Das Kalenderblatt 101225

24/12/2010 - 14:58 von WM | Report spam
Krieg der Frösche und der Màuse (22)

Small wonder, then, that Hilbert was upset when Weyl joined the
Brouwerian camp.

Hilbert's response was to develop an entirely new approach to the
foundations of mathematics with the ultimate goal of establishing
beyond doubt the consistency of the whole of classical mathematics,
including arithmetic, analysis, and Cantorian set theory. With the
attainment of that goal, classical mathematics would be placed
securely beyond the destructive reach of the intuitionists. The core
of Hilbert's program was the translation of the whole apparatus of
classical mathematical demonstration into a simple, finitistic
framework (which he called “metamathematics”) involving nothing more,
in principle, than the straightforward manipulation of symbols, taken
in a purely formal sense, and devoid of further meaning. {{Solch
sinnlose Symbolik vertreten heute immer noch ein paar
rückwàrtsgewandte Zeitgenossen und behaupten, ohne sie könne kein
Mensch und kein Beweis sinnvoll existieren.}} Within metamathematics
itself, Hilbert imposed a standard of demonstrative evidence stricter
even than that demanded by the intuitionists, a form of finitism
rivalling (ironically) that of Kronecker. The demonstration of the
consistency of classical mathematics was then to be achieved by
showing, within the constraints of strict finitistic evidence insisted
on by Hilbert, that the formal metamathematical counterpart of a
classical proof in that system can never lead to an assertion
evidently false, such as 0 = 1.

Hilbert's program rested on the insight that, au fond, the only part
of mathematics whose reliability is entirely beyond question is the
finitistic {{da hat er zweifellos Recht}}, or concrete part: in
particular, finite manipulation of surveyable domains of distinct
objects including mathematical symbols presented as marks on paper.
Mathematical propositions referring only to concrete objects in this
sense Hilbert called real, concrete, or contentual propositions, and
all other mathematical propositions he distinguished as possessing an
ideal, or abstract character. (Thus, for example, 2 + 2 = 4 would
count as a real proposition, while there exists an odd perfect number
would count as an ideal one.) Hilbert viewed ideal propositions as
akin to the ideal lines and points “at infinity” of projective
geometry. {{Verfehlter Analogischluss.}} Just as the use of these does
not violate any truths of the “concrete” geometry of the usual
Cartesian plane, so he hoped to show that the use of ideal propositions
—even those of Cantorian set theory—would never lead to falsehoods
among the real propositions, that, in other words, such use would
never contradict any self-evident fact about concrete objects.
Establishing this by strictly concrete, and so unimpeachable means was
thus the central aim of Hilbert's program. [...]

Weyl soon grasped the significance of Hilbert's program, and came to
acknowledge its “immense significance and scope”. Whether that program
could be successfully carried out was, of course, still an open
question. But independently of this issue Weyl was concerned about
what he saw as the loss of content resulting from Hilbert's
thoroughgoing formalization of mathematics. “Without doubt,” Weyl
warns, “if mathematics is to remain a serious cultural concern {{diese
Pràmisse kann ein echter Mathematiker nicht aufgeben}}, then some
sense must be attached to Hilbert's game of formulae.“

[John L. Bell: "Hermann Weyl", Staford Encyclopedia of Philosophy
(2009)]
http://plato.stanford.edu/entries/weyl/index.html

Gruß, WM
 

Lesen sie die antworten

#1 Bobo
24/12/2010 - 16:21 | Warnen spam
WM wrote:
Small wonder, then, that Hilbert was upset when Weyl joined the
Brouwerian camp.

Hilbert's response was to develop an entirely new approach to the
foundations of mathematics with the ultimate goal of establishing
beyond doubt the consistency of the whole of classical mathematics,
including arithmetic, analysis, and Cantorian set theory. With the
attainment of that goal, classical mathematics would be placed
securely beyond the destructive reach of the intuitionists. The core
of Hilbert's program was the translation of the whole apparatus of
classical mathematical demonstration into a simple, finitistic
framework (which he called “metamathematics”) involving nothing more,
in principle, than the straightforward manipulation of symbols, taken
in a purely formal sense, and devoid of further meaning.



WMs Kommentar: "Solch sinnlose Symbolik vertreten heute immer noch ein
paar rückwàrtsgewandte Zeitgenossen und behaupten, ohne sie könne kein
Mensch und kein Beweis sinnvoll existieren."

Du hast offenbar nicht verstanden worum es geht. Ich habe den Eindruck,
dass Du (obiges) Englisch nicht verstehst.


Bobo

Ähnliche fragen