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Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement.
The mobs of the great cities add just so much to the support of pure government as sores do to the strength of the human body. It is the manners and spirit of a people which preserve a republic in vigor. A degeneracy in these is a canker which soon eats to the heart of its laws and constitution.
Nothing shall warp me from the belief that every man is a lover of truth. There is no pure lie, no pure malignity in nature. The entertainment of the proposition of depravity is the last profligacy and profanation. There is no skepticism, no atheism but that. Could it be received into common belief, suicide would unpeople the planet.
Our experience hitherto justifies us in trusting that nature is the realization of the simplest that is mathematically conceivable. I am convinced that purely mathematical construction enables us to find those concepts and those lawlike connections between them that provide the key to the understanding of natural phenomena. Useful mathematical concepts may well be suggested by experience, but in no way can they be derived from it. Experience naturally remains the sole criterion of the usefulness of a mathematical construction for physics. But the actual creative principle lies in mathematics. Thus, in a certain sense, I take it to be true that pure thought can grasp the real, as the ancients had dreamed.
You perceive, now, that these things are all impossible except in a dream. You perceive that they are pure and puerile insanities, the silly creations of an imagination that is not conscious of its freaks - in a word, that they are a dream, and you the maker of it. The dream-marks are all present; you should have recognized them earlier.