My interest in Russell’s The Analysis of Matter came originally from the perspective it offers in sorting through the problem of mind. Recently, thanks to a couple of papers linked to on twitter by @LogicalAnalysis, I learned about its connection with recent work on Structural Realism in the context of philosophy of science. (This continues a long-standing tradition for me where philosophy of mind serves as a “gateway drug” leading to the exploration of a wide range of philosophical concerns - btw this analogy was inspired by this parody poster).
Very roughly, Russell said that while we only have access to percepts (units of phenomenal perception), and lack access to external objects, these percepts do lie at the end of causal chains which link them to counterparts in the world. He argued that because of this linkage, the structure of our percepts is shared with that of the counterparts, allowing us to draw inferences about this structure.
In the 2001 “Is Structural Realism Possible” Stathis Psillos discusses the Russellian view as one of several paths toward Structural Realism (SR). Its construction from a starting point of bottom-up empiricism marks it in Psillos’ language as an “upward path” toward SR. The “downward path” characterizes approaches which look to save a broader scientific realism from objections by limiting the realism to certain of the mathematically described structural portions of the theories.
James Ladyman has a very nice SEP entry on SR, which lays out the contemporary research. In his taxonomy, he distinguishes “epistemic” SR (which includes Russell’s given its basis in concerns about knowledge of the external world) from “ontic” SR. Some versions of the latter (including Ladyman's own work) look to make the case that structure is all there is; that is, they take an anti-realist approach to non-structural elements of physical world (e.g. objects), rather than just taking an agnostic approach based on the epistemic difficulties of knowing about them.
While I have a lot more to read on this topic, I have an initial suspicion that both approaches to SR have a shortcoming which has to do with causation. Russell invokes causal relations as giving rise to structure, but doesn’t provide details regarding how causation works. Without more to the story, he is apparently left with the claim that we can make the appropriate inferences based on logico-mathematical structure. And this left him open to a logically based criticism due to M.H.A. "Max" Newman (1928 – see discussion in this article by Demopoulos and Friedman). This criticism argues that the inference based only on structure fails because a set of relations among any set of units with sufficient cardinality can be shown to be consistent with it (for a paper which concludes this argument lacks force, see a 2003 article by Ionnis Votsis).
When it comes to ontic SR, I think it is pretty clear that the formal mathematical structures in physical theories don’t provide a theory of causation, and such a theory will need to invoke additional ontology (e.g. properties, objects) to serve as part of its explanatory apparatus. Therefore, we can’t conclude structure is “all there is.” Psillos has another paper which, in part, argues against ontic SR from this basis.