(source) 
Niels Henrik Abel
(5 Aug 1802  6 Apr 1829)

Science Quotes by Niels Henrik Abel (12 quotes)
La théorie des séries infinies en général est justqu’à présent très mal fondée. On applique aux séries infinies toutes les opérations, come si elles aient finies; mais cela estil bien permis? Je crois que non. Où estil démonstré qu/on ontient la différentielle dune série infinie en prenant la différentiaella de chaque terme. Rien n’est plus facile que de donner des exemples où cela n’est pas juste.
Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible? I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so.
Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible? I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so.
— Niels Henrik Abel
[About Gauss’ mathematical writing style] He is like the fox, who effaces his tracks in the sand with his tail.
— Niels Henrik Abel
[In reply to a question about how he got his expertise:]
By studying the masters and not their pupils.
By studying the masters and not their pupils.
— Niels Henrik Abel
Cauchy is mad, and there is no way of being on good terms with him, although at present he is the only man who knows how mathematics should be treated. What he does is excellent, but very confused…
— Niels Henrik Abel
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 Henrik Abel
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 Henrik Abel
It [analysis] lacks at this point such plan and unity that it is really amazing that it can be studied by so many people. The worst is that it has not at all been treated with rigor. There are only a few propositions in higher analysis that have been demonstrated with complete rigor. Everywhere one finds the unfortunate manner of reasoning from the particular to the general, and it is very unusual that with such a method one finds, in spite of everything, only a few of what many be called paradoxes. It is really very interesting to seek the reason.
In my opinion that arises from the fact that the functions with which analysis has until now been occupied can, for the most part, be expressed by means of powers. As soon as others appear, something that, it is true, does not often happen, this no longer works and from false conclusions there flow a mass of incorrect propositions.
In my opinion that arises from the fact that the functions with which analysis has until now been occupied can, for the most part, be expressed by means of powers. As soon as others appear, something that, it is true, does not often happen, this no longer works and from false conclusions there flow a mass of incorrect propositions.
— Niels Henrik 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 Henrik Abel
The following theorem can be found in the work of Mr. Cauchy: If the various terms of the series u_{0} + u_{1} + u_{2} +... are continuous functions,… then the sum s of the series is also a continuous function of x. But it seems to me that this theorem admits exceptions. For example the series
sin x  (1/2)sin 2x + (1/3)sin 3x  …
is discontinuous at each value (2m + 1)π of x,…
sin x  (1/2)sin 2x + (1/3)sin 3x  …
is discontinuous at each value (2m + 1)π of x,…
— Niels Henrik Abel
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 Henrik Abel
There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general and it is extremely peculiar that such a procedure has led to so few of the socalled paradoxes.
— Niels Henrik Abel
With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series the sum of which has been rigorously determined. In other words, the things which are the most important in mathematics are also those which have the least foundation.
— Niels Henrik Abel
Quotes by others about Niels Henrik Abel (3)
No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young man's game. … Galois died at twentyone, Abel at twentyseven, Ramanujan at thirtythree, Riemann at forty. There have been men who have done great work later; … [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. … A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.
In his wretched life of less than twentyseven years Abel accomplished so much of the highest order that one of the leading mathematicians of the Nineteenth Century (Hermite, 18221901) could say without exaggeration, “Abel has left mathematicians enough to keep them busy for five hundred years.” Asked how he had done all this in the six or seven years of his working life, Abel replied, “By studying the masters, not the pupils.”
Abel has left mathematicians something to keep them busy for five hundred years.
See also:
 5 Aug  short biography, births, deaths and events on date of Abel's birth.
 Niels Henrik Abel and his Times, by Arild Stubhaug and Richard R. Daly.  book suggestion.