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Das Kalenderblatt 110111

10/01/2011 - 09:48 von WM | Report spam
Krieg der Frösche und der Màuse (39)

SECOND ACT OF INTUITIONISM

Admitting two ways of creating new mathematical entities: firstly in
the shape of more or less freely proceeding infinite sequences of
mathematical entities previously acquired (so that, for decimal
fractions having neither exact values, not any guarantee of ever
getting exact values admitted); secondly in the shape of mathematical
species, i.e. properties supposable for mathematical entities
previously acquired, satisfying the condition that if they hold for a
certain mathematical entity, they also hold for all mathematical
entities which have been defined to be 'equal' to it, definitions of
equality having to satisfy the conditions of symmetry, reflexivity and
transitivity. [...]

Theorems holding in intuitionistic, but not in classical, mathematics
often originate from the circumstance that for mathematical entities
belonging to a certain species the inculcation of a certain property
imposes a special character on their way of development from the basic
intuition; and that from this compulsory special character properties
ensue which for classical mathematics are false. Striking examples are
the modern theorems that the continuum does not split, and that a full
function of the unit continuum is necessarily uniformly continuous.
{{Reelle Funktionen, die nicht auf dem ganzen Kontinuum definiert
sind, sind nicht überall stetig. [W. Mückenheim: "Mathematik für die
ersten Semester", Oldenbourg, München, 2. Auflage (2010) ISBN:
978-3-486-58913-9, p. 199]
http://www.oldenbourg-wissenschafts...1845646.de
}}

[L.E.J. Brouwer: "Lectures on Intuitionism - Historical Introduction
and Fundamental Notions" (1951), Cambridge University Press (1981)]
http://www.marxists.org/reference/s...rouwer.htm

Gruß, WM
 

Lesen sie die antworten

#1 Ralf Bader
10/01/2011 - 10:22 | Warnen spam
WM wrote:


Theorems holding in intuitionistic, but not in classical, mathematics
often originate from the circumstance that for mathematical entities
belonging to a certain species the inculcation of a certain property
imposes a special character on their way of development from the basic
intuition; and that from this compulsory special character properties
ensue which for classical mathematics are false. Striking examples are
the modern theorems that the continuum does not split, and that a full
function of the unit continuum is necessarily uniformly continuous.
{{Reelle Funktionen, die nicht auf dem ganzen Kontinuum definiert
sind, sind nicht überall stetig. [W. Mückenheim: "Mathematik für die
ersten Semester", Oldenbourg, München, 2. Auflage (2010) ISBN:
978-3-486-58913-9, p. 199]
http://www.oldenbourg-wissenschafts...1845646.de
}}



Gröhöhöööhl, Mückenheim, bilden Sie sich etwa ein, "uniformly continuous"
bedeute "überall stetig"? Und der Satz über die uniform
continuity "intuitionistischer" Funktionen habe etwas zu tun mit dem
Gekrampfe über Stetigkeit in Ihrem Machwerk für die ersten Semester?

Neueste Forschungsergebnisse aus deutschen Spitzenhochschulen. Heute von
Prof. Dr. Wolfgang Mückenheim, Mathematikkoryphàe der FH Augsburg, aus
seiner Postille "Physical constraints of numbers": "Even some single
numbers smaller than 2^10^100 ... do not exist."

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