Mathematics and Physical Reality

Russell argues that what physics actually tells us about the physical world is its abstract relational structure — the pattern of causal and spatiotemporal relations among events — rather than the intrinsic nature of the events themselves. The equations of physics specify how quantities relate to and change in terms of one another; they do not say what those quantities are in themselves. This "structural realism" implies that the real content of physical knowledge is mathematical structure, which explains why mathematics is the natural language of physics without requiring any mysterious pre-established harmony between abstract thought and physical reality.

A consequence of Russell's structural picture is that the intrinsic nature of physical events — what they are in themselves, apart from their relational structure — is unknown and perhaps unknowable. We know how electrons behave; we do not know what they are. This is not a failure of physics but its nature: physics is the science of structure, and structure is all that is publicly observable and communicable. What electrons are in themselves might be mental — Russell toys with this idea in The Analysis of Mind and in his later work — or something else entirely. The structural description is complete; the metaphysical question is open.

Russell's account of mathematics and physics raises a deep question: if mathematics describes abstract structure and physics instantiates that structure in the physical world, what is the status of mathematical objects? Russell's answer oscillates between Platonism — mathematical objects exist independently of minds — and logicism's deflationary view that mathematical "objects" are logical constructions with no robust independent existence. In his best moments, Russell is agnostic: mathematical truths hold necessarily, they are not about any particular physical or mental things, and their applicability to the world follows from the fact that the world has structure — but the ultimate metaphysical ground of this structural necessity remains obscure.

The essays "Mathematics and the Metaphysicians" and "On Scientific Method in Philosophy" in Mysticism and Logic (1918) develop Russell's account of physics and structure. His later "structural realism" is most fully developed in The Analysis of Matter (1927), which responds to the implications of Einstein's relativity.