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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.
158 I What Makes It Visual? The first quotation from Hilary Putnam is in lecture 2 in The Many Faces of Realism, and the other two are found in lecture 1. 29 Richard Rorty limns the ironist’s literary skill in Contingency, Irony, and Solidarity. 30 Lionel March and I enjoy talking about design at the Moustache Cafe in West Los Angeles, over a long lunch and a good bottle of wine. Neither of us is keen on scruples or conscientiousness—there are no proprieties when it comes to conversation and design. I have transcribed one of his wonderfully Kierkegaardian insights. Rote results blunt creative experience, too—they’re the stuff of scruples and appeal only to the conscientious. It would be nice to have lunch again to see where all of this goes.
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( 2006 Massachusetts Institute of T
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Contents Acknowledgments ix INTRODU
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INTRODUCTION: TELL ME ALL ABOUT IT
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3 Answer Number One—What Do You S
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9 Answer Number Two—Three More Wa
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11 Answer Number Two—Three More W
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33 Trying to Be Clear Points aren
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35 Trying to Be Clear and twenty-fo
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37 Trying to Be Clear things look.
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39 Trying to Be Clear on correspond
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41 Tables, Teacher’s Desks, and R
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43 Tables, Teacher’s Desks, and R
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45 It Always Pays to Look Again In
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47 Background always more when you
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49 Background Getting it first and
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51 Background and not These are the
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53 Background and count as I please
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55 Background and time. The way Eul
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57 Background them apart—a triang
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59 Background without ‘‘agreeme
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62 I What Makes It Visual? (2) What
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64 I What Makes It Visual? Whenever
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66 I What Makes It Visual? calculat
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68 I What Makes It Visual? And my r
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72 I What Makes It Visual? Table 1
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80 I What Makes It Visual? But what
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86 I What Makes It Visual? I see a
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306 II Seeing How It Works Table 12
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308 II Seeing How It Works initial
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310 II Seeing How It Works Kinemati
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312 III Using It to Design (3) desi
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314 III Using It to Design does mor
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316 III Using It to Design that I d
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322 III Using It to Design Figure 1
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332 III Using It to Design Then the
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338 III Using It to Design But this
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340 III Using It to Design but occa
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342 III Using It to Design numbers
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344 III Using It to Design to start
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346 III Using It to Design The rule
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348 III Using It to Design Everythi
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352 III Using It to Design Let’s
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354 III Using It to Design ever hav
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356 III Using It to Design (2) Furt
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360 III Using It to Design with the
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362 III Using It to Design are equi
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Notes Introduction 1. S. Pinker, Th
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393 Notes to pp. 48-50 Langer’s d
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395 Notes to pp. 51-55 19. H. L. Dr
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397 Notes to p. 156 9. C. Alexander
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399 Notes to p. 304 of lowest-level
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401 Notes to p. 305 24. The ‘‘n
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403 Notes to pp. 306-387 35. G. Sti
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405 Notes to pp. 388-389 M. A. Bode
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407 Notes to p. 389 The essence in
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410 Index Aristotle, 81, 167, 337 A
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412 Index creativity and (cont.) re
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414 Index greatest lower bound, 224
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416 Index maximal elements and basi
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418 Index Quine, W. V., 57, 58, 210
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420 Index shape grammars (cont.) an
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422 Index Van Gogh, V., 403n1 Vanto
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