16/04/2025

Folha: 'Henri Poincaré and Bertrand Russell, a duel of giants'

Reproduction of Marcelo Viana's column in Folha de São Paulo.

On one side, Bertrand Russell (1872–1970): mathematician, logician, philosopher, intellectual and activist, an English aristocrat and member of the House of Lords, who married four times and lived to be 98 years old.

On the other hand, Henri Poincaré (1854–1912): one of the greatest mathematicians of all time, physicist, engineer, philosopher, from the French intellectual bourgeoisie —his cousin Raymond was president of France—, with fragile health throughout his life, which made him a victim of bullying at school.

Russell wanted to improve Cantor's set theory, to rid it of the contradictions resulting from the use of infinite sets . Poincaré said that "future generations will consider set theory a disease from which they have managed to rid themselves," adding that "actual infinity does not exist: what we call infinity is merely the endless possibility of creating new objects, no matter how many objects already exist."

For Russell, " mathematics and logic are the same thing," since "all mathematics follows logical premises and uses only concepts that can be defined in logical terms." Poincaré responded that "it is through logic that we prove, but it is through intuition that we discover" and "logic is sterile unless it is irrigated by intuition."

It is clear that they were destined to be adversaries . What is surprising is that Poincaré remained silent for so long, even after Russell published the first volume of his book "Principles of Mathematics" in 1903.

But when, in March 1906, Russell presented his ideas in the scientific journal Proceedings of the London Mathematical Society, Poincaré launched an attack: in an article published in the journal Revue de Méthaphysique et Morale, he dismantled Russell's theory. It was the beginning of a duel that lasted for years.

A respectful duel. Russell was a British lord, and it was said of Poincaré that "among the scientists who lived in the last century, only he achieved the miracle of never having made a single enemy, a single person hostile to him in science." But that didn't stop the two adversaries from exchanging blows in the realm of ideas.

In September of the same year, Russell did not hesitate to play on his opponent's turf: in a publication in the Revue he stated that "Monsieur Poincaré dislikes it because he does not understand it." Poincaré rebutted this in an article in the same magazine, to which Russell responded in 1908 in the American Journal of Mathematics. Back in the Revue, in 1909, Poincaré published "The Logic of Infinity," to which Russell replied in 1910 with "The Theory of Logical Types."

They could have continued like this for years, if it hadn't been for the fact that Poincaré fell ill and, due to complications following surgery, died shortly afterwards.

To read the full article, visit the newspaper's website.