One of the principle things that has always stood as an
impediment to man's understanding of nature is the great
complexity of what can be observed. While it is often fairly
easy to notice patterns in natural phenomena, they are seldom
repeated in exactly the same way in all their details, even to
whatever precision our observations permit us (example?). Thus
it can seem hopeless in principle to predict anything without
being able to understand everything. (While we have gone a long
way to show how this point is refuted, in our work on
dimensional analysis, it is interesting to recognize the
importance it is assigned in many philosophical systems, notably
eastern ones, which are currently experiencing a *naissance*
of popularity, in part fueled by enthusiasm for this premise.)
What was needed, for physics to have a beginning, was a way to
circumscribe some of the complexity of nature, to ``draw an
imaginary line'' around some part of the natural world that is
small enough to be comprehended, and to exclude or to simply
encapsulate the rest of the complexity of the universe in some
way that can be addressed separately and after the part we wish
to consider. Yet this exclusion of most of the complexity, if
it is to be useful, must be done in such a way that the *quantitative accuracy* of the prediction is not also thrown away
in the process (companion example?).

This is not something that we have accomplished in our
discussion so far. Recall what we *have* done. We have
understood the most basic level of the patterns in nature when
we recognize the ability to *compare* properties of objects,
and have made that ability precise and useful with the notion of
measurement. That was the first step in creating a concrete and
useful language. Then in noting that not all the different
kinds of comparisons were independent, we were led to the notion
of *dimensions*, and through these recognized the redundency
that we have described under the name of *scaling
relations*. The most important thing learned in the process of
creating these aspects of language was the concept of *dimensional homogeneity*. This starts with the premise, that we
understand a thing by describing its properties, and that we
describe them by telling how to compare them to standards. From
there we are led necessarily to the conclusion that, because any
*question* is necessarily asked in terms of some set of
comparisons, and any *answer* must necessarily also be
phrased in terms of the same comparisons, there are only certain
kinds of answers possible to any given question. We adopted the
habit of expressing these question/answer pairs in the form of
equations, with the question-comparisons we wish to predict on
one side, predicted to be ``equal to'' some eligible
answer-comparisons, based on properties we know, written on the
other. The content of the principle of dimensional homogeneity
is that the two terms written on either side of the ``equality''
symbol, though they have far different origins and
interpretations, must have *the same dimensions*. Finally,
in realizing how very few dimensions there are that are
independent, and relating the properties that we know of an
object to comparisons in terms of those dimensions, we obtained
some predictive power over the general sizes and the possible
scaling relations of the eligible answers, in terms of those
properties. The last step is, of course, that of *dimensional analysis*.

It happens that in the course of these developments, we have built a concrete way of asking and answering questions that lends itself naturally to Newton's problem of circumscribing the complexity of nature, because in our language of equations to express questions and answers, we have naturally divided our description of the world into two parts, that which we wish to know but do not (the question side), and that which we assume we can measure in order to identify which system we are considering, but which is not obviously related to what we have asked (the answer side). What Newton was able to do was find a very useful class of questions and answers for which, not only did the known and the unknown divide into distinct halves of an equation, but so also did the complexity and the simplicity.

Here is how:

Thu Aug 31 12:01:42 CDT 1995