"mathematical entities such as sets and other structures are part of the physical world and not therefore mysterious abstract objects”.
At least this position “suggest[s] a kind of Pythagoreanism” to L & R. However, the fusing of mathematical entities with the physical world doesn't seem altogether Platonic - even if it is Pythagorean; though it may express L & R's position very well. (It's not Platonic because Plato's prime concern was the abstract and atemporal realm of mathematics; not numbers or maths as they are instantiated in the physical world.)
- “abandoning the distinction between the abstract structures employed in models and the concrete structures that are the objects of physics”.
L & R go on to say that such “abstract structures employed in models” are the “objects of physics” if such a distinction is indeed abandoned. In L & R's case, we can say that abstract structures are the things or individuals of physics. In other words, if we erase abstract structures from the picture of physics - we have nothing. Though does it follow that abstract structures are everything?
“it is often not at all obvious whether a theoretical term refers to a concrete entity or a mathematical entity”.
L & R then express a position which one would imagine many people have aimed at L & R themselves. They say that
(i) There are only relations and no relata. (ii) There are relations in which things are primary, and their relations are secondary. (iii) There are relations in which relations are primary, while things are secondary. (iv) There are things such that any relation between them is only apparent.
“best sense that can be made of the idea of a relation without relata is the idea of a universal”.
Ladyman, James, Ross, Don. (2007) Every Thing Must Go: Metaphysics Naturalised.
Russell, Bertrand. (1912) The Problems of Philosophy (see the 'The World of Universals' chapter).