Life stands before me like an eternal spring with new and brilliant clothes.

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.

Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully,but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.

The enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it.

We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.

There have been only three epoch-making mathematicians, Archimedes, Newton, and Eisenstein.

Response, when asked how he came upon his theorems.

Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems.

Theory attracts practice as the magnet attracts iron.

That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.

Mathematics is the queen of the sciences and arithmetic [number theory] is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations, she is entitled to first rank.

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.

When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again.

If others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries.

I have had my results for a long time: but I do not yet know how I am to arrive at them.

Does the pursuit of truth give you as much pleasure as before? Surely it is not the knowing but the learning, not the possessing but the acquiring, not the being-there but the getting there that afford the greatest satisfaction. If I have exhausted something, I leave it in order to go again into the dark. Thus is that insatiable man so strange: when he has completed a structure it is not in order to dwell in it comfortably, but to start another.

Sin^{2} φ is odious to me, even though Laplace made use of it; should it be feared that sin^{2} φ might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin(φ^{2}), well then, let us write (sin φ)^{2}, but not sin^{2} φ, which by analogy should signify sin (sin φ)

Mathematicians stand on each other's shoulders.

I believe you are more believing in the Bible than I. I am not, and, you are much happier than I.

I have a true aversion to teaching. The perennial business of a professor of mathematics is only to teach the ABC of his science; most of the few pupils who go a step further, and usually to keep the metaphor, remain in the process of gathering information, become only Halbwisser [one who has superficial knowledge of the subject], for the rarer talents do not want to have themselves educated by lecture courses, but train themselves. And with this thankless work the professor loses his precious time.

I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession.

It may be true that people who are merely mathematicians have certain specific shortcomings; however that is not the fault of mathematics, but is true of every exclusive occupation. Likewise a mere linguist, a mere jurist, a mere soldier, a mere merchant, and so forth. One could add such idle chatter that when a certain exclusive occupation is often connected with certain specific shortcomings, it is on the other hand always free of certain other shortcomings.

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