classical number is a thought-process dealing not with spatial relations but with visibly limitable and tangible units, and it follows naturally and necessarily that the classical knows only the "natural" (positive and whole) numbers, which on the contrary olay in our western mathematics a quite undistinguished part in the midst of complex, hypercomplex, non-archimedean and other number-systems. on this account, the idea of irrational numbers the unending decimal fractions of our notation was unrealizable within the greek spirit. euclid says and he ought to have been better understood that incommensurable lines are "not related to one another like numbers." in fact, it is the idea of irrational number that, once achieved, separates the notion of number from that of magnitude, for the magnitude of such a number (π, for example) can never be defined or exactly represented by any straight line. moreover, it follows from this that in considering the relation, say, between diagonal and side in a square the greek would be brought up suddenly against a quite other sort of number, which was fundamentally alien to the classical soul, and was consequently feared as a secret of its proper existence too dangerous to be unveiled. there is a singular and significant late-greek legend, according to which the man who first published the hidden mystery of the irrational perished by shipwreck, "for the unspeakable and the formless must be left hidden for ever".
every philosophy has hitherto grown up in conjunction with a mathematic belonging to it. number is the symbol of causal necessity. like the conception of god, it contains the ultimate meaning of the world-as-nature. but the actual number with which the mathematician works, the figure, formula, sign, diagram, in short the number-sign which he thinks, speaks or writes exactly, is (like the exactly-used word) from the first a symbol of these depths, something imaginable, communicable, comprehensible to the inner and the outer eye, which can be accepted as representing the demarcation. the origin of numbers resembles that of the myth. primitive man elevates indefinable nature-impressions (the "alien," in our terminology) into deities, numina, at the same time capturing and impounding them by a name which limits them. nature is the numerable, while history, on the other hand, is the aggregate of that which has no relation to mathematics hence the mathematical certainty of the laws of Nature, the astounding Tightness of Galileo's saying that nature is "written in mathematical language," and the fact, emphasized by kant, that exact natural science reaches just as far as the possibilities of applied mathematics allow it to reach.
thoughts, rebuttals and denunciations are welcome.
