From Osher Doctorow
Now compare the various other types of "derivatives", including the
Lie derivative of funcitons, vector fields, differential forms, and
tensors, with the simplicity of the Probable Influence/Causation (PI)
"derivative", that is to say (Probable) Causation P(A-->B) or P' (A--
B):
1) P(A-->B) = 1 + y - x, y = P(AB), x = P(A)
2) P' (A-->B) = 1 + y - x, y = P(B), x = P(A), P(B) < = P(A)
Equations (1) and (2) are in an obvious sense linear or antilinear
(linear in the second argument, antilinear in the first), which is
especially true for (2) and can be shown to be true for (1) when (1)
and (2) coincide, etc.
Most important, + or - only are used in PI, while the various
generalized "derivatives" of other types used various multiplications
together with additions or subtractions. For example, if L_X or LX
for short is the Lie derivative with respect to X:
3) LX(w) = i_X dw + d(i_X w)
for differential form w.
So the other types of "derivatives" introduce greater complexity and
complications, but also they typically introduce geometry as "coding"
for Causation. A typical such introduced "geometric Causation" is
nonzero curvature of space introduced via nonzero Lie brackets or
connections. The "object" so introduced is "supposed" intuitively to
account for Causation, but the code involves not only + and - but
multiplication and/or division and other things. It is roughly
analogous to using complex numbers to solve real equations and then
forgetting to translate back to real equations, or worse still not
knowing how to translate back.
Osher Doctorow
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