On the other side of the screen, it all looks so easy.
A good proof is one that makes us wiser.
Most likely, logic is capable of justifying mathematics to no greater extent than biology is capable of justifying life.
The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument.
What binds us to space-time is our rest mass, which prevents us from flying at the speed of light, when time stops and space loses meaning. In a world of light there are neither points nor moments of time; beings woven from light would live "nowhere" and "nowhen"; only poetry and mathematics are capable of speaking meaningfully about such things.
A proof only becomes a proof after the social act of "accepting it as a proof".
Of the properties of mathematics, as a language, the most peculiar one is that by playing formal games with an input mathematical text, one can get an output text which seemingly carries new knowledge. The basic examples are furnished by scientific or technological calculations: general laws plus initial conditions produce predictions, often only after time-consuming and computer-aided work. One can say that the input contains an implicit knowledge which is thereby made explicit.
The same rebellion, the same impatience, the same anger that exists in the hearts of the dark people in Africa and Asia is existing in the hearts and minds of 20 million black people in this country who have been just as thoroughly colonized as the people in Africa and Asia.
Her mind is an unquiet one, words and thoughts and impulses constantly crashing into each other.
Diaghilev was the first to notice good character dancers and that sort of thing.
I believe in miracles. . . prayers that are answered and healing hands.