Algebraic models for understanding: coordinate systems and cognitive empowerment

We identify certain formal algebraic models affording understanding (including positional number systems, conservation laws in physics, and spatial coordinate systems) that have empowered humans when we have augmented ourselves using them. We survey how, by explicit mathematical constructions, such algebraic models can be algorithmically derived for all finite state systems and give examples illustrating this, including coordinates for the rigid symmetries of a regular polygon, and recovery of the decimal expansion and coordinates arising from conserved quantities in physics. By accepting D. Haraway's (1991) 'ironic myth' of the ourselves as cyborgs, one opens the door to empowerment of humans responsible and in control of their hybrid natures as beings augmented by both cognitive and physical tools. Indeed we argue that biological systems share with the subjects of all sciences of the 'artificial' (H. Simon (1969)), an essential quality of contingent, conscious or unconscious design. Coordinate systems derived by algebra or computation for affordance of the understanding and manipulation of physical and conceptual worlds are thus a 'natural' step in use of 'tools' by biological systems as they/we learn to modify selves and identities appropriately and dynamically. The mathematics of constructing formal models for understanding is explained, application to familiar examples and to the Noether-Rhodes theory of conservation laws, and extensions to handling formal relations and analogies between models are discussed.