The way we will treat this complexity is representative of the way it has been treated in much of history, and this is a good place to make a point about progress in physics and the nature of the difference between mathematics and physics.

We have already said that the process of physics is one of looking for patterns in nature and trying to invent ways to describe them that are useful. At any stage in this process, it is a sure bet that there are aspects of what we say that we really don't understand very deeply. Some of them lead to confusions that take decades to resolve, and others to confusions that we still don't understand. But, recognizing that, we try anyway to formulate descriptions and rules for making guesses, and we check them against what happens in nature to see if they predict correct answers. It is a tautology that nature is consistent, because whatever happens is what happens. If we can make correct predictions about what happens using rules we have invented, then even if the rules are not complete (which is almost always the case) or worse yet, if we know that there are points where they are internally contradictory if used in certain combinations (which is also not at all unusual), the fact that they give right answers is taken to suggest that they may be the crude approximations of better rules, which become more complete and more consistent as they better approximate the real structures of nature.

In those cases, with appropriate cautions about what we don't know, to suggest where to look for mistakes if we predict something that doesn't work, we are willing to use the rules to generate answers, and work to refine them as we learn more, making them more detailed where possible and finding origins to verify some of the parts that previously were just guesses. When we finally come up with any set of a few of the rules that can be used with guaranteed consistency, in any combination, we call it a part of mathematics.

Thus, the standards for what can be called mathematically sensible are much more restictive than those that delineate what a physicist is willing to use. On the other hand, mathematics exists in the minds of people, and nature seems to exists with or without the special accident of people, and since all of mathematics has its origins at some level in attempts made by people to describe their experiences with nature, it seems appropriate to regard mathematics as simply the highest and most clear-headed refinement of physical modeling. Therefore, mathematics can certainly never ``prove'' anything about nature; it simply can prove the requirements for consistency within its own models. When a mathematical proof about something we thought was true contradicts something we see, all that tells us (assuming the proof is really properly constructed) is that we are somehow using the wrong model, and that we have to invent another one, and hopefully at some point all the mathematics to describe it.

This point, made en passant, illustrates an increadibly important division that should be made among the various notions of ``what is true''. If it were more generally used, it would obviate a great deal of unnecessary confusion caused by bad language.

Recall, from the chapter on language, that the notion of internal consistency-checking done within mathematics is called proof, by mathematicians and (at long last) by physicists. Properly, this is the only real notion of proof that makes any sense. It applies only within the domain of logic, and refers to this checking of the consistency of rules, including the rules of the logic itself, once these have been created.

The separate notion of checking an answer against nature is called validation of the model and the proof. A disagreement of the answer is alone not enough to tell you which part of the analysis is not useful, whether it is the things that are assumed to exist, the rules that are assumed to govern them, or even the logic itself, that tells how the rules are to be applied. Any of these could be responsible for the disagreement. And, an agreement is not enough to tell you that you have done everything correctly. As with anything else, it is possible to generate what seem to be good models that can be understood with consistent rules of logic, get many correct results (according to nature), and think the entire construction is valid, only to find later that some other result is wrong.

Thus, the notions of proof and validity should be understood, each in its proper context. Proof in the sense of mathematics is exceedingly useful, because often we are close enough to correct by the time such a result can be constructed that the proven conclusion is a very good guess at the right answer. It can tell us where to look for new insights, and it can help us find deeply buried mistakes in our observations. But proof does not apply to the natural world, except through the filter of validation, which necessarily is limited by confidence and the extent of our experiences, and always subject to change. Conversely, the fact that we cannot prove something, even in the worst case that we know that there are inconsistencies in our methods, may not be a fatal affliction, because one can sometimes be right in cases, without being able to be right all of the time. The only requirement is that one be able to reliably identify the cases, and retain an acute awareness of the division between what one does and does not understand.

Much of what we describe below is a prime example of using things and getting increadibly good answers, in spite of the fact that they are not really mathematically understood at the time they are first used. In particular, this notion of the set of all imaginable histories has been historically one of the richest grounds for the prediction of right answers and the subsequent creation of catastrophic mistakes, whose resolution requires the invention of new areas of mathematics. This did not just happen once, but repeatedly as each new application was invented that used the idea of the set of all imaginable histories. And, in fact, the latest versions of these confusions are still at the forefronts in both mathematics and physics. With this caveat, and a promise not to state deep and subtle points as if they were obvious, we forge ahead.


Thu Aug 31 12:01:42 CDT 1995