The penultimate paper of "Bit-string Physics: A Finite and Discrete Approach to Natural Philosophy" discusses an attempted revival of "Relation Arithmetic" with which Russell and Whitehead had planned to cap off their Principia Mathematica in its final volume.
Of Relation Arithmetic, Russel said:
"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. For this reason, if for no other, I am sorry that the theory of relation-arithmetic has been largely unnoticed."
-- " My Philosophical Development" by Bertrand Russell
An example of going astray in attempting to understand the empirical world is when people attempt to combine incommensurable quantities in their calculations, not understanding the structure of the relations between the quantities.
Ordinarily, programming languages treat units, as I/O formats for dimensions, as an afterthought -- independent of type checking. However, what if we saw numbers themselves as embodying relational structure, as intended by Russell, thereby unifying the notion of "type checking" with the notion of "number"? Might then the power of dimensional analysis be brought to bear, in a mathematically rigorous way, on the relatively ad hoc notions of "type", hence problematic areas such as the object relational impedance mismatch?