数学家传记
大卫·希尔伯特在几何学方面的工作在该领域的影响仅次于欧几里得。对欧几里得几何公理的系统研究使大卫·希尔伯特提出了21条这样的公理，并分析了它们的意义。他在数学和物理学的许多领域都做出了贡献。
大卫·希尔伯特的父亲Otto Hilbert是一位法官的儿子，这位法官是高级枢密顾问。Otto是一位县法官，娶了Maria Therese Erdtmann，她是柯尼斯堡商人Karl Erdtmann的女儿。Maria对哲学、天文学和素数着迷。Otto Hilbert有一个兄弟是律师，另一个兄弟是一所文理中学的主任。Otto晋升为高级法官后，他和Maria搬到了柯尼斯堡的Kirchenstrasse 13号，这就是弗洛伦斯·南丁格尔·大卫度过大部分童年的家。他受到父亲的严格管教，父亲是一个按固定模式生活的人，每天走同样的路线，每年只离开柯尼斯堡一次去度一年一度的家庭假期。大卫是他父母的第一个孩子，也是唯一的儿子。他六岁时，妹妹Elsie出生了。
通常开始上学的年龄是六岁，但大卫直到八岁才进入他的第一所学校，即皇家弗里德里希学校。几乎可以肯定，他的母亲在他八岁之前一直在家教他。弗里德里希学校，也称为弗里德里希学院，有一个低年级部，大卫在那里读了两年，然后于1872年进入弗里德里希学校的高级中学。尽管这所学校被认为是柯尼斯堡最好的学校，但重点放在拉丁语和希腊语上，数学被认为不太重要。弗里德里希学校完全不教授科学。学习的主要方法是让学生记忆大量材料，而大卫对此并不特别擅长。对于一个后来对数学产生巨大影响的人来说，也许令人惊讶的是，他在学校并不出色。在后来的生活中，他形容自己在弗里德里希学校时是一个“迟钝而愚蠢”的男孩。虽然这些话中无疑有谦虚的成分，但它们可能反映了希尔伯特对自己学生时代的感受。1879年9月，他从弗里德里希学校转到威廉高级中学，在那里度过了他最后一年的学业。这里更注重数学，老师们鼓励原创性思维，这在弗里德里希学校是没有的。希尔伯特快乐得多，他在所有科目上的表现都有所提高。他的数学获得了最高分，他的期末报告写道：
对于数学，他总是表现出非常活跃的兴趣和敏锐的理解力：他以非常令人满意的方式掌握了学校教授的所有材料，并能够准确而巧妙地应用它。
从威廉高级中学毕业后，他于1880年秋季进入柯尼斯堡大学。在第一学期，他学习了积分学、行列式理论和曲面曲率等课程。然后，按照当时德国的传统，在第二学期他去了海德堡，在那里听了拉扎勒斯·福克斯的讲座。回到柯尼斯堡开始1881-82学年时，希尔伯特听了海因里希·马丁·韦伯关于数论和函数论的讲座。1882年春天，赫尔曼·闵可夫斯基在柏林学习后回到柯尼斯堡。希尔伯特和赫尔曼·闵可夫斯基，后者也是一名博士生，很快成为亲密的朋友，他们对彼此的数学进步产生了强烈影响。费迪南德·冯·林德曼于1883年被任命到柯尼斯堡接替海因里希·马丁·韦伯，阿道夫·赫维兹于1884年春天被任命为那里的编外教授。阿道夫·赫维兹和希尔伯特成为亲密的朋友，这是希尔伯特数学发展的另一个重要因素，而费迪南德·冯·林德曼成为希尔伯特的学位论文导师。他于1884年12月11日参加了他的学位论文Über invariante Eigenschaften specieller binärer Formen, insbesondere der Kugelfunctionen Ⓣ（关于不变性质，特别是二元形式，尤其是球谐函数）的口试。费迪南德·冯·林德曼曾建议希尔伯特研究某些代数形式的不变性质，而希尔伯特在设计一种费迪南德·冯·林德曼未曾设想的方法时表现出极大的独创性。赫尔曼·闵可夫斯基在阅读了论文后，写信给希尔伯特（见[8]）：
我非常感兴趣地研究了你的工作，并为那些可怜的不变量在消失之前必须经历的所有过程感到高兴。我没想到在柯尼斯堡能获得这样一个好的数学定理。
1885年2月7日，他在一次公开辩论中为两个命题进行了答辩。希尔伯特选定的命题之一是关于物理学的，另一个是关于哲学的。这是他博士学位的最后阶段，随后学位正式授予。获得博士学位后的一个月里，他参加并通过了国家考试（Staatsexamen），从而有资格在文理中学任教；他还参加了费迪南德·冯·林德曼关于尤里乌斯·普吕克的线几何和索菲斯·李的球几何的几何课程，也听了阿道夫·赫维兹关于模函数的讲座。阿道夫·赫维兹建议希尔伯特去莱比锡进行学术访问，与菲利克斯·克莱因交谈。他采纳了这个建议，前往莱比锡并听了菲利克斯·克莱因的讲座。他还结识了Georg Pick和爱德华·斯图迪。菲利克斯·克莱因建议希尔伯特和爱德华·斯图迪都应访问埃尔朗根，与保罗·哥尔丹讨论他们的研究，后者是不变量理论方面的顶尖专家。然而，那次访问当时并未成行。菲利克斯·克莱因随后告诉爱德华·斯图迪和希尔伯特，他们应该去巴黎。两人于1886年初前往，希尔伯特是在三月底。菲利克斯·克莱因曾指示他们应该拜访哪些巴黎数学家，他们照办了，并轮流写信给菲利克斯·克莱因讲述他们的经历。他们最早拜访的数学家之一是儒勒·昂利·庞加莱，后者几天后回访了他们。这两位年轻访客互相大声朗读写给菲利克斯·克莱因的信，以免两人告诉他同样的事情。他依次回复了每一封信，明确表示他对他们一视同仁。在巴黎，卡米耶·若尔当为希尔伯特和爱德华·斯图迪设宴，George-Henri Halphen、Amédée Mannheim和让·加斯东·达布应邀出席。在这次场合，法国数学家出于对德国客人的礼貌都讲德语，而德国客人后来向菲利克斯·克莱因抱怨说，数学交谈非常肤浅。他们对与Pierre Bonnet的会面也感到失望，觉得他太老了，不适合进行数学讨论。他们似乎相处得最好的是夏尔·埃尔米特。尽管他们觉得他很老（他64岁），但他“格外友好和好客”，并讨论了不变量理论的大问题。由于他们发现这次访问特别有用，几天后他们又回到夏尔·埃尔米特家中进行了第二次拜访。显然，希尔伯特在巴黎期间的心思完全在数学上，他没有写下任何观光的事情。访问快结束时，他生病了，可能还思乡。当然，到1886年春天，他回到德国时精神很好。在回柯尼斯堡的路上，他访问了哥廷根，当时菲利克斯·克莱因即将就任讲席，他在那里遇到了赫尔曼·阿曼杜斯·施瓦茨。希尔伯特告诉赫尔曼·阿曼杜斯·施瓦茨他接下来要去柏林，后者提醒他利奥波德·克罗内克可能会冷淡接待他。然而，希尔伯特描述他在柏林受到的欢迎非常友好。
希尔伯特继续回到柯尼斯堡，准备提交他关于不变量理论的教授资格论文（Habilitation）论文。他还必须在阿尔贝蒂娜的主礼堂发表就职演讲，在希尔伯特提供的两个选项中，他被要求发表The most general periodic functions的演讲。菲利克斯·克莱因曾告诉希尔伯特，柯尼斯堡可能不是他habilitate的好地方，但希尔伯特乐意这样做。他写信给菲利克斯·克莱因（例如见[8]）：-
我很满足，满心欢喜地决定选择柯尼斯堡。与费迪南德·冯·林德曼教授，尤其是与阿道夫·赫维兹的经常交往，不仅对我有利、令我振奋，而且同样有趣。柯尼斯堡远离各种事物这一缺点，我希望明年能通过再次进行一些旅行来克服，也许那时我能见到保罗·哥尔丹先生。
他从1886年到1895年在柯尼斯堡任职，1892年之前是Privatdozent，之后任编外教授一年，1893年被任命为正教授。他在柯尼斯堡取得任教资格后所说的那次旅行发生在1888年[126]：-
……他于1888年3月出发，前往德国几个主要数学中心旅行，包括柏林、莱比锡和哥廷根。在一个月的时间里，他与大约二十位数学家交谈，从中获得了对全国当前研究兴趣的令人振奋的概览。
在柏林，他会见了利奥波德·克罗内克和卡尔·魏尔斯特拉斯，他们向年轻的希尔伯特展示了关于未来的两种截然不同的观点。接着，在莱比锡，他终于见到了保罗·哥尔丹[126]：-
……两人一拍即合，因为他们都最爱谈论数学。
希尔伯特在哥廷根逗留了八天后返回柯尼斯堡。1892年10月12日，他与自己的远房表妹Käthe Jerosch结婚；他们有一个儿子Franz Hilbert，生于1893年8月11日。
1892年，赫尔曼·阿曼杜斯·施瓦茨从哥廷根搬到柏林，接任卡尔·魏尔斯特拉斯的讲席，菲利克斯·克莱因想将空缺的哥廷根讲席提供给希尔伯特。然而菲利克斯·克莱因未能说服他的同事，海因里希·马丁·韦伯被任命为该讲席。当海因里希·马丁·韦伯三年后搬到斯特拉斯堡的讲席时，菲利克斯·克莱因可能并不太不高兴，因为这次他成功实现了任命希尔伯特的目标。因此，在1895年，希尔伯特被任命为哥廷根大学数学讲席，在那里他继续教书直到职业生涯结束。
希尔伯特在1900年后在数学界的杰出地位意味着其他机构想引诱他离开哥廷根，1902年，柏林大学向希尔伯特提供了拉扎勒斯·福克斯的讲席。希尔伯特拒绝了柏林讲席，但在此之前他利用这一提议与哥廷根讨价还价，说服他们设立一个新讲席，将他的朋友赫尔曼·闵可夫斯基带到哥廷根。
如上所述，希尔伯特的第一项工作是关于不变量理论，1888年，他证明了他著名的基定理。二十年前，保罗·哥尔丹使用高度计算的方法证明了二元形式的有限基定理。将保罗·哥尔丹的工作推广到多于两个变量的系统的尝试失败了，因为计算困难太大。希尔伯特本人起初试图遵循保罗·哥尔丹的方法，但很快意识到需要一种新的攻击路线。他发现了一种全新的方法，以完全抽象的方式证明了任意数量变量的有限基定理。尽管他证明了有限基的存在，但他的方法并没有构造出这样的基。
希尔伯特向Mathematische Annalen提交了一篇证明有限基定理的论文。然而，保罗·哥尔丹是Mathematische Annalen的不变量理论专家，他觉得希尔伯特的革命性方法难以欣赏。他审阅了这篇论文，并将自己的意见寄给了菲利克斯·克莱因：——
问题不在于形式……而在于更深层的地方。希尔伯特不屑于按照形式规则来表述他的思想，他认为只要没有人反驳他的证明就够了……他满足于认为他的命题的重要性和正确性已经足够。……对于《Annalen》这样一部综合性著作来说，这是不够的。
希尔伯特通过他的朋友阿道夫·赫维兹得知了保罗·哥尔丹写给菲利克斯·克莱因的信，于是希尔伯特自己以强硬的措辞写信给菲利克斯·克莱因：——
……我不准备修改或删除任何内容，关于这篇论文，我以最大的谦逊说，只要没有针对我的推理提出明确且无可辩驳的反对意见，这就是我的最终答复。
当菲利克斯·克莱因收到来自希尔伯特和保罗·哥尔丹的这两封信时，希尔伯特还是一名助理讲师，而保罗·哥尔丹则是公认的不变量理论世界领先专家，也是菲利克斯·克莱因的密友。然而，菲利克斯·克莱因认识到了希尔伯特工作的重要性，并向他保证，该工作将原封不动地出现在Annalen中，事实也确实如此。
希尔伯特在后来的一篇论文中进一步阐述了他的方法，该论文再次提交给了Mathematische Annalen，而菲利克斯·克莱因在阅读手稿后，写信给希尔伯特说：-
我毫不怀疑，这是《数学年刊》有史以来发表的最重要的普通代数著作。
1893年，仍在柯尼斯堡的希尔伯特开始撰写一部关于代数数论的ZahlberichtⓉ（一份数论报告）。German Mathematical Society在该学会于1890年成立三年后，请求撰写这份重要报告。Zahlbericht（1897年）是恩斯特·爱德华·库默尔、利奥波德·克罗内克和理查德·戴德金工作的精彩综合，但也包含了希尔伯特本人的大量思想。当今‘类域论’这一主题的所有思想都包含在这部著作中。Rowe在[124]中这样描述这部著作：-
……并非传统意义上的报告，而是一项原创性研究，揭示出希尔伯特绝非仅仅是一位才华横溢的专家。……他不仅综合了先前研究的结果……还塑造了新的概念，这些概念在此后许多年里影响了代数数论的研究进程。
希尔伯特为Zahlbericht所写序言中的一段摘录，是我们收藏在THIS LINK的Quotes by and about Hilbert中的引文7。
希尔伯特在几何方面的工作在该领域的影响仅次于欧几里得。对欧几里得几何公理的系统研究使希尔伯特提出了21条这样的公理，并分析了它们的意义。他于1899年出版了Grundlagen der Geometrie，将几何置于形式公理化的框架中。该书不断以新版本出现，对推动数学的公理化方法产生了重大影响，而公理化方法一直是整个20世纪该学科的主要特征之一。
对Grundlagen der Geometrie及希尔伯特其他著作的书评见THIS LINK。
关于Grundlagen der Mathematik的更多内容见THIS LINK。
希尔伯特著名的23个巴黎问题挑战（今天仍然挑战）数学家去解决基本问题。希尔伯特的著名演讲《数学问题》是在巴黎举行的第二届国际数学家大会上发表的。这是一篇对下个世纪数学充满乐观的演讲，他认为开放问题是该学科活力的标志：-
确定问题对数学科学总体进步的极大重要性……是不可否认的。……[因为]只要一个知识分支提供过剩的此类问题，它就保持其活力。……每个数学家当然都分享……这样的信念：每个数学问题必然能够严格解决……我们在内心听到不断的呼喊：问题就在这里，寻求解答。你可以通过纯粹思考找到它……
希尔伯特的问题包括连续统假设、实数的良序、哥德巴赫猜想、代数数幂的超越性、波恩哈德·黎曼假设、约翰·彼得·古斯塔夫·勒热纳·狄利克雷原理的扩展等等。许多问题在本世纪得到解决，每当其中一个问题被解决时，都是数学界的一件大事。
关于希尔伯特问题的更多信息见THIS LINK。
希尔伯特的名字常常通过希尔伯特空间的概念而被最好地记住。厄文·卡普兰斯基在[2]中写道，解释了希尔伯特导致这一概念的工作：-
希尔伯特在大约1909年在积分方程中的工作直接导致了20世纪泛函分析（数学中集体研究函数的分支）的研究。这项工作也为他在无限维空间上的工作奠定了基础，后来称为希尔伯特空间，这一概念在数学分析和量子力学中有用。利用他在积分方程上的结果，希尔伯特通过他关于动理气体理论和辐射理论的重要论文为数学物理的发展做出了贡献。
许多人声称，1915年希尔伯特在阿尔伯特·爱因斯坦之前发现了广义相对论的正确场方程，但从未声称优先权。然而，文章[54]表明这种观点是错误的。在这篇论文中，作者令人信服地表明，希尔伯特于1915年11月20日提交了他的文章，比阿尔伯特·爱因斯坦提交包含正确场方程的文章早五天。阿尔伯特·爱因斯坦的文章于1915年12月2日发表，但希尔伯特论文的校样（日期为1915年12月6日）不包含场方程。
正如54的作者所写：-
在其论文的印刷版中，希尔伯特添加了对阿尔伯特·爱因斯坦决定性论文的引用，并承认了后者的优先权：“由此得到的引力微分方程，在我看来，与阿尔伯特·爱因斯坦在其后期论文中建立的宏伟广义相对论理论是一致的”。如果希尔伯特只将日期行改为“1915年11月20日提交，[1915年12月2日之后任意日期，即阿尔伯特·爱因斯坦决定性论文的日期]修订”，那么后来就不会出现优先权问题。
1934年和1939年，Grundlagen der MathematikⓉ（数学基础）的两卷出版，旨在导向一种“证明论”，即对数学一致性的直接检验。库尔特·弗雷德里希·哥德尔1931年的论文表明这一目标是不可能的。
希尔伯特对数学的许多分支都有贡献，包括不变量、代数数域、泛函分析、积分方程、数学物理和变分法。他的数学才能被他的第一位学生奥托·布卢门塔尔[30]很好地总结道：-
在分析数学才能时，必须区分创造新概念从而产生新类型思维结构的能力，与感知更深层次联系和潜在统一性的天赋。就希尔伯特而言，他的伟大之处在于一种极其强大的洞察力，能穿透问题的深处。他的所有著作都包含来自遥远领域的例子，只有他才能 discern 出与手头问题的相互关联和联系。由此，综合，他的艺术作品，最终被创造出来。就新思想的创造而言，我会把赫尔曼·闵可夫斯基置于更高位置，而在古典伟人中，卡尔·弗里德里希·高斯、埃瓦里斯特·伽罗瓦和波恩哈德·黎曼。但就穿透性洞察力而言，只有少数最伟大者能与希尔伯特匹敌。
希尔伯特的学生有赫尔曼·外尔、著名的世界象棋冠军伊曼纽·拉斯克和恩斯特·策梅洛。但名单还包括许多其他著名名字，包括威廉·阿克曼、费利克斯·伯恩斯坦、奥托·布卢门塔尔、理查·科朗特、哈斯凯尔·加里、马克斯·登、Rudolf Fueter、阿尔弗雷德·哈尔、乔治·哈梅尔、埃里希·赫克、Earle Hedrick、恩斯特·黑林格、爱德华·卡斯勒、Oliver Kellogg、赫尔穆特·克内泽尔、奥托·纽格包尔、艾尔哈德·施密特、胡戈·施泰因豪斯和高木贞治。
1930年希尔伯特退休，但仅仅几年后，在1933年，当纳粹上台、犹太讲师被解职时，哥廷根的生活彻底改变了。到1933年秋天，大多数人已经离开或被解职。希尔伯特虽然已经退休，但仍一直在讲一些课。在1933-34年冬季学期，他每周讲一次几何基础。他讲完这门课程后，就再也没有踏进研究所。1942年初，他在哥廷根散步时摔倒并摔断了手臂。这使他完全无法活动，这似乎是他事故一年后去世的一个主要因素。
希尔伯特获得了许多荣誉。1905年，Hungarian Academy of Sciences特别表彰了希尔伯特。他于1910年被授予鲍耶奖，并于1928年当选为伦敦皇家学会会士。1930年，希尔伯特退休，柯尼斯堡市授予他荣誉市民称号。他发表了一次演讲，结尾的六个著名词语展现了他对数学的热情以及他致力于解决数学问题的一生：-
Wir müssen wissen, wir werden wissen - 我们必须知道，我们将会知道。
参见THIS LINK中的引文3。
1939年，他获得了瑞典科学院颁发的约斯塔·米塔格-莱弗勒奖。他与埃米尔·皮卡分享了该奖。希尔伯特于1901年当选为伦敦数学会荣誉会员，1942年当选为德国数学会荣誉会员。
关于描述希尔伯特性格和爱好的引文，见THIS LINK中的5和10。
David Hilbert's father, Otto Hilbert, was the son of a judge who was a high ranking Privy Councillor. Otto was a county judge who had married Maria Therese Erdtmann, the daughter of Karl Erdtmann, a Königsberg merchant. Maria was fascinated by philosophy, astronomy and prime numbers. Otto Hilbert had a brother who was a lawyer and another who was the director of a Gymnasium. After Otto was promoted to become a senior judge, he and Maria moved to 13 Kirchenstrasse in Königsberg and this was the home in which David spent much of his childhood. He had a strict upbringing by his father who was a man who lived his life to a standard pattern, always walking the same way every day and only leaving Königsberg once a year for the annual family holiday. David was his parents' first child and only son. He was six years old when his sister Elsie was born.
The usual age for someone to begin schooling was six but David did not enter his first school, the Royal Friedrichskolleg, until he was eight years old. It is almost certain that his mother taught him at home until he was eight. The Friedrichskolleg, also known as the Collegium Fridericianum, had a junior section which David attended for two years before entering the gymnasium of the Friedrichskolleg in 1872. Although this was reputed to be the best school in Königsberg, the emphasis was on Latin and Greek with mathematics considered as less important. Science was not taught at all in the Friedrichskolleg. The main approach to learning was having pupils memorise large amounts of material, something David was not particularly good at. Perhaps surprisingly for someone who was to make a gigantic impact on mathematics, he did not shine at school. In later life he described himself as a "dull and silly" boy at the Friedrichskolleg. Although doubtless there is modesty in these words, nevertheless they probably reflect Hilbert's own feeling about his school days. In September 1879 he transferred from the Friedrichskolleg to the Wilhelm Gymnasium where he spent his final year of schooling. Here there was more emphasis on mathematics and the teachers encouraged original thinking in a way that had not happened at the Friedrichskolleg. Hilbert was much happier and his performance in all his subjects improved. He received the top grade for mathematics and his final report stated:-
For mathematics he always showed a very lively interest and a penetrating understanding: he mastered all the material taught in the school in a very pleasing manner and was able to apply it with sureness and ingenuity.
After graduating from the Wilhelm Gymnasium, he entered the University of Königsberg in the autumn of 1880. In his first semester he took courses on integral calculus, the theory of determinants and the curvature of surfaces. Then following the tradition in Germany at this time, in the second semester he went to Heidelberg where he attended lectures by Lazarus Fuchs. Returning to Königsberg for the start of session 1881-82, Hilbert attended lectures on number theory and the theory of functions by Heinrich Weber. In the spring of 1882, Hermann Minkowski returned to Königsberg after studying in Berlin. Hilbert and Minkowski, who was also a doctoral student, soon became close friends and they were to strongly influence each others mathematical progress. Ferdinand von Lindemann was appointed to Königsberg to succeed Heinrich Weber in 1883 and Adolf Hurwitz was appointed as an extraordinary professor there in the spring of 1884. Hurwitz and Hilbert became close friends, another friendship which was important factor in Hilbert's mathematical development, while Lindemann became Hilbert's thesis advisor. He received his oral examination on 11 December 1884 for his thesis entitled Über invariante Eigenschaften specieller binärer Formen, insbesondere der Kugelfunctionen Ⓣ. Lindemann had suggested that Hilbert study invariant properties of certain algebraic forms and Hilbert showed great originality in devising an approach that Lindemann had not envisaged. Minkowski, after reading the thesis, wrote to Hilbert (see [8]):-
I studied your work with great interest and rejoiced over all the processes which the poor invariants had to pass through before they manage to disappear. I would not have supposed that such a good mathematical theorem could have been obtained in Königsberg.
On 7 February 1885 he defended two propositions in a public disputation. One of Hilbert's chosen propositions was on physics, the other on philosophy. This was the final stage of his doctorate, which was then duly awarded. He spent the month following the award of his doctorate taking, and passing, the Staatsexamen so that he was qualified to teach in a Gymnasium, and he also attended Lindemann's geometry course on Plücker's line geometry and Lie's sphere geometry, and he also attended Hurwitz's lectures on modular functions. Hurwitz suggested that Hilbert make a research visit to Leipzig to speak with Felix Klein. Taking this advice, he went to Leipzig and attended Klein's lectures. He also got to know Georg Pick and Eduard Study. Klein suggested that both Hilbert and Study should visit Erlangen and discuss their research with Paul Gordan who was the leading expert on invariant theory. However, the visit did not take place at that time. Klein then told both Study and Hilbert that they should visit Paris. They both went in early 1886, Hilbert at the end of March. Klein had given them instructions as to which of the Paris mathematicians they should visit and they did as he told them, alternately writing to Klein about their experiences. One of the first mathematicians they visited was Henri Poincaré who returned their visit a few days later. The two young visitors read their letters to Klein out loud to each other so that they would not both tell him the same things. He replied to each in turn, making clear that he was treating them equally. In Paris, Camille Jordan gave a dinner for Hilbert and Study to which George-Henri Halphen, Amédée Mannheim and Gaston Darboux were invited. On this occasion the French mathematicians all spoke German out of politeness to their German guests who complained to Klein afterwards that the mathematical conversation had been very superficial. They were also disappointed with their meeting with Pierre Bonnet whom they felt was too old for mathematical discussions. The mathematician with whom they seemed to get on best was Charles Hermite. Although they considered him very old (he was 64), he was "extraordinarily friendly and hospitable" and discussed the big problems of invariant theory. Since they had found their visit especially useful, they returned to Hermite's home for a second visit a few days later. It is clear that Hilbert's thoughts were entirely on mathematics during his time in Paris and he wrote nothing of any sightseeing. Towards the end of his visit he suffered an illness and was probably homesick. Certainly by the spring of 1886 he was in good spirits as he returned to Germany. On his way back to Königsberg he visited Göttingen, where Klein was about to take up the chair, where he met Hermann Amandus Schwarz. Telling Schwarz that he was next going to Berlin, Hilbert was advised to expect a cold reception by Leopold Kronecker. However, Hilbert described his welcome in Berlin as very friendly.
From Berlin, Hilbert continued back to Königsberg where he prepared to submit his habilitation paper on invariant theory. He also had to give an inaugural lecture in the main auditorium of the Albertina and, from the two options offered by Hilbert, he was asked to deliver the lecture The most general periodic functions. Klein had told Hilbert that Königsberg may not be a good place for him to habilitate but Hilbert was happy to do so. He wrote to Klein (see for example [8]):-
I am content and full of joy to have decided myself for Königsberg. The constant association with Professor Lindemann and, above all, with Hurwitz is not less interesting than it is advantageous to myself and stimulating. The bad part about Königsberg being so far away from things I hope I will be able to overcome by making some trips again next year, and perhaps then I will get to meet Herr Gordan.
He was a member of staff at Königsberg from 1886 to 1895, being a Privatdozent until 1892, then as Extraordinary Professor for one year before being appointed a full professor in 1893. The tour that he spoke about after habilitating at Königsberg happened in 1888 [126]:-
... he set off in March 1888 on a tour of several leading mathematical centres in Germany, including Berlin, Leipzig, and Göttingen. During the course of a month, he spoke with some twenty mathematicians from whom he gained a stimulating overview of current research interests throughout the country.
In Berlin he met Kronecker and Weierstrass who presented the young Hilbert with two rather different views of the future. Next, in Leipzig, he finally met Paul Gordan [126]:-
... the two hit it off splendidly, as both loved nothing more than to talk about mathematics.
Hilbert spent eight days in Göttingen before returning to Königsberg. He married his second cousin, Käthe Jerosch, on 12 October 1892; they had one son Franz Hilbert born on 11 August 1893.
In 1892 Schwarz moved from Göttingen to Berlin to occupy Weierstrass's chair and Klein wanted to offer Hilbert the vacant Göttingen chair. However Klein failed to persuade his colleagues and Heinrich Weber was appointed to the chair. Klein was probably not too unhappy when Weber moved to a chair at Strasbourg three years later since on this occasion he was successful in his aim of appointing Hilbert. So, in 1895, Hilbert was appointed to the chair of mathematics at the University of Göttingen, where he continued to teach for the rest of his career.
Hilbert's eminent position in the world of mathematics after 1900 meant that other institutions would have liked to tempt him to leave Göttingen and, in 1902, the University of Berlin offered Hilbert Fuchs's chair. Hilbert turned down the Berlin chair, but only after he had used the offer to bargain with Göttingen and persuade them to set up a new chair to bring his friend Minkowski to Göttingen.
As we saw above, Hilbert's first work was on invariant theory and, in 1888, he proved his famous Basis Theorem. Twenty years earlier Gordan had proved the finite basis theorem for binary forms using a highly computational approach. Attempts to generalise Gordan's work to systems with more than two variables failed since the computational difficulties were too great. Hilbert himself tried at first to follow Gordan's approach but soon realised that a new line of attack was necessary. He discovered a completely new approach which proved the finite basis theorem for any number of variables but in an entirely abstract way. Although he proved that a finite basis existed his methods did not construct such a basis.
Hilbert submitted a paper proving the finite basis theorem to Mathematische Annalen. However Gordan was the expert on invariant theory for Mathematische Annalen and he found Hilbert's revolutionary approach difficult to appreciate. He refereed the paper and sent his comments to Klein:-
The problem lies not with the form ... but rather much deeper. Hilbert has scorned to present his thoughts following formal rules, he thinks it suffices that no one contradict his proof ... he is content to think that the importance and correctness of his propositions suffice. ... for a comprehensive work for the 'Annalen' this is insufficient.
However, Hilbert had learnt through his friend Hurwitz about Gordan's letter to Klein and Hilbert wrote himself to Klein in forceful terms:-
... I am not prepared to alter or delete anything, and regarding this paper, I say with all modesty, that this is my last word so long as no definite and irrefutable objection against my reasoning is raised.
At the time Klein received these two letters from Hilbert and Gordan, Hilbert was an assistant lecturer while Gordan was the recognised leading world expert on invariant theory and also a close friend of Klein's. However Klein recognised the importance of Hilbert's work and assured him that it would appear in the Annalen without any changes whatsoever, as indeed it did.
Hilbert expanded on his methods in a later paper, again submitted to the Mathematische Annalen and Klein, after reading the manuscript, wrote to Hilbert saying:-
I do not doubt that this is the most important work on general algebra that the 'Annalen' has ever published.
In 1893 while still at Königsberg Hilbert began a work Zahlbericht Ⓣ on algebraic number theory. The German Mathematical Society requested this major report three years after the Society was created in 1890. The Zahlbericht (1897) is a brilliant synthesis of the work of Kummer, Kronecker and Dedekind but also contains a wealth of Hilbert's own ideas. The ideas of the present day subject of 'Class field theory' are all contained in this work. Rowe, in [124], describes this work as:-
... not really a Bericht in the conventional sense of the word, but rather a piece of original research revealing that Hilbert was no mere specialist, however gifted. ... he not only synthesized the results of prior investigations ... but also fashioned new concepts that shaped the course of research on algebraic number theory for many years to come.
An extract from Hilbert's Preface to Zahlbericht is quote 7 in our collection Quotes by and about Hilbert at THIS LINK.
Hilbert's work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance. He published Grundlagen der Geometrie in 1899 putting geometry in a formal axiomatic setting. The book continued to appear in new editions and was a major influence in promoting the axiomatic approach to mathematics which has been one of the major characteristics of the subject throughout the 20th century.
Reviews of Grundlagen der Geometrie and other of Hilbert's books are at THIS LINK.
More about Grundlagen der Mathematik is at THIS LINK.
Hilbert's famous 23 Paris problems challenged (and still today challenge) mathematicians to solve fundamental questions. Hilbert's famous speech The Problems of Mathematics was delivered to the Second International Congress of Mathematicians in Paris. It was a speech full of optimism for mathematics in the coming century and he felt that open problems were the sign of vitality in the subject:-
The great importance of definite problems for the progress of mathematical science in general ... is undeniable. ... [for] as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality. ... every mathematician certainly shares ..the conviction that every mathematical problem is necessarily capable of strict resolution ... we hear within ourselves the constant cry: There is the problem, seek the solution. You can find it through pure thought...
Hilbert's problems included the continuum hypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more. Many of the problems were solved during this century, and each time one of the problems was solved it was a major event for mathematics.
For more information about Hilbert's problems see THIS LINK.
Today Hilbert's name is often best remembered through the concept of Hilbert space. Irving Kaplansky, writing in [2], explains Hilbert's work which led to this concept:-
Hilbert's work in integral equations in about 1909 led directly to 20th -century research in functional analysis (the branch of mathematics in which functions are studied collectively). This work also established the basis for his work on infinite-dimensional space, later called Hilbert space, a concept that is useful in mathematical analysis and quantum mechanics. Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations.
Many have claimed that in 1915 Hilbert discovered the correct field equations for general relativity before Einstein but never claimed priority. The article [54] however, shows that this view is in error. In this paper the authors show convincingly that Hilbert submitted his article on 20 November 1915, five days before Einstein submitted his article containing the correct field equations. Einstein's article appeared on 2 December 1915 but the proofs of Hilbert's paper (dated 6 December 1915) do not contain the field equations.
As the authors of [54] write:-
In the printed version of his paper, Hilbert added a reference to Einstein's conclusive paper and a concession to the latter's priority: "The differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein in his later papers". If Hilbert had only altered the dateline to read "submitted on 20 November 1915, revised on [any date after 2 December 1915, the date of Einstein's conclusive paper]," no later priority question would have arisen.
In 1934 and 1939 two volumes of Grundlagen der Mathematik Ⓣ were published which were intended to lead to a 'proof theory', a direct check for the consistency of mathematics. Gödel's paper of 1931 showed that this aim is impossible.
See THIS LINK.
Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. His mathematical abilities were nicely summed up by Otto Blumenthal, his first student [30]:-
In the analysis of mathematical talent one has to differentiate between the ability to create new concepts that generate new types of thought structures and the gift for sensing deeper connections and underlying unity. In Hilbert's case, his greatness lies in an immensely powerful insight that penetrates into the depths of a question. All of his works contain examples from far-flung fields in which only he was able to discern an interrelatedness and connection with the problem at hand. From these, the synthesis, his work of art, was ultimately created. Insofar as the creation of new ideas is concerned, I would place Minkowski higher, and of the classical great ones, Gauss, Galois, and Riemann. But when it comes to penetrating insight, only a few of the very greatest were the equal of Hilbert.
Among Hilbert's students were Hermann Weyl, the famous world chess champion Emanuel Lasker, and Ernst Zermelo. But the list includes many other famour names including Wilhelm Ackermann, Felix Bernstein, Otto Blumenthal, Richard Courant, Haskell Curry, Max Dehn, Rudolf Fueter, Alfred Haar, Georg Hamel, Erich Hecke, Earle Hedrick, Ernst Hellinger, Edward Kasner, Oliver Kellogg, Hellmuth Kneser, Otto Neugebauer, Erhard Schmidt, Hugo Steinhaus, and Teiji Takagi.
In 1930 Hilbert retired but only a few years later, in 1933, life in Göttingen changed completely when the Nazis came to power and Jewish lecturers were dismissed. By the autumn of 1933 most had left or were dismissed. Hilbert, although retired, had still been giving a few lectures. In the winter semester of 1933-34 he gave one lecture a week on the foundations of geometry. After he finished giving this course he never set foot in the Institute again. In early 1942 he fell and broke his arm while walking in Göttingen. This made him totally inactive and this seems to have been a major factor in his death a year after the accident.
Hilbert received many honours. In 1905 the Hungarian Academy of Sciences gave a special citation for Hilbert. He was awarded the Bolyai Prize in 1910 and elected a fellow of the Royal Society of London in 1928. In 1930 Hilbert retired and the city of Königsberg made him an honorary citizen of the city. He gave an address which ended with six famous words showing his enthusiasm for mathematics and his life devoted to solving mathematical problems:-
Wir müssen wissen, wir werden wissen - We must know, we shall know.
See quote 3 at THIS LINK.
In 1939 he was awarded the Mittag-Leffler prize by the Swedish Academy of Sciences. He shared this Prize with Émile Picard. Hilbert was elected an honorary member of the London Mathematical Society in 1901 and of the German Mathematical Society in 1942.
For quotes which describe Hilbert's personality and hobbies see 5 and 10 at THIS LINK.
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