Re: Quantum Gravity Via Expansion-Contraction 24.1: Fundamental Equations and Further Unusual Properties of Exponentials

From Osher Doctorow mdoctorow@xxxxxxxxxxx

Where else but in physics and mathematics can somebody develop so many
convincing arguments, for example that exponentials are more useful
than logarithms, and yet so little changes in such long periods of
time? Frankly, there is one other place - politics. Whether
Republican or Democrat Administrations, you can argue till hell freezes
over and if it doesn't agree with the Conformist Elite (typically the
State Department and other similar Bureaucracies), nothing is likely to
change.

But aren't exponentials and logarithms just reflections of each other
around the main diagonal in the Cartesian plane, at least in the common
representation? Or to generalize this, aren't "duals" of many types,
such as S and T duals in Superstring Theory, more or less "the same
thing in different form"? But so are forward and past directions in
time, and try going backward in time some nice sunny day! Between a
quantity and its composition inverse, or between a quantity and its
"dual", there is often an infinity of differences!

It is true that the name "Cauchy functional equation" is used for both
exponential and logarithmic functional equations, of forms:

1) f(xy) = f(x) + f(y) (logarithmic)
2) f(x + y) = f(x)f(y) (exponential)

But the name "Cauchy" doesn't imply that they are the same or even
similar other than in using addition and multiplication in arguments
and ranges. In fact, the obsession with composition of functions
(functions of type f(g(x)), written f o g(x) ) obscures the fact that
equations (1) and (2) are quite significantly and meaningfully
different in the particular way that they use addition and
multiplication! And if you rotate an exponential function around the
1st-third quadrant diagonal, you change a very fast increasing curve
(exponential) to a very slow increasingf curve (logarithm).

On functional equations, look up the names Aczel and Dhombres on the
internet.

Osher Doctorow

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