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