This year marks the 80th anniversary of the death of Felix Hausdorff, which is a perfect excuse to dedicate a couple of posts to talk about the inextricable link between mathematics and the human condition, because in the vital journey of this brilliant German mathematician, everything from the most abstract mathematics to the most intense emotional circumstances came together, especially in his terrible end where Hausdorff gave a supreme example of dignity.
Felix Hausdorff was born in Breslau in 1868. He studied mathematics and astronomy in Leipzig, Freiburg and also in Berlin. Although his early mathematical works fall into the category of applied mathematics – in his case, astronomy and optics – Hausdorff ended up as a “pure mathematician”. And there are perhaps no better adjectives to describe most of his mathematical production than those usually applied to Borges’ fictions: “imaginary”, “paradoxical”, “ironic”, “labyrinthine”.
Certainly, the Hausdorffian pinnacle of the labyrinthine is his concept of dimension. With it, he enriched the classical concept and allowed a better classification of objects according to it. Thus, the fractals, labyrinthine objects par excellence, made so famous and popular by Benoît Mandelbrot in the last quarter of the 20th century, are precisely described as sets whose Hausdorff dimension is not a natural number.
Hausdorff also considered the antecedent of what is nowadays called “inaccessible cardinals”. These infinite sets are mental entelechies with an unmistakable sense of irony. The characteristic that determines them is their enormous immensity; but this amorphous gigantism makes them so improbable that it is not known whether they really exist. That is the irony of it: being so enormous in size, no one would have thought that the mind’s eye would have such difficulty in seeing them!
And it is not only the labyrinthine or the ironic that we find in Hausdorff’s mathematics; the contradictory is also the main protagonist. Certainly, the Hausdorffian summit of the contradictory is the description he made in his book Principles of Set Theory of the paradoxical decomposition of a spherical surface, the origin of the de-construction that ten years later the Poles Banach and Tarski would make of a solid sphere, and which makes it possible to divide it into pieces -five, for example- and obtain, by fitting them together, two spheres identical to the starting one; or divide a pea into pieces, suitably designed, so that by rearranging them in a suitable way we can obtain a solid sphere the size of the Sun. It is the mathematical version of the Gospel multiplication of the loaves and fishes.
Hausdorff had other intellectual interests besides mathematics. As a teenager he wanted to study music and become a composer and, although his professional career later took a different path, he composed some pieces and was always an accomplished pianist.
Under the pseudonym Paul Mongré, Hausdorff wrote poetry, philosophical essays and also a satirical play. His literary output was mainly concentrated in the decade 1896-1906. Philosophically, he was strongly influenced by Nietzsche and Schopenhauer, and postulated the advantage of a certain elitist individuality over egalitarian societies. Hausdorff used to break up the serious philosophical discourse of his books with, shall we say, less elevated reflections on selfishness, hedonism, love, passion, Mozart’s music, or hypnosis – it is not difficult to see the influence of Freud in his writings. One of his aphorisms states: “When we have no woman to love, we love humanity, science or eternity […] Idealism, which always points to the lack of something better, is a substitute for eroticism”.
In order for the reader to appreciate Hausdorff’s poetry, here is one of his poems, entitled Infinite Melody (Unendliche Melodie):
Striding on trembling planes,
Swinging the brazen primal sound,
To dancing wet smoke rings
Arching and widening the soul
On edges and corners and sides
Do not push and force the gaze,
Striding on trembling planes,
That swing the brazen primal sound,
No clinging to details,
A private singing
Without origin springing sound
Shapeless swimming and gliding,
Striding on trembling planes…
It was his theatrical play, however, that was most successful. It shares its title with a drama by our own Calderón, El médico de su honra, although Hausdorff’s approach is rather more satirical and wild: the play tells the story of a Prussian architect, an idealist, who, having seduced the wife of a state councillor, has to fight a duel with him. But when the appointed day and time arrived, the duel had to be called off due to the alarming state of drunkenness of both contestants and their respective witnesses. As a result of the scandal, the councillor loses his job but ends up reconciled with his wife. The play was performed in Berlin and Hamburg and, according to local chronicles, was warmly received (here you can see a present-day performance, in German, of course).
Hausdorff was a professor at the universities of Leipzig (1902-1910), Greifswald (1913-1921) and Bonn (1910-1913 and 1921-1935). He retired from the latter in March 1935; he was 67 years old at the time, and as he himself had predicted a few years earlier, things were beginning to change in Germany.
In the second part of this entry, I will tell how tragically these differences affected Hausdorff.
References
Czyż, J., Paradoxes of measures and dimensions originating in Felix Hausdorff’s ideas, World Scientific, Londres, 1994.
Durán, Antonio J., Pasiones, piojos, dioses … y matemáticas, Destino, Barcelona, 2009.
Durán, Antonio J., La poesía de los números, RBA, Barcelona, 2010.
Segal, S.L., Mathematicians under the nazis, Princeton University Press, Princeton, 2003.
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