Bricoleur's Algebra

Why this definition and not another? Who decides what we spend our time on? And how do we defend against the accusation that abstract algebra is a song and dance designed-through otherwise unmotivated definitions-around things we already know, like arithmetic and linear algebra?

A definition is, by definition, a construction going from general to specific. We need to place on this construction a filter that we can all agree on. A familiar one already exists: canonical construction. Canonical means making no arbitrary choices within the construction once the components are given. If the goal is to be specific yet canonical, we can go a step further.

Metacanonical construction adds the criterion that the selection of which canonical constructions matter must itself be non-arbitrary. A familiar device offers one implementation: mirrors. We canonically pick out canonical constructions that lie entirely within the mirrored space of a mirror-that is, a correspondence. As it turns out, the objects produced by such constructions are precisely the ones that stand out most in importance.

The organizing principle of algebra is then to move from the general to the specific without making any arbitrary choices. Here, we demonstrate this approach first for binary operations, then associativity, and finally groups.