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of science. For in fact, in areas of well-developed craft practices, technological developments typically preceded and gave rise to advances in scientific understanding, not vice versa. The steam engine is a good example. It was developed by mechanics who observed the relations between volume, pressure, and temperature. This was at a time when theoretical scientists were tied to the caloric theory of heat, which later turned out to be a conceptual dead end. The success of the steam engine contributed to the development of what we now call classical thermodynamics. This history provides a nice illustration of a point made by Aristotle:

Lack of experience diminishes our power of taking a comprehensive view of the admitted facts. Hence those who dwell in intimate association with nature and its phenomena are more able to lay down principles such as to admit of a wide and coherent development; while those whom devotion to abstract discussions has rendered unobservant of facts are too ready to dogmatize on the basis of a few observations.11

Many inventions capture a reflective moment in which some worker has made explicit the assumptions that are implicit in his skill. In a beautiful article, the cognitive scientists Mike Eisenberg and Ann Nishioka Eisenberg give real pedagogical force to this idea, and draw out its theoretical implications. They offer a computer program to facilitate making origami, or rather Archimedean solids, by unfolding these solids into two dimensions. But they then have their students actually make the solids, out of paper cut according to the computer’s instructions. “Computational tools for crafting are entities poised somewhere between the abstract, untouchable world of software objects and the homey constraints of human dexterity; they are therefore creative exercises in making conscious those aspects of craft work . . . that are often more easily represented ‘in the hand’ than in language.”12 It is worth pausing to consider their efforts, as they have implications well beyond mathematics instruction.

In our early work with HyperGami, we often ran into situations in which the program provided us with a folding net that was mathematically correct—i.e., a technically correct unfolding of the desired solid—but otherwise disastrous. . . . Here, we are trying to create an approximation to a cone—a pyramid on a regular octagonal base. HyperGami provides us with a folding net that will, indeed, produce a pyramid; but typically, no paper crafter would come up with a net of this sort, since it is fiendishly hard to join together those eight tall triangles into a single vertex. In fact, this is an illustrative example of a more general idea—the difficulty of formalizing, in purely mathematical terms, what it means to produce a “realistic” (and not merely technically correct) solution to an algorithmic problem derived from human practice.

I take their point to be that a realistic solution must include ad hoc constraints known only through practice, that is, through embodied manipulations. Those constraints cannot be arrived at deductively, starting from mathematical entities. These experiments with origami help us to understand why certain aspects of mechanical work cannot be reduced to rule following.

When I first starting working in the bike shop, after quitting the think tank, I would come home from work and my wife would sniff at me. She’d say “carbs” or “brakes” as she learned to identify the various solvents used in cleaning different parts of a motorcycle. Leaving a sensible trace, my workday was at least imaginable to her. But while the filth and odors were apparent, the amount of head scratching I’d done since breakfast was not. Mike Rose writes that in the practice of surgery, “dichotomies such as concrete versus abstract and technique versus reflection break down in practice. The surgeon’s judgment is simultaneously technical and deliberative, and that mix is the source of its power.”13 This could be said of any manual skill that is diagnostic, including motorcycle repair. You come