Shape as Memory Storage

In a sequence of books, 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

memoryless

. 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:

Invariants cannot act as memory stores

. In consequence, I conclude that geometry, from Euclid to Einstein has been concerned with

memorylessness

. In fact, 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 computers 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 Einstein), 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 book

Symmetry, Causality, Mind

(MIT Press, 630 pages); and the mathematical structure is elaborated in my book

A Generative Theory of Shape

(Springer-Verlag, 550 pages).