Shape

309 Background
To be sure, Mill would agree that freedom (liberty) to observe and to say what you see
is key throughout education ‘‘in the most enlarged sense of the term.’’ Education isn’t
about morality and good and bad manners, or testing what you know. Good answers
and right behavior aren’t set in advance—they’re always a surprise. This isn’t a question
of logic but of unfolding experience. That’s why I’m so keen on shapes and rules.
My rules let me focus on this while going freely to that, without having to represent
anything in between. They let me see ‘‘what parts that thing is composed of’’ in new
ways as I go on. Only my rules aren’t Mill’s. His rules encourage getting in the right
frame of mind. This is important enough—and for me, it involves marshaling an array
of schemas x fi y to define rules. That’s the main job of part III. Then using these rules
to calculate with shapes puts observing and inventing in a working relationship—I do
the one to do the other.
The quotations from Franz Reuleaux that link ‘‘Invention’’ and ‘‘Thought’’ are in
The Kinematics of Machinery. 44 In keeping with this, Reuleaux divides design into direct
and indirect synthesis. He prefers the latter to the former, although this seems inevitable
when invention is thought—sooner or later, the one is bound to be organized in
the same way as the other.
Diagram of the Synthetic Processes
The importance of this part of the subject is so great that I have thought it worth while to add the
accompanying diagram in the hope that it may make the connection between the different synthetic
methods somewhat more clear to the reader.