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June 24, 2006
Pairs
Mark Dominus has been reading Whitehead & Russell's ‘Principia Mathematica’. He makes the point that there is a certain amount of duplication involved on account of sets and relations being considered separately, despite being essentially the same. The concept of an ordered pair is needed to treat relations as being sets, and this had not yet been developed. Furthermore, pairing itself can be defined in terms of set operations.
There are two things that come to mind. The first is that the concept of a relation was apparently an important one for Russell. I can't say I've read any of his works, so I'll have to go on a scrap from Ray Monk:
Russell's theme throughout [his 1900 book on Leibniz] is that Leibniz's mistaken views on metaphysics can be traced to a mistaken assumption about propositions […] that every proposition has a subject and a predicate. The assumption […] leads, Russell thought, to a belief that all truths are of this form, which, in turn, leads one to imagine the world as consisting of only two kinds of ‘thing’: objects […] and properties […]. It was of enormous philosophical importance, Russell believed, to acknowledge that, as well as objects and properties, the world contains relations. Propositions such as ‘John is taller than his father’ […] must be understood, not as the predication of a property to an object, but as the assertion of a relation between two objects.
But relations can now be cast as properties of compound objects defined by a general pairing operation. In the example given, the property would be “having a first part taller than the second part” and the compound object would be “John and then his father”. This translation involves a certain amount of violence to the language of the proposition; but these things often do. I'm less interested in the validity of this manoeuvre than in how much this sort of thing is supposed to matter. I don't know how Russell responded to the redefinition of relations in terms of sets: perhaps his philosophical project was by then sufficiently de-railed to make further tinkering pointless. But if it is philosophically important that there are relations then it must also be important that they can be defined in more primitive terms. The idea that set theory is ontology could sit very comfortably within this line of thought.
The second is to do with the ordered pairs themselves. This is something I was almost going to get into earlier, with regard to Badiou's distaste for accounts of numbers as certain classes of algebraic structure that adhere to particular axioms. The objection seems to be that an axiomatic definition may be able to say what can be done with numbers, but never says what numbers are: if for no other reason than that a set of axioms may be satisfied in more than one way. (Hence the hyperreals of non-standard analysis count as a “great revenge” against Peano's axioms.) Whereas giving the make up of a number as a particular sort of set does say was the number is in itself, in the way that Badiou wants.
Ordered pairs ought to be an easier case than numbers, and the same issue crops up. (Incidentally, Badiou's account of surreal numbers in ‘Number & Numbers’ relies on pairing. He presumably thought it too primitive an operation to be worth discussing in itself.) It is normal to define the pair (a, b) as the set {{a}, {a, b}}. This obeys the rule that (a, b) = (c, d) if and only if a = c and b = d. To get the individual objects out of the pair again, the projections fst and snd can be defined by:
However, this is not the only way of defining pairs. Trivially there is the reverse definition, with the pair (a, b) defined as {{b}, {a, b}}. There are others. But how do we know that some way of defining pairs is valid? Well, a pairing operator must give pairs uniquely, and there must be definitions of the projections such that fst(a, b) = a and snd(a, b) = b in all cases.
So we have a situation where we have two styles of answer to the question: What is an ordered pair? One where we give the make-up of a set that can used as a pair; and one where we have axioms which we expect any definition for pairs to fulfil. But to my mind the first way isn't answering this question at all. If there are other ways of defining pairs then this one way cannot tell you what a pair is. Perhaps this comes from my being a programmer, but to me it answers a different question: If sets are the means that you have to hand, how can you implement ordered pairs? Whilst the axioms may not make a window into the soul of a pairing operation, it does show what something has to do in order to be a pair.