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Feb23-12, 02:27 AM
P: 6,863
Quote Quote by IsometricPion View Post
Therefore, in order to obtain any data representable as numbers or collections of categories thereof one must compare the quantity(ies) of interest to a reference value (or set thereof in the case of multiple categories).
No. I just thought of a counter example. Twenty questions.

I'm located in a spot on the earth. By asking me yes-no questions, you can figure out my latitude and longitude. Am I on land? Yes. Do I see taxicabs? Yes. Are they green? No. Are they yellow? No. Do I see water? Yes.

With each question, you can eliminate parts of the vector space. The fact that I see tax cabs and they are not yellow, means that I'm not in Manhattan. Now if you can ask enough questions, you can figure out my location and convert to GPS coordinates.

Note that you've figured out my GPS coordinates without actually measuring my latitude and longitude or doing any reference comparisons at all. You can show that no reference comparisons were done, because you can play this game without knowing anything about latitude and longitude at all, and it's the same game that you can play with things that are *not* vector spaces (i.e. words in a dictionary).

One other way of thinking about it is that you can specify points in a vector space as the interaction of subsets of that vector space, which allows you to specify a point in that space without reference to basis vectors at all.