Quotes Tagged "knowledge"
Intuition must be taken seriously in the absence of substantial theoretical understanding, but once such theoretical understanding begins to take shape, prior intuitive judgments carry little weight unless they have been endorsed by the progress of theory. The greater one's theoretical understanding, the less weight one may assign to untutored judgment. All this applies equally well to the case of appeals to intuition in philosophy. We sometimes hear philosophers speak of some intuitions as 'merely' driven by theory, and thus to be ignored. While it is certainly true that judgments driven by bad theories are not to be taken seriously, the solution is not to try to return to some pure state of theory independent judgment, before the fall, as it were; rather the solution is to get a better theory. Intuition in the absence of theory does not count for nothing, especially if no credible theory is available. But this is not to award high marks to intuitive judgment before the arrival of successful theory, let alone after, when the initially low value of such judgment drops still lower.
Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value. The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them.