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When I was reading Kant's Critique, I got the sense that he'd sort of found a formula for calling something a dimension. Space seems to arise out of an infinity of extension. Time seems to arise from an infinity of duration.

More recently, with fractal geometry, it seems like there's another kind of infinity: complexity. Can complexity, or "zoom" be thought of as a dimension?

Is it valid to think of infinity in terms of dimensions?

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Is there any chance I might be able to persuade you to unpack this concern a bit further? –  Joseph Weissman Apr 13 '12 at 17:16
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I was wondering if you had some particular fractal idea in mind, perhaps? What is 'zoom' for instance? –  mixedmath Apr 14 '12 at 6:12
    
A dimension in the broadest sense is something quantifiable. some dimensions are infinite. –  Jo Rijo Nov 15 '12 at 16:31
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Closing for the time being pending some clarification of the concern -- maybe we could explore the connection with Kant a little bit further? –  Joseph Weissman Nov 15 '12 at 16:54
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closed as not a real question by Joseph Weissman Nov 15 '12 at 16:53

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I'm not sure exactly what your question is, so I'm going to interpret it in my own fashion:

Roughly, dimension first arose when Descarte discovered how to coordinatise space. Once you've noted 3 numbers serve to identify a point in space, you can generalise to say, 5 numbers, and that identifies a point in a new space, which is now 5-dimensional.

This seems naive at first, and not really that interesting. Are we just playing games with a list of numbers? But it turns out traditional geometrical terms can be applied to this context. Stuff like curvature, length, angle etc. So it does become interesting.

It turns out that the idea of dimension can be applied to different kinds of spaces and algebras for example topological spaces or rings (a specific type of algebra), these have varying definitions. These spaces or algebras can be finite, so in a sense dimension has nothing to do with having an infinite number of points.

Now dimension as traditionally conceived in space is an intrinsic notion, you don't have to go out of the space to measure it. For example you can conceive of a line in and of itself and not lying in some other space as opposed to say a line that you could draw on the flat surface of a table. A better way to think of the latter is that you've taken your line in and of itself and embedded it (thats the traditional mathematical term) on the table.

You get a fractal when this embedding is complicated in a certain definate sense, that is its self-similar. Fractal dimension measures the complexity of this embedding. So for our example it lies between 1, the dimension of what you're placing - the line, and 2 the dimension of what you're placing into - the plane.

For fractal dimension to differ from one you will need an infinite number of points, otherwise you couldn't hope to get self-similarity as you keep zooming in.

Whats this got to do with Kant I have no idea. I understand he was interested in space and time, and he posited them as forms of our intuition. That is space & time are not out there, but in us: we view the world though space & time spectacles. I wasn't aware that he was interested in the idea of dimension, perhaps you can point out the passage where he descibes it?

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You can define fractals intrinsically too, as metric spaces, so that their topological properties do not come from an embedding. This is the preferred definition nowadays anyway. –  Ron Maimon Apr 13 '12 at 23:12
    
yes, you can do that. I slanted my discussion the way I did to emphasise that there was a choice of dimension depending on how you wanted to look at the situation. Topologically or fractally. –  Mozibur Ullah Apr 14 '12 at 1:45
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