Animated Logical Graphs • 15

In George Spencer Brown’s Laws of Form the relation between the primary arithmetic and the primary algebra is founded on the idea that a variable name appearing as an operand in an algebraic expression indicates the contemplated absence or presence of any expression in the arithmetic, with the understanding that each appearance of the same variable name indicates the same state of contemplation with respect to the same expression of the arithmetic.

For example, consider the following expression:

Cactus Graph (a(a))

We may regard this algebraic expression as a general expression for an infinite set of arithmetic expressions, starting like so:

Cactus Graph Series (a(a))

Now consider what this says about the following algebraic law:

Cactus Graph Equation (a(a)) =

It permits us to understand the algebraic law as saying, in effect, that every one of the arithmetic expressions of the contemplated pattern evaluates to the very same canonical expression as the upshot of that evaluation.  This is, as far as I know, just about as close as we can come to a conceptually and ontologically minimal way of understanding the relation between an algebra and its corresponding arithmetic.

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This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

7 Responses to Animated Logical Graphs • 15

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