My complaint is that Russell said non-mathematical language was inaccurate, but he went on to use non-mathematical language regardless. I'd appreciate it if someone can can translate his law of causality into a differential equation. Thanks.
It is impossible to state this accurately in non-mathematical language; the nearest approach would be as follows: "There is a constant relation between the state of the universe at any instant and the rate of change in the rate at which any part of the universe is changing at that instant, and this relation is many-one, i.e. such that the rate of change in the rate of change is determinate when the state of the universe is given." If the "law of causality" is to be something actually discoverable in the practice of science, the above proposition has a better right to the name than any "law of causality" to be found in the books of philosophers.
Source: Russell, Bertrand. On the Notion of Cause.