Quantum Gravity Via Expansion-Contraction 24.2: The Differences Between Cauchy Exponential and Cauchy Logarithmic Equations

From Osher Doctorow mdoctorow@xxxxxxxxxxx

Exponential functions are generated by Cauchy's "Exponential" Equation:

1) f(x + y) = f(x)f(y)

while logarithmic functions are generated by the corresponding Cauchy
"Logarithmic" Equation:

2) f(xy) = f(x) + f(y)

Since the former relates to Probable Influence/Causation or Riccati and
the latter relates to Shannon/Renyi/Hartle information/entropy, it
might be wondered why there should be such a big difference between
physical and other applications of these two composition inverses
(composition inverses in the sense that exp(log(x)) = x for x > 0 and
log(exp(x)) = x for all real x), so exp o log(x) = log o exp(x) = x).

The reason and explanation for the big difference is that functions in
general include subtypes with very considerable dependence on their
ranges and domains beyond their mere enumeration or "magnitudes".

To be very specific in relation to exponentials and logarithms, recall
that in Probable Influence/Causation, subtraction is fundamental to
(Probable) Causation, while in Conditional Probability division is
fundamental to what replaces or comes sometimes close to Causation,
namely Conditional Probability. The respective equations are:

3) P(A-->B) = 1 + y - x
4) P(B|A) = y/x

where 0 < = y < = x < = 1 (further details about x and y needn't
concern us here, although for those who are interested, it turns out
that y = P(AB) and x = P(A) for P(A) not 0 in (4)).

There is a "catch" or deep implicit relationship in all this, namely
that subtraction is the inverse of addition and multiplication is the
inverse of division, and all four are not cases of composition of
functions in the ordinary sense, since functions f, g have respective
operations f + g, f - g, fg, f/g if g is not 0 at a point, f o g, g o f
as appropriately defined. One could regard f + g as a sort of f o g,
but this would be in a very trivial sense of replacing + by o as
symbols, which is not what new operations are supposed to do.

So we should arguably look at +/- versus multiplication/division to
compare PI and Conditional Probability, and now we begin to notice some
rather curious things about exponentials versus logarithms. We have:

5) exp(x + y) = exp(x)exp(y), exp(x - y) = exp(x)/exp(y)
6) log(xy) = log(x) + log(y), log(y/x) = log(y) - log(x)

The domain of exp is all real x, y, but the domain of log is not all
real x, y since if xy (x times y) is negative, or 0, then log(xy) is
not defined, and analogously for log(y/x).

But if Probable Influence/Causation is correct, then expressions of
form x + y or x - y are fundamental and xy or y/x are not fundamental,
and according to (5) and (6), only exp is defined on all real
expressions of form x + y, x - y with x and y both real.

Isn't there a counter-argument, namely that the right hand sides of (5)
and (6) seem to reverse this reasoning because addition and subtraction
characterize the right hand side of (6) but not of (5)? No. Algebra
tends to obscure the fact that the domain of a function has a
"priority" over its range geometrically, since if you have a graph
without a domain axis you haven't the faintest idea "where you are".
If you have a known domain (e.g., x) axis but not a known range-onto or
range-into axis, you can always try "normalizing" or taking into
consideration the whole graph picture, and you aren't "lost".

Osher Doctorow

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