The exceptional derivative signals itself in mathematical discourse by an italicized
lower case e. [7] The
numerical value of e is 2.718218...(to infinity). It goes on
and on forever, like pi. The curve made up of e is smooth in contour
and sigmoid (S) in shape, and it relates Y to X, as follows:
Y = eX. That is, Y equals
2.718218....if X is 1; the square of e if X is 2; the cube of e
if X is 3....and so forth to infinity. In the latter expressions, e to
some power of X is a function of X, meaning that e has a
variational relationship to Y. But what makes e so very special to
us is that its derivative equals the function. In other words, when we look at
a sigmoid curve, we see what we would see if we could actually plot
Y = eX at a single point. Thus what we
represent in perceptual space as a S-curve has validity for what we
can't actually see at the point. [7a]
Imagine now that we have undertaken the task of exploring geometric figures.
Assume that, like Riemann, we don't know the rules in advance and that our only
metering device is e. To assist our imagery, envisage e 's as a
string of pearls. Assume that we can bend and flex the string, increase or decrease the
number of our e 's but can neither break nor stretch the string.
Okay, now suppose we come upon a flat surface and find two points, A and B.
What would be the shortest distance between them? Remember, we must base
our answers only on what we can measure. Gauged by our string of pearly
e
's, the shortest path is the least number of e 's between A and B.
Suppose the distance from A to B is 12 pearls.
Now imagine that we put our
string of pearls around some body's neck. Clearly, a path of 12 pearls remains a path of 12 pearls
even though the new surface (neck) has a different shape from the first. Or with e 's as a gauge, we can
relate a flat surface to a curved surface merely by finding their equivalents--the number of e 's. In our imaginary universe,
round and square thus become variants of a common theme.
Let's return to the flat surface. This time, imagine that we run the hypotenuse of a large
right triangle from A to B. When we lay the pearls
between A and B, we find that the string fits very loosely on the hypotenuse.
Even though we can always count pearls, we cannot gauge the hypotenuse very
accurately because of the poor fit.
Suppose though that we reduce the
number of e 's in our string and proportionally shrink the triangle; we
continue to make the string and the triangle smaller and smaller but always
remembering to take measurements by counting e 's. As we proceed, the
slack between the string and the hypotenuse becomes smaller and smaller and
eventually becomes so small as to be insignificant. Despite all the
proscriptions of tradition, the hypotenuse--which looks like a straight
line--has begun to approach a limit, relative to our pearls. We don't want
to assert that the hypotenuse and our pearly e 's are "the same thing."
If we do, we create the intolerable contradiction that the hypotenuse and the
S-curve look alike, which we can see is false. But at infinity, the one
is part of the other. The measurable part in question is little old e. Thus points
in the hypotenuse have a measurable feature in common with points in our
string of e 's. And that measurable feature is curvature.
Riemann also arrived at such an inference, although in a much more general, inclusive
and rigorous way. "About any point," he discovered, "the metric [measurable]
relationships are exactly the same as about any other point." A straight line,
a flat surface or a rectilinear space consists of the same fundamental elements as
a curve, circle or sphere; and magnitudes--round or square (pie or
cornbread)--"are completely determined by measurements of curvature." Riemann
showed that "flatness in smallest parts" represents "a particular case of those
[geometric figures] whose curvature is everywhere constant." Flatness turned
out to be zero curvature. In other words, a geometric universe constructed from
elementary measurable relationships among points becomes an infinite
continuum of curvatures, positive and negative, as well as zero.
Riemann demonstrated that "the propositions of geometry are not derivable from
general concepts of quantity, but...only from experience." Out the window went the
absolute prohibition against dealing with infinitely small regions of straight
lines.[8] Out went the notion
that parallel lines never! never! cross. Out went the universal dogma of even
our own day that the shortest distance between two points is absolutely always the
straight line.
