Today we face a problem that involves two difficult to satisfy conditions. On the one hand we have to find a way for computer assisted verification of mathematical proofs. This is necessary, first of all, because we have to stop the dissolution of the concept of proof in mathematics. On the other hand, we have to preserve the intimate connection between mathematics and the world of human intuition. This connection is what moves mathematics forward and what we often experience as the beauty of mathematics.
Vladimir VoevodskyRelated topics
beauty computer concept connection dissolution experience face find forward hand human intimate intuition mathematics necessary preserve problem proof stop way world today verification conditions movesRelated quotes
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.
Richard Courant
I know nothing of philosophical philanthropy. But I know what I have seen, and what I have looked in the face in this world here, where I find myself. And I tell you this, my friend, that there are people (men and women both, unfortunately) who have no good in them-none. That there are people whom it is necessary to detest without compromise. That there are people who must be dealt with as enemies of the human race. That there are people who have no human heart, and who must be crushed like savage beasts and cleared out of the way.
Charles Dickens
Individuals who have been wronged by unlawful racial discrimination should be made whole; but under our Constitution there can be no such thing as either a creditor or a debtor race. That concept is alien to the Constitution's focus upon the individual. ...To pursue the concept of racial entitlement - even for the most admirable and benign of purposes - is to reinforce and preserve for future mischief the way of thinking that produced race slavery, race privilege and race hatred. In the eyes of government, we are just one race here. It is American.
Antonin Scalia
Theater of Cruelty means a theater difficult and cruel for myself first of all. And, on the level of performance, it is not the cruelty we can exercise upon each other by hacking at each other's bodies, carving up our personal anatomies, or, like Assyrian emperors, sending parcels of human ears, noses, or neatly detached nostrils through the mail, but the much more terrible and necessary cruelty which things can exercise against us. We are not free. And the sky can still fall on our heads. And the theater has been created to teach us that first of all.
Antonin Artaud
I am obliged to interpolate some remarks on a very difficult subject: proof and its importance in mathematics. All physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.
G. H. Hardy
