Luke asked:

I have been reading Bertrand Russell's Introduction to Mathematical Philosophy,and I am stuck on
his discussion of Frege's definition of the concept of number.

As a visual example, Russell talks about putting things into bins according to the relation of similarity.
For example, I note that there is a one-to-one correspondence between my socks and my feet. So, I
should put my pair of socks and my pair of feet in the same bin. In this bin, we can also put my hands,
my gloves, my friend Eddie's hands, my friend Jenny's eyes, each married couple, and in fact any
collection that comes in a pair. This bin will be, as Russell says, a collection with an infinite number of
members, and each of these members is a collection with 2 members. We label (define) this bin as
the number 2.

So here is where I start getting confused...Russell defines a number as "the set of all classes that are
similar to the given class". (Here class essentially means 'collection'.) For an example, the number 2
is defined as the class of couples. I think my confusion is over what Russell means here by "the given
class". He phrases it another way "The number of a class is the class of all those classes which are
similar to it." What is meant by "it"? Which class is "it" referring to?

I am trying to sort this definition out in terms of the bins analogy. We assembled bins filled with
collections that are similar to each other, and labeled them 'two' or 'three' or whatever the case may
have been. But Russell then defines the number of the bin as the collection of the collections that are
similar to the bin, not to each other. My confusion is that the bin has an infinite number of members,
so its members are not similar to it, but to each other. (For example, the number of the bin of couples
is the set of all couples, and there are an infinite amount of couples. They are similar to each other,
not to the whole bin.) It seems to me that this definition of number leads to every number being
infinite.

I think that the key to my understanding of this is the point which we define the number 2 to be the bin
containing all couples. It seems that the class of all couples is not the same as the number 2. (for
example, that bin has infinitely many members). Russell says essentially that of course defining 2 as
the class of couples feels strange at first, but this strange feeling goes away. The bin containing all
couples is a certainty whereas the number 2 is a "metaphysical entity about which we can never feel
sure that it exists". Therefore it becomes natural to deal instead with the class of couples. I think my
problem might be that the strange feeling has not gone away yet, and I could use some further
discussion to help see why it should.

============

Well, notice that on p. 18 of Russell, B. Introduction to Mathematical Philosophy(London: George
Allen & Unwin 1930) he states, "the number of a class is the classof all those classes that are similar
to it". So first, you must be very very careful of your terminology here. It's not the set. Second, class
does not mean "collection", and that is your basic problem. Russell very specifically states that this is
incorrect; see p. 12, for example, where he says that he will speak of a "class" instead ofa
"collection".

The bin containing all couples has this similarity between couples: they all have two members. That is
their onesimilar characteristic: that of having two elements. That one characteristic holds over an
infinite number of specific instances, and that is the pointof Russell's conception of the class.

So one might say, employing Russell's intensivedefinition (p. 12), that the "defining property" of the
infinite-sized class (p. 13: "a class and a defining characteristic of it are practically interchangeable")
of things with two members: couples, is twoness, which is the class-idea, the "number": two. The "bin"
is precisely that defining property, no more and no less. Therefore, the class: couples: twoness: two;
isprecisely identical with that bin, and with the number two.

I mean, you're getting lost in the details. It's just a way of saying, "What do all sets of two things have
in common? Hey, there are two of them! So we'll define the number two by just saying: what they
have in common isthat number." That's really it. Really. The only confusing thing is that Russell is
taking that as a definitionof number, which sort of turns things around from the normal way of
thinking of it, which is: the number two describesthat there are two things. He's just saying, no, it
doesn't do that, what's really happening is that we getthe number from our intuition, if you want to
think of it that way, that what all those things have in common is that there are two of them. So
realizing that we have that intuitionof number afterseeing all the couples is the "strange feeling",
because we usually think that the number is first,as a description. You see?

Steven Ravett Brown