in human constructions a particular cause has a particular effect; a particular intention brings to pass a particular object; but this is all; we see no reciprocity... In Divine constructions the object is either design or object as we choose to regard it -- and we may take at any time a cause for an effect, or the converse -- so that we can never absolutely decide which is which.

He wrote this specifically to describe the unthinkable perfection of the mathematical relations among the heavenly bodies; the mutuality of their interactions would seem to have no distinction between effect and cause. At its most grossly simplistic, one might ask, are molecules seen moving toward each other because of gravity, or is gravity seen because molecules are moving toward each other?

Today, as I look again at Ellen Langer's Mindfulness, I'm reminded of one of the major principles I first took from it-- that exclusive focus on an outcome could destroy the possibility of thinking of an appropriate solution. I had, initially, equated her assertion to John Holt's observation that students who had been trained to produce answers rarely considered the process by which the answers had been produced, and so were incapable of devising new solutions and were, besides, easily ruined by the slightest of deviations from their mindlessly-applied answer-generating processes. John Holt described that a student trained to produce answers will, rather than reason out the problem and proceed to a sensible result, flail about blindly with whatever tool comes to hand (or mind) and disgorge a flood of possible answers hoping that some authority will eventually approve one of them-- and this description originally helped me form my understanding that my role as teacher was not to provide approval for answers, but to empower students to evaluate their own responses without needing verification from anyone but themselves.

Today, however, thinking about my own process in the research on this site, I'm reminded of something I noticed myself as a direct consequence of this line of thought: when a problem is well-structured, its answer is inevitable. When you have fully understood the problem you'll invariably discover that you already have its solution-- or to put it most plainly, any problem is merely the description of its solution.

The simplest example is 1 + 1 = 2. As a "math problem", we tend to think of this problem as a sequential action: I add 1 to 1 and the result is 2. Most of us later learn to understand that this problem can be viewed as a statement of equivalence-- the equal-sign symbol tells us that the expression on one side is identical to the expression on the other-- but we may still think of a problem as linear, not reciprocal. A problem, considered linearly, is something we "do" for the purpose of "getting" a solution; once the solution has been "achieved" then the problem itself is superfluous and should be abandoned. We have our Answer. But reciprocally, a problem is its solution. It is impossible to conceive the expression "1 + 1" without simultaneously recognizing it to be "2". Cause cannot be separated from effect.

Today I'm simply reminding myself of the truth of reciprocity. I want to learn absolute pitch, and if the problem I'm trying to solve can be stated as "absolute pitch learning process = absolute pitch ability", then I must keep in mind that my goal is not merely to find the fastest way to the right side of the equation, whether or not my result makes sense, but to develop the left side to the point where the right side becomes not only self-evident but inevitable. It is crucial to recognize that, if this is an expression of equivalence, what is true of the ability must be true of the learning process, and vice versa. If absolute musical ability is the result of an autistic-like brain malfunction, then inducing an autistic-like brain malfunction will produce absolute pitch. If absolute listeners perceive pitches as musical phonemes, then learning to perceive pitches as musical phonemes must contribute to learning absolute pitch.

The principle of reciprocity reminds me, in sum, that if I succeed in determining what absolute pitch truly is, and how it is naturally learned, then not only is learning it myself inevitable, but what I have learned shall be richly self-evident.