现代数学观，何处寻？

俄罗数学家根据上世纪活半叶出版发行的数学文献（共计542篇）多年研提炼出一种“现代数学观”，事实其实，不容争辩。

注：现代数学观是什么（说法），请见本文附件。

袁萌陈启清 4月28日

附件 :

Chapter 3

completely by the “stationary” set-theoretic view of a function as a set which prevails nowadays. “Itis aformal set-theoretic model of the intuitive idea of a function, a model that captures an aspect of the idea, but not itsfull signi cance” [133, p. 20]. We recall in this regard that if s,t∈[0,1] then (s+ t)= s + t, 0=0 , 1=1 ,

and, moreover, t = 0 for allt in some interval [0,h], where h is a strictly positive real (every nonzero positive in nitesimal will do). The presence of such a “numerical”function is an outright contradiction or, to put it mildly, a harbinger of antinomy. Thesecircumstances callforclarifying, immediatelyandexplicitly,theconcepts and means we use as well as specifying the foundations we rest them on. Aswehavealreadymentioned, in nitesimalanalysisacquiresjusti cationwithin the set-theoretic stance. More exactly, it appears that the ideas of the naive nonstandard set theory we have presented above can be placed on the same (and so, equally solid) foundations as cantorian set theory or, strictly speaking, the axiomatic set theories “approximating the latter from below.” In order to bring into focus the relations between mathematical analysis and set theory, the following statements are worth comparing:

Analysis ... is the science of the in nite itself. Leibniz Mathematical analysis is just the science of the in nite. This old de nition lives through ages. Luzin SET THEORY, an area of mathematics which studies the general properties of sets, primarily, of in nite sets. The Great Encyclopedic Dictionary

Consequently, the very notion of the in nite intertwines analysis and set theory quite tightly. At the same time we should never forget that the classical articles by Cantor appeared two centuries after the invention of calculus.The attempt at grounding mathematics on set theory could be compared with a modern method of building erection, rack mounting, when a house is assembled starting with upper stores, “from attic to cellar.” By the way, this technology requires that the footing of the building to be erected has been laid before the rack mounting begins. Likewise, the initial footing of mathematical analysis is a product of the material and mental activities of mankind. The present-day mathematics leans its basic parts on set theory. In other words, the set-theoretic foundation has been oated under the “living quarters” of mathematics. Only the future will reveal what is going to happen next. By Set-Theoretic Formalisms of In nitesimal Analysis

now we may just state that the process continues of erecting the edice of future mathem

atics and that this process is fraught with drastic changes. Aggravation of the state of

the art, collision of opinions, and a t struggle of ideas are faithful witnesses of rapid development. A collection of quoations to follow (far from claiming for completeness) will illustrate the process of polarization of views now in progress.

Pro Contra

After an initial period of distrust the newly created set theory made a triumphal inroad in all elds of mathematics. Its in uence on mathematics of the present century is clearly visible in the choice of modern problems and in the way these problems are solved. Applications of set theory are thus immense. Kuratowski and Mostowski [254, p. v]

Itis claimed that the theory of sets is important for the progress of science and technology, while presenting one of the most recent achievements in mathematics. In actuality, the theory of sets has nothing to do with the progress of science and technology nor it is one of the most recent achievements of mathematics. Pontryagin [400, p. 6]

Part of the creation of Georg Cantor is, of course, set theory, and some of this is now taught in high school and earlier. This is another of the domains of mathematicsthat many persons thought could never be of the remotest practical use, and how wrong they were. Elementary sets even nd their application in little collections of murder mysteries. Set theory has well-known connections with computer programs and these a ect an untold number of practical projects. Young [533, p. 102]

Mathematics, based on Cantor set theory, changed to mathematics of Cantor set theory.... Contemporary mathematics thus studies a construction whose relation to the real world is at least problematic.... This makes the role of mathematics as a scienti c and useful method rather questionable. Mathematics can be degraded to a mere game played in some speci c arti cial world. This is not a danger for mathematics in the future but an immediate crisis of contemporary mathematics. Vopˇenka [513] Concluding the preliminary discussion we emphasize that only now, after dispelling the illusion that it is possible to provide some nal “absolute” foundation for in nitesimal analysis (as well as for the whole of mathematics) by the set-theoretic or whatever stance, we may proceed with exposing some available implementations of this project.

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