On Infinite (1930-1939)

"Space is not only infinite in the sense that nowhere wit hin it does one come to an end, but at every place it is, so to speak, infinite in the inward direction. A point may be more and more exactly fixed only through a process of division extending to infinity, from level to level. This stands in contrast with the calm, completed existence of space for intuition. For the quality that fills it, space is the principie of separation, is that which makes possible the variability of the qualitative, but also is separation and, at the sarne time, contact, continual coherence, so that no piece may be 'chopped off, as if by an axe', from another. Therefore no actual spatial thing can ever be adequately given because it unfolds its 'inner horizon' in a process of ever newer and more precise experiences that unfolds to infinity." (Hermann Weyl, "Levels of Infinity", 1930)

"The infinite is accessible to mind and intuition in the form of a field of possibilities opening to infinity, as with the always further continuable sequence of numbers; [...]" (Hermann Weyl, "Levels of Infinity", 1930)

"The prototype of all infinite processes is repetition. […] Our very concept of the infinite derives from the notion that what has been said or done once can always be repeated." (Tobias Dantzig, "Number: The Language of Science", 1930)

"Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory." (Hermann Weyl, "Levels of Infinity", cca. 1930)

"Roughly it amounts to this: mathematical analysis as it works today must make use of irrational numbers (such as the square root of two); the sense if any in which such numbers exist is hazy. Their reputed mathematical existence implies the disputed theories of the infi nite. The paradoxes remain. Without a satisfactory theory of irrational numbers, among other things, Achilles does not catch up with the tortoise, and the earth cannot turn on its axis. But as Galileo remarked, it does. It would seem to follow that something is wrong with our attempts to compass the infinite." (Eric T Bell, "Debunking Science", 1930)

"In pure mathematics the maximum of detachment appears to be reached: the mind moves in an infinitely complicated pattern, which is absolutely free from temporal considerations. Yet this very freedom - the essential condition of the mathematician’s activity – perhaps gives him an unfair advantage. He can only be wrong – he cannot cheat." (Kytton Strachey, "Portraits in Miniature", 1931)

"A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details." (Hermann Weyl, "Unterrichtsblätter für Mathematik und Naturwissenschaften", 1932)

"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is, finite means." (Hermann Weyl, "Mind and Nature", 1934)