and he took him well beyond what would be expected at school, encouraging him to undertake research. Despite his Jewish upbringing, Kronecker was given Lutheran religious instruction at the Gymnasium which certainly shows that his parents were openminded on religious matters.

Kronecker became a student at Berlin University in 1841 and there he studied under Dirichlet and Steiner. He did not restrict himself to studying mathematics, however, for he studied other topics such as astronomy, meteorology and chemistry. He was especially interested in philosophy studying the philosophical works of Descartes, Leibniz, Kant, Spinoza and Hegel. After spending the summer of 1843 at the University of Bonn, which he went to because of his interest in astronomy rather than mathematics, he then went to the University of Breslau for the winter semester of 1843-44. The reason that he went to Breslau was certainly because of his interest in mathematics because he wanted to study again with his old school teacher Kummer who had been appointed to a chair at Breslau in 1842.

Kronecker spent a year at Breslau before returning to Berlin for the winter semester of 1844-45. Back in Berlin he worked on his doctoral thesis on algebraic number theory under Dirichlet's supervision. The thesis, On complex units was submitted on 30 July 1845 and he took the necessary oral examination on 14 August. Dirichlet commented on the thesis saying that in it Kronecker showed:-

... unusual penetration, great assiduity, and an exact knowledge of the present state of higher mathematics.

It may come as a surprise to many Ph.D. students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.

Jacobi had health problems which caused him to leave Königsberg, where he held a chair, and return to Berlin. Eisenstein, whose health was also poor, lectured in Berlin around this time and Kronecker came to know both men well. The direction that Kronecker's mathematical interests went later had much to do with the influence of Jacobi and Eisenstein around this time. However, just as it looked as if he would embark on an academic career, Kronecker left Berlin to deal with family affairs. He helped to manage the banking business of his mother's brother and, in 1848, he married the daughter of this uncle, Fanny Prausnitzer. He also managed a family estate but still found the time to continue working on mathematics, although he did this entirely for his own enjoyment.

Certainly Kronecker did not need to take on paid employment since he was by now a wealthy man. His enjoyment of mathematics meant, however, that when circumstances changed in 1855 and he no longer needed to live on the estate outside Liegnitz, he returned to Berlin. He did not wish a university post, rather he wanted to take part in the mathematical life of the university and undertake research interacting with the other mathematicians.

In 1855 Kummer came to Berlin to fill the vacancy which occurred when Dirichlet left for Göttingen. Borchardt had lectured at Berlin since 1848 and, in late 1855, he took over the editorship of Crelle's Journal on Crelle's death. In 1856 Weierstrass came to Berlin, so within a year of Kronecker returning to Berlin, the remarkable team of Kummer, Borchardt, Weierstrass and Kronecker was in place in Berlin.

Of course since Kronecker did not hold a university appointment, he did not lecture at this time but was remarkably active in research publishing a large number of works in quick succession. These were on number theory, elliptic functions and algebra, but, more importantly, he explored the interconnections between these topics. Kummer proposed Kronecker for election to the Berlin Academy in 1860, and the proposal was seconded by Borchardt and Weierstrass. On 23 January 1861 Kronecker was elected to the Academy and this had a surprising benefit.

Members of the Berlin Academy had a right to lecture at Berlin University. Although Kronecker was not employed by the University, or any other organisation for that matter, Kummer suggested that Kronecker exercise his right to lecture at the University and this he did beginning in October 1862. The topics on which he lectured were very much related to his research: number theory, the theory of equations, the theory of determinants, and the theory of integrals. In his lectures [1]:-

He attempted to simplify and refine existing theories and to present them from new perspectives.

For the best students his lectures were demanding but stimulating. However, he was not a popular teacher with the average students [1]:-

Kronecker did not attract great numbers of students. Only a few of his auditors were able to follow the flights of his thought, and only a few persevered until the end of the semester.

Berlin was attractive to Kronecker, so much so that when he was offered the chair of mathematics in Göttingen in 1868, he declined. He did accept honours such as election to the Paris Academy in that year and for many years he enjoyed good relations with his colleagues in Berlin and elsewhere. In order to understand why relations began to deteriorate in the 1870s we need to examine Kronecker's mathematical contributions more closely.

We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers. However the topics he studied were restricted by the fact that he believed in the reduction of all mathematics to arguments involving only the integers and a finite number of steps. Kronecker is well known for his remark:-

God created the integers, all else is the work of man.

Kronecker believed that mathematics should deal only with finite numbers and with a finite number of operations. He was the first to doubt the significance of non-constructive existence proofs. It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature. Another consequence of his philosophy of mathematics was that to Kronecker transcendental numbers could not exist.

In 1870 Heine published a paper On trigonometric series in Crelle's Journal, but Kronecker had tried to persuade Heine to withdraw the paper. Again in 1877 Kronecker tried to prevent publication of Cantor's work in Crelle's Journal, not because of any personal feelings against Cantor (which has been suggested by some biographers of Cantor) but rather because Kronecker believed that Cantor's paper was meaningless, since it proved results about mathematical objects which Kronecker believed did not exist. Kronecker was on the editorial staff of Crelle's Journal which is why he had a particularly strong influence on what was published in that journal. After Borchardt died in 1880, Kronecker took over control of Crelle's Journal as the editor and his influence on which papers would be published increased.

The mathematical seminar in Berlin had been jointly founded in 1861 by Kummer and Weierstrass and, when Kummer retired in 1883, Kronecker became a codirector of the seminar. This increased Kronecker's influence in Berlin. Kronecker's international fame also spread, and he was honoured by being elected a foreign member of the Royal Society of London on 31 January 1884. He was also a very influential figure within German mathematics [1]:-

He established other contacts with foreign scientists in numerous travels abroad and in extending to them the hospitality of his Berlin home. For this reason his advice was often solicited in regard to filling mathematical professorships both in Germany and elsewhere; his recommendations were probably as significant as those of his erstwhile friend Weierstrass.

Although Kronecker's view of mathematics was well known to his colleagues throughout the 1870s and 1880s, it was not until 1886 that he made these views public. In that year he argued against the theory of irrational numbers used by Dedekind, Cantor and Heine giving the arguments by which he opposed:-

... the introduction of various concepts by the help of which it has frequently been attempted in recent times (but first by Heine) to conceive and establish the "irrationals" in general. Even the concept of an infinite series, for example one which increases according to definite powers of variables, is in my opinion only permissible with the reservation that in every special case, on the basis of the arithmetic laws of constructing terms (or coefficients), ... certain assumptions must be shown to hold which are applicable to the series like finite expressions, and which thus make the extension beyond the concept of a finite series really unnecessary.

Lindemann had proved that π is transcendental in 1882, and in a lecture given in 1886 Kronecker complimented Lindemann on a beautiful proof but, he claimed, one that proved nothing since transcendental numbers did not exist. So Kronecker was consistent in his arguments and his beliefs, but many mathematicians, proud of their hard earned results, felt that Kronecker was attempting to change the course of mathematics and write their line of research out of future developments. Kronecker explained his programme based on studying only mathematical objects which could be constructed with a finite number of operation from the integers in Über den Zahlbergriff Ⓣ in 1887.

Another feature of Kronecker's personality was that he tended to fall out personally with those whom he disagreed with mathematically. Of course, given his belief that only finitely constructible mathematical objects existed, he was completely opposed to Cantor's developing ideas in set theory. Not only Dedekind, Heine and Cantor's mathematics was unacceptable to this way of thinking, and Weierstrass also came to feel that Kronecker was trying to convince the next generation of mathematicians that Weierstrass's work on analysis was of no value.

Kronecker had no official position at Berlin until Kummer retired in 1883 when he was appointed to the chair. But by 1888 Weierstrass felt that he could no longer work with Kronecker in Berlin and decided to go to Switzerland, but then, realising that Kronecker would be in a strong position to influence the choice of his successor, he decided to remain in Berlin.

Kronecker was of very small stature and extremely self-conscious about his height. An example of how Kronecker reacted occurred in 1885 when Schwarz sent him a greeting which included the sentence:-

He who does not honour the Smaller, is not worthy of the Greater.

Here Schwarz was joking about the small man Kronecker and the large man Weierstrass. Kronecker did not see the funny side of the comment, however, and never had any further dealings with Schwarz (who was Weierstrass's student and Kummer's son-in-law). Others however displayed more tact and, for example, Helmholtz who was a professor in Berlin from 1871, managed to stay on good terms with Kronecker.

The Deutsche Mathematiker-Vereinigung was set up in 1890 and the first meeting of the Association was organised in Halle in September 1891. Despite the bitter antagonism between Cantor and Kronecker, Cantor invited Kronecker to address this first meeting as a sign of respect for one of the senior and most eminent figures in German mathematics. However, Kronecker never addressed the meeting, since his wife was seriously injured in a climbing accident in the summer and died on 23 August 1891. Kronecker only outlived his wife by a few months, and died in December 1891.

We should not think that Kronecker's views of mathematics were totally eccentric. Although it was true that most mathematicians of his day would not agree with those views, and indeed most mathematicians today would not agree with them, they were not put aside. Kronecker's ideas were further developed by Poincaré and Brouwer, who placed particular emphasis upon intuition. Intuitionism stresses that mathematics has priority over logic, the objects of mathematics are constructed and operated upon in the mind by the mathematician, and it is impossible to define the properties of mathematical objects simply by establishing a number of axioms.