   In the period leading up to his own abandonment of idealism, Russell was already pursuing a research program involving the foundations of arithmetic (see, e.g., AMR). This work, along with his earlier work on the foundations of geometry (see EFG), had convinced him of the importance of relations for mathematics. However, he found that one category of relations, viz., asymmetrical transitive relations, resisted any such reduction to the properties of the relata or the whole formed of them. These relations are especially important in mathematics, as they are the sort that generates series. Consider the relation of being taller than, and consider the fact that Shaquille O’Neal is taller than Michael Jordan. It might be thought that this relation between O’Neal and Jordan can be reduced to properties of each: O’Neal has the property of being 7′2″ tall, and Jordan has the property of being 6′6″ tall, and the taller than relation in this case is reducible to their possession of these properties. The problem, according to Russell, is that for this reduction to hold, there must be a certain relation between the properties themselves. This relation would account for the ordering of the various height properties, putting the property of being-6′8″-tall in between that of being-7′2″-tall and that of being-6′6″-tall. This relation among the properties would itself be an asymmetrical and transitive relation, and so the analysis has not rid us of the need for taking relations as ultimate. Another hypothesis would be that there is such an entity as the whole composed of O’Neal and Jordan, and that the relation between the two men is reducible to some property of this whole.  Russell’s complaint was that since the whole composed of O’Neal and Jordan is the same as the whole composed of Jordan and O’Neal, this approach has no way to explain what the difference would be between O’Neal’s being taller than Jordan and Jordan’s being taller than O’Neal, as both would seem to be reduced to the same composite entity bearing the same quality (see POM, 221–26).