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.

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    See Newton's second law of motion : the dv/dt term is the "second derivative" of the function s(t) (space traversed as function of time) i.e. "the rate of change of the rate of change". – Mauro ALLEGRANZA Jun 5 '15 at 22:27
  • @MauroALLEGRANZA - Thanks. That totally makes sense. You pinpointed exactly what confused me. I thought it was a misprint. – George Chen Jun 6 '15 at 0:44
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    Rate of change of position is velocity. Rate of change of velocity is acceleration. It's not clear from this snippet what accelerating "part" Russel was talking about, but possibly individual particles, if "universe" has its general meaning here. I sort of associate this with gyroscopes that maintain their directions thus demonstrating a connection to the universe at large. But I'm not sure where that association comes from. – Cheers and hth. - Alf Jun 7 '15 at 1:41
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    How about "Russel's Law of Causality"? It's not like "speculated" or "proposed" or "remarks on" or "formulation of" distinguishes this one among other "Russel's Law of Causality". The adjective isn't doing any work other than editorializing. – ben rudgers Jun 7 '15 at 16:25
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    Extreme caution must be taken to avoid misrepresenting Russell's ideas. – George Chen Jun 8 '15 at 8:26

My mathematical interpretation of Russel's statements is as follows:

 U(t) = U(o) + K{[d^2P(i)]/dt^2}

U(t) - Current state of universe
U(o) - Initial state(condition) of universe
K - constant of proportionality
d^2P(i) - second derivative of part(i) of the universe
dt^2 - second derivative with respect to time

I hope this helps.

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