For those interested in higher dimensions, I've just grabbed a link
from wikipedia:
* Christopher E. Heil, A basis theory primer, 1997.
http://www.math.gatech.edu/~heil/papers/bases.pdf
Well, a mathematician needs to _understand_ (as opposed to what I
would call a "knowledge base - inference engine disconnect"), and
visualization is a metaphor for understanding, not the understanding
itself.
What "visualization" actually often means, is an _unsound reduction_
of a general sophisticated notion to a simple model, without ever
realizing that the model contradicts the notion, but instead
supplementing this "heuristical" model with additional mental
discipline at "rough corners".
On 10/12/07, Eliezer S. Yudkowsky <[EMAIL PROTECTED]> wrote:
> Benjamin Goertzel wrote:
> >
> > Well ... going beyond imaginary numbers... how do *you* do mathematics
> > in quaternionic and octonionic algebras? Via visualization?
> > Personally, I can sorta visualize 4D, but I I suck at visualizing
> > 8-dimensional space, so I tend to reason more abstractly when thinking
> > about such things...
>
> Just visualize it in N-dimensional space, then let N go to 8.
>
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