Felix Klein

Chapter 1 · A Chair at Twenty-Three

Düsseldorf, Plücker, and a rapid start

1849 – 1872 · Düsseldorf · Bonn · Göttingen · Paris · Erlangen

Felix Klein was born at Düsseldorf in 1849. He went to Bonn intending to be a physicist and became assistant to Julius Plücker, who worked on both physics and geometry. Plücker died in 1868 leaving a geometrical work unfinished, and Klein — twenty years old — completed it, which is how his career began.

He took his doctorate that year, then went to Berlin and to Paris. In Paris he met Sophus Lie, the Norwegian, and the two spent months together thinking about transformations and the groups they form; it was the most formative intellectual friendship of his life. The Franco-Prussian War interrupted it and sent him home.

In 1872, at twenty-three, he was appointed to a chair at Erlangen. A new professor was expected to produce an inaugural essay setting out his programme, and Klein's — the Vergleichende Betrachtungen über neuere geometrische Forschungen, comparative considerations on recent geometrical research — is the document he is remembered for.

It is worth being clear that he was very young, in a minor post, addressing a problem everybody agreed was real. Nineteenth-century geometry had split. There was projective geometry, affine geometry, the geometry of the sphere, and after Bolyai and Lobachevsky there were geometries in which Euclid's parallel assumption simply failed. They had different results and different methods, and nobody could say how they related — or whether some were more legitimate than others.

Why this matters

Mathematics can fragment, and when it does somebody has to say how the pieces relate; the demand is as real as any technical problem.

A very young professor with an inaugural essay to write. What would you ask him?

Ask Klein

• “What had gone wrong with geometry by 1872?”
• “What did you and Sophus Lie talk about in Paris?”
• “How does a twenty-year-old finish a dead professor's book?”
• “Were the new geometries thought legitimate at the time?”
• “What is an inaugural programme supposed to do?”

Chapter 2 · What the Transformations Leave Alone

The Erlangen programme, 1872

1872 · Erlangen

The idea is one sentence long and it reorganised a subject.

A geometry, Klein said, is not a collection of facts about figures. It is the study of the properties that a particular family of transformations leaves unchanged. Fix the family, and you have fixed the geometry.

The consequences follow immediately. Ordinary school geometry is what survives sliding, turning and reflecting: lengths, angles, areas, and every relation built from them. Allow enlargements as well, and lengths stop being meaningful while shapes, angles and ratios survive — that is the geometry of similarity, which is why every triangle with the same angles behaves alike. Allow the transformations of projection, as a draughtsman does, and even angles go, while the property of lying on a straight line and the crossing of lines survive. Allow any continuous deformation at all and almost everything goes except connectedness.

So the geometries are not rivals. They are a hierarchy, ordered by how much freedom their transformations have: the more you allow, the less survives, and the fewer things there are to prove.

It also reframes what a mathematician is doing. Instead of asking what is true of this figure, ask what is preserved. That question turned out to be portable far beyond geometry — it is how physicists now think about conservation laws, and how a great deal of twentieth-century mathematics is organised.

And almost nobody noticed. The essay was printed in a small edition at Erlangen and attracted little attention for roughly twenty years. It became famous partly because Klein, who was by then eminent and had a talent for institutions, arranged for it to be reprinted and translated.

Why this matters

Asking what a transformation preserves, rather than what it does, is one of the most portable ideas in mathematics — and it began as a young man's job talk.

One sentence that ordered a subject. What would you ask him?

Ask Klein

• “What is a geometry, on your account?”
• “How do the different geometries end up in an order?”
• “Why ask what is preserved rather than what is true?”
• “Why did nobody notice for twenty years?”
• “Where else does this way of thinking apply?”

Chapter 3 · The Race with Poincaré

And what it cost

1881 – 1883 · Leipzig

In 1881 Klein, then at Leipzig, read a note by a young and almost unknown Frenchman named Henri Poincaré on a class of functions with remarkable symmetry properties. Poincaré called them Fuchsian, after Lazarus Fuchs. Klein wrote to him immediately to say the name was wrong and that he himself had been working in the area.

What followed was a correspondence of about two years and one of the great competitive episodes in mathematics. Both men were racing towards the same set of results about automorphic functions. The letters are outwardly civil and unmistakably a contest; each announces progress, and each is plainly working flat out to get there first.

Poincaré was five years younger, faster, and — as Klein came to recognise — better. He got the central theorem.

Klein worked himself to a standstill. In 1882 he suffered what was described as a collapse: exhaustion, depression, and an inability to work. He was thirty-three. He recovered enough to function, but by his own account he never regained his powers as a research mathematician, and his significant original work is essentially all before it.

He was candid about this. It is unusual, and worth noticing, that a man of his eminence said openly that he had burned himself out at thirty-three and that the research life was over.

What he did next is the answer to the question of what a mathematician does when the mathematics stops. He became the greatest organiser the subject has had.

Why this matters

A frank account of burnout at thirty-three from a mathematician who then found a second and arguably larger role.

A two-year race, lost, and a collapse. What would you ask him?

Ask Klein

• “What were you and Poincaré racing towards?”
• “What actually happened to you in 1882?”
• “Was the race worth what it cost?”
• “How do you carry on when the research stops?”
• “What did you think of Poincaré afterwards?”

Chapter 4 · Building Göttingen

Hilbert, engineers, and the first women

1886 – 1913 · Göttingen

Klein moved to Göttingen in 1886 and spent nearly forty years turning it into the mathematical centre of the world.

He did it deliberately and with methods that were unusual for a mathematician. He recruited: most importantly David Hilbert, in 1895, over considerable opposition. He built a mathematical reading room where the journals were open and people could work — a small thing that changed how a department functioned. He created a collection of mathematical models, plaster and string constructions of surfaces, so that ideas could be handled.

He cultivated industry and engineering, raising money from firms and founding institutes for applied mechanics and applied electricity, on the argument that mathematics that touches nothing eventually stops being asked interesting questions. Many colleagues found this vulgar.

And he supported the admission of women at a time when German universities did not admit them. In 1895 he supervised Grace Chisholm Young, an Englishwoman who had been unable to take a doctorate in England, and she became the first woman to receive a doctorate in mathematics in Germany by the regular examination process. He supervised Mary Frances Winston, an American, shortly after, and later supported Emmy Noether's presence at Göttingen against the faculty.

The result was that between about 1895 and 1933 Göttingen was where you went. Hilbert, Minkowski, Landau, Courant, Noether, Weyl — and visitors from everywhere, including nearly every significant American mathematician of the period.

It ended in 1933, eight years after Klein's death, when the Nazi dismissals emptied it in a matter of months.

Why this matters

Building the conditions for other people's work is a form of mathematical achievement, and Göttingen is the clearest case of it.

Forty years spent making a place rather than a theorem. What would you ask him?

Ask Klein

• “How do you build a mathematical institute?”
• “Why did you fight to bring Hilbert?”
• “Why did your colleagues object to the engineers?”
• “Why did you support the admission of women?”
• “What makes a place somewhere people want to come?”

Chapter 5 · Elementary Mathematics from an Advanced Standpoint

The campaign nobody expected

1900 – 1925 · Göttingen · Rome

In the last quarter of his life Klein turned to school teaching, which was widely regarded as a subject beneath a research mathematician's notice.

His diagnosis was specific. A student arrives at university having learned school mathematics one way, is taught university mathematics in a way that appears to have no connection to it, and then — if they become a teacher — reverts entirely to the school version and teaches as they were taught. He called it the double discontinuity. The consequence is that advances in mathematics never reach schools at all.

His remedy was a set of lecture courses, published as Elementary Mathematics from an Advanced Standpoint, going back over the mathematics a teacher will actually teach and showing what it looks like from above: why the school rules are true, where they come from, what they are special cases of. The books are still read.

He campaigned for the inclusion of functions and of the calculus in the school curriculum, on the ground that they were the organising ideas of the subject and that leaving them out left school mathematics a heap of techniques. In 1908 he was made chairman of the first International Commission on the Teaching of Mathematics, which is the ancestor of the international bodies that exist now.

He signed the Manifesto of the Ninety-Three in 1914, defending German conduct at the start of the war, which damaged his international standing and which he later regretted.

He died at Göttingen in 1925, aged seventy-six, having spent thirty years making a place for other people to work and twenty arguing that mathematics teachers deserved better than they were getting.

Why this matters

The gap between what a subject knows and what its schools teach is a real problem, and it took a first-rank mathematician to say so and be listened to.

A research mathematician who spent his last decades on schools. What would you ask him?

Ask Klein

• “What is the double discontinuity?”
• “Why should a teacher see school mathematics from above?”
• “Why fight to get functions into schools?”
• “Did your colleagues think this was a waste of you?”
• “You put your name to the war manifesto of 1914. Why?”

What Klein changed

The Erlangen programme of 1872 is how mathematicians relate the geometries to one another, and its central move — describe a subject by what its transformations leave unchanged — travelled far beyond geometry into physics and much of twentieth-century mathematics. Göttingen under Klein and Hilbert was the mathematical centre of the world for four decades, and his support for women there opened doors that had been formally shut. His analysis of the double discontinuity between school and university mathematics is still the standard diagnosis in teacher education.

A debate that continues

The Erlangen programme was little noticed for around twenty years and became famous partly because Klein reissued it; how much it shaped the mathematics of the 1870s and 1880s as opposed to describing it afterwards is arguable. His work overlaps substantially with Sophus Lie's on continuous transformations and the two men's contributions are not cleanly separable. He signed the Manifesto of the Ninety-Three in 1914, which damaged his standing abroad.

Keep exploring — ask Klein

• “Is a property that survives no transformation a property at all?”
• “Can building the conditions for others' work count as one's own achievement?”
• “Why does what a subject knows take so long to reach the people teaching it?”

Related lives

• Carl Friedrich Gauss — Few, But Ripe
• Leonhard Euler — Who Gave Us The Notation
• Euclid — Who Made Proof The Standard

Related themes

Transformations and symmetry · Geometry after Euclid · Mathematics education