Who's On First?

What is an ordered pair? One explanation is that it is a mapping: the ordered pair of elements (a, b) is viewed as a mapping from some set of indices of cardinality 2 onto the set {a, b}. Since the names of the domain elements (the indices) are arbitrary, we can presumably call them whatever we wish--say, {i, j}.

What then makes (a, b) different from (b, a)? "The mapping assigns, in the one case, say, a to the argument i, and in the other case, j". But this seems to get us nowhere-- don't we now need a way of specifying which case is the (a, b) case and which is the (b, a)? If we don't know which of i and j comes first, how can we tell which form of the mapping identifies b or as first in the ordered pair?

We might try to restrict the domain of the mapping to a set of 2 elements which are ordered, say, {0, 1}, so that we can say (b, a) is the mapping in which the smaller element 0 gets mapped onto b, and (a, b ) is the mapping in which 0 gets mapped onto a. But now the domain of the mapping must itself be structurable as an ordered pair, say, (0, 1), and we find ourselves again presupposing the very concept which was to be explained.

Choose a pair of symbols which are themselves symmetrical about the vertical, say, 'O' and 'V', and suppose we use a period rather than the comma to separate them. Imagine (O . V) represented by large wooden symbols, so large we could walk around it and see it "from the other side" as (V . O). Surely, there is something which remains the same in either view and is yet distinct from {O, V}. What is it?

How can we represent the orderedness without making a commitment in the representation itself to some particular order? Certainly no device such as claiming that (a, b) is "really" {a, {b}} will do, for there is nothing (other than an arbitrary naming decision) which requires us to take a as the 'first' element in (a, b) or which requires us to bracket the second rather than the first element.

Users of some particular notation may tell us that they construct the notion of order from the notion of an unordered set by using (a, b) as the name for the set {a, {b}} but that does not tell us which element they take to be "first" in that notation, and even if the users told us that it is a rather than b which is "first", we still cannot tell whether they mean by "first" what we mean by "first" -- even if elsewhere, we can establish semantic agreement for this term, it might still have a different meaning for them just in this context.

The slogan "Look to the use, not the meaning" creates a limit here, beyond which there is no meaning and therefore no difference in meaning.