Das Kalenderblatt 090806

05/08/2009 - 22:46 von WM | Report spam
[...] the meaning of a term or concept is contained in those
operations which are performed in making application of the term or
concept to relevant situations.
I can only report that as a matter of personal analysis I find this
operational aspect at the bottom of all meaning, but my inference is
that other persons go through similar processes [...] we may perhaps
say that self-consistency is in some way intimately connected with
real things. [...] The accepted method of proving that some system of
postultes does not conceal some contradiction is to exhibit some
"real", "existing" system which satisfies the postulates. Nothing
further in the way of proof or analysis is felt to be necessary; the
feeling that actually existing things are not self-contradictory is so
elemental as almost to constitute a definition of what we mean by self-
consistent. Now when we are concerned with "things" we are evidently
concerned with some form of experience, so that we may make an even
broader statement and say that experience is not self-contradictory.
[...] it is once obvious that the operational technique automatically
secures to mathematics the sine qua non of self-consistency, for
operations actually carried out, whether physical or mental, are a
special form of experience, so that any mathematical concept or
argument analyzed into actual operations must have the self-
consistency of all experience. [...]

P.W. Bridgman: "A physicist's second reaction to Mengenlehre", Scripta
Mathematica, Vol. II, 1934.

Gruß, WM
 

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#1 Albrecht
06/08/2009 - 08:35 | Warnen spam
On 5 Aug., 22:46, WM wrote:
[...] the meaning of a term or concept is contained in those
operations which are performed in making application of the term or
concept to relevant situations.
I can only report that as a matter of personal analysis I find this
operational aspect at the bottom of all meaning, but my inference is
that other persons go through similar processes [...] we may perhaps
say that self-consistency is in some way intimately connected with
real things. [...] The accepted method of proving that some system of
postultes does not conceal some contradiction is to exhibit some
"real", "existing" system which satisfies the postulates. Nothing
further in the way of proof or analysis is felt to be necessary; the
feeling that actually existing things are not self-contradictory is so
elemental as almost to constitute a definition of what we mean by self-
consistent. Now when we are concerned with "things" we are evidently
concerned with some form of experience, so that we may make an even
broader statement and say that experience is not self-contradictory.
[...] it is once obvious that the operational technique automatically
secures to mathematics the sine qua non of self-consistency, for
operations actually carried out, whether physical or mental, are a
special form of experience, so that any mathematical concept or
argument analyzed into actual operations must have the self-
consistency of all experience. [...]

P.W. Bridgman: "A physicist's second reaction to Mengenlehre", Scripta
Mathematica, Vol. II, 1934.

Gruß, WM




Genau das ist es. Wir haben keine verlàsslichen Mittel um die
Konsistenz eines Systems ab einem gewissen Komplexitàtsgrad
zuverlàssig zu prüfen. Vielleicht liegt es daran, dass ab einem
ewissen Punkt die Mittel selbst auf das System zurückgreifen müssen.
Ein Computerprogramm das Formeln checkt muss eben selber diese
Mathematik anwenden, die es überprüft.

Gödel hat diesen Umstand formal nachgewiesen.

Die einzige Sicherheit, die uns damit bleibt ist die Konsistenz
unserer Wirklichkeit. Da all unsere Erfahrung und all unser Wissen
sowieso nur aus unserer Umwelt, unserer Wirklichkeit stammen kann,
làsst sich der Begriff "Konsistenz" in dem hier gebrauchten Sinne
wahrscheinlich sogar direkt auf etwas wie "der Wirklichkeit analoge
Strukturen" zurückführen.

Dies alles bedeutet letzenendes, dass eine Mathematik, die sich nicht
ausschließlich auf den in unserer Wirklichkeit nachvollziehbaren
Begriff von Anzahlen stützt, Teile enthàlt, deren Richtigkeit
angezweifelt werden muss. Man erhàlt eine Mathematik, die teils
richtig und teils falsch ist. Eine solche Mathematik kann aber nicht
das Ziel sein. Das Wesen der Mathematik ist es, zutreffende
Voraussagen machen zu können. Eine Mathemati, die das nicht mehr
leistet ist Schwachsinn.

Gruß
Albrecht

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