Das Kalenderblatt 090808

07/08/2009 - 22:42 von WM | Report spam
The ordinary diagonal Verfahren I believe to involve a patent
confusion of the program and object aspects of the decimal fraction,
which must be apparent to any who imagines himself actually carrying
out the operations demanded in the proof. In fact, I find it difficult
to understand how such a situation should have been capable of
persisting in mathematics. Doubtless the confusion is bound up with
the notion of existence; the decimal fractions are supposed to "exist"
whether they can be actually produced and exhibited or not. But from
the operational point of view all such notions of "existence" must be
judged to be obscured with a thick metaphysical haze, and to be
absolutely meaningless from the point of view of those restricted
operations which can be allowed in mathematical inquiry.
This repudiation of the conventional proof by the diagonal Verfahren
of the non-denumerability of the non-terminating decimals will be
found to be very similar in spirit, although not in detail, to the
argument in Bentley's book [Linguistic Analysis of Mathematics]. It
may be worth while to record that the argument above was reached by me
independently of Bentley [...]
One can obviously say that all the rules for writing down
nonterminating decimals formulatable by the entire human race up to
any epoch in the future must be denumerable [...]
I do not know what it means to talk of numbers existing independent
of the rules by what they are determined; operationally there is
nothing corresponding to the concept. If it means anything to talk
about the existence of numbers, then there must be operations for
determining whether alleged numbers exist or not, and in testing the
existence of a number how shall it be identified except by means of
the rules?

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

Gruß, WM
 

Lesen sie die antworten

#1 Rainer Willis
08/08/2009 - 02:18 | Warnen spam
WM schrieb:

[...]

One can obviously say that all the rules for writing down
nonterminating decimals formulatable by the entire human race up to
any epoch in the future must be denumerable [...]



Da hat er völlig recht, wir können nicht alle Produktionsregeln
aufschreiben. Auch die Zahlen selbst nicht, aber was soll das beweisen?
Dass es sie nicht gibt? Und wenn er "the entire human race" bemüht, ist
das entweder Polemik oder ein Missverstàndnis des Unterschieds zwischen
mathematischer und physikalischer Existenz. Wohl eher Letzteres:

I do not know what it means to talk of numbers existing independent
of the rules by what they are determined; operationally there is
nothing corresponding to the concept. If it means anything to talk
about the existence of numbers, then there must be operations for
determining whether alleged numbers exist or not, and in testing the
existence of a number how shall it be identified except by means of
the rules?



Wàre er doch als hochspezialisierter Handwerker nur bei seiner Physik
geblieben, da reichen numerische Approximationen völlig aus.

Nehmen wir beispielsweise die Gravitationskonstante, dafür gibt es AFAIK
keinen geschlossenen Ausdruck, sie làsst sich nur experimentell
bestimmen. Existiert die Zahl deshalb nicht?

Bridgman schrieb seinen Beitrag vor 75 Jahren, also relativ kurz nach
den ersten Versuchen, die Mengenlehre axiomatisch zu begründen.
Man kann einem gestandenen Physiker durchaus nachsehen, dass er der
Argumentation damals nicht folgen wollte, selbst Mathematiker taten sich
schwer damit.
Aber heute ...?

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



Vielleicht hab ich einen Hinweis verpasst, aber gibt es den
vollstàndigen Text im Netz?

Gruß, WM



Gruß Rainer

Ähnliche fragen