It is good to understand a certain point about the way humans beings think about nature. We always derive something less fundamental from something more fundamental, but we discover them the other way around.

This can be said another way. The first point is more or less a tautology, or could be considered a definition of what it means for one idea to be more fundamental than another. Since any correct idea is required to predict the way things in nature actually happen, and since the whole reason physics works is that a given physical system only behaves in one way, all correct descriptions of some phenomenon must predict the same results. Therefore, the only distinction that can make one idea more fundamental than others is that it both includes the predictions they make, and also goes on beyond them.

In physics, this point of view is carried to a quite pure extreme. Whenever two explanations about some part of nature can correctly account for different things, but each of them can explain things that the other cannot, there really is no ground for considering one of them more fundamental than the other. Rather, both of them appear as isolated descriptions that apply to different special cases. If one of them is really to be considered more fundamental than another, it must be able to account for everything that the other does, as well as reach beyond it. In other words, it must imply all of the results of the other, and in that sense contain it as a special part.

On the other hand, that is never the way we learn about nature. We catch hold of nature's various patterns any way we can, usually only very approximately at first, and usually only in special cases. Then once we are sure we have recognized something and made some inroads to understanding it, we look more closely and try to tie it together with other special cases, to find the vein of consistency that underlies all of them, if one exists. But almost by construction of our thoughts, we can never derive the more fundamental underlying pattern from one or even a few of its applications. If we could, it would not be a more fundamental way to organize our thinking, but merely a consequence of what we had before. Thus the only way to discover the underlying principles from their various manifestations is a process of enlightened guessing and trial and error, often carried forth only by the leaps of intuition of single people at any one time.

This makes learning physics a curious experience. Each person has to learn it the same way we have learned it as a population, beginning to see the patterns by recognizing them in our experience, and then following the various trails of ideas that have been invented to make their underlying consistency visible. At each step into the more fundamental, ideas are introduced that seem strange, because from any few of their manifestations, they do not have to be the way they are at all. But then, once we see how they work, and how they tie together the clutter of applications that led us to them, we have a temptation to look back and say ``Of course! How else could it have been?'' as if somehow the consequences made the principles obvious, instead of the principles' having been constrained by the requirement that they make the consequences inevitable.

These are good things to keep in mind as we take a large step into more fundamental ideas, and introduce the Principle of Least Action. Nothing in what we have said so far requires that such a principle exist, or that it take the form it does. Therefore, many of the ideas will be new, and we will not be able to ``show'' that the world must be this way. On the other hand, it is possible to make fairly simple observations about nature, and advance some reasonable motivations, that make our results plausible and indeed almost natural. And once we have found the principle, in any way at all, we can work backward and show how it requires all of the results that have gone before it, as well as many more, and ties them all together into a single remarkably simple set of ideas.

So, in keeping with our general policy of starting with the simplest and most general, but therefore also most imprecise descriptions of the patterns of nature, and working to refine them, to make them more powerful but also easier to use, we look at what we have achieved so far and ask what could be done to improve it.


Thu Aug 31 12:01:42 CDT 1995