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THE SET OF ALL PATHS

First and formost, it might be good to recall just what a huge selection process we are asking from our physical law of minimization. Fig. 6.6

shows just a few of the paths that the authors can imagine for a falling object, and there are obviously infinitely more than these. A little needs to be said about just how big an infinity that is.

Given any imagined path, it is always possible to construct from it another imaginable path by just supposing a little difference at some point on the path, as indicated by the little bump in fig. 6.6.

Independently, we could make yet another path by adding another little bump somewhere else along its length. This new little bump is not ``more of'' the previous little bump, but something entirely independent of it.

Now, if we were standing in a room, and someone asked us how many different ways there were to change our location, we would say that there are infinitely many, because we can move by an arbitrarily small amount in any direction. But this is a fairly simple infinity. If we take a small step toward the north wall, and then another small step toward the north wall, we can simply say that the second motion was ``more of'' the first one. To make a new motion that is not ``more of'' the old one, we have to move either toward the east or west walls or toward the ceiling. Once we have tried each of these new motions, anything else we could do is just ``more of'' something we have already done. This limitation, that we can sample all of the possible independent ways of changing our position in a room by trying three different ones, is what we mean when we say that the space in the room is three-dimensional. Certainly there may be an infinite collection of different places to move to, distinguished by how far we moved in the three directions, but it only takes three measurements to identify any such change.

Compare that to the case of the paths. Once we have added one little bump somewhere, we can always add ``more of'' the same little bump at the same place, or we can add an independent new bump at any one of the infinitely many other places available along the path. Thus, to describe this set of all imaginable changes in path, not only must the number of descriptions be infinite because of the smallness of the changes we are free to make, but even the dimension of that set of changes must be infinite, because it would require infinitely many such numbers to label the change. So this set of all imaginable paths that we treat so glibly in our discussions should be understood as a very big collection, and the proper treatment of it has been a difficult but also fruitful area of mathematics for more than a century.




Next: ONE CAVEAT: MATHEMATICS Up: THE PRINCIPLE OF Previous: RELATION TO THE


desmith@
Thu Aug 31 12:01:42 CDT 1995