From: Mitchell Porter (mitchtemporarily@hotmail.com)
Date: Sat Jan 28 2006 - 09:29:43 MST
On 1/26/06, pdugan <pdugan@vt.edu> wrote:
>From Chris' first online suicide note: "Oh and BTW, the mind is a maximum
>hypersurface and thought a trajectory on it and the amygdala and
>hippocampus
>are Hopf maps of it."
>
>Does this sentence contain a pearl that might prove useful to the AGI
>community? I think an expoundation from some of more math-savvy members of
>the
>list would be a fitting eulogy.
After studying his blog, this is the best I can manage:
He thought that the human brain is wired to index memories
seven-dimensionally, seven dimensions being optimal "because" the
hypersurface area of an n-sphere is maximized at n=7 (here I use the
topological naming convention, in which n is the dimensionality of the
surface, so a 7-sphere inhabits 8-dimensional space). The seven dimensions
correspond to the states of the six cortical layers and of the thalamus.
Mathematically, he then proposes a decomposition of local coordinates on the
7-sphere (via the "quaternionic Hopf bundle") into a 4-sphere and a
3-sphere. The first four coordinates are to be used for space-time indexing
by the hippocampus, the last three coordinates are to be used by the
amygdala for indexing according to Wilhelm Wundt's three dimensions of
emotion (pleasure, arousal, dominance, in Mehrabian's recent formulation).
I have been unable to motivate the focus on n-spheres to my own
satisfaction, but some adaptation of Douglas Matzke's "corob theory" *might*
do it. Matzke talks about using random points in a high-dimensional
hypercube as naming tokens; the higher the dimensionality, the better their
properties from the perspective of error correction (greater spread, more
uniform spread). *If* you were restricted to using uniformly normalized
vectors drawn from a Euclidean space, you *might* find that unit 8-vectors
(which form a 7-sphere) are optimal for a similar reason. But I cannot see
that McKinstry himself made this argument anywhere.
Also, if one were combining the states of seven disjoint brain subsystems to
define an abstract state space, you'd think that the appropriate way to
combine them would be via Cartesian product, but that would produce the
Euclidean space R^7, not the hypersphere S^7. I might be able to contrive
some further way to save McKinstry's hypothesis of optimality, utilizing
coordinate maps which are locally R^7, but I leave that as an exercise for
the newly formed McKinstry Institute (see the Wikipedia page). For the sake
of completeness, I'll also note that while McKinstry started out wanting to
apply hyperspace geometry to abstract state spaces (common enough, even in
AI; see "VC dimension"), in his final year he became interested in the
*physical* hyperspaces of string theory, where Hopf maps really have found
some application, so evidently he started thinking about this space of
representational states as being encoded in fundamental physical variables.
And how does this relate to his AI enterprise, the Mindpixel Project?
Mindpixel had accumulated a list of thousands of propositions, the
"mindpixels", which were to be a sample of the space of all possible
thoughts, and indexed by a single parameter, consensus probability of truth.
(He started using Google query probability as a ranking parameter at the
end.) I think that eventually he wanted to devise a seven-dimensional
indexing system as interface to his archive of mindpixels, along the lines
of his brain theory, which would then be a component of an AGI.
http://www.mindpixel.com/chris/2005/08/my-motivation-for-creating-new-unified.html
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