Shape as Memory Storage - Ambient Intelligence for Scientific Discovery - page 81

Shape as Memory Storage

Michael Leyton

DIMACS & Rutgers University, USA
[e-mail]

1 New Foundations to Geometry

In a sequence of topics, I have developed new foundations to geometry that are
directly opposed to the foundations to geometry that have existed from Euclid to
modern physics, including Einstein. The central proposal of the new foundations
is this:

SHAPE ≡ MEMORY STORAGE

Let us see how this contrasts with the standard foundations for geometry that
have existed for almost three thousand years. In the standard foundations, a
geometric object consists of those properties of a figure that do not change
under a set of actions. These properties are called the invariants of the actions.
Geometry began with the study of invariance, in the form of Euclid's concern
with congruence , which is really a concern with invariance (properties that do
not change). And modern physics is based on invariance. For example, Einstein's
principle of relativity states that physics is the study of those properties that
are invariant (unchanged) under transformations between observers. Quantum
mechanics studies the invariants of measurement operators.
My argument is that the problem with invariants is that they are memory-
less . That is, if a property is invariant (unchanged) under an action, then one
cannot infer from the property that the action has taken place. Thus I argue: In-
variants cannot act as memory stores. In consequence, I conclude that geometry,
from Euclid to Einstein has been concerned with memorylessness .Infact,since
standard geometry tries to maximize the discovery of invariants, it is essentially
trying to maximize memorylessness. My argument is that these foundations to
geometry are inappropriate to the computational age; e.g., people want comput-
ers that have greater memory storage, not less.
As a consequence, I embarked on a 30-year project to build up an entirely new
system for geometry - a system that was recently completed. Rather than basing
geometry on the maximization of memorylessness (the aim from Euclid to Ein-
stein), I base geometry on the maximization of memory storage . The result is a
system that is profoundly different, both on a conceptual level and on a detailed
mathematical level. The conceptual structure is elaborated in my topic Symme-
try, Causality, Mind (MIT Press, 630 pages); and the mathematical structure
is elaborated in my topic A Generative Theory of Shape (Springer-Verlag, 550
pages).