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“My work in the future must be devoted entirely to pure mathematics in its abstract meaning. I shall apply all my strength to bring more light into the tremendous obscurity which one unquestionably finds in analysis. It lacks so completely all plan and system that it is peculiar that so many have studied it. The worst of it is, it has never been treated stringently. 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 do few of the so-called paradoxes. It is really interesting to seek the cause.
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. I have examined several of them, and been so fortunate as to make this clear. …I have had to be extremely cautious, for the presumed theorems without strict proof… had taken such a stronghold in me, that I was continually in danger of using them without detailed verification.”

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Niels Henrik Abel

Letter to Christoffer Hansteen (1826) as quoted by Øystein Ore, Niels Henrik Abel: Mathematician Extraordinary (1957) & in part by Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) citing Œuvres, 2, 263-65

“Lety5 - ay4 + by3 - cy2 + dy - e = 0be the general equation of the fifth degree and suppose that it can be solved algebraically,—i. e., that y can be expressed as a function of the quantities a, b, c, d, and e, composed of radicals. In this case, it is clear that y can be written in the formy = p + p1R1/m + p2R2/m +…+ pm-1R(m-1)/m,m being a prime number, and R, p, p1, p2, etc. being functions of the same form as y. We can continue in this way until we reach rational functions of a, b, c, d, and e. [Note: main body of proof is excluded]
…we can find y expressed as a rational function of Z, a, b, c, d, and e. Now such a function can always be reduced to the formy = P + R1/5 + P2R2/5 + P3R3/5 + P4R4/5, where P, R, P2, P3, and P4 are functions or the form p + p1S1/2, where p, p1 and S are rational functions of a, b, c, d, and e. From this value of y we obtainR1/5 = 1/5(y1 + α4y2 + α3y3 + α2y4 + α y5) = (p + p1S1/2)1/5,whereα4 + α3 + α2 + α + 1 = 0.Now the first member has 120 different values, while the second member has only 10; hence y can not have the form that we have found: but we have proved that y must necessarily have this form, if the proposed equation can be solved: hence we conclude that
It is impossible to solve the general equation of the fifth degree in terms of radicals.
It follows immediately from this theorem, that it is also impossible to solve the general equations of degrees higher than the fifth, in terms of radicals.”

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Niels Henrik Abel

A Memoir on Algebraic Equations, Proving the Impossibility of a Solution of the General Equation of the Fifth Degree (1824) Tr. W. H. Langdon, as quote in A Source Book in Mathematics (1929) ed. David Eugene Smith

“On the whole, I do not like the French as well as the Germans; the French are extremely reserved toward strangers… Everybody works for himself without concern for others. All want to instruct, and nobody wants to learn. The most absolute egotism reigns everywhere. The only thing the French look for in strangers is the practical; no one can think except himself, he is the only one who can produce anything theoretical. This is the way he thinks and so you can understand it is really difficult to be noticed, particularly for a beginner.”

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Niels Henrik Abel

Letter to Bernt Michael Holmboe (ca. 1826) as quoted by Øystein Ore, Niels Henrik Abel: Mathematician Extraordinary (1957)

“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.”

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Niels Henrik Abel

“It is readily seen that any theory written by Laplace will be superior to all produced of lower standing. It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils.”

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Niels Henrik Abel

Marginal note in his mathematical notebook (ca. 1826) as quoted by Øystein Ore, Niels Henrik Abel: Mathematician Extraordinary (1957)

“On the whole, Divergant series are the work of the Devil and it's a Shame that one dares base any Demonstration on them. You can get whatever result you want when you use them, and they have given rise to so many Disasters and so many Paradoxes. Can anything more horrible be conceived than to have the following oozing out of you:
0 = 1 - 2n + 3n - 4n + etc.
where n is a whole Number? Risum teneatis amici.”

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Niels Henrik Abel

Laughter retains friends
Letter to Bernt Michael Holmboe (Jan. 1826) as quoted by Arild Stubhaug, Niels Henrik Abel and his Times: Called Too Soon by Flames Afar (2013)