From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Mon Mar 11 2002 - 12:54:28 MST
Ken Clements wrote:
>
> No, and it is not just that you stated the ratio backwards, it is much more fundamental. First, you have to understand that a circle is defined as the locus of points that are all equidistant from a given point (the center). Points are theoretical; they are not dots on paper, they are not places in any real space. Points have no width or length or thickness (dimensionless). They are part of the tautological mental
> construct we call mathematics. When you "draw a circle" on a piece of paper, the actual mathematical circle is in your mind. What is on the paper is a pattern of ink blobs that all have physical dimensions not associated with mathematical points. If you do a good job drawing, the length of your figure divided by the widest distance across it will be something close to pi, but will not equal it. A circle is not a physical
> thing.
Just to be really pedantic, a circle in your mind does not occupy some
abstract mathematical space; it occupies a retinotopic pixel map in your
visual cortex. The actual visual map in the brain is stretched near the
center and shrunk near the edges, so if the imagined circle were visualized
off-center, it wouldn't actually *be* a circle in the physical substrate.
-- -- -- -- --
Eliezer S. Yudkowsky http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence
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