现代数学观,何处寻?
现代数学观是送什么?根据是什么?
袁萌
Fjian :
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 signicance” [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 innitesimal 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
Analysis ... is the science of the innite itself. Leibniz Mathematical analysis is just the science of the innite. This old denition lives through ages. Luzin SET THEORY, an area of mathematics which studies the general properties of sets, primarily, of innite sets. The Great Encyclopedic Dictionary
Consequently, the very notion of the innite 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 Innitesimal 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 inuence 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]
It is 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 mathematics that 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 aect 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 scientic and useful method rather questionable. Mathematics can be degraded to a mere game played in some specic articial world. This is not a danger for mathematics in the future but an immediate crisis of contemporary mathematics. Vopenka [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 innitesimal 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.