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233 How Rules Work When I Calculate In the first stage, the rule A fi B is used to see. It works as an observational device. If A can be embedded in C in any way at all—no matter what’s gone on before— then C has a part like A. In the second stage, the rule changes C in accordance with A and B. Now it’s a constructive device. But once something is added, it fuses with what’s left of C and may or may not be recognized again. The shapes (parts) in the rule are lost once it’s applied. There’s no record (memory) of the part identified in stage 1 or of the part added in stage 2. Calculating always starts over with an undifferentiated shape. And notice also that the two stages are intentionally linked via the same transformation t. This completes the visual analogy tðAÞ fi tðBÞ < A fi B The rule A fi B is a convenient example of the kind of thing that’s supposed to happen. The rule tðAÞ fi tðBÞ is the same and works for the given shape C. This is one way of thinking about context. It’s evident when I transpose the terms in the analogy—the part of C that’s subtracted is to A as the part that’s added is to B. But the analogy stops there. It doesn’t go on. It ends with A and B, and the transformation t. There are neither finer divisions nor additional relationships in A or B. The parts of shapes are no less indefinite because they’re in rules, and they aren’t differentiated when rules are tried. The formal details of rule application shouldn’t obscure the main idea behind rules. Observation—the ability to divide shapes into definite parts—provides the impetus for meaningful change. This relationship is nothing new, but it’s no less important for that. As I’ve already said, it’s seeing and doing, the behaviorist’s stimulus and response, and Peirce’s habitual when and how. And James has it at the center of reasoning. This is worth repeating. And the art of the reasoner will consist of two stages: First, sagacity, or the ability to discover what part, M, lies embedded in the whole S which is before him; Second, learning, or the ability to recall promptly M’s consequences, concomitants, or implications. The twin stages in the reasoner’s art are formalized in the stages given to apply rules. In fact, it’s the same kind of analogy contained in rules themselves. I can show it in this way tðAÞ : tðAÞ fi tðBÞ < sagacity : learning or, alternatively, so tðAÞ a C : ðC tðAÞÞ þ tðBÞ < sagacity : learning in terms of what the rule does as it’s used. Every time a rule is tried, sagacity and learning—redescription and inference—work jointly to create a new outcome. And it’s sagacity that distinguishes rules the most as useful devices for calculating. Rules divide shapes anew as they change in an unfolding process in which parts are picked out, combine, and fuse. But learning isn’t remembering how shapes are divided into parts—
234 II Seeing How It Works that can change anytime. It’s finding rules and remembering ones as examples of what to see and do. This makes a huge difference when embedding isn’t identity. I can know how to go on without knowing anything for sure about the world. It’s using your eyes to decide what to do next. (The root value of observation in calculating is reinforced when James quotes John Stuart Mill—‘‘The observer is not he who merely sees the thing which is before his eyes, but he who sees what parts that thing is composed of.’’ And again, for me this changes every time a rule is tried. But there’s more. For Mill, observing and inventing are alike. This makes a neat metaphor for rules and a good heuristic in design.) Trying It Out The way rules are used to calculate is clear in an easy example. The rule produces the shapes in this ongoing series when I start calculating with the square In particular, the third shape in the series is produced in two steps in the following way and in three steps in this way
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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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70 I What Makes It Visual? more tha
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72 I What Makes It Visual? Table 1
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74 I What Makes It Visual? but neve
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76 I What Makes It Visual? may be f
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78 I What Makes It Visual? formula
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80 I What Makes It Visual? But what
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82 I What Makes It Visual? they pro
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84 I What Makes It Visual? The one
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86 I What Makes It Visual? I see a
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88 I What Makes It Visual? But this
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90 I What Makes It Visual? contains
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92 I What Makes It Visual? and othe
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94 I What Makes It Visual? The lett
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96 I What Makes It Visual? and corr
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98 I What Makes It Visual? (Is hh i
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100 I What Makes It Visual? and the
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102 I What Makes It Visual? There
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104 I What Makes It Visual? to the
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106 I What Makes It Visual? Table 2
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108 I What Makes It Visual? is trie
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110 I What Makes It Visual? artifac
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112 I What Makes It Visual? and thi
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116 I What Makes It Visual? looks l
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120 I What Makes It Visual? when th
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122 I What Makes It Visual? until t
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124 I What Makes It Visual? The fir
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126 I What Makes It Visual? What Sc
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128 I What Makes It Visual? Here, g
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130 I What Makes It Visual? be desc
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132 I What Makes It Visual? An addi
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134 I What Makes It Visual? forget
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136 I What Makes It Visual? that bl
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138 I What Makes It Visual? Why is
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140 I What Makes It Visual? see wha
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142 I What Makes It Visual? to calc
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144 I What Makes It Visual? How cav
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146 I What Makes It Visual? togethe
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148 I What Makes It Visual? and six
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150 I What Makes It Visual? and its
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152 I What Makes It Visual? these h
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154 I What Makes It Visual? So how
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156 I What Makes It Visual? (TRI(XY
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158 I What Makes It Visual? The fir
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160 II Seeing How It Works was thre
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162 II Seeing How It Works and lowe
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164 II Seeing How It Works Back to
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166 II Seeing How It Works Divide e
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168 II Seeing How It Works There ar
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170 II Seeing How It Works while th
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172 II Seeing How It Works there ar
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174 II Seeing How It Works This has
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176 II Seeing How It Works And noti
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178 II Seeing How It Works looks ex
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180 II Seeing How It Works meaning.
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284 II Seeing How It Works applies
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286 II Seeing How It Works the same
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288 II Seeing How It Works in terms
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290 II Seeing How It Works It’s e
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292 II Seeing How It Works If a rul
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294 II Seeing How It Works But then
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296 II Seeing How It Works And, cer
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298 II Seeing How It Works are all
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300 II Seeing How It Works combine
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302 II Seeing How It Works In fact,
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304 II Seeing How It Works next pap
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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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318 III Using It to Design don’t
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320 III Using It to Design fix the
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322 III Using It to Design Figure 1
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324 III Using It to Design —divid
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326 III Using It to Design Perhaps
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328 III Using It to Design I seem t
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332 III Using It to Design Then the
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334 III Using It to Design and ‘
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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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358 III Using It to Design that wor
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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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364 III Using It to Design and in f
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366 III Using It to Design to the p
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368 III Using It to Design rules th
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370 III Using It to Design x fi bð
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372 III Using It to Design These we
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374 III Using It to Design easy to
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376 III Using It to Design and spir
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378 III Using It to Design I said I
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380 III Using It to Design include
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382 III Using It to Design x fi x i
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384 III Using It to Design It’s g
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386 III Using It to Design take car
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388 III Using It to Design wealth o
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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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