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.
The total number of Dirichlet’s publications is not large: jewels are not weighed on a grocery scale.
The Infinite is only a manner of speaking.
By explanation the scientist understands nothing except the reduction to the least and simplest basic laws possible, beyond which he cannot go, but must plainly demand them; from them however he deduces the phenomena absolutely completely as necessary.
I have the vagary of taking a lively interest in mathematical subjects only where I may anticipate ingenious association of ideas and results recommending themselves by elegance or generality.
The higher arithmetic presents us with an inexhaustible store of interesting truths – of truths, too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties.
When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again.
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 may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation.
If others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries.
Response, when asked how he came upon his theorems.
His second motto: Thou, nature, art my goddess; to thy laws my services are bound.
There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
The enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it.
Sophie Germain proved to the world that even a woman can accomplish something in the most rigorous and abstract of sciences and for that reason would well have deserved an honorary degree.
To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings.
Arc, amplitude, and curvature sustain a similar relation to each other as time, motion, and velocity, or as volume, mass, and density.
Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems.
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.
I believe you are more believing in the Bible than I. I am not, and, you are much happier than I.