Riemann, Bernhard - On Inspiration and inner speech
Type of Spiritual Experience
A description of the experience
Jacques Hadamard – The Psychology of Invention in the mathematical field
Whenever nationalistic passions come into play, one will find tendentious interpretations of the facts. In a rather detailed article [Revue des Mondes 1915] Duhem depicted German scientists, especially mathematiciens, as lacking intuition or even deliberately setting it aside. It is especially hard to understand how he can characterise in that way Bernhard Riemann who is undoubtedly one of the most typical examples of an intuitive mind.
His argument rests on the claim that for certain proofs, 'series' used to be used in preference to 'integrals' and the use of 'series' looks more logical and the use of integrals more intuitive. Perhaps there is some [backing for the argument] as Weierstrass – a most evident logician – used 'series' whilst Cauchy or Hermitte used integrals – though this was also true of Riemann. When Poincare compared Weierstrass and Riemann he concluded that Riemann is typically intuitive and Weierstrass typically logical. ...................................
Bernhard Riemann, whose extraordinary intuitive power we have already mentioned, has especially renovated our knowledge of the distribution of prime numbers, also one of the most mysterious questions in mathematics. He has taught us to deduce results in that line from considerations borrowed from the integral calculus : more precisely, from the study of a certain quantity, a function of a variables which may assume not only real, but also imaginary values. He proved some important properties of that function, but pointed out several as important ones without giving the proof. At the death of Riemann, a note was found among his papers, saying
'These properties of (s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it."
We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one. The question concerning that last one remains unsolved as yet, though, by an immense labour pursued throughout this last half century, some highly interesting discoveries in that direction have been achieved. It seems more and more probable, but still not at all certain, that the "Riemann hypothesis" is true. Of course, all these complements could be brought to Riemann's publication only by the help of facts which were completely unknown in his time; and, for one of the properties enunciated by him, it is hardly conceivable how he can have found it without using some of these general principles, no mention of which is made in his paper.
In general, Riemann's intuition, as Poincare observes, is highly geometrical; but this is not the case for his memoir on prime numbers, the one in which that intuition is the most powerful and mysterious: in that memoir, there is no important role of geometrical elements.