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Courant said much more much better in much less words: There seems to be a great danger in the prevailing overemphasis on the deductive-postulational character
by stiff 14y ago
Courant said much more much better in much less words:
There seems to be a great danger in the prevailing overemphasis on the deductive-postulational character of mathematics. True, the element of constructive invention, of directing and motivating intuition, is apt to elude a simple philosophical formulation; but it remains the core of any mathematical achievement, even in the most abstract fields. If the crystallized deductive form is the goal, intuition and construction are at least the driving forces. A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms, without motive or goal. The notion that the intellect can create meaningful postulational systems at its whim is a deceptive half-truth. Only under the discipline of responsibility to the organic whole, only guided by intrinsic necessity, can the free mind achieve results of scientific value…To establish once again an organic union between pure and applied science and a sound balance between abstract generality and colourful individuality may well be the paramount task of mathematics in the immediate future.
- jordigh 14y agoCourant is kinda saying the opposite here, by de-emphasising the need for purely abstract formalisms, or by saying that they have to go hand-in-hand with applications, or are worthless without applications. As a numerical analyst, he of course thinks "applications" of mathematics are tantamount to its existence. The thesis in the linked article is that we need to emphasise the importance of purely abstract formalisms, not only in the context of possible applications.
- stiff 14y agoWhat I meant is that he described the same problem in a more complete way, not that he just said the same thing more nicely. There are infinitely many formal systems and after all we choose only some particular ones, what guides this choice? Isn't the existence of models in the real world one of the most important factors in choosing which formalism is valuable to develop? Courant how I understand him is saying that the tension between the two approaches is vital to the development of mathematics. There is also a nice lecture [1] from Vladimir Arnold, who was very much on the side of intuition, where he raises an interesting point that is related: in todays mathematics all the praise goes to the people who prove theorems, where perhaps the person who first _stated_ an theorem that is interesting and relevant deserves at least as much credit. [1] http://www.msri.org/web/msri/online-videos/-/video/showVideo/2714 http://www.msri.org/web/msri/online-videos/-/video/showVideo...