Up to now our series on people who were regarded as “crackpots” but who turned out to be right has not included any mathematicians. That is because most issues in mathematics can be resolved quite quickly. The logic of a claimed proof can normally be checked and either verified or debunked beyond doubt by examining its logic.

Not that mathematics is without its controversies. These have mostly arisen when a mathematician introduced a new level of abstraction whose validity was disputed. Here are a few classic examples just to give a flavour:

- 1545: Gerolamo Cardano introduced imaginary numbers which were further developed by his disciple Rafaello Bombelli, but they were not widely accepted until the work of Leonhard Euler in the eighteenth century.
- 1830: Nikolai Lobachevsky introduced hyperbolic geometry which was an early example of non-euclidean geometry, but his work was rejected by the St. Petersburg Academy of Sciences. Gauss and Bolyai had similar ideas at the same time but it was not until later when Riemann introduced more general non-euclidean geometries that the validity of the work started to be accepted.
- 1852: Ludwig Schläfli classified the six regular polytopes in four dimensions but his manuscript was rejected by the Austrian Academy of Sciences and then by the Berlin Academy of Sciences and was not published in full until after his death.
- 1873: Georg Cantor analysed transfinite numbers but his work was opposed by a number of mathematicians who believed that such entities do not exist. Leopold Kronecker was especially critical and prevented publication of the work in
*Crelle’s Journal.* - 1888: David Hilbert’s proof of his basis theorem was heavily criticised for being non-constructive. Paul Gordan who had originated the problem was particularly unimpressed and refused to accept the solution.
- 1945: When Saunders MacLane and Samuel Eilenberg tried to publish their seminal work on category theory it was at first rejected as being devoid of content.

Some of these examples could be justified as cases of “crackpots” who were right, and there are others, but a more striking story is that of Roger Apéry. His mathematics was initially rejected, not because it was too abstract, but rather because his colleagues would not believe that it could be right.

Roger Apéry was a Greek-French mathematician born in 1916. As a working mathematician in France he took a rebellious political and philosophical position that was not liked by his contemporaries. When he was asked to join the Bourbaki team who were famously compiling an encyclopedia of mathematics under the pseudonym, Apéry declined because he saw mathematics as a more individual pursuit. This led to his isolation from French mathematicians.

At the age of 61 Apéry was an undistinguished mathematician suffering from the dislike of his colleagues and his own problems with alcoholism. At such a late age mathematicians are not normally expected to produce ground breaking results so it is easy to understand the level of skepticism that greeted his announcement of a proof that the number ζ(3) is irrational.

The incredulity heightened when it was seen that the method of proof was very basic and used methods that could have been understood by mathematicians such as Euler who died nearly 200 years earlier. Many claimed proofs of old problems are rejected today at a glance simply because an experienced mathematician “knows” that methods that are too elementary can not solve the problem. All such avenues should have been explored before.

In 1978 he presented a lecture on his proof at the Journées Arithmétiques de Marseille which was greeted with doubt and disbelief. Each step he wrote on the blackboard appeared to be a remarkable identity that his audience considered unlikely to be true. When someone asked him “where do these identities come from?” he replied “They grow in my garden.” obviuosly this did not boost anyone’s confidence.

Nevertheless, a few mathematicians recognised that there was something significant in the proposed proof. They checked the identities numerically and found that they did indeed seem to hold. It was not long before the full validity of Apéry’s work was confirmed and the skeptics were forced to eat their words.

There are those who would argue that category theory is still devoid of content😉

Apery’s paper is a classic now and is cited countless number of times. Perhaps he would be immortal just for his very simple proof of a single result.