For each member of α there
is a member of
β.
The
classes α &
β
fall into couples.
This
is similar with a
proposition of say physics,
e.g. “they join & form couples when
they are brought together”.
But this is just
not what is meant.
We mean something that follows from what there exists in these
classes.
And we have an image of them
, something like this
:
If now we say for this there is this, for this there is this
etc., this sounds as if we said something about the dots;
like “this belongs to this etc.” whereas
we
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are saying words &
gestures to put them into couples.
And this
is a way of finding whether they have
the same
number
And now we must say that there are many different phenomena
of equality of
number or
of having a certain
number
Just as having a length & having equal lengths.
Let me remin
d you of the problem “are these two
rods
now of the same length.”
Take the definition you have to give of this expression when the rods have
to be measured & on the other hand when you use this
difference.
“These bodies have the same weight”
etc.
Now consider: “There are as many
gra
ins of sand in this heap as in the
other”.
How do we know this?
(This is no psychological question.)
Now suppose we said we test it by connecting the classes one-one;
then the question is: how shall we know that we have
connected them?
For there are several utterly different criteria.
But further what shall we say in the cases where no such connection is
possible?
What about saying then that the members of the two
similar classes still fall into couples??
Is this now an explanation?
For when we gave it we thought of it as reducing the statement
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of numeric equality to simpler terms.
The falling into couples was an image which in some cases was most natural
namely in those in which there was the possibility of joining terms into
couples.
But in fact it wasn
't at all the only aspect of
numeric equality.
The term “having the same
number” in fact
suggests a different aspect.
I mean this
Having the same
number can be
interpreted as having the same one of these schemata.
Of course this aspect too is only natural in a very limited
number of cases.
Aspect of
stars.
The explanation, that two classes have the same
number if they fall into couples, is
really taken from
¤ a case like
∙ –––––––– ∙ ∙ –––––––– ∙ ∙ –––––––– ∙ ∙ –––––––– ∙
|
“The pentagram has twice as many points
as the
6.
pentagon”
.
Demonstration
Timelessness of the demonstrated
proposition:
blue & red = purple
.
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