Niels H Abel - Collected Quotes

"The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations." (Niels H Abel, "Memoir on algebraic equations, proving the impossibility of a solution of the general equation of the fifth degree", 1824)

"I shall devote all my efforts to bring light into the immense obscurity that today reigns in Analysis. It so lacks any plan or system, that one is really astonished that so many people devote themselves to it - and, still worse, it is absolutely devoid of any rigour." (Niels H Abel, "Oeuvres", 1826)

"In analysis, one is largely occupied by functions which can be expressed as powers. As soon as other powers enter - this, however, is not often the case - then it does not work any more and a number of connected, incorrect theorems arise from false conclusions." (Niels H Abel, [Letter to Christoffer Hansteen] 1826)

"One can divide the entire circumference of the lemniscate into m equal parts by ruler and compass alone if m is of the form 2^n or a prime of the form 2^n + 1, or if m is a product of numbers of these two kinds. This theorem, as one sees, is exactly the same as the theorem of Gauss for the circle." (Niels H Abel, "Euvres completes de Niels Henrik Abel", 1881)

"If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation." (Niels H Abel) 

"It appears to me that if one wishes to make progress in mathematics, one should study the masters and not the pupils." (Niels H Abel)

"The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes." (Niels H Abel) 

"With the exception of the geometric series, there does not exist in all of mathematics a single infinite series whose sum has been determined rigorously." (Niels H Abel)