数学机器人与火星人相遇了
我们的数学机器人原不不认识火星人。
但是,今年五一节,他们两人驻在互联网上相遇了。
我们4人预祝大家节日愉快
!
袁萌 陈启清
4月30日
附件:
现代数学观
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 edi ce of future mathematics 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]
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 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. 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 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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