The world is not an asymmetric graph

Abstract

1. Might the world be a mathematical structure? My question is not: might the world have a mathematical structure, but might it be identical with one? There are two general ways of interpreting this question. If it is whether, given that mathematical structures are abstract objects, the world is an abstract object, the answer has to be negative. Think of the world as wholly concrete (space, time and the spatio-temporal denizens of the world) or as a mix of the concrete and the abstract (if we include universals, laws, propositions and the like), but whichever of these is the case, the world is not purely abstract, as a formal structure is. One might claim, however, that the world is a structure in the sense that it instantiates a structure and is nothing else. In other words, all there is to the existence of the world is its being a case of a structure. Another case of the same structure might be a visual representation of it. If the structure that is the world were simple enough to draw, one might draw it and say, ‘This is what the world is’. That is how I understand Randall Dipert’s thesis that the world is an asymmetric graph (Dipert 1997). He claims to have proved ‘for the first time in the history of philosophy’ (1997: 349) that, employing the branch of mathematics known as graph theory, the distinctness of relata can be established ‘through relations alone’, i.e. that, in a nutshell, it is coherent – and indeed true – to suppose that everything that exists is relational in nature. This means, he contends, that Aristotle was wrong to have denied the possibility of pure relationalism in the description of what there is. Alexander Bird (2007: Ch. 6) picks up Dipert’s theory and uses it to argue, not that the entire world is purely relational in nature, but that at the ‘fundamental level’, whatever that may be, there is nothing but a world of pure powers, all defined relationally in terms of their stimulus and manifestation properties which are themselves pure powers defined similarly. Bird believes that graph theory provides a convincing response to the objection that a purely relational world generates either a vicious circle or a vicious infinite regress. [Elsewhere (Oderberg, 2011) I argue for the defensibility of the