Any Ideas ? (II)

(4)  The free invention of structures

            To stick for a moment with Good Old Dad--   I recall, a couple of grades later, when he demystified the notion of the obscurely named “imaginary numbers”, the “square root of minus one” which, thitherto, had seemed some sort of deliberate paradox, like the “sound of one hand clapping” or the  “difference between a duck”.   He drew the “complex plane” – not at all complex, a sample fits nicely on a table top – and characterized the imaginary-unit i  as a rotation through a right angle.

            Not only was the demonstration delightful in itself, it opened up the notion of *positing* something that was not there before, sufficiently well-defined that we can calculate with it, and  by luck or insight  capable of yielding a variety of interesting results:  creating, in effect, a new world.  Thus groups, rings, fields, topological spaces, etc. Thus math.

            A note at this point: the individual structures such as finite group, commutative ring, etc., though each wonderful in its own special way, we shall not count individually as Ideas.  Rather, each is comparable to a species in biology; its algebra, to the anatomy, physiology, and ethology of the beast in question.  (The analogy might be taken further, loose as it is: “group” might be a genus with “finite group” and “infinite Abelian group” as species; the genus “ring” splits smartly along the line of commutative or not, a taxonomic faultline  not a priori predictable, and thus like the surprises of biological systematics. And – so – forth.)

[Since originally writing the above, I have grown jaundiced with the notion of simply positing anything.  Our posits may stand at the head of the logical deductive queue, but if they are any good at all, they have been derived from experience, then dressed up in axiomatic costume for the photographer.  A subject treated here:
"You Choose:  A Minimum Axiomatization for Reality".

[Continued here]