戈特洛布·弗雷格Gottlob Frege

戈特洛布·弗雷格的父母是Alexander Frege和Auguste Bialloblotzky。Alexander Frege是维斯马一所女子高中的校长，弗雷格就出生在那个城镇。维斯马位于德国北部，坐落在波罗的海的一个小湾上。自1803年以来，它一直由梅克伦堡-什未林州管理，但在弗雷格出生时，该镇仍被瑞典声称拥有，瑞典从1648年《威斯特伐利亚和约》到1803年一直控制着它。据认为，弗雷格的母亲来自一个并非原籍该地区的家庭，但可能是波兰裔。

弗雷格在维斯马长大，就读于当地的文理中学，在那里由Leo Sachse授课。几乎可以肯定，弗雷格是听从Sachse的建议才选择去耶拿大学继续深造，总的来说，Sachse对他的年轻学生有很大影响（例如见[48]和[54]）。弗雷格为生活在梅克伦堡州而自豪，他热爱梅克伦堡公爵家族，并且肯定相信这种政府形式而不是民主选举的政府。欧洲这一地区即将经历一个政治大变革时期，事件在1866年开始迅速进展。然而，在审视这些事件之前，我们应该注意到，1866年是Alexander Frege，即弗雷格的父亲去世的那一年。

事实上，推动事件发展的政治变革是奥托·冯·俾斯麦于1862年成为普鲁士首相。俾斯麦认为，普鲁士的领导作用最好通过统一梅克伦堡等德意志各邦来实现，但奥地利反对这一路线。由此引发的1866年七周战争中，梅克伦堡站在普鲁士一边对抗奥地利，普鲁士的胜利导致1867年成立北德意志联邦，梅克伦堡为其成员。当弗雷格于1869年前往耶拿大学时，欧洲的政治格局已经改变，而在他于该校学习的两年间，更多的变化即将发生。普鲁士带领德意志各邦在1870-71年的普法战争中战胜法国，1871年，以普鲁士的威廉一世为皇帝的德意志帝国（German Reich）成立。

弗雷格由恩斯特·阿贝和K Fischer授课。在耶拿大学学习两年后，弗雷格于1871年继续深造，进入哥廷根大学，在那里学习了数学、物理学、化学和哲学课程。他于1873年在哥廷根获得博士学位，学位论文为Über eine geometrische Darstellung der imaginären Gebilde in der Ebene Ⓣ（论平面中虚结构的几何表示），他在文中试图为几何学的一部分奠定基础。该论文在他获得博士学位的同一年于耶拿出版。在恩斯特·阿贝的支持下，他于1874年向耶拿大学提交了他的教授资格论文（Habilitation） Rechnungsmethoden, die sich auf eine Erweitung des Grössenbegriffes gründen Ⓣ（基于域大小概念增强的计数方法），这实质上是一部关于阿贝尔群和不变量理论的著作，并于同年5月被任命为耶拿的数学Privatdozent。他在那里度过了余下的职业生涯，安静地从事自己的工作，与学生的接触极少，与同事的接触也极少。然而，Rudolf Eucken在哲学系与弗雷格共事超过40年，与他有着密切的科学联系。Eucken——像伯特兰·罗素和萨特一样——是少数获得诺贝尔文学奖（1908年）的哲学家之一。在弗雷格发表任何主要著作之前，他的母亲于1878年去世。

弗雷格是现代符号逻辑的创始人之一，他提出数学可归约为逻辑的观点。他讲授数学的各个分支，特别是解析几何、微积分、微分方程和力学，尽管他在逻辑领域之外的数学出版物很少。他关于逻辑哲学、数学哲学和语言哲学的著作极为重要。他曾说过：-

每一位优秀的数学家至少是半个哲学家，每一位优秀的哲学家至少是半个数学家。

1879年，弗雷格出版了他的第一部重要著作Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens Ⓣ（概念文字，一种模仿纯粹思想语言的算术公式）。A George和R Heck在[45]中写道：-

1879年，他以极其清晰、严谨和技术上的卓越，首次提出了他关于理性辩护的构想。实际上，这或许是有史以来对逻辑学所作的最伟大的单一贡献，无论如何，它是自亚里士多德以来最重要的进展。第一次，人们能够对涉及含有多重嵌套普遍性表达式的句子的演绎推理进行深入分析。此外，他提出了一个逻辑系统，在其中这类论证可以得到清晰的表达：这是自欧几里得以来我们对公理系统理解的最重要发展。

在这部著作中，弗雷格首次提出了我们今天会认为包含否定、蕴涵、全称量化、真值表思想等内容的逻辑系统，但今天无法辨认的是弗雷格所使用的记号。对于蕴涵$A \Rightarrow B$，弗雷格使用了一种跨两行书写的记号，其中$A$写在$B$下面一行。不难看出为什么他的记号没有流传下来，但我们不应让这一点在任何方面减损他成就的重要性。Begriffsschrift出版后，同年弗雷格再次在恩斯特·阿贝的支持下晋升为耶拿的编外教授，但总体而言，他杰出的工作为他带来的认可少得令人惊讶。似乎很少有人能够认识到这一里程碑式出版的重要性。然而，与他后来的论著相比，Begriffsschrift收到了六篇评论：来自埃拉斯穆斯·赖因霍尔德 Hoppe、约翰·维恩、保罗·塔内里、Kurd Lasswitz、Karl Michaëlis和恩斯特·施勒德。不过，前三篇评论表明其作者对弗雷格的论著不感兴趣，而后三篇尽管有一些批评，却更为同情（更多细节见[81]）。

有理由追问，是什么促使弗雷格产生了革命性的BegriffsschriftⓉ（概念文字）。他想要有一种精确的方式来陈述结果并证明它们，因为他意识到使用日常语言的困难，日常语言必然是不精确且有歧义的。他在该著作的序言中声明，他想“借助纯逻辑”来证明算术的基本真理。这一目标使弗雷格成为第一个充分发展逻辑主义主要论题的人，即数学可归约为逻辑。然而，我们应当注意，他只将该论题应用于数论和实分析。他的下一部主要著作Die Grundlagen der ArithmetikⓉ（算术基础），出版于1884年，写作目的是实现他在较早著作序言中已明确阐述的目标，并给出算术的公理化理论。

在GrundlagenⓉ（《算术基础》）开头设定自己的议程之后，弗雷格考察了先前数学家对两个基本问题的贡献：-

数是什么？算术真理的本质是什么？

事实上，他以卓越的清晰性推翻了此前回答这些问题的所有尝试。本文的读者或许会惊讶地得知，在弗雷格之前所有定义“数”的尝试都包含逻辑错误。这确实正是他所表明的，因为这些较早的定义把“数”的观念与“多数”的观念混淆了。多数“二”指两个对象的集合，例如两把椅子、两支铅笔、两座房子等。然而，数“二”是所有“多数二”实例的类，因此是“多数的多数”，而未能认识到这一点所犯的逻辑错误意味着，在弗雷格的Grundlagen之前，没有人能够给出“数”的逻辑上正确的定义。弗雷格随后给出了他自己关于算术基本概念的定义，这些定义纯粹基于逻辑，并由此再次运用纯逻辑推导出算术的基本定律。Dummett写道[39]：-

这部著作甚至对那些对数学哲学毫无兴趣的人也极具吸引力，因为在其中呈现了许多对整个哲学都具有重要意义的观念。

数学家和哲学家对Grundlagen的反应如何？人们或许会期待极大的兴趣，但这并未成为现实。Grundlagen只收到了一篇评论，而且是由格奥尔格·康托尔撰写的。格奥尔格·康托尔对这本杰出的书有何看法？Dummett写道，这篇评论[2]：-

……是格奥尔格·康托尔写的一篇极具敌意的文章，这位数学家的思想与弗雷格最为接近，他在对弗雷格的书施以完全不公正的蔑视之前，根本没有费心去理解它。

Grundlagen是一部非技术性著作，没有使用符号，只给出了证明的梗概，弗雷格将其视为实现自己目标的第一步，即定义一个精确的逻辑框架，用以建立算术的基本概念并推导出算术规则。尽管他对Grundlagen的反响极为失望，但在随后的几年里，他还是写了许多文章，打磨并扩展了他实施该项目所需的思想。Dummett称这些为：

……一系列出色的哲学文章，他在其中阐述了自己的逻辑哲学。

让我们简要地看其中一篇，即1892年发表的Über Sinn und BedeutungⓉ（论意义与重要性）。在这篇文章中，他给出了著名的论证，表明意义和指称是不同的。他的例子涉及金星，在人们意识到二者都是金星之前，它被称为“昏星”和“晨星”。弗雷格论证道：“昏星”=“晨星”与“昏星”=“昏星”不具有相同的意义，所以“昏星”与“晨星”不具有相同的意义。然而“昏星”和“晨星”指称同一个对象，所以“昏星”的指称与其意义是不同的。

1893年，Die Grundgesetze der Arithmetik, Volume1Ⓣ（算术的基本定律，第1卷）问世，其中弗雷格建立了一个形式逻辑系统，其推理规则比他较早的著作Begriffsschrift更多。弗雷格用一组直观的公理将算术公理化，并且对他先前只给出梗概的数论结果，现在给出了形式证明。本卷的主旨是发展数论的规则，而在后续各卷中，弗雷格打算将工作扩展到实数。他对其较早著作缺乏反响的痛苦失望，明确表现在第1卷的序言中，他在那里抱怨其他作者不熟悉他的观念。他一定曾希望，这部他视为自己最伟大成就的著作的第1卷会得到好评，但除了朱塞佩·皮亚诺的一篇评论外，它被同时代人忽视了。

弗雷格没有让先前缺乏反响的情况使他偏离自己设定的任务，决定推迟出版他计划的三卷本中的第二卷。在此期间，弗雷格被任命为耶拿的常任名誉教授，这一职位由与恩斯特·阿贝密切相关的卡尔·蔡司基金会资助。事实上，在Die Grundgesetze der Arithmetik第一卷出版十年之后，第二卷才问世。这第二卷给出了弗雷格对实数的发展，他直接从整数构造实数，而没有走先定义有理数的路线。他现在感到的苦涩在这一卷中清晰可见，他对早期数学家工作的攻击变得辱骂性（以前从未如此），并且有明显迹象表明他在回击那些他认为忽视了他贡献的人。他特别强烈批评了格奥尔格·康托尔和理查德·戴德金的无理数理论。在这部作品写成之后、出版之前，弗雷格发现这一卷和第一卷基于不一致的公理。

当The Basic Laws of Arithmetic第二卷在印刷厂时，弗雷格收到了伯特兰·罗素的一封信（1902年6月16日）。伯特兰·罗素非常谦逊地指出，伯特兰·罗素悖论在弗雷格的公理系统中给出了矛盾。两人之间往来了许多信件之后，弗雷格修改了他的一条公理，并在书的附录中解释说，这样做是为了恢复系统的一致性。然而，有了这条修改后的公理，第一卷的许多定理不再成立，弗雷格肯定知道这一点。他可能从未意识到，即使有了修改后的公理，这个系统也是不一致的，因为这一点直到弗雷格去世后才由斯坦尼斯瓦夫·雷斯涅夫斯基证明。

人们常常看到这样的说法：由于伯特兰·罗素指出的不一致，弗雷格的工作毫无价值。事实上这远非真相，人们必须把弗雷格视为对数学基础做出过有史以来最重要贡献之一的人。事实上，从许多方面来看，伯特兰·罗素在其History of Western Philosophy:-中写道的是正确的：

尽管[弗雷格的]发现具有划时代的意义，但在我1903年引起人们注意他之前，他完全没有得到承认。

弗雷格的短期影响是通过朱塞佩·皮亚诺、路德维希·维特根斯坦、Husserl、Carnap和伯特兰·罗素的工作产生的。然而，从长远来看，弗雷格对哲学逻辑的发展产生了重大影响，这位似乎被同时代人基本忽视的人，在二十世纪下半叶被许多人热切阅读，尤其是在他的著作被翻译成英文之后。

人们经常读到的另一种说法是，弗雷格在收到伯特兰·罗素的信后如此沮丧，以至于放弃了研究。这并非完全没有根据，而且他确实从未出版The Basic Laws of Arithmetic计划中的第三卷，这是肯定的，但尽管他确实变得非常沮丧，原因却远比这复杂得多。他沮丧的另一个因素是他的妻子玛格丽特去世。弗雷格娶了玛格丽特·利泽贝格，但他们没有孩子。弗雷格和他的妻子确实收养了一个儿子阿尔弗雷德，他后来成为一名工程师，但在弗雷格的妻子于1904年去世后，他似乎更深地陷入了自我封闭。

德国的政治局势使他痛苦。正如我们已经提到的，弗雷格是统一前德国各邦所实行旧式君主制的坚定信仰者。在德意志帝国，除了大多不民主的邦议会之外，还有一个民主选举的议会。弗雷格不喜欢向民主的转变，随着社会主义者获得权力，他更加憎恶它。他攻击了大多数数学家同行，远远超出了专业批评的范围。例如，同样在耶拿任教的卡尔·约翰尼斯·托马，受到了弗雷格严厉的人身攻击。他似乎对各种各样的人大发雷霆，他的日记显示了他对法国人、天主教徒和犹太人的深切仇恨。

弗雷格于1917年从耶拿的教授职位上退休。从1904年到退休期间，他没有发表任何东西（如果人们不算他对卡尔·约翰尼斯·托马等数学家同行发表的尖刻争论性攻击的话）。伯特兰·罗素曾邀请他在1912年剑桥的一次数学大会上发表演讲，但弗雷格拒绝邀请的回复显示了他沮丧的精神状态。这不是人们会从那个早先生动地意识到自己天才、并完全相信自己辉煌思想会得到承认的人那里期望得到的回复。然而，弗雷格在1918年又开始发表重要文章，对思想的本质做出了贡献。这些出版物具有他早期作品的新鲜感，表明多年来困扰他的沮丧至少部分地减轻了。1923年，弗雷格得出结论，他整个职业生涯大部分时间所设定的目标，即把算术建立在逻辑之上，是错误的。他转而决定，人们必须把整个数学建立在几何学之上。他开始研究这些想法，但到去世时进展不大。他没有发表任何关于这些想法的东西。

我们引用了许多对弗雷格天才的赞誉，但让我们以再一个赞誉结束。Weiner在[20]中写道：-

弗雷格的著作对当代思想产生了深远影响。他革命性的新逻辑是现代数理逻辑的起源——这一领域不仅对抽象数学重要，对计算机科学和哲学也很重要。

Gottlob Frege's parents were Alexander Frege and Auguste Bialloblotzky. Alexander Frege was the head of a girls' high school in Wismar and it was in that town that Gottlob was born. Wismar, in northern Germany, is situated on an inlet of the Baltic Sea. It had been administered by the Mecklenburg-Schwerin state since 1803 but at the time when Gottlob was born there, the town was still claimed by Sweden, the country which had controlled it from the Peace of Westphalia in 1648 until 1803. It is thought that Gottlob's mother was from a family which did not originate in that area but was probably of Polish origin.

Gottlob grew up in Wismar, attending the local Gymnasium where he was taught by Leo Sachse. It was almost certainly through following Sachse's advice that Frege chose to go to the University of Jena to continue his studies and in general Sachse had a large influence on his young student (see for example [48] and [54]). Frege was proud to live in the state of Mecklenburg, he loved the ducal house of Mecklenburg, and certainly believed in this form of government rather than a democratically elected one. A period of great political change in this part of Europe was approaching and events began to move quickly in 1866. Before looking at these events, however, we should note that 1866 was the year in which Alexander Frege, Gottlob's father, died.

In fact the political change which set events in motion was Otto von Bismarck becoming prime minister of Prussia in 1862. Bismarck saw that Prussia's leading role would be best served by the unification of the German states such as Mecklenburg, but Austria opposed this course. The resulting Seven Weeks' War in 1866 saw Mecklenburg side with Prussia against Austria and the Prussian victory led to the setting up of the North German Confederation, with Mecklenburg as a member, in 1867. When Frege went to the University of Jena in 1869 it was a politically changed Europe, and during the two years that he studied there more changes were to take place. Prussia led the German states to victory over France in the Franco-Prussian War of 1870-71 and in 1871 the German Reich (German Empire), with William I of Prussia as its emperor, came into existence.

At Jena Frege was taught by Ernst Abbe and K Fischer. After his two years of study at the University of Jena, Frege continued his education in 1871 entering the University of Göttingen where he studied courses in mathematics, physics, chemistry and philosophy. He received his doctorate in 1873 from Göttingen for a dissertation Über eine geometrische Darstellung der imaginären Gebilde in der Ebene Ⓣ, in which he tried to lay down foundations for a portion of geometry. The thesis was published at Jena in the same year that he was awarded his doctorate. Supported by Abbe, he presented his habilitation Rechnungsmethoden, die sich auf eine Erweitung des Grössenbegriffes gründen Ⓣ, essentially a work on abelian groups and invariant theory, to the University of Jena in 1874 and was appointed as a Privatdozent in mathematics in Jena in May of that year. He taught there for the rest of his career, carrying out his job quietly with minimal contacts with his students and colleagues. However Rudolf Eucken was a colleague of Frege's for more than 40 years in the faculty of philosophy with whom he had close scientific contacts. Eucken - like Russell and Sartre - was one of the few philosophers who were awarded the Nobel Prize in Literature (1908). Before Frege had published any of his major pieces of work, his mother died in 1878.

Frege was one of the founders of modern symbolic logic putting forward the view that mathematics is reducible to logic. He lectured on all branches of mathematics, in particular analytic geometry, calculus, differential equations, and mechanics, although his mathematical publications outside the field of logic are few. His writings on the philosophy of logic, philosophy of mathematics, and philosophy of language are of major importance. He once said:-

Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.

In 1879 Frege published his first major work Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens Ⓣ . A George and R Heck write in [45]:-

In 1879, with extreme clarity, rigour and technical brilliance, he first presented his conception of rational justification. In effect, it constitutes perhaps the greatest single contribution to logic ever made and it was, in any event, the most important advance since Aristotle. For the first time, a deep analysis was possible of deductive inferences involving sentences containing multiply embedded expressions of generality. Furthermore, he presented a logical system within which such arguments could be perspicuously represented: this was the most significant development in our understanding of axiomatic systems since Euclid.

In this work Frege presented for the first time what we would recognise today as a logical system with negation, implication, universal quantification, essentially the idea of truth tables etc., but what would not be recognisable today is the notation which Frege used. For the implication $A \Rightarrow B$, Frege used a notation which was placed across two lines with $A$ being written on the line below $B$. It is not hard to see why his notation has not survived, but we should not allow this in any way to diminish the magnitude of his achievement. The publication of the Begriffsschrift was followed in the same year by Frege's promotion, again supported by Abbe, to Extraordinary Professor at Jena but on the whole his remarkable work led to surprisingly little recognition for him. Very few people seemed to be able to appreciate the importance of this landmark publication. However, in contrast to his later treatises, the Begriffsschrift received six reviews: from Reinhold Hoppe, John Venn, Paul Tannery, Kurd Lasswitz, Karl Michaëlis and Ernst Schröder. The first three of these reviews, however, show that their authors are uninterested in Frege's treatise, while the latter three, despite some criticisms, are more sympathetic (see [81] for more details).

It is reasonable to ask what prompted Frege to produce the revolutionary Begriffsschrift Ⓣ. He wanted to have a precise way of stating results and of proving them, for he realised the difficulties of using ordinary language which was necessarily imprecise and ambiguous. He stated in the Preface to the work that he wanted to prove the basic truths of arithmetic "by means of pure logic". This aim makes Frege the first to fully develop the main thesis of logicism, that mathematics is reducible to logic. However, we should note that he only applied the thesis to number theory and real analysis. His next major work Die Grundlagen der Arithmetik Ⓣ, published in 1884, was written to achieve the aim that he had clearly set out in the Preface to the earlier work and present an axiomatic theory of arithmetic.

After setting his agenda at the start of the Grundlagen Ⓣ, Frege looked at the contributions made by previous mathematicians to two fundamental questions:-

What are numbers? What is the nature of arithmetical truth?

In fact he demolishes all earlier attempts to answer these questions with brilliant clarity. Perhaps it will come as a surprise to readers of this article to learn that all attempts to define "number" before Frege contained logical errors. Indeed this is precisely what he showed, for these earlier definitions had confused the idea of "number" with that of "plurality". The plurality "two" refers to a collection of two objects, for example two chairs, two pencils, two houses, etc. The number "two" is, however, the class of all instances of the "plurality two" and so is a "plurality of pluralities" and the logical error which had been made in not recognising this meant that before Frege's Grundlagen nobody had managed to give a logically correct definition of "number". Frege then went on to give his own definitions of the basic concepts of arithmetic based purely on logic, and from these he deduced, again using pure logic, the basic laws of arithmetic. Dummett writes [39]:-

The work is fascinating even for those quite uninterested in the philosophy of mathematics, since in the course of it many ideas are presented which are of significance for the whole of philosophy.

What was the reaction to the Grundlagen from mathematicians and philosophers? One might have expected an enormous amount of interest, but this did not materialise. The Grundlagen only received a single review and that was by Cantor. What did Cantor think of this brilliant book? Dummett writes that the review [2]:-

... was a devastatingly hostile one by Georg Cantor, the mathematician whose ideas were the closest to Frege's, who had not bothered to understand Frege's book before subjecting it to totally unmerited scorn.

The Grundlagen was a non-technical work, written without symbolism and with only sketches of proofs, which Frege saw as a first step towards the realisation of his goal of defining a precise logical framework in which to set up the basic concepts of arithmetic and to deduce the rules of arithmetic. Although he was extremely disappointed at the reaction to the Grundlagen nevertheless in the following years he wrote a number of articles which polished and extended the ideas which he would need to carry out his project. Dummett calls these:-

... a series of brilliant philosophical articles in which he elaborated his philosophy of logic.

Let us look briefly at one of these, namely Über Sinn und Bedeutung Ⓣ published in 1892. In this he gives his famous argument to show that sense and reference are distinct. His example concerns the planet Venus which was known as "the evening star" and as "the morning star" before it was realised that both were Venus. Frege argues: "the evening star" = "the morning star" does not have the same sense as "the evening star" = "the evening star" so "the evening star" does not have the same sense as "the morning star". However "the evening star" and "the morning star" refer to the same object so the reference of "the evening star" is distinct from its sense.

In 1893 Die Grundgesetze der Arithmetik, Volume1 Ⓣ appeared in which Frege set up a formal logical system with more rules of inference than that of his earlier work the Begriffsschrift. Frege axiomatized arithmetic with an intuitive collection of axioms, and proofs of number theory results which he had only sketched earlier he now gave formally. The main thrust of this volume was to develop the rules of number theory and in the later volumes Frege intended to extend the work to the real numbers. His bitter disappointment at the lack of reaction to his earlier work shows explicitly in the Preface to Volume 1 where he complains about other authors being unfamiliar with his ideas. He must have hoped that this first volume of what he viewed would be his greatest achievement would be well received, but except for one review by Peano, it was ignored by his contemporaries.

Frege, who had not allowed the previous lack of reaction to divert him from the tasks that he had set himself, decided to delay publication of the second of his three proposed volumes. During this period Frege was appointed ordinary honorary professor at Jena, a post funded by the Carl Zeiss Foundation with which Abbe was closely associated. In fact it would be ten years after the publication of Volume 1 of Die Grundgesetze der Arithmetik before Volume 2 appeared. This second volume gives Frege's development of the real numbers which he constructed straight from the integers without taking the route of first defining the rational numbers. The bitterness which he now felt shows clearly in this volume with his attacks on the work of earlier mathematicians being abusive (which it had never been before) and there were clear signs that he was hitting back at those he felt had ignored his contributions. In particular he strongly criticised Cantor's and Dedekind's theories of irrational numbers. After the work was written, but before it was published, Frege discovered that this volume, and Volume 1, were based on inconsistent axioms.

While Volume 2 of The Basic Laws of Arithmetic was at the printers Frege received a letter (on 16 June 1902) from Bertrand Russell. Russell pointed out, with great modesty, that the Russell paradox gave a contradiction in Frege's system of axioms. After many letters between the two, Frege modified one of his axioms and explains in an appendix to the book that this was done to restore the consistency of the system. However with this modified axiom, many of the theorems of Volume 1 do not go through and Frege must have known this. He probably never realised that even with the modified axiom the system is inconsistent since this was only shown by Lesniewski after Frege's death.

One often sees it stated that Frege's work was worthless because of the inconsistency pointed out by Russell. In fact this is far from the truth and one must view Frege as the person who made one of the most important contributions to the foundations of mathematics that has ever been made. In fact in many ways Russell is correct when he wrote in his History of Western Philosophy:-

In spite of the epoch-making nature of [Frege's] discoveries, he remained wholly without recognition until I drew attention to him in 1903.

Frege's influence in the short term came through the work of Peano, Wittgenstein, Husserl, Carnap and Russell. In the longer term, however, Frege has become a major influence on the development of philosophical logic and the man who seems to have been largely ignored by his contemporaries has been avidly read by many in the second half of the twentieth century, particularly after his works were translated into English.

Another statement that one often reads is that Frege was so depressed after Russell's letters that he gave up research. This is not entirely without foundation and it is certainly true that he never published the intended third volume of The Basic Laws of Arithmetic, but although he did indeed become very depressed the reasons are far more complex than this. Another factor in his depression was the death of his wife Margarete. Frege had married Margarete Lieseberg but they never had any children. Frege and his wife did adopt a son, Alfred, who went on to become an engineer, but after Frege's wife died in 1904 he seemed to sink more deeply into himself.

The political situation in Germany distressed him. Frege, as we have mentioned, was a firm believer of the old style monarchy which operated in the German States before the unification. In the German Empire there was a democratically elected parliament, in addition to the mostly undemocratic state parliaments. Frege disliked the move to democracy, and detested it even more as the socialists gained power. He attacked most of his fellow mathematicians going far beyond professional criticism. For example Thomae, who also taught at Jena, came in for severe personal attacks from Frege. He seemed to lash out at a wide variety of people and his diary shows a deep hatred of the French, of Catholics, and of Jews.

Frege retired from his professorship in Jena in 1917. He had published nothing between 1904 and the time he retired (if one discounts bitter argumentative attacks published against fellow mathematicians such as Thomae). Russell had invited him to address a mathematical congress in Cambridge in 1912 but Frege's reply, declining the invitation, shows his depressed state of mind. This was not the reply one would have expected from the man who had earlier been vividly aware of his own genius and had a total belief that his brilliant ideas would be recognized. However, Frege began to publish important articles again in 1918 with contributions to the nature of thoughts. These publications have the freshness of his earlier work and show that the depression which had gripped him for many years had, at least partially, lifted. In 1923 Frege came to the conclusion that the aim he had set himself throughout most of his career, namely to found arithmetic on logic, was wrong. He decided instead that one had to base the whole of mathematics on geometry. He began to work on these ideas but had not progressed far by the time of his death. He published nothing on these ideas.

We have quoted many tributes to Frege's genius, but let us end with one more. Weiner writes in [20]:-

Gottlob Frege's writings have had a profound influence on contemporary thought. His revolutionary new logic was the origin of modern mathematical logic - a field of import not only to abstract mathematics, but also to computer science and philosophy.