In the last chapter, we made our first inroads toward sketching
the order we can see in the events around us, by creating the
process of dimensional analysis and identifying the notion of a
scaling relation. Our methods were somewhat haphazard, in that
we cobbled tools together as we discovered we could make them,
with little attention to what we were seeing in the tools
themselves. For instance, we ``discovered'' that it is possible
to make the abstraction to the notion of dimension from the fact
that *we found we were able to make certain kinds of
substitutions*, of one ruler for another ruler, or one clock for
another clock, that kept all the important functions of rulers
and clocks the same, independently of any of the details of how
they were made or how they worked internally.

Having in this way ``discovered'' that it was possble to make measurements, and having given names to ``the quantity that was measured'' (as in, ``length'', ``mass'', ``time'', etc), in other words having named the kind of substitution that constituted the measurement, we found that there were scaling relations among the results of various measurements. For instance, because all things fall toward earth ``in the same way'', there is always a certain relation between the distance they are allowed to fall and the time they take to cross it, while there is a different but still precise relation between the length of a clock pendulum and the time it takes to complete a swing.

The introduction of the possibility of measurement and the
resulting observation that there are scaling relations are
useful observations, but one can imagine that their power could
have ended there, short of our full process of dimensional
analysis, because *there could conceivably have been* any
number of measureable ``properties'' (or kinds of
interchangeability, or ``dimensions'') none of which had
anything to do with any other. Certainly there is no reason to
suspect otherwise, especially given our process of
``discovering'' dimensions essentially by accident, by finding
that certain kinds of substitutions can be made transparently to
the effect we care about.

Yet what we found was far simpler and more powerful than what
could have been imagined. Notice that all of the dimensional
properties that we discovered, no matter how or through what
process, always turned out to be reducible (in other words,
equivalent) to some combination of measurements of three of
them, such as mass, length and time. There is no magic in these
particular three; we could as well have reduced to comparisons
of mass, speed and time, or many other possibilities. The
important point is that *a properly chosen set of three is
enough*. This result is remarkable in two ways. It is
obviously interesting that three very simple dimensions, among
the earliest abstractions people learned to make, turned out to
be complete for reducing others. Since there was no ``method''
for seeking dimensions, it is interesting that nonetheless a
complete set was found. The remarkable point in this is that
the dimensions were so obvious, even when we didn't understand
well what they ``meant''. The second and more remarkable aspect
is that, among all of the kinds of dimensionful measurements
that have been made since then, *no new dimensions have been
needed* beyond the original three to reduce them. The
completeness of this very small set of dimensions for reducing
all dimensional comparisons, and therefore all scaling relations
that have been observed, made it possible to connect the various
scaling relations with the process we have called dimensional
analysis, whereby it is possible to predict the scaling
behavior, and therefore estimate the magnitude, of dimensionful
properties that have not been measured yet, from any arbitrary
but sufficient knowledge of dimensionful properties that have,
such as the period of a pendulum from its length and the fact
that it swings in the earth's gravity, which can be
characterized through other means.

In this chapter we set out to account for another remarkable aspect of nature, which is a certain kind of increadible repeatability. At the outset, it is not at all obvious where such repeatability could originate, or that it has anything to do with scaling relations. We will find, though, that they are both expressions of essentially the same thing, and what we have discovered that accounts for the repeatability turns out to suggest a far deeper order than expected, and probably accounts for the existence of scaling relations as well.

- STARLIGHT AND LIFE
- DIMENSIONS AND SCALING REVISITED
- THE CONSTANT h/
- THE SPEED OF LIGHT
- NEWTON'S CONSTANT
- PLANCK'S OTHER OBSERVATION
- MISSING STEPS
- EQUALLY OBSCURE REPEATABILITY

Thu Aug 31 12:01:42 CDT 1995