In "My Philosophical Development", of Principia Mathematica Part IV "Relation Arithmetic", Bertrand Russell laments:
"I think relation-arithmetic important, not only as an interesting generalization, but because it supplies a symbolic technique required for dealing with structure. It has seemed to me that those who are not familiar with mathematical logic find great difficulty in understanding what is meant by 'structure', and, owing to this difficulty, are apt to go astray in attempting to understand the empirical world[emphasis JAB]. For this reason, if for no other, I am sorry that the theory of relation-arithmetic has been largely unnoticed."
However, the ultimate project of Principia Mathematica was directed at "the empirical world" in the conclusion of PM: Part VI "Quantity". "Quantity" consists of 3 sections the last of which, section "C", is about "Measurement" in terms of a generalization of the concept of number (section "A"), to include units of measurement (mass, length, time, etc.) as commensurable (dimensioned) quantities ("B" "Vector-Families").
Yet, other than *314:
"Relational real numbers are useful in applying measurement by means of real numbers to vector-families, since it is convenient to have real numbers of the same type as ratios."
I see nothing in Part VI that references anything like "relation numbers" as defined in Part IV.
Has this intriguing project, initiated in the conclusion of Principia Mathematica, been subsumed by subsequent work on the notion of "number" for the purpose of quantitative measurement in such a way that we might look for a new foundation for computer science based not on type theory but on commensurability ("vector-families")?