Saturday, August 10, 2013

Enrico Betti: an anniversary

Enrico Betti was an Italian mathematician and politician who was born in October 1823, i.e. 190 years ago, and died on August 11th, 1892 – we will have an anniversary tomorrow.

He is most famous for a 1871 paper on topology that explained the Betti numbers – a term that was later coined by Henri Poincaré – which I used in my fairy-tale about the Euler characteristic. While Betti was a one-hit wonder of a sort, his life was pretty interesting.

He was born in Pistoia, Tuscany, in 1823 and his father died when he was a baby. His mother may really be credited for all of his early education. However, mathematics and physics is probably what he only studied at the University of Pisa under Ottaviano-Fabrizio Mossotti – where he got a degree at the age of 23.

Mossotti (famous for some work on correcting coma and spherical but not chromatic aberation, and for some ideas about the physics of the brain) brought Betti not only to some secrets of mathematics and physics but also to the military. At that time, Italy was getting close to independence and unification, ideas held dear by Mossotti and Betti. Mossotti himself established the Tuscany University Batallion. Betti joined the unit, trained using the Dismounted Soldier Training System ;-), and fought in two battles against Austria and France that were protecting their own interests, not quite compatible with the independence and unity of Italy.

Betti would work as an assistant in the University of Pisa after he got degrees but years later, he would be hired as a high school teacher in his hometown of Pistoia as well as Florence. However, when he was 34, he was finally appointed as a professor of algebra in Pisa. He was able to travel a lot and befriend numerous leading mathematicians, including Riemann.

In Pisa, he would later switch from algebra to the job of the chair of analysis and higher geometry.

In 1859, there was a war against Austria. Two years later, unified Italy was finally established. At that time, this unified Italy didn't include some irrelevant towns such as Rome and Venice ;-) but it was already there. One more year later, Betti would become a member of the Parliament and he would even serve in the government of the new country.

One more year later, in 1863, Riemann visited Betti and the friendship got some new energy. Betti would go through lots of jobs and functions in the national politics and the university politics (he chaired mathematical physics after Mossotti, celestial mechanics, and analysis and geometry, as I mentioned, at various moments). Also, Betti loved classical culture as well as Euclid whose work he helped to translate.

The Italian politics was violent and exciting at those times. In 1866, Venice would finally join the new Italian Kingdom. Rome was also supposed to be overtaken but the troops of one country, France, did everything they could to defend the thesis that Rome had nothing whatsoever to do with Italy. ;-) Finally, the Italian soldiers succeeded in 1870 and turned Rome into the new capital. In 1874, Betti would become an undersecretary of state for education of this "already pretty credible" Italian state. He missed the academic life, meditation, and close friends. However, he didn't learn the lesson because in 1874, he became a senator and he would miss the noble philosophical purpose of the academic life again. When I read these quotes, I tend to think that he was just bullshitting, trying to look more idealistic than he was, and he probably enjoyed life in politics quite a lot.

Over the years, Betti has actually done much more mathematical work than just the Betti numbers. He would prove some theorems in the Galois theory – e.g. the closure of the Galois group under multiplication. Some of these proofs contained obscurities and errors, however. He also showed how elliptic functions were useful to write a solution to the quintic equation.

Influenced by the 1863 visit of Riemann, Betti did some work in theoretical physics, in particular potential theory and elasticity. Betti has obviously done various things that were characteristic for his lifetime but I suppose that in mathematics and physics, we will always remember him as the originator of the Betti numbers and nothing else.


  1. yes, where the Betti numbers beautifully relate to the number of critical points of a Morse function :)

  2. The vast majority of people, even scientists and mathematicians, have no hits.

  3. I don't know in exactly what way Betti introduced Betti numbers. However, it was certainly Emmy Noether (in 1925, at a dinner in Brouwer's house) who showed that they are the dimensions of singular homology groups of polyhedra and that it is better to study these groups rather than just Betti numbers. In fact, it was Noether who first defined singular homology groups and named them "Betti groups".

    The name "Betti homology" is still used for singular homology but very rarely and it seems only by algebraic geometers (perhaps because Italians played such a big role in the development of classical algebraic geometry).

    Of course the critical points of a Morse function on a smooth manifold convey vastly more information about the manifold than just its Betti numbers.