Das Kalenderblatt 091002

01/10/2009 - 19:23 von WM | Report spam
Dedekind tried to describe an infinite class by saying that it is a
class which is similar to a proper subclass of itself. [...] I am to
investigate in a particular case whether a class is finite or not,
whether a certain row of trees, say, is finite or infinite. So, in
accordance with the definition, I take a subclass of the row of trees
and investigate whether it is similar (i.e. can be co-ordinated one-to-
one) to the whole class! (Here already the whole thing has become
laughable.) It hasn’t any meaning; for, if I take a "finite class" as
a subclass, the attempt to co-ordinate it with the whole class must eo
ipso fail: and if I make the attempt with an infinite class – but
already that is a piece of nonsense, for if it is infinite, I cannot
make an attempt to co-ordinate it. [...] An infinite class is not a
class which contains more members than a finite one, in the ordinary
sense of the word "more". If we say that an infinite number is
greater than a finite one, that doesn't make the two comparable,
because in that statement the word "greater" hasn’t the same meaning
as it has say in the proposition 5 > 4!

The form of expression "m = 2n correlates a class with one of its
proper subclasses" uses a misleading analogy to clothe a trivial sense
in a paradoxical form. (And instead of being ashamed of this
paradoxical form as something ridiculous, people plume themselves on a
victory over all prejudices of the understanding). It is exactly as
if one changed the rules of chess and said it had been shown that
chess could also be played quite differently.

When "all apples" are spoken of, it isn’t, so to speak, any concern of
logic how many apples there are. With numbers it is different; logic
is reponsible for each and every one of them.

Mathematics consists entirely of calculations.

[L. Wittgenstein: "Philosophical Grammar", Basil Blackwell, Oxford
[entnommen aus einer Sammlung von E.D.Buckner: "THE LOGIC
MUSEUM" (2005)]

Gruß, WM

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#1 Peter
01/10/2009 - 21:54 | Warnen spam
On 1 Okt., 19:23, WM wrote:

Dedekind tried to describe an infinite class by saying that it is a
[L. Wittgenstein: "Philosophical Grammar", Basil Blackwell, Oxford

Ich helfe Ihnen einmal noch wesentlich schönere Kommentare
von Ludwig Wittgenstein zu Dedekind und diesem Thema zu finden:

Lesen Sie "Philosophie für Mathematiker", Wittgensteins
Vorlesungen 1932/33, Abschnitt 4, die von Alice Ambrose
aufgezeichnet wurden. Das Original erschien 1979 bei
Blackwell in Oxford, es gibt aber auch eine deutsche

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