Monday, March 27, 2006
The idea of "dimensions" is interesting. It's such a powerful concept. A dimension in linear algebra corresponds to an extra "basis vector" that you need to reconstruct the entire space. If you can make every point in the space from linear combinations of n vectors, then the space is at most n-dimensional. (If those n vectors are linearly independent, then the space is exactly n-dimensional.) We often think of dimensions in a spatial sort of way, but in something like Principal Components Analysis, the dimensions are not necessarily spatial at all. They're more like, how many factors do you need to explain the behavior of the system. Where a "factor" is some principle at the level just beneath the system itself; the level that creates the system. In that sense, there are way more dimensions in the universe than just the spatial dimensions of string theory. For example, there is the dimension of "how happy am I right now", or the dimension of, "how far along is my computer in downloading some files". And then dimensions build on dimensions. When there is some pattern, and that pattern has parameters within which it remains coherent, those parameters are dimensions. But if another pattern contains that original pattern as a constituent, it might "compress" a lot of the parameters from the constituent pattern by causing them to covary, and that creates a new dimension.