It is neither the traditions of teaching nor the deficiency of appropriate textbooks that explains the minor educational influence of the modern ideas of nonstandard analysis. The real point is that the place in science of the concept of infinitesimal has changed drastically.
The roots of analysis stem from the atomistic ideas of antiquity. The concept of atom as an indivisible material particle was combined with that of the ultimate imaginable constituent of the ideals about the Universe. These dual views were reflected in the primary mathematical concepts of Euclid's Elements: A point, an atom of geometry, is that which has no parts; whereas a monad, an atom of arithmetic, is that by virtue of which each of the things that exist is called one.
The views of the world in the age of the microscope and telescope gave rise to the two forms of differential and integral calculus. Lebniz's monadology and Newton's method of prime and ultimate ratios reflect the dual nature of the ancient ideas of the microcosm. Robinson's nonstandard analysis has splendidly united and completed the development of the two-millennia old conceptions, paving the best way to the classical calculus.
The present-day physical views have little in common with the atomism of the ancients. We perceive the laws of the microworld within the framework of quantum mechanics and the uncertainty principle alien to Aristotle's logic. Omnipresent is the process of discretization and constructivization of applied mathematics which is connected with the prominent role of the technologies based on physical binary devices.
Mathematics must constantly fit itself to the common paradigms of science. Robinson's nonstandard analysis crowns the old-fashioned ideas of the ancient atomism. Mankind will never waste out its intellectual treasures. Therefore, nonstandard analysis in some form will be the “analysis of the future” as forecasted by Gödel. Nevertheless, there are no grounds to think that the calculus of Newton and Leibniz will play a key role in the formation of the outlook of the future generations.
The low demand for the contemporary infinitesimal analysis is not connected only with the lack of new textbooks and the conservatism and ignorance of tutors. In fact, the problem relates to the status of the classical calculus rather than the modern approaches of nonstandard set theory.
The main reason of stagnation in teaching is the diminishing of vitality of what is taught.
S. Kutateladze
May 23, 2009
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