Shape

157 Background
you think you’re not. The interesting thing is to figure out what kind of calculating it
is. That’s what I’ve been trying to do with shapes and rules.
There’s no escaping a short list of miscellaneous stuff. Ralph Waldo Emerson
does little minds in ‘‘Self-Reliance,’’ 21 and Jacques Barzun describes the rubber-band
effect in A Stroll with William James. 22 Nelson Goodman’s definition of relative discreteness
in Languages of Art is neither consistent nor elastic. 23
John Dewey dismisses logical analysis and morphological method in teaching in
How We Think. 24 But Barzun finds Dewey’s emphasis on the scientific method, even
with his nascent logic of inquiry, a prime example of ‘‘preposterism.’’ It puts recapitulation
ahead of thinking and structure before calculating.
Dewey’s plan is thus another piece of preposterism. When a good mind has done its work, idiosyncratically,
it will no doubt submit the results to others in Dewey form. But this is no warrant
for believing or requiring that the end[s] serve as a prescription of the means. 25
‘‘That expectation is pre-post-erous, the cart before the horse.’’ Instead Barzun looks to
James.
James . . . understands the child’s mind (and the adult’s, for that matter) as quite other. It is not an
engine [computer] chugging away in regular five-stroke motion [there are five steps in Dewey’s scientific
method]; it is an artist mind; it works by jumps of association and memory, by yielding to
esthetic lures and indulging private tastes—all in irregular beats of attention, in apparent wanderings
out of which some deep sense of rationality rises to consciousness. There is no formula, for
the trained or the untrained. 26
There are no set answers when it comes to education. Training may help to get
started—there’s plenty of it at home and in school—but soon enough reasoning will
follow artistic lines. This is what happens to figures when Ludwig Wittgenstein calculates.
And I’ve tried to show that visual calculating encourages the creative wanderings
of the artist mind. The artist mind is free to see and do in the unstructured flow of experience,
to engage it directly without deciding beforehand what surprises it holds.
That’s why there are shapes and rules. And it’s why embedding makes a real difference.
Wittgenstein adds shapes, so that lines fuse, and worries about what this entails
for mathematics in Remarks on the Foundations of Mathematics. 27 I said earlier that going
from combinatorics to phenomenology was something like the shift from identity to
embedding, so that i exceeds 0. It strikes me now that this may be the same for Wittgenstein
when he moves on from the Tractatus to the Investigations—he’s going from
points to lines. The value of the dimension i may explain some aspects of the split.
It certainly helps to explain why calculating with symbols and calculating with shapes
are different. William F. Hanks suggests that structure is an outcome of action, and that
teaching and learning can occur in situations where master and apprentice act according
to their separate representations of what’s going on in his foreword to Jean Lave
and Etienne Wenger’s book Situated Learning. 28 This sounds like calculating according
to rules that apply to shapes in terms of embedding. More accurately, though, representations
(descriptions) as stand-ins for shapes aren’t necessary to calculate.