For each member of α there is a member of β.
     The classes α & β fall into couples.      This 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

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 remind 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 grains 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 give it we thought of it as reducing the statement

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

pentagon”. Demonstration
     Timelessness of the demonstrated proposition:

blue & red = purple.