Sometimes a difficult problem can be solved by lifting into a higher-dimensional space --- generalizing, adding complexity, transforming, and then simplifying back at the end. This is a strikingly counterintuitive approach. Are we not seeking to reduce complexity? But lifting can work wonders when our original challenge is solvable only in an extremely subtle and non-obvious way, because it's a special case of a much larger problem --- a larger problem that has more structure and therefore more handles to get a grip on.
Think about the shadow cast by a moving three-dimensional object with a complicated shape. It could be almost impossible to predict the changes of that shadow, if we are only allowed to observe and think about the shadow itself. But if we can figure out the shape of the real object from evidence that the shadow provides us, then the problem to understand the shadow's motions becomes simpler.
The big magic happens when we realize that, for many complicated tasks, there need not be a unique "real object" at all --- and that the "shadow problem" we must solve can be extended into higher dimensional spaces in many different ways. So we can choose a lifting strategy deliberately to make our shadow problem turn into something easier to solve. We thus take advantage of the freedom that comes from going to a richer space of possibilities.
This method of "lifting" applies to numerous mathematical puzzles, and analogous ideas can be used on challenges in other areas of life. Often our struggles can be eased if we see them as special cases within larger situations --- and if we can escape from our immediate contexts to view ourselves in that larger universe....
Monday, May 31, 1999 at 12:02:30 (EDT) = Datetag19990531