Do mathematics have a relation to reality or are they only a mathematical symbol?
Mathematics. . . is a bit like discovering oil. . . . But mathematics has one great advantage over oil, in that no one has yet. . . found a way that you can keep using the same oil forever.
We decided that 'trivial' means 'proved'. So we joked with the mathematicians: We have a new theorem- that mathematicians can prove only trivial theorems, because every theorem that's proved is trivial.
The general mental qualification necessary for scientific advancement is that which is usually denominated "common sense," though added to this, imagination, induction, and trained logic, either of common language or of mathematics, are important adjuncts.
Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.
We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it.
The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization.
Mathematics is not yet capable of coping with the naïveté of the mathematician himself.
There is a very good saying that if triangles invented a god, they would make him three-sided.
What affected me most profoundly was the realization that the sciences of cryptography and mathematics are very elegant, pure sciences. I found that the ends for which these pure sciences are used are less elegant.
The extreme possibilities are the most illuminating.
Mathematics is a place where you can do things which you can't do in the real world.
Mathematics is one of the deepest and most powerful expressions of pure human reason, and, at the same time, the most fundamental resource for description and analysis of the experiential world.
Berkshire was built on the eternal verities: basic mathematics, basic horse sense, basic fear, and basic diagnosis of human nature to make predictions regarding human behavior. We stuck to the basics with a certain amount of discipline and it has worked out quite well.
The tree is made by nature, mathematics by people. And combining the two is creating this beautiful alliance between humanity and nature. That's why my forests are mathematical expansion systems, all of them.
Poincaré [was] the last man to take practically all mathematics, pure and applied, as his province. . . . Few mathematicians have had the breadth of philosophic vision that Poincaré had, and none in his superior in the gift of clear exposition.
The infinitely competent can be uncreative.
Theoretically, I grant you, there is no possibility of error in necessary reasoning. But to speak thus "theoretically," is to uselanguage in a Pickwickian sense. In practice, and in fact, mathematics is not exempt from that liability to error that affects everything that man does.
There is a logic of language and a logic of mathematics.
I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.