Mathematics is enough for me

At Charendon, André Bloch was a patient of the renowned French psychiatrist Henri Baruk. In his book Des hommes comme nous, Baruk writes about Bloch:

Every day for forty years, this man sat at his table in a small corridor leading to his room, without leaving his desk except to eat, in the afternoon. He spent his time writing algebraic or mathematical signs on scraps of paper, or reading and annotating mathematical books whose level was that of the great specialists in this field… At half past six he would close his notebooks and books, eat his supper, and immediately return to his room, fall into bed and sleep until the next morning. While other patients constantly claimed his freedom, he was completely happy studying his equations and keeping his correspondence up to date.

When his doctor advised him: “You should go out in the yard and walk like the rest”, he replied: “Mathematics is enough for me”.

He was extremely polite, caring for his nurses who held him in high esteem: “An ideal patient”. From Charendon he corresponded with many mathematicians, especially G. Valiron, H. Cartan, J. Hadamard, G. Pólya, E. Picard, P. Montel and G. Mittag-Leffler.

Bloch’s main result is really surprising. Consider the set of functions \(f(z)=z+a_2z^2+a_3z^3+\cdots\) analytic for \(|z|<1\). We have that \(f(0)=0\) and \(f'(0)=1\), so that \(f\) transforms a small neighborhood of \(0\) bijectively into another small neighborhood of \(0\). Therefore there exists a power series \(g(z)=z+b_2z^2+\cdots\) convergent in a small neighborhood of \(0\) such that \(f(g(z))=z\). One might imagine that the radius of convergence of \(g(z)\) is going to be larger than a certain amount, say \(1/20\), whatever \(f\) is under the above conditions. But this is not true: the function $$f(z)=\frac{\varepsilon}{2}\Bigl[\Bigl(\frac{1+z}{1-z}\Bigr)^{1/\varepsilon}-1\Bigr]=z+\frac{1}{\varepsilon}z^2+\cdots$$ is analytic on \(|z|<1\) and does not take the value \(-\varepsilon/2\), so the radius of convergence of \(g\) in this case is \(<\varepsilon/2\). 

What Bloch proves is that a disc of radius \(>1/72\) does exist, but not necessarily with centre at the origin, so that the inverse of \(f\) exists in this disc and is analytic.

Theorem (Bloch): There exists a constant \(b>\frac{1}{72}\) with the following property: if \(f\) is a holomorphic function on the unit disk \(|z|< 1\) such that \(|f'(0)|=1\), then there exists a disk of radius \(b\) and an analytic function \(\varphi\) defined on this disk such that \(f(\varphi(z))=z\) for all \(z\) on this disk.

The bound \(\frac{1}{72}\) is not the best possible one. The number \(B\) defined as the supremum of all \(b\) for which the theorem is true is called Bloch’s constant. The best known bounds for \(B\) are $$0.433025\approx\frac{\sqrt{3}}{4}+2\cdot 10^{-4}\le B\le \frac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)\sqrt{1+\sqrt{3}}}\approx 0.471862$$ The upper bound is due to Ahlfors and Grunsky, who conjecture that their upper bound is the true value of \(B\).

Bloch is also the origin of a philosophical principle in Mathematics: $$\text{ Nihil est in infinito quod non prius fuerit in finito.}$$ Nothing happens in the infinite that does not happen before in the finite.

For example, his theorem is the finite version of Valiron’s theorem:

Theorem (Valiron): If \(f\) is a non-constant integer function, there exists a disk \(D\) of arbitrarily large radius and an analytic function \(\varphi\) on \(D\) such that \(f(\varphi(z))=z\) for all \(z\in D\). 

And Picard’s theorem

Theorem (Picard): If \(f\) has an essential singularity at \(a\), then in every open set containing \(a\) the function \(f\) takes  all complex values infinitely many times, except for at most  one single value.

corresponds by Bloch’s principle with

Theorem (Schottky): For every \(M>0\) and \(r\in(0,1)\) there exists a constant \(C>0\) such that the following implication holds: If \(f\) is holomorphic in the unit disk, \(|f(0)|\le M\) and the range of \(f\) omits the value \(0\) and \(1\), then \(\sup_{|z|\le r}|f(z)|\le C\).

Why was Bloch in an asylum? The story we now know comes from several sources.

George Valiron, in 1949, commented on the life of André Bloch as follows:

Bloch was born in Besançon, France, on 20 November 1893. Carrus, a professor of analysis, had recognised André’s talent and suggested that he and his brother Georges, born less than eleven months after him, should go to the École Polytechnique. The following year, in October 1910, I had both brothers in my class. André was already showing his interest in the abstract properties to which he later contributed so significantly. But he hardly spoke at all and did not bother to prepare for his exams. Georges was more communicative and perhaps as good a mathematician as his brother. Georges was at the top of the class and clearly the best in written exams. André was last in my class of eleven students. But he was lucky to have Vessiot in the oral exam. Vessiot recognised André’s aptitude and gave him 19 out of 20 points in the oral exam. In October he and his brother entered the École Polytechnique, André with number 151 and Georges with 229. In 1914 they had to leave because of the war.

Toward 1920 or 1921, I learned about the drama that made André a recluse. I heard about it from a former pupil in Besançon. André’s brother Georges had been released from military service and had returned to the Polytechnique on 7 October 1917. André was on convalescence leave towards the end of 1917. On 17 November 1917, at the end of this leave and three days before his 24th birthday, he killed his brother Georges, his uncle and his aunt in a stroke of madness.

Declared mentally ill, he was confined to the “house of health” in Saint-Maurice, where he remained until his death on 11 October 1948.

Additional information on Bloch’s life is provided by H. Cartan and J. Ferrand. The following is part of their account:

André Bloch was born on 20 November 1893 in Besançon. His parents were of Alsatian origin, his father was a watchmaker in town; his parents died prematurely. Raised in Besançon with his two brothers, André was a brilliant student at the local Lycée, and in 1912 he passed the competitive entrance examination to the École Polytechnique at the same time as his younger brother Georges, born on 13 October 1894.

After a year of military service, followed by their first year of study at the École Polytechnique, the two brothers were mobilised in 1914. André was assigned to General De Castelnau’s barracks in Nancy as a second lieutenant in the artillery. After months at the front, he fell from the top of an observation post during a bombardment. The severe shock involved several hospital stays, interrupted by periods of convalescence, and rendered him unfit for active service. The brother, Georges, received a head wound and lost an eye. In October 1917 he resumed his studies, which had been interrupted by the war.

In this period the incomprehensible drama described by Valiron took place: on 17 November 1917, in the course of a meal at the family’s flat on Boulevard Courcelles in Paris, André killed Georges (by stabbing him with a knife, according to rumours among André’s fellow students), as well as his uncle and aunt (whose names we do not publish out of consideration for the family). He then ran screaming into the street and let himself be arrested without resistance. (It seems that this painful affair was hushed up at the time and was not published in the press. Even the companions of the Bloch brothers did not know the details of the drama. It must be borne in mind that France was at war and the murderer was an officer on leave). Judged not responsible for his actions, André was confined in the Saint-Maurice psychiatric hospital, also called the Charenton house, in the suburbs of Paris. He did not leave the hospital for 31 years: on 21 August 1948, ill with leukaemia, he entered the Sainte-Anne hospital in Paris for an operation. He died there on 11 October 1948, just when the mathematician Benjamin Amira from Jerusalem had come to visit him.

What were his motivations? The most authoritative voice on this subject is that of his doctor. According to him Bloch exhibited a morbid rationality. He killed his relatives to carry out what he saw as a duty of eugenics. He had to eliminate a branch of his family that he considered defective. Much later, at the end of his life, his other younger brother came to visit him from Mexico. André asked detailed questions about the whole family. But the next day, he told Baruk: It’s a question of mathematical logic. There have been mental illnesses in my family. The destruction of the whole branch follows as a consequence. I started my task at the time of that famous meal. I haven’t finished yet. I wanted to know how things were going. To the doctor’s protest, he added – you use emotional language. Above that is mathematics and its laws. You know well that my philosophy is inspired by pragmatism and absolute rationality. I have applied the example and principles of a famous mathematician of Alexandria, Hypatia.

Cartan and Ferrand wonder what Bloch could be referring to when he speaks of Hypatia. There is nothing that we know of Hypatia that could link her to this madness. They conclude that perhaps Bloch is referring to a passage in a novel about Hypatia’s life published in 1853 by C. Kingsley. Written some 1400 years after Hypatia’s death, what Kingsley said about Hypatia’s feelings has no credibility whatsoever.

He was right about one thing: there were mentally ill people in her family.

 

Learn more

I have worked from several sources, an article in the journal Mathematical Intelligencer, unfortunately hijacked by a publisher:

Douglas M. Campbell, Beauty and the beast: the strange case of André Bloch, Math. Intelligencer 7 (1985) 36–38. (https://doi.org/10.1007/BF03024484)

A few years later in the same journal Henri Cartan and Jacqueline Ferrand provide further information, clarifying some of the points in Campbell’s article:

Henri Cartan and Jacqueline Ferrand, The case of André Bloch, Math. Intelligencer 10 (1988) 23–26. (https://doi.org/10.1007/BF03023847)

More complete, containing also a list of Bloch’s published works and freely accessible, only in French, we have

Henri Cartan and Jacqueline Ferrand, Le cas André Bloch, Cahiers du séminaire d’histoire des mathématiques, 9, Univ. Paris VI, Paris (1988) 210–219.

I have also used the book

Reuben Hersh and Vera John-Steiner, Loving + hating mathematics, Princeton University Press, Princeton NJ, 2011 Challenging the myths of mathematical life.

There is a video that tries to explain the context that can make it easier to understand Bloch’s motivations: Murder by Numbers.

On this occasion, the featured image is one of Jim Denevan’s compositions. It is intended to illustrate Bloch’s theorem.