We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything; nowadays we think nothing. Already the distance-concept is logically arbitrary; there need be no things that correspond to it, even approximately.
There is an astonishing imagination, even in the science of mathematics. ... We repeat, there was far more imagination in the head of Archimedes than in that of Homer.
Next you'd see a raft sliding by, away off yonder, and maybe a galoot on it chopping. . . you'd see the ax flash and come down-you don't hear nothing; you see the ax go up again, and by the time it's above the man's head then you hear the k'chunk!-it had took all that time to come over the water.
What's all this whining about the environment? They're always talking about 'stop the clearcuts.' I mean do the math people. If we were out of trees then we wouldn't have any clearcuts to be complaining about now would we?
The Prussian Academy of Sciences is a fast-track academic institute that requires a proactive, hands-on-type individual to overthrow the Newtonian conception of the universe. The successful candidate will have an excellent command of mass, energy, space, time and some maths. Bonus paid upon completion of proofs
I had a feeling once about Mathematics - that I saw it all. Depth beyond depth was revealed to me - the Byss and Abyss. I saw - as one might see the transit of Venus or even the Lord Mayor's Show - a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go.
But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.
But the creative principle resides in mathematics. In a certain sense, therefore, I hold true that pure thought can grasp reality, as the ancients dreamed.