Scientific Representation: Paradoxes of Perspective

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2 Scientific Representation: Paradoxes of Perspective

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4 Scientific Representation: Paradoxes of Perspective Bas C. van Fraassen CLARENDON PRESS OXFORD

5 1 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York Bas C. van Fraassen 2008 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2008 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Laserwords Private Limited, Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd, King s Lynn, Norfolk ISBN

6 for Janine Blanc Peschard and the memory of Dina Landman van Fraassen

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8 Preface When I began to rewrite the Locke Lectures I gave in Oxford in 2001 I found the comments I received leading me into new paths, some quite unexpected. My main concern had been, and remained, the possibilities for an empiricist version of structuralism in philosophy of science. But how I came to conceive of those possibilities was altered first of all by closer contact with the revivals of transcendentalist and neo-kantian thought, and secondly by the lively and growing interest I encountered in technology, instruments, and experimental practices both in contrast with, and complementary to, philosophical reflection on scientific theories. Not all of my debts will appear explicitly in the text. I am thankful especially for the great good fortune I had to meet Isabelle Peschard, my main interlocutor on both these subjects. My thanks also to the writings, helpful discussion, and correspondence of Michel Bitbol, Michael Friedman, Alan Richardson, and Thomas Ryckman for more insight into the neo-kantian tradition, and to Mieke Boon, Nancy Cartwright, Margaret Morrison, and Hans Radder for discussions of instruments and experimentation. These debts are in addition to many debts, accumulated with respect to structural realism and surrounding topics, to Jeffrey Bub, Jeremy Butterfield, Otávio Bueno, Chris Fuchs, Hans Halvorson, James Ladyman, Bradley Monton, Carlo Rovelli, and Simon Saunders. On the subject of representation in the sciences, as will be quite clear, I have substantial debts to Ronald Giere and Paul Teller. In addition, Bradley Monton and Paul Teller went through an early manuscript version with a fine tooth comb and made many valuable detailed comments on the text. I am painfully aware that I owe more debts to more people than I can relate here. But my greater debt, beyond words, is to Isabelle Peschard. The main addition to the locke lectures comes in Part Two, on measurement as representation, which is also the more technical part of the book. While in any logical ordering this material precedes the third and fourth parts, the less patient reader may wish to skip ahead to them.

9 viii PREFACE The generous sabbatical and leave policy of Princeton University, a splendid year at the Center for Advanced Study in the Behavioral Sciences in Stanford, and the hospitality enjoyed at All Souls and Magdalen Colleges in Oxford, the CREA (Centre de Recherche en Epistémologie Appliquée) in Paris, and the University of Twente in the Netherlands, as well as Senior Scholar Award SES from the National Science Foundation, made this work possible. My special thanks to Ralph Walker, Jean Petitot, and Philip Brey for making me welcome at their institutions in Oxford, Paris, and Twente respectively. Bas C. van Fraassen

10 List of Figures 2.1. Reflection and Refraction Perspective Altimetry Window and Checkerboard Dürer, TheDraughtsmenoftheLute Frame of Reference VS. Perspective Speed in Perspective Cross Ratio Invariance Geometry of the Rainbow Image Categories Measurement Schema Coherence of Measurement Adequacy as Symmetry Putnam s Paradox Copernicus s Model of Retrograde Motion Failure of Supervenience 293

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12 Contents Preface List of Figures vii ix Introduction: the picture theory of science 1 Part I. Representation 1. Representation of, Representation As 11 The value of distortion 12 How does a representation represent? 15 What s in a photo? 20 What is a representation then? 22 Appearance to the intellect: illumination as embedding 29 In conclusion Imaging, Picturing, and Scaling 33 Modes of representation 33 What distinguishes a picture? 36 Mathematical imagery, distortion through abstraction 39 Scale models and virtuous distortion 49 Conclusion about imaging and scaling Pictorial Perspective and the Indexical 59 Pictorial perspective and the Art of Measuring 60 Perspective versus Descartes s frames of reference 66 Mapping and perspectival self-location 75 What is in a map? 82 Visual perspective and the metaphor 84 Concluding empiricist postscript 86

13 xii CONTENTS Part II. Windows, Engines, and Measurement 4. A Window on the Invisible World (?) 93 Instrumentation s diversity of roles 94 Engines of creation: engendering new phenomena 100 The microscope s public hallucinations 101 Objections to this view of observation by instruments 105 Experimentation s diversity of roles The Problem of Coordination 115 Coordination: a historical context 116 The problem of coordination reconceived 121 Mach on the history of the thermometer 125 Poincaré s analysis of time measurement 130 Observables coordinated: two morals Measurement as Representation: 1. The Physical Correlate 141 Physical conditions of possibility for measurement 141 General theory of measurement 147 What is not measurement Measurement as Representation: 2. Information 157 What is measurement number-assigning? 158 The scale as logical space 164 Data models and surface models 166 The over-arching concept for measurement 172 What is a measurement outcome? 179 Relating the views from above and from within 184 Part III. Structure and Perspective 8. From the Bildtheorie of Science to Paradox 191 The Bildtheorie controversy 191 Representation: the problem for structuralism 204

14 CONTENTS xiii 9. The Longest Journey: Bertrand Russell 213 Prolegomena to Russell s conversion to structuralism 213 Russell s structuralist turn 217 Conclusion Carnap s Lost World and Putnam s Paradox 225 Carnap: Der Logische Aufbau der Welt 225 Putnam s Paradox 229 Staying with Putnam: the Paradox dissolved An Empiricist Structuralism 237 What could be an empiricist structuralism? 237 The fundamental remaining problem for a structuralist view of science 239 The two main dangers for an empiricist 244 The problem in concrete setting revisited and dissolved 253 Return to our epistemological question 261 Part IV. Appearance and Reality 12. Appearance vs. Reality in the Sciences 269 Appearance and reality: the real and unreal problem 270 Appearance versus reality at the birth of modern science 270 Three putative completeness criteria 276 Appearance vs. reality: A deeper Criterion 280 Phenomena versus appearances 283 Three-faceted representation Rejecting the Appearance from Reality Criterion 291 The supervenience of mind challenge 292 The Great Leibnizian Escape move 296 The quantum mechanics challenge 297 Exploring the case of quantum mechanics 300 Supervenience? 304 An empiricist view 304

15 xiv CONTENTS APPENDICES Appendix to CH 1. Models and theories as representations 309 Appendix to CH 6. Quantum peculiarities: fuzzy observables 312 Appendix to CH 7. Surface models and their embeddings 315 Appendix to CH 13. Retreat (?) from The Scientific Image 317 Notes to Appendices 320 Bibliography 322 Notes 345 Index 399

16 Introduction: the picture theory of science The Bildtheorie picture theory of science formed the frame for much discussion and controversy among physicists in the decades around the year In retrospect, it had close connections with philosophers attempts to forge what we would now call a structuralist view of science. The debates and projects of those years foreshadowed the debates some half a century later, closer to our time, over scientific realism and structural realism. To understand science we need to approach it from many directions. I will focus on one aspect that I take to be central to the scientific enterprise: representation of the empirical phenomena, by means of artifacts, both physical and mathematical. The position that success in this respect is the defining aim of the empirical sciences is an empiricist theme from which I will not depart. In fact, this focus on representation fits the empiricist theme very well. While we are now, it seems to me, much more demanding in what we expect of philosophical accounts of science, I will take the Bildtheorie as exemplar and inspiration. But there will be some major differences between the view of science that I shall advocate and what was then known under that name. First of all, the Bildtheorie view was phrased in terms of mental pictures. Thus Boltzmann formulates the goal of physical theory as constructing a picture of the external world that exists purely internally (Boltzmann 1905, 77; 1974, 33) and called the product of scientific theorizing an inner picture or mental construction ( inneres Bild, gedankliche Konstruction ; Boltzmann 1974, 106; ). In an Encyclopedia Britannica article he spelled it out:

17 2 INTRODUCTION On this view our thoughts stand to things in the same relation as models to the objects they represent. The essence of the process is the attachment of one concept having a definite content to each thing, but without implying complete similarity between thing and thought; for naturally we can know but little of the resemblance of our thoughts to the things to which we attach them. (Boltzmann 1974, 214.) I will have no truck with mental representation, in any sense. 1 The view Boltzmann expresses here, a view in philosophy of mind or language, has nothing to contribute to our understanding of scientific representation not to mention that it threw some of the discussion then back into the Cartesian problem of the external world, to no good purpose. Scientific representation is as Boltzmann s own examples in that article amply show by means of artifacts both concrete (graphs, scale models, computer monitor displays, and the like) and abstract (mathematical models, needed especially when the infinite on infinitesimal play a role). It is on these artifacts, their use, and the characteristics that are germane to the roles they play in this use, that we must focus. The reservations about how the represented must be like its representation, which Boltzmann expresses about thought, pertain equally to these artifacts, and are pertinent to any view of how a science relates to its domain of application. 2 Secondly, it is not only to our understanding of theories and their models that representation is relevant. The achievement of theoretical representation is mediated by measurement and experimentation, in the course of which many forms of representation are involved as well. Scientific representation is not exhausted by a study of the role of theory or theoretical models. To complete our understanding of scientific representation we must equally approach measurement, its instrumental character and its role. I will argue that measuring, just as well as theorizing, is representing. The representing in question also need not be, and in general is not, a case of mimesis; rather, measuring locates the target in a theoretically constructed logical space. In this respect I shall make common cause with views currently found in philosophy of technology. Thirdly, the analysis of measurement as well as of the conditions of use for theoretical models can be completed only through a reflection on indexicality. Since at least the time of Poincaré, Einstein, and Bohr it is a commonplace that a measurement outcome does not display what the measured entity is like, but what it looks like in the measurement set-up.

18 INTRODUCTION 3 That point does not go nearly far enough. It serves, however, to announce the introduction of relationality, perspective, intensionality, intentionality, and the essential indexical into the discussion of science, though it stops far short of presenting their full role. Debates in philosophy of science take place in the context of much wider tensions and oppositions in epistemology and metaphysics. When, in Part Three, I come to the theme of structuralism I will begin with protagonists in the historical Bildtheorie debate, and show how it was already implicitly challenged by an apparent paradox at the very heart of its conception of science. I will then take the message about the role of indexicality in scientific representation to the paradoxes that beset, bedeviled, or otherwise preoccupied Bertrand Russell, Rudolf Carnap, Hilary Putnam, and David Lewis. For my part I will propose an empiricist structuralism, in contrast to structural realism, as a view of science that can stand up to these challenges. Then, in the last part, I will address the troubling relations that the appearances bear to the world s scientific image. This, so far, is the general outline of what I am setting out to do. But it may help to add here something about my own starting point. I try to be an empiricist, and as I understand that tradition (what it is, and what it could be in days to come) it involves a common sense realism in which reference to observable phenomena is unproblematic: rocks, seas, stars, persons, bicycles.... Empiricism also involves certain philosophical attitudes: to take the empirical sciences as a paradigm of rational inquiry, and to resist the demands for further explanation that lead to metaphysical extensions of the sciences. There is within these constraints a good deal of leeway for different sorts of empiricist positions. For my part, specifically, I add a certain view of science, that the basic aim equivalently, the base-line criterion of success is empirical adequacy rather than overall truth, and that acceptance of a scientific theory has a pragmatic dimension (to guide action and research) but need involve no more belief than that the theory is empirically adequate. 3 While this will undoubtedly shape my discussion, I have tried to write as much as possible of this book in a way that does not trade on the differences between this view of science ( constructive empiricism ) and its contraries ( scientific realisms ). What scientific representation is and how it works is everyone s concern, and there we may find a large area where more general philosophical differences need make no difference. 4

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20 PART I

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22 Representation Aristotle was trading on a typical ambiguity when he wrote Tragedy is a representation of a serious, complete action which has magnitude, in embellished speech... ; represented by people acting and not by narration.... (Poetics 49b25). We can read his statement as describing either the poet s activity or the poet s product. It is in the activity of representation that representations are produced. This is not an accidental equivocation. We lose our topic altogether if we attempt to ask what is a representation? and tacitly take just one or the other aspect into account; for in fact we cannot understand either in isolation. That applies to scientific representation as well. There is a vast and recently rapidly increasing literature on representation both in general and in philosophy of science. Let me express at once my accord with the approach advocated by Mauricio Suarez: I propose that we adopt from the start a deflationary attitude and strategy towards scientific representation, in analogy to deflationary or minimalist conceptions of truth, or contextualist analyses of knowledge. [...] Representation is not the kind of notion that requires a theory to elucidate it: there are no necessary and sufficient conditions for it. We can at best aim to describe its most general features This does require some decisions about what we can and cannot take for granted, to be used in this description, as already understood. While not trying to define representation or to reduce it to something else, we will have to place it in a context where we know our way around. The first question to broach is this: how, or to what extent, is representation related to resemblance or likeness-making? This venerable question occupies the first two chapters. Not all, but certainly many forms of representation do trade on likeness, likeness in some respects, selective likeness.that is not what makes them representations; it is part of what defines them as the sort of representation they are, and may figure in what constitutes success. But even these tend to trade equally on unlikeness, distortion, addition. A representation is made with a purpose or goal in mind, governed by criteria of adequacy pertaining to that goal, which guide its means, medium, and selectivity. Hence there is even in those cases no general valid inference

23 8 PART I : REPRESENTATION from what the representation is like to what the represented is like overall. Not surprisingly, empiricist views of science will differ from scientific realist views on where they locate the selective likeness and unlikeness. The second question concerns perspective. The third chapter will examine and elaborate on the enigmatic but now oft-seen contention that scientific representation is perspectival. In their general use, the words perspective and perspectival are largely metaphorical. The literal use appears only when we assert, for example, that artists in the Renaissance began to draw and paint in perspective. 2 As I shall elaborate, the pertinent point about this technique is Albrecht Dürer s: drawing in perspective is a measurement technique. 3 The art of perspectival drawing is an art of measuring. It is a technique for rendering a systematically selective likeness, yielding information in desired respects, and it provides an initial paradigm for measurement in general. We are perennially plagued by the shifting uses and senses of even our most common terms. In the long history of tension between physics and astronomy before the modern period, saving the phenomena refers to the appearances of the celestial bodies and their motions to the astronomer, that is, in the outcomes of the astronomer s measurements. Those celestial bodies and their motions are one and all observable, unlike e.g. the postulated crystalline spheres. But when Kant takes on this terminology of appearance and phenomenon, he entwines their meanings so deeply in his transcendentalist philosophy that we find ourselves as it were in a different language. While I have not done so before, I will here make a terminological distinction between these two words, though neither will have Kant s meaning. Phenomena will be observable entities (objects, events, processes). Thus observable phenomenon is redundant in my usage. Appearances will be the contents of observation or measurement outcomes. The celestial motions of concern to the ancient or medieval astronomer were all phenomena, in my sense, but Copernicus insisted that they had confused what those phenomena are like with how they appear to the earthly observer. Thus he distinguished, in my terms, the appearances (in the measurements made by the astronomers) from the phenomena (which they observed and measured). Regimenting the terminology in this way, or in any way at all, will chafe on some common usage. But it will align with other usage no less common. For example, combustion, St. Elmo s fire, lightning, and the aurora borealis

24 PART I : REPRESENTATION 9 are all commonly named as phenomena; and whereas Giorgione was so called because of his size, Mars is called the Red Planet not because of its color but because of its reddish appearance as seen or photographed from the Earth. 4 With this regimentation we will have available a distinct terminology to honor the insight that what measurement shows is not directly what the measured is like but how it appears in that particular measurement set-up. There are other techniques besides perspectival drawing, such as those of the cartographer, that can provide a paradigm for our conception of measurement and modelling. Perspective comes in there as well, though in a quite different way, when we examine the conditions for the possibility of use of these representations, for prediction and manipulation in practice. The lessons drawn from these seminal examples illuminate not only the painter s art but the construction and use of models in science and technology. Much of Part I will come by way of prolegomenon to further study, marshalling telling cases to illuminate representation and what will be its use? Firstly, to remove the blinders that could focus us naïvely on the idea that what is represented is simply like what is presented in the representation. Equally, to save us from the opposite error, of assuming a total independence of the represented from the content of its representation. Likeness in contextually selective fashion is important to scientific practice. The world, the world that our science is of, is the world depicted in science, and what is depicted there, is the content of its theoretical representations; but there is less to this equation than meets the eye and thereby hangs a tale....

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26 1 Representation Of, Representation As The most naïve view of representation might perhaps be put something like this: A represents B if and only if A appreciably resembles B. Vestiges of this view, with assorted refinements, persist in most writing on representation. Yet more error could hardly be compressed into so short a formula. (Goodman 1976: 3 4) Nelson Goodman was quite right to say so. The budget of examples and counter-examples that prove it we will look at some below have largely been with us since almost the beginning of philosophy. But there must be a reason if the idea he disparages, that resemblance is crucial to representation, is so persistently seductive. That perfect likeness is an ideal pursued in visual imagery, at least, has much historical support. Pliny described the painter Zeuxis grapes as so lifelike that birds tried to eat them while Zeuxis in turn was fooled by Parrhasios who painted a curtain with such trompe l oeil perfection that Zeuxis asked him to pull the curtain aside in order to show his painting. 1 We can cite the art of our time as well: hyperrealism in recent painting, such as Donald Jacot s or Jacques Bodin s, is surely admired in part for excellence of that kind. So while representation cannot be equated with the presentation of a likeness, and resemblance to what is represented is not crucial to representation as such, resemblance does play a role inviting our attention.

27 12 PART I: REPRESENTATION The value of distortion 2 That even visual, pictorial or plastic, representation is not a matter of producing accurate copies exactly like their originals is already clear in Plato s dialogues. Socrates and his young students once had a visitor from Elea whom they invited to take over Socrates usual role in their ongoing seminar. The visitor agreed, choosing Theaetetus as interlocutor, and soon has him tied in knots on the subject of representation. There are real things, says the Eleatic Stranger, and there are images of real things some are artifacts like paintings or sculptures, others natural like dreams, shadows, or reflections (Sophist 265e 266d). Aren t these images copies of what they are images of? Theatetus had already expressed a view on copy-making: What in the world would we say a copy is, sir, except something that s made similar to a true thing and is another thing that s like it? (Sophist 239d 240a). But does that apply to images in general? The Stranger reminds him that a sculptor may need to distort in order to represent something successfully. To make an exact likeness with respect to shape would require preserving the proportions of length, breadth, and depth of the original, and the colors of the parts as well. Those who sculpt or draw very large works don t do that: If they reproduced the true proportions..., the upper parts would appear smaller than they should, and the lower parts would appear larger, because we see the upper parts from farther away.... (Sophist 235d 236a). We may wonder whether Plato is referring to actual examples generally discussed in Athens. There is a story, though its provenance not entirely clear, related by Ernst Gombrich, concerning two sculptors of the fifth century BC: The Athenians intending to consecrate an excellent image of Minerva upon a high pillar, set Phidias and Alcamenes to work, meaning to chuse the better of the two. Aclamenes being nothing at all skilled in Geometry and in the Optickes made the goddesse wonderfull faire to the eye of them that saw her hard by. Phidias on the contrary... did consider that the whole shape of his image should change according to the height of the appointed place, and therefore made her lips wide open, her nose somewhat out of order and all the rest accordingly... when these two images were afterwards brought to light and compared, Phidias was in great danger to have been stoned by the whole multitude, until the statues were at

28 REPRESENTATION OF, REPRESENTATION AS 13 length set on high. For Alcamenes his sweet and diligent strokes being drowned, and Phidias his disfigured and distorted hardnesse being vanished by the height of the place, made Alcamenes to be laughed at, and Phidias to be much more esteemed. (Gombrich 1960: 191) As Roger Shepard (1990) has studied and richly illustrated in our own time, this point is general, and does not just apply to sculptors of images to be seen from below. Even in an ordinary drawing, if you want two differently oriented parallelograms to look congruent, you have to make one larger than the other. It seems then that distortion, infidelity, lack of resemblance in some respect, may in general be crucial to the success of a representation. 3 This does not rule out that resemblance in some other respect may be required. Yet even when that is the case and it may be a special case the choice of those respects in which resemblance or a specific kind of distortion is required, and those for which just anything at all will do, will have to be seen as crucial as well. There must be a cautionary tale here for how we are to understand scientific representation. It may be natural to take a successful representation to be a likeness of what it represents but much hinges here on what the criteria of success were when it was made. One sort of success is precisely what Copernicus was taken to have as against Ptolemy: that his theory displayed the real structure of the cosmos. 4 In general though, can we infer from success of a representation, in respects that we can directly appreciate, to the conclusion that it bears a structural resemblance to what is represented? The examples of how distortion may be crucial to successful representation (in view of the purpose of the representing) should certainly give us pause. But now we are running ahead of the story. Caricature and misrepresentation Successful representation may require deliberate departures from resemblance. It does not follow that likeness will always be irrelevant to successful representation. Certainly the Eleatic Stranger s example does not show that, since even the sculptor of statues placed on high must ensure resemblance in some respect. But now consider another side to the role of distortion. Misrepresentation is a species of representation after all: a caricature of Mrs. Thatcher may

29 14 PART I: REPRESENTATION misrepresent her as draconian, but it certainly does represent her, and not her sister or her pet dragon or whatever else she may have. Yet even if we take the caricature to represent her because of some carefully introduced resemblance there, we can declare it a misrepresentation by insisting that it represents her as something she is not. A caricature may represent a rather tall man as short (as a well known cartoon depicts Supreme Court Justice Clarence Thomas as very small compared to the chair he occupies), but it represents that man, and not someone that it resembles more as to height. A caricature misrepresents on purpose, to convey a message that is clear enough in context but is to be gleaned in a quite indirect fashion. My two examples are both visual representations, but they are not visually accurate, nor does their visual inaccuracy serve to produce a visually accurate appearance to a properly placed eye (as in the Sophist s sculptor s case), and yet, neither is their inaccuracy accidental. 5 So distortion departure from resemblance which may be crucial to accurate representation in certain cases, is in other cases the vehicle of effective misrepresentation. Resemblance in some particular respect may be the vehicle of reference: we recognize the caricature as being of Mrs. Thatcher because of resemblance in certain respects. It may also be the means of attribution or misattribution of some characteristic: we take the caricature to represent her as draconian because of some likeness to a dragon, which is actually an unlikeness to her. She is represented, and she is represented as thus or so: the drawing is of her, and depicts her as thus or so. 6 But a list of likenesses and unlikenesses does not tell us this much why, for example, is this not a caricature of a dragon as Thatcherian? Let us look at another drawing, say, Spott s drawing of Bismarck as a peacock. 7 Is this drawing a misrepresentation? There we broach a question of truth that certainly is not settled in terms of visual accuracy or inaccuracy even if both reference and attribution were effected by selective uses of resemblance and non-resemblance. The judgment whether this is an illuminating caricature that conveys a truth about him, or amounts to a falsehood, a lie, a misrepresentation, is not settled by geometrical relations between the line drawing and Bismarck s appearance to the eye. If we just focus on resemblance in some respect as the core notion in representation then it is at best puzzling that distortion might be needed for effective representation. But if the resemblances are just a means to an

30 REPRESENTATION OF, REPRESENTATION AS 15 end there is no puzzle. The sculptor wants the object he makes to have a certain use, and he chooses the way in which the proportions of the object are related to those of the original the ways in which they are like and the ways in which they are unlike so as make that use possible. There have (of course!) been efforts aimed at naturalizing representation, such as Fred Dretske s account of information-bearing as correlation, but these tend to founder specifically on the issue of misrepresentation. 8 Misrepresentation is a species of representation. If the relationship X represents Y were to lie in a resemblance or correlation or other such structural relation between the two, what would misrepresentation be? Suppose I say that the caricature that depicts Mrs. T as draconian misrepresents her. Then my assertion has as first part that it does represent Mrs. T as draconian, but as second part not only that it is unlike her with respect to shape, but that it depicts her as something she is not. To say that it misrepresents her with respect to shape is to say that rather than resembling her it depicts her as resembling (being like) something which, according to me, she is not like. To say that it is a caricature, however, is to say that it purveys an interpretative attribute, something that the picture can convey only by drawing on a social context, not just on what is in the picture taken by itself. So what is accuracy? The evaluation of a representation as accurate or inaccurate is highly context-dependent. A subway map, for example, is typically not to scale, but only shows topological structure. Relative to its typical use and our typical need, it is accurate; with a change in use or need, it would at once have to be classified as inaccurate. Similarly, in one political context, or relative to a certain kind of evaluation, a caricature may rightly be judged to be accurate, in another misleading or blatantly false. How does a representation represent? We confront here the general question of how an item such as a picture can correctly represent, misrepresent, caricature, flatter, or revile its subject. Note well that the answer will not be also an answer to the question of what representation is. The question here addressed arises rather if we take it for granted that something is playing the representational role, and want to know just how it plays that role.

31 16 PART I: REPRESENTATION Nelson Goodman s Languages of Art, the twentieth century s seminal text on the matter, characterizes even pictures as being like statements, depicting a subject (the referent) as thus or so (as with a predicate). 9 By drawing the of/as distinction I was already more or less following him in this view. That something is a picture of Bismarck does not imply that Bismarck was in every respect the way he looks in the picture, it does depict Bismarck as thus or so. Goodman translates this into: Depiction is [also] predication. Since pictures denote things they are, in that respect, like names; but since they depict those things as thus or so, they are also predicates applied to what they denote. The content of this picture is the same sort of thing or should be talked about in the same sort of way as the content of a predicate or a sentence applying a predicate to the subject of that painting. To say that X represents Y as F, that is taken to have as guiding example something like Snow is white represents snow as colored Putting these two points together it seems quite apt to liken a picture to a sentence, rather than to either a name or a predicate. For in a sentence we typically also see a subject to which a predicate is being applied the subject may be real enough, but it may or may not be correctly characterized. If the sentence is complex, we may distinguish respects in which the predication is accurate and ways in which it is not, just as we do with a painting. Denotation, as Goodman understands it, is of something real, and the relation is extensional. Pictorial predication need meet neither of the conditions placed on denotation. If the painting depicts Thackeray as the author of Waverley we cannot infer that Waverly exists, nor that it had a single author if it does. But in addition, if Scott is the author of Waverley we cannot even say that the painting depicts Thackeray as Scott, except as a joke. Neither predicates nor depicts as is an extensional term. 10 Goodman s most controversial thesis is that denotation and predication are also in the case of pictures entirely independent of each other. The denotation of a picture no more determines its kind, he writes, than the kind of picture determines its denotation. 11 (He uses the kind terminology to distinguish pictures that predicate differently.) This thesis is controversial since it is hard to accept that a picture could fail to convey anything correct or true about something and still be a picture of that thing. 12 Instead, we would typically think that we can identify what it is a

32 REPRESENTATION OF, REPRESENTATION AS 17 picture of by looking at how things are depicted in it! But there are limiting cases where this can be drawn into doubt. 13 Undeniably, though, Goodman has brought into the limelight the strong analogy: a picture is a picture of something, and depicts that something as thus or so, and so is in that respect similar to how a verbal description is a description of something, and describes that something as being thus or so. We need now to see how we can go beyond this analogy. One way in which Goodman did bring in resemblance was through the intricate notion of exemplification. If a hardware store clerk or interior decorator shows you color swatches or fabric samples, those exemplify the property of interest in the following sense: they both have and refer to that property (Goodman 1976: ;Elgin1996: ). Obviously their use is to represent to you what your wall or floor will look like if you choose the corresponding paint or carpet. Representation by exemplification involves likeness, but much more than likeness. In these examples visual resemblance plays a role: in fact, the color swatch has the same color, and in that crucial respect resembles, the paint that it represents to the customer. This only works if the relevant visual resemblance is highlighted in some way, has a status unlike that of the many other resemblances that is the point of taking exemplification of the relevant property, rather than mere instantiation, as the vehicle by which representation is achieved. 14 The relevant visual resemblance must be highlighted in some way in that context, so as to bestow that status here we have strayed from semantics into pragmatics. But do not equate even this beautifully articulated relationship with picturing or imaging! As Goodman later pointed out, the word word both refers to and has the property of being a word, so it exemplifies that property, but is not a picture or image. (Goodman : 419) Asymmetry of representation There is an asymmetry in representation that resemblance does not have. This is a much repeated point, made to show that resemblance is not the right criterion for representation. Resemblance could not be the crucial clue to representation, it is said, for even if representation did require resemblance to its target, the target would then resemble its representation but not represent it. While resemblance is indeed not the right criterion, the argument from asymmetry is not all that strong.

33 18 PART I: REPRESENTATION First of all, we do tend to use terms like resemble and looks like and such asymmetrically in certain contexts, because the subject of a statement tends to select the focus or contrast. Of Rosemary s baby they said He has his father s eyes. Hard to think of their saying about the father He has his baby s eyes. The retort may be that literally the noticed resemblance goes both ways; but literally may here just mean if you ignore the context. Probably we can find a context in which someone may, without oddity, assert not that a given picture is an exact likeness of me but that I am its exact likeness, or something to that effect. But that would not show that resembles is symmetric in general, let alone in every context in which it comes in. That literally resemblance must go both ways that literally speaking it is both reflexive and symmetric, while representation is neither is most likely based on the simple idea that to resemble (in some respect) is to have a property (of the pertinent kind) in common. But that is too simplistic a construal anyway. The production of a photo involves a collapsing of shades of color and of three-dimensional spatial structure into two dimensions. If that counts as pertinent resemblance, then this is a relation of homomorphism rather than isomorphism, yet central to modeling. That A is a homomorphic image of B certainly does not entail that B is that of A. 15 But this contrary point too is weaker than it seems. If A is a homomorphic image of B then there is a reduction of B, modulo some equivalence relation, to which A is isomorphic so we might say that in this respect B resembles A just as A resembles B. So there is no strong argument, as far as I can see, based on any clear asymmetry to banish resemblance from our topic, nor one to make it relevant to representation in general. What does remain, as needs to be emphasized, is that certain modes or forms of representation (but not all) do trade on selective (and not arbitrary) resemblances for their effect, efficacy, and usefulness, and that this typically goes in one direction only. Resemblance in discord with representation In examples of picturing, of visual representation, resemblance tends to spring to the eye. Both reference and attribution can be achieved in other ways, however, even in visual representation, rather than by means of carefully selected resemblances. And conversely, even there, what is represented, and how it is represented, cannot in general be deduced simply by

34 REPRESENTATION OF, REPRESENTATION AS 19 attending to resemblances and non-resemblances. To show this, Goodman mentions the painting of the Duke of Wellington which everyone agreed resembled the Duke s brother much better. Socrates argues the point by means of a look at the extreme case, where the resemblance is greatest, in his discussion with Cratylus about verbal representation: I quite agree with you [Cratylus] that words should as far as possible resemble things, but I fear that this dragging in of resemblance... is a shabby thing, which has to be supplemented by the mechanical aid of convention with a view to correctness. (Cratylus 435c) Socrates thought experiment leading to this remark has a quite contemporary ring, if we replace gods (as is usual now) with mad scientists. In his discussion with Cratylus, correctness and accuracy are being characterized in terms of greater resemblance: [... ] in pictures you may either give all the appropriate colors and figures, or you may not give them all some may be wanting or there may be too many or too much may there not? [... ] And he who gives all gives a perfect picture or figure, and he who takes away or adds also gives a picture or figure, but not a good one. (Cratylus 431c) Here the mere resemblance view of representation appears on Socrates lips, somewhat surprisingly; but Socrates himself will soon put us right on this. When Cratylus draws the parallel with language too simplemindedly, the discussion immediately shifts into a subtler gear. Socrates replies, in partial contradiction to the above: [... ] I should say rather that the image, if expressing in every point the entire reality, would no longer be an image. Let us suppose the existence of two objects. One of them shall be Cratylus, and the other the image of Cratylus, and we will suppose, further, that some god makes not only a representation such as a painter would make of your outward form and color, but also creates an inward organization like yours, having the same warmth and softness, and into this infuses motion, and soul, and mind, such as you have, and in a word copies all your qualities, and places them by you in another form. Would you say this was Cratylus and the image of Cratylus, or that there were two Cratyluses? Cratylus I should say there were two Cratyluses.

35 20 PART I: REPRESENTATION Socrates Then you see, my friend, that we must find some other principle of truth in images.... Do you not perceive that images are very far from having qualities which are the exact counterpart of the realities which they represent? (Cratylus 432a d) That Cratylus does not grant that the copy, made by the god to duplicate Cratylus entirely, is an image of Cratylus shows at the very least that resemblance is not sufficient to make for representation. But the example shows much more we need to explore it in detail. What s in a photo? Just to assert of something that it is a representation, or that it represents, or that it represents something, is woefully elliptic and invites obscurity and confusion. Our full locution must in the general case be at least of form X represents Y as F, as in the caricature represents (depicts) Mrs. Thatcher as draconian. 16 Here X is a representation, Y its referent, and F a predicate that X depicts Y as instantiating. The as... locution is fascinating, a difficult topic in the analysis of language, and we will need to carefully distinguish how its intensionality is connected with the relationality and intentionality of representation. But this form too is still too brief to allow all needed distinctions. Why didn t Cratylus agree that the god would have made an image of him? The god would, in Socrates example, have made a perfect replica, more perfect than any statue made by a human sculptor. The replica would certainly, if properly displayed, have created the appearance of Cratylus being there. So as far as the later classification in the Sophist goes, the god would have acted both as likeness-maker and as appearance-creator. Yet Cratylus demurs: the god would not have made an image of him but would have made another Cratylus. What are we to make of this intuition? A more contemporary example will show the inherent ambiguity in play here about what represents what, with what, and to whom. Imagine: I have acquired a famous photograph of the Eiffel Tower, Au Pont de l Alma by Doisneau. It hangs on my wall, but I scan it and print the scanned image. This print is an image too what does it represent? The Eiffel Tower seen from the Pont de l Alma, or the famous photograph?

36 REPRESENTATION OF, REPRESENTATION AS 21 There is no single immediately and obviously right answer; there couldn t be. It depends on what I do with the thing. If I send it to you from Paris as a postcard, with the single note Wish you were here!, then it is itself a photo of the Eiffel Tower. If I insert it into the book I am writing about photography, then it represents the famous photo by Doisneau. There are still other possibilities. 17 In other words, if it is an image of something at all then what it is an image of depends on the use, on what I use it to represent. So the question what does it represent? must in this case be taken as elliptic for what is it being used to represent? This term use can assimilate make and take : the caricaturist made the caricature to depict Mrs. Thatcher as draconian while I, seeing the caricature, take it to depict Mrs. Thatcher as draconian, and display it to yousoastodepicthertoyouasdraconian.theremayalsobeadisparity: the boy in William Golding s Free Fall made drawings of hills and forests but his teacher takes the drawings as pornographic. All of this falls under use in the general sense in which we say that in pragmatics we are meant to study the relation not just of symbols to things, but the three-term relation between symbol, user, and thing. Our fuller locution shortened in different ways depending on what is taken for granted in context must be Z uses X to depict Y as F. 18 I have put this in individualistic terms: you or I can use something to represent something, though of course to communicate or convey anything at all that way be it factual information, feeling, intention, or command we must let each other in on how something is being represented. (As Goodman put it, the most probative question is not what is art? but when is art?) Since communication presupposes community to some significant extent, this will be possible only in a context where some modes of representation are already held and understood in common. Within discussions in which the institutionalization of relevant symbolry is already taken for granted, the point about use may be put in terms of function or role. And although these terms make sense also for an individual creation of a symbol, the role will generally be one specifiable within a manifold of roles, a system of representation, a language of art. Thus Ned Block writes What any representation represents, and how it represents... depends on the system of representation within which it functions. (Block 1983: 511)

37 22 PART I: REPRESENTATION My point is not just that what represents what is relative to a system of representation. Rather my point is that you can t tell for sure whether you are looking at arepresentationatalljust by looking.... One has to determine how the thing functions. (Ibid.: 512) 19 Relativity to systems, languages, recalls Goodman of course; and this applies very well to pictures, as is well enough understood when different styles of representation are studied. 20 This notion of system, or of function if understood as a role in a system, sounds still quite impersonal, but we must understand it in terms of pragmatics, referring to contexts of use, broadly construed. The contextsensitivity does not go away when (in different ways) Nelson Goodman and Ned Block say that you need to know which system of representation the item belongs to. For since the same item will in different contexts belong to different such systems, we then need the relevant contextual factors which determine that. What is really to be emphasized here is the way in which individuals and groups, though relying on some pre-existing communally understood form of representation (a universal qualification of all communal activity), create new representations and new modes of representation. When Descartes created his method of coordinates, it is not as if he was just using an already extant way of representing spatial shapes and motion. But it is true that in his initiative, to use known numerical equations in this way, he bestowed a role on already familiar equations that they had not had before. Unlike a moment s poetic depiction quickly lost to history, this act engendered a mode of representation fundamental to all subsequent science. What is a representation then? Look back now at Socrates, Cratylus, and the god they imagine. Did the god make an image of Cratylus or did he not make a representation of anything, but a clone? That depends. Cratylus was too hasty in his response! Did this god go on to display what he made to the Olympic throng as a perfect image of Greek manhood? Or did he display it as an example of his prowess at creature-making? Or did he do neither, but press the replica into personal service, since he couldn t have Cratylus himself?

38 REPRESENTATION OF, REPRESENTATION AS 23 What is represented, and how it is represented, is not determined by the colors, lines, shapes in the representing object alone. Whether or not A represents B, and whether or not it represents the represented item as C, depends largely, and sometimes only, on the way in which A is being used. Use must here be understood to encompass many contextual factors: the intention of the creator, the coding conventions extant in the community, the way in which an audience or viewer takes it, the ways in which the representing object is displayed, and so forth. To understand representation we must therefore look to the practice of representing, to how representation is a matter of use; and this involves attention first of all to the users in a broad sense of use. That is the main thing to be concluded both from our discussion of caricature and misrepresentation, and from the Cratylus and Eiffel Tower photo examples. There is no representation except in the sense that some things are used, made, or taken, to represent some things as thus or so. I do not advocate a theory of representation, and this could not possibly be offered as such since that would be circular. But if I did, I think this would be its Hauptsatz. 21 What does this exclude from the category of representations? That depends of course on precisely what used, made, or taken means. And that in turn depends on what is required if this Hauptsatz is to solve or dissolve puzzles about representation. (For example, the puzzles that result if one begins with the thought that resemblances likenesses, correlations will determine, by themselves, what the representor represents.) What we can conclude, at least, is that use, in the appropriate sense, must determine the selection of likenesses and unlikenesses which may, in their different ways, play a role in determining what the thing is a representation of, and how it represents that. Moreover, the selection cannot be mute: in the pertinent context, this selection and the precise role it plays, the selection must be salient, so the use must be such as to highlight that selection. 22 The use is what bestows the relevant role or function on the item used. There are uses of the terms use, make, take which imply no intentionality: the car uses gasoline, the tornado took the life of my neighbor, the Ice Age glaciers made these valleys. But our puzzles about what representation is do not disappear unless use and its cognates are understood here in the sense in which they presuppose intentional activity.

39 24 PART I: REPRESENTATION That said, I will just write use for use, make, ortake understood in this sense. If that Hauptsatz is understood in this sense, then it places some immediate limits on the range of representation. Firstly, at least if taken entirely literally, it has no room for the notion of mental images or mental representations, whether taken to be brain states or something more ephemeral for no such things, if they exist at all, are used or put to use, ortaken in one way or another. 23 At least, not in the relevant sense: we can conceive of a brain surgeon bestowing a representational role on the patients brain states, but not of a person bestowing roles on his or her own brain states or, presumably, on whatever could count as mental states. 24 Secondly, this conception leaves no room for representation in nature, in the sense of naturally produced representations that have nothing to do with conscious or cognitive activity or communication. The Eleatic Stranger gave a whole series of examples of copies, but it is not clear that they are all images in the sense of representations: Things in dreams, and appearances that arise by themselves during the day. They re shadows when darkness appears in firelight, and they re reflections when a thing s own light and the light of something else come together around bright, smooth surfaces and produce an appearance that looks the reverse of the way the thing looks from straight ahead. [... ] And what about human expertise? We say housebuilding makes a house itself and drawing makes a different one, like a human dream made for people who are awake. (Sophist 266c d) Most shadows and reflections that occur in nature are not being used or taken, let alone made, by anyone to do anything. (The Balinese shadow puppet theater is an exception.) So by our Hauptsatz, they are generally not representations. Nor is the track left in sand by an ant in some desert long, long ago in a galaxy far, far away.... not even if it has the shape of our word Coca Cola. 25 A black mark on a rock does not refer, represent, or mean anything unless it has a role, or has bestowed on it a role, in some practice no matter whether it is a simple stroke or a complex pattern. Nor is it sufficient that it has the sort of shape, coloring, etc. that would place it in a certain role if encountered or produced in a certain cultural context, by persons belonging or assimilated there, if in fact it does not bear any relation to such a context.

40 REPRESENTATION OF, REPRESENTATION AS 25 But a natural object can represent, just as it can play other roles, namely if we bestow such a role on it. Imagine I am using a stone, found on the ground, to hammer in a tent peg. I am using it as a hammer it is my hammer now, I have bestowed the hammer role on it. The hammers we buy, in contrast, are manufactured precisely to play this role they are manufactured artifacts. The stone was not made for that, and it is not an object that I created, constructed, or assembled. Nevertheless it is now a physical object with a function that is to say, an artifact. There is an analogy here to objets trouvés, natural objects made, without physical modification, into works of art. All of this applies mutatis mutandis when I use the stone to represent, for example, a certain stateman s heart: I bestow a role on the stone for it to play, I give the stone a function for it to serve. What is in a photo? What is in a picture? This question has the misleading form of What is in a box? We won t get much further by taking this form at face value and giving an answer with the correlative form, such as What is in the representation is its content. That is just a verbal answer, conveying nothing by itself. To call an object a picture at all is to relate it to use. As an analogous example we can think of Herbert Mead s reflections on the teacup (McCarthy 1984). If there were no people there would be no teacups, even if there were teacup-shaped objects. For there are teacups implies there are things used to drink tea from which in turn implies there are tea-drinkers. By ignoring the contextuality of representation, the fact that we are dealing with about-ness, and that what the representation is about is a function of its use, we could land ourselves in useless metaphysical byways. If we were to ask What is in a picture? while taking the picture simply to be the physical object and with no relation to anything that can bestow meaning, the answer would have to be Nothing! The notion of use, the emphasis on the pragmatics rather than syntax or semantics of representation in general, I will give pride of place in the understanding of scientific representation. But does that exclude too much? That a particular person at a specific time uses or takes or presents something to represent something else is a very local event. Could it really be a general condition on representation that something so specific has to happen? In a comment on similar intentional views of what constitutes representation, Mauricio Suarez suggests that it will hamstring the idea that theories represent:

41 26 PART I: REPRESENTATION on the intentional account of representation a theory cannot represent a phenomenon that hasn t been observed. For a theory cannot be intended for a phenomenon that hasn t yet been established. (Suarez 1999: 82) 26 The objection, if valid, would not just apply to a theory or a model, it would imply that nothing can be intended (let alone used) to represent something that has not entered our acquaintance, or something that we do not know to exist. The objection is presumably not that there can t be a representation of something that we have not already encountered. A meteorological model, found for example on a weather forecasting website, does in fact provide us with a representation more or less accurate, or not accurate at all of the weather in the next five or ten days. Is the objection then that we cannot be said to use or take or present this meteorological model to represent the coming week s weather? But we do use it, and the viewers so take it. So could the objection rather be that this model could have been or provided such a representation although it could not have been to use Suarez s exact words intended for the actual meteorological phenomena, which are still in the future? It does not seem so, it seems that it was intended precisely to represent those (as yet unknown) phenomena. Relation, intention, intension Representation is a relational notion, if we go by the form of assertions that attribute this status. Tragedy is a representation of an action.... Here is a painting of a picnic on the grass, the statue over there represents justice, that graph depicts the growth of a bacteria colony. In each case we have a subject term, a relational predicate, and a term for the second relatum. But we are not dealing with something as simple as a relation of physical contact or impact or proximity. First of all, the second relatum may not be real. In fact, to say that something is a painting of a picnic does not at all imply that there is a real picnic which that painting depicts. The Mona Lisa is a portrait of Mary Magdalene : this assertion purports to mention two real things and a relationship between them. Perhaps it does. But the important point is that the form by itself cannot reveal this. For the assertion may be true still if it turns out that Mary Magdalene was not a real historical character. Even if she was, the Mona Lisa could at best depict how Da Vinci imagined her to be, which we can t necessarily equate with depicting her. 27

42 REPRESENTATION OF, REPRESENTATION AS 27 The of that marks the relation of representing object to what is represented is like the one familiar from Brentano s characterization of mental acts as directed, intentional. Intentionality we also see in semantic discourse when we say that Zeus is the name of a god, for example. Representation is intentional in the sense of relating to epistemic intention, in the sense of being about something, in just the way that reference (by someone) and predication (by someone) are. But just as thought can be directed in this sense at what is not present, not experienced, not known, or even non-existent, so can any use of something to represent something. By so using it, the user bestows a role, the role of representing such and such as thus or so. If for example I draw a graph and present it as representing the rate of bacterial growth under certain conditions, then by virtue of that very act, what the graph represents is the bacterial growth rate under those conditions period. It is equally apt to say that I represented that growth rate as thus or so and it would be apt to say that if, instead of drawing the graph, I had displayed the equation of which the graphed function is a solution. And so forth, mutatis mutandis, for the case in which I display a function that has such graphs as output for inputs about ambient conditions of bacteria colonies, or state a theory that describes a family of such functions with a further free variable for the type of bacteria.... Given this intentionality it is perhaps not surprising that in the case of representation, the relations can change with context of use. The very same object or shape can be used to represent different things in different contexts, and in other contexts not represent at all. 28 The expression A represents B when used all by itself is misleading. It is easy to get into confusions when the relational character of a term is suppressed. To illustrate: every woman is a daughter and every daughter is a woman, so why is being a daughter not the same as being a woman? Precisely because to be a daughter is to be daughter of someone. Analogously, to represent something is to represent something as thus or so. The complexities appear in force when we extend these assertions to the threeplace relation A represents B as C. Simone de Beauvoir depicted herself as a dutiful daughter, but not as a dutiful woman. 29 All and only creatures with hearts are creatures with kidneys yet to represent something as having a heart is not the same as representing it as having kidneys. And so forth. This opacity, the resistance to substitutivity of identity, is a mark not only of the intentionality of thought, but of intensionality in discourse. 30

43 28 PART I: REPRESENTATION We see this in modal contexts: 9 = the number of planets, but it does not follow that the number of planets is necessarily greater than 7. Also in oratio obliqua: the mathematics teacher who taught us that 7 < 9 did not tell us that 7 is smaller than the number of planets. 31 Representation of an object as the evening star is an activity that is intentional, in the way that mental acts are traditionally said to be, precisely because to do so is not the same as representing it as the morning star even though that is the very same object. So assertions to the effect that something represents, are intensional. This is primarily a point about language, but is closely related to the point that representation itself (the activity) is intentional, both in Brentano s sense and in the common sense of the term. Ordinary discourse does not mark the distinctions we are making here, or not very well. In analytic philosophy language has been regimented to some extent to do so. Thus He said of Mrs. Thatcher that she was draconian asserts a relation of the speaker to the real Mrs. Thatcher, while He said that Mrs. Thatcher was draconian does not, in this regimented form of discourse. If Mrs. Thatcher was the Prime Minister, the first sentence implies He said of the Prime Minister that she was draconian but the second does not imply that. The role here given to the of locution marks an artificial verbal distinction (even if not without roots in prior general usage), but such artifice can be useful when confusion threatens. 32 Where do the intensionality and intentionality come from? To understand this is as important for representation in science as in the arts. The answer lies of course in our Hauptsatz, that there is no representation except in the sense that some things are used, made, or taken, to represent some things as thus or so. But even this does not suffice by itself, for it does not make explicit what all is involved in the use, by way of value, purpose, aim, and yes, intention. Ronald Giere spells this out concisely for scientific representation and one further contextual factor, purpose: If we think of representation as a relationship, it should be a relationship with more than two components. One component should be the agents, the scientists who do the representing. Because scientists are intentional agents with goals and purposes I propose explicitly to provide a space for purposes in my understanding of representational practices in science. So we are looking at a relationship with roughly the following form: S uses X to represent W for purposes P. (Giere 2006: 60)

44 REPRESENTATION OF, REPRESENTATION AS 29 But the point is quite general. The spelling out can only go so far, because the notion of representing has (suffers from?) variable polyadicity: for every such specification we add there will be another one. Appearance to the intellect: illumination as embedding We have been concentrating mainly on appearance to the senses, but when viewing a painting or movie, reading a story, or watching a computer simulation, we may well ask I can see what is happening, but what is really going on? Quite often however we do not even get to this question: our active, agile imaginations have already supplied, assumed, or conjectured a pattern behind the displayed events. On other occasions we press to find or construct a model in which the random or puzzling appearance is sublimated in a well-behaved structure, of which it is the surface. In such a case we arrive at a representation of the original as embedded in a larger whole, and thereby made to satisfy certain demands of the intellect. There certainly are examples of this in the visual arts. A picture of an apparently levitating woman may be immediately recognizable as of the Virgin Mary because the figure s feet are on a crescent moon and she is surrounded by jubilant angels. The embedding structure presents the story in which the event is embedded. But something similar can be said of the event depicted in Jacques-Louis David s Oath of the Horatii, though it does not draw on such pervasively common knowledge. The lamentations of the seated women in the background show how much more is going on than the event depicted taken in itself; they help to give meaning to what is happening through an embedding in a larger story. 33 But the most straightforward examples come from the history of modeling in mechanics. Think for a moment of a swinging pendulum. Determinism requires that if the same state occurs at different times, it is always followed by the same succeeding states. At its lowest point, the bob has its maximum speed but location and speed can be the same at two different moments, and the location a moment later be different. If we keep only those two factors in our description, we have an apparent violation of determinism. The remedy is simple: enlarge the description by replacing speed by velocity, which is speed + direction. The velocity at

45 30 PART I: REPRESENTATION the lowest point is not always the same; when it is, so are the immediately succeeding locations. All these features are still in the domain of kinematics. Descartes s famous aim for mechanics was that it should be deterministic but only have quantities of extension, that is, kinematic parameters. When Leibniz and Newton each came up with counter-examples, they also introduced new, non-kinematic (dynamic) parameters to fill out the picture. The apparently indeterministic kinematic behavior is embedded in a model that has additional parameters such as masses and forces which is deterministic after all. These examples provide the pattern for hidden variable interpretations of apparent indeterminism. Reichenbach argued that such added parameters might not correspond to anything real, and that physics could forego satisfying the demand for determinism. But as a good empiricist, he offered this as methodological advice: neither demand them nor ban them from modeling. The touch stone would be the usefulness of such models for empirical prediction. For us, the point here is simply that, for a given purpose, the best representation might well be one that embeds its target in a larger structure. So we can add addition of surplus structure to distortion and the trading on selective unlikenesses, to our catalogue of means for representation achieved by departures from mimesis. Hence, again, there is no universally valid inference from what the best representation is like to what the represented is like. In conclusion A scientific, technical, or artistic representation is an artifact. As such, it is both an object or event in nature, that we can regard purely through the physicist s or chemist s or mathematician s eyes. But it is at the same time something constituted as a cultural object, through its role or function, bestowed upon it in practice. Just what the representation is, or what is represented and how, is not determined entirely and often enough, hardly at all either by what is in the natural object or by its physical or structural relations to other things. When resemblance is the vehicle of representation, for example, the representation relation derives from selective resemblances and selective

46 REPRESENTATION OF, REPRESENTATION AS 31 non-resemblance, but what the selections are must be somehow highlighted. If the selection or the highlighting is indicated by signs placed in the artifact itself, these too need to be meaningful to play their role, and so the task of identification is pushed back but reappears as essentially unchanged. Thus what determines the representation relationship, with all its polyadicity, can at best be a relation of what is in it to factors neither in the artifact itself nor in what is being represented. In the examples and puzzles here examined, the extra factors characterized use, practice, and context, and these form the proper basis for generalization there. That is not the end of the matter, representation is not to be subjected to definition: it is inexhaustible as a subject.

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48 2 Imaging, Picturing, and Scaling Despite the arguments against taking resemblance to be the clue to representation, there are many cases where what the representation refers to, or what it attributes, is conveyed effectively by displaying a salient resemblance. 1 Examples come readily from the visual and plastic arts, but there are relevant representational techniques commonly used in the sciences as well. Resemblance comes in, not when we are answering the question What is representation?, but rather when we address How does this or that representation represent, and how does it succeed? In the case of representation by pictures, scale models, diagrams, and maps, and many other examples, the initial answer to the latter is By selective resemblance and selective (even systematic) non-resemblance. To fix terminology that I have used informally so far let us reserve the terms imaging and imagery for those cases. Modes of representation Everything resembles everything else in many ways, so an effective use of resemblance must always be selective. This can only be effective if the selection itself is understood or conveyed. That point applies more generally: even if resemblance is not the vehicle, whatever features of the representing entity are instrumental to the representing must be somehow highlighted there. It is not sufficient for the representor just to have them! Secondly, even if we recognize a role played by a resemblance, this resemblance need not be with respect to any visually or even perceptually detectable features. Additionally, resemblance need not consist in sameness of properties, but can also be at higher levels. This was a theme emphasized and elaborated especially by Wilfrid Sellars in his discussion of theories

49 34 PART I: REPRESENTATION and models. 2 Resemblance may consist in having properties in common, or instead in having properties that have properties in common with relevant properties in what is represented. That is, the representor may have properties which form a structure resembling a structure formed by the properties of the represented, and so on up the hierarchy of types. We should honor these distinct categories with distinct names. Representation in general has under it the special case of representation that trades for its success on some (specific) resemblance, or on multiple resemblances. For this special case I will use the terms image, imaging, and imagery which in turn has under it the sort of imaging that manages to represent in part by trading on some visually detectable resemblance visual imagery. 3 When Galileo introduced his primary qualities as the properties to be solely considered as primitive in scientific description, he both narrowed and extrapolated from this base of visually detectable resemblances. The list that became more or less standard in his century comprised just the quantities of spatial and temporal extension and their combinations hence of space, time, and motion, the kinematic quantities but were not qualified by the limitations of human perception. So let us set imaging which manages to represent in part by trading on resemblances with respect to kinematic quantities side by side with visual imagery. Call it kinematic imagery. Trading on resemblance is a very broad category, so we may come across other sorts still besides visual and kinematic imagery. These as well may admit of both simple and higher level (structural) varieties. Overlapping these categories of representation that trade on selective resemblances lies still a further salient case, which shares some crucial features found in visual perspective, a development which in art we associate specifically with Renaissance painting. Perspective involves (as we shall explore further below) such features as occlusion, marginal distortion, texture-fading. For cases of imagery in which such features of perspective are present I ll use the terms picture and picturing these can include cases of kinematic and visual but perhaps also still further forms of imagery. Obviously none of these categories are hereby precisely defined, we will have to explore all of them further. But there are two caveats we must emphasize. The first, already noted several times now, relates to the level of resemblance, and the second to the reality or non-reality of what is represented.

50 IMAGING, PICTURING, AND SCALING 35 Resemblance, as I said, can be higher order: the spatial structure of a set of letters on a page may be the same as the temporal structure of a set of events named by those letters. The use of visual or kinematic imagery to depict things that are not visual or kinematic is rife, and not excluded by our notion of imagery. For the resemblance of some structure to a visually recognizable structure may be precisely on the level of structure, not on the level of features that only visible things can have. This point may threaten to trivialize the notion of imagery. For it does not take much by way of intellectual gymnastics to find some minimal relevant resemblance in any two things at all. But the threat does not seem to me very serious, for actualizing it would quickly turn into something no one could really take seriously. 4 Does the word animal resemble anything that is not a word, does it resemble anything having to do with animals? Perhaps so, but it will not be anything that plays a role in the representation of animals by that word. The Cratylus attempt to see verbal representation as hinging on resemblances of not just words but syllables and even letters to what is represented, I take to be a choice bit of Socratic irony, that reduces the entire idea to absurdity. Second caveat: a representation may be of something non-existent, non-actual. So a representation that trades on resemblance may in fact not resemble any real thing, any person living or dead, or actually occurring event. Le Déjeuner sur l herbe is immediately seen as a picture of a picnic because of a certain kind of resemblance to real scenes of the picnic-kind, but it is still a painting of something that did not happen. 5 The Judgment of Paris, whether by Rubens, Renoir, or as Nazi propaganda by Ivo Saliger, trades for its success on resemblances. Not resemblance to Paris or to the goddesses, who may not exist, nor even to any real scene that ever happened, but on resemblances to human judgments, bodies, clothes, and the settings of such human scenes. So to call something an image, a visual image, or a picture will not imply anything about the reality of what it depicts. 6 This is a point obviously of some importance to an empiricist view of scientific modeling as representation. That an image trades on resemblance, on any level, does not imply that it resembles what it represents, nor that there is something that it resembles, nor even that there exists something that it represents. Fundamental to the understanding of representation in all contexts is this fact, that images which represent something unreal have their importance, their role, their effect in the context in which they function.

51 36 PART I: REPRESENTATION What distinguishes a picture? At first blush, it is easy to see what is in a picture such as that familiar one, of a picnic, in curiously dishabillé condition. What is in Le Déjeunersur l herbe? A picnic. But how does that distinguish the picture from a verbal expression or icon? After all, what does the phrase Le Déjeuner sur l herbe describe? A picnic. Despite the sameness in answer to these questions, there is a difference. As I have regimented the terms, above, picturing is a case of imagery, that is, representation that trades on resemblance, distinguished by bearing hallmarks of perspective. So it is the latter we should now try to isolate. One popular response is that a picture can t help but be specific. A sentence that says that George is in the garden need have no information about whether he is standing, sitting, or lying down. But a picture of George in the garden, it is said, can t very well be neutral on that. Dominic Lopes calls this a myth of specificity. 7 In fact a picture need not be determinate in any particular way. A picture of a written word might not be of any specific word; the word might be illegible in the picture. A picture of George in the garden might just show his hand emerging out of some shrubbery, without revealing his posture. A few brush strokes can suffice to depict George as in the garden, in tears, or in love. But there are two ways in which pictures are peculiar in the way they represent. Both ways have to do with what is accessible to vision. The first relates to resemblance, the second does not. First of all, as Dominic Lopes emphasizes: pictures are unlike other sorts of representation precisely because there is a way in which they do literally look like what they depict. Pictures are physical objects; among their physical properties there is a privileged set of visual properties, those by which pictures represent their subjects such as line, shading, color, and visible texture. But note how this point is to be qualified. The line, shading, color, and texture in the picture may not match, and in fact generally must not match, those of the represented scene. This is essentially the same point as the Eleatic Stranger s about large statues on high pedestals: to create a life-likeness, to create the right appearance, distortion is needed. That applies to color as well as shape or proportion. The technique of chiaroscuro was accordingly developed in the Renaissance along with perspective. This

52 IMAGING, PICTURING, AND SCALING 37 is only the beginning of the systematic mis-matching necessary in a picture to make it look like the original. In addition, as we have seen, on many counts the resemblances and non-resemblances can easily give wrong clues both to what is depicted and what is attributed to it. A picture can misrepresent its subject, attributing properties to it that it does not have, whether by accident or on purpose. To that extent the idea that pictures represent because they just look like their subjects is indeed a mistake. All representation is selective and the selectivity is crucial to what is depicted, but for pictures, the selection is subject to very specific constraints. The selection is not a choice simply to render some aspects and be non-committal as to others. That sort of choice is also made when we describe something in words. The crucial difference here we come to the second distinguishing point appears when we notice that in picturing, the selection of one aspect may force the picture not to include certain others. There is first of all occlusion, which is closely related to perspective: to depict a situation from a given point entails that some objects will be in front of, and hence hide, certain other objects. Thus revealing what things are like from one angle is incompatible with simultaneously revealing the values of certain other parameters. This point has been exploited well in technical description of painting. But the topic has also been explored as crucial to machine design and drawing (cf. Rothbart 2003: 242 4). John Hyman introduced the terms occlusion shape and relative occlusion sizes : an occlusion shape of an object is its outline, relative to a line of sight. 8 It is the shape of an object as seen from a particular point of view. The occlusion size of an object is the area that the object occludes from view from a particular viewpoint. It depends on the actual size of the object and its distance from the observer. As the observer moves, the occlusion shapes and the occlusion sizes of the objects around him change. But this is not something subjective. These terms concern the shapes and sizes that are projected from a particular viewpoint on a plane perpendicular to the line of sight. As Hyman points out, we may be mistaken about them and our mistake can be corrected by measurement and geometrical calculation. Dominic Lopes develops this notion so as to distinguish picturing from other modes of representation. Revealing what things are like from one angle is incompatible with simultaneously revealing the values of certain other parameters. Besides occlusion there are also:

53 38 PART I: REPRESENTATION Grain: more distant objects, textures are not as finely depicted as near; and in fact there is a minimum to the fine-grainedness of a given picture. Angle: even with multiple views, there is a limit to the number of angles and distances from which an object can be depicted Marginal distortion: this derives from the limitation of the view, which can take in only a circumscribed range, and picturing reaches its limits of reproduction near the limits of that range. Marginal distortion is clearest in the pinhole camera picture that ordinary photos do not show this, is precisely because the camera, like the Sophist s sculpture of large statues, adjusts the copy so as to create a more faithful appearance. Angle and marginal distortion are topics that belong under the heading of specifically spatial perspective, and we will discuss that further below. These characteristics are, for example, the ones drawn on in Ronald Giere s account of astronomical observation, to argue that what is gained from the instruments is perspectival. 9 All four of the characteristics mentioned are plausibly grouped under the heading of perspectivity, but we cannot hope for an explicit definition here. The notion of perspective is undoubtedly what Wittgenstein called a cluster-concept, so that no specific set of characteristics is both necessary and sufficient for its application. 10 Lopes introduces an important further characteristic to the cluster. He adapts here some terminology from Ned Block and Daniel Dennett. 11 A representation is committal with respect to some property F if it represents its subject as having that property, also if it represents it as not having that property, and not otherwise. If it simply does not go into the matter of whether the subject has that property, then it is inexplicitly non-committal with respect to F. But finally, a representation is explicitly non-committal with regard to this property if it represents its subject as having some property (or properties) that preclude the representation from being committal in that respect. It is crucial to the notion of picturing that being committal in one respect will preclude being committal in some other respect in the sense that it will force being explicitly non-committal. This point elaborates on perspectivity: perspectives can be aufgehoben in a higher unity, if I may use an expression from a much earlier time, but they cannot be simply combined as parts of a third perspective.

54 IMAGING, PICTURING, AND SCALING 39 Does this allow us to differentiate picturing from describing? Given a particular style of representation, it may not be possible to add more to a picture and still let it remain a single picture as opposed to a gallery. But what about collage? And what about cubism? Or impossible pictures, like Escher s? Lopes maintains, and this seems correct, that although different styles differ in precisely what they make possible in this respect, the same point about the explicitly non-committal will apply mutatis mutandis.there is still in each case a choice to represent the subject as thus or so, and this precludes representing it as having certain other properties, which could have been selected for depiction in another picture in the same style. 12 What is still not obvious, in the catalogue of characteristics we have now gone through, is that they capture the notion of perspective. When we think of a picture as being drawn from one point of view (the location of the eye and direction of vision), we are attending to its alternatives: thinking of it as set in a horizon of other perspectives on the same objects. Occlusion is connected with this only if we have a sense that it can be varied so that other objects come to light, other objects are occluded. Similarly with what the picture is non-committal about: this is connected with perspective only if we can imagine a shift in what is excluded and included. It would be a mistake to concentrate on what is actually in a picture, taken by itself, if we want to say what it is for an image to be perspectival. 13 Thus, as to the hallmarks of perspective: the characteristics listed will not suffice if applied piecemeal. The content of the picture must be related to a horizon of alternatives that we can think of as coming from different points of view, if these characteristics are to count as marks of perspectivity and the explicit or implicit reference to such a horizons of alternatives is what is most important in the concept of perspectivity. 14 Mathematical imagery, distortion through abstraction Visual imagery and kinematic imagery are so-called because of the category of features with respect to which they trade on selective resemblance. Mathematical imagery, on the other hand, is so-called first of all because it is imagery i.e. representation trading on selective resemblance and

55 40 PART I: REPRESENTATION secondly because the representor is a mathematical object. While there is no implication therein of perspectivity, mathematical imaging too involves in general necessary or inevitable distortion, in both simple and subtle ways. Of course the story is apocryphal, that a professional gambler funded a mathematician to analyze horse-racing, and was thoroughly unhappy with the report which began Let each horse be a perfect sphere, rolling along a Euclidean straight line.... But is that so far from real examples of mathematical modeling? Consider the example in dimensional analysis used to model the motion of a cloud of small, electrically charged oil droplets in air under the influence of an electric and a gravitational field reminiscent of Millikan s famous experiment to measure the charge of the electron which begins with the simplifying assumptions the oil in the droplets is in thermodynamic equilibrium with the oil vapor in the air, and no further evaporation or condensation occurs the oil droplets are so small that surface tension effects dominate the distortion of the droplets by the forces acting on it the electric charge is distributed with spherical symmetry over each droplet and so forth, such as that the droplets acceleration is negligible.... If we are to understand mathematical modeling in general, we had better see such simplifying assumptions granted to be most likely false in fact as the norm. The question, though, is whether this is just a matter of human limitations, inessential to mathematical modeling as such, or whether distortion of any sort is inevitable in principle. Mathematical statuary It will have been obvious that the criteria for distinguishing pictures from other visual representations do not imply two-dimensionality. The word picture no doubt connotes, in common usage, representation on a plane surface. But a statue, as we saw at once in the Sophist example, is subject to the sort of distortion practiced in painting to produce a visually faithful image. There is necessary occlusion and the statue is explicitly non-committal with respect to certain features that we don t even think about in the case of painting. That is easily seen when we compare, say, the Venus of Milo with the Anatomical Venus in Florence s Museo della Specola.

56 IMAGING, PICTURING, AND SCALING 41 If we are to bring these concepts to the study of scientific representation we must look to how they can be applied more generally beyond painting, drawing, photography, holography, or sculpture. Descartes s analytic geometry, Newton s and Leibniz s differential and integral calculus, and the subsequent developments in descriptive geometry and analysis provide, on an abstract level, resources for representation so perfect that they tend to engender oblivion to the distortions on which they trade and oblivion as well to the necessary sacrifices of perfection in practice. To counteract this, let us begin with a Cartesian dream of abstract perfection, and then consider how abstraction itself blurs the real. When Galileo said that the Book of Nature is written in the language of mathematics, he was referring to geometry and geometric figures; shortly afterward Descartes founded analytic geometry in which these figures can be equally represented by numerical functions. That was an enormous step, in which our spatially structured world came to be represented algebraically and one might equally say that almost all its qualities were so to speak spatialized. Let me illustrate this by introducing the notion of a mathematical statue. 15 Here is a man, we ll call him Kurtz, who stands at the precise intersection of the Equator and the Greenwich meridian. He is of a certain height, no more than 2 meters from head to toe. Our task: to construct a statue of this man, as accurately as possible with respect to size and shape. No plaster concoction will do him great justice. Let us define a function K as follows. Its domain is the set of triples of real numbers x,y,z. ThevalueofK equals always 1 or 0, with this condition: K(x,y,z) = 1 if and only if Kurtz s body occupies a region including latitude x, longitude y and distance z meters from the center of the Earth. This function has value 1 on a region that precisely fits Kurtz s body. In analytic geometry, this function describes a solid, three-dimensional figure, and indeed, within analytic geometry there is not much difference between figure and function. I offer this to you as a statue, invisible to be sure, but more accurate with respect to size and shape than any plaster or bronze could be. Such mathematical statues are the objects on which the new scientists of the modern era practiced their craft. Of course they are much more

57 42 PART I: REPRESENTATION versatile than I have indicated yet. 16 Thermodynamic study lets in a fourth parameter: K(x,y,z,T) = 1 if and only if Kurtz s body occupies a region including latitude x, longitude y and distance z meters from the center of the Earth, and T is the temperature in degrees Kelvin at that point. No Earthly museum contains a statue with this internal temperature correspondence to Mr. Kurtz! Kinematics lets in still more: K(x,y,z,T,t) = 1 if and only if at time t, Kurtz s body occupies a region including latitude x, longitude y and distance z meters from the center of the Earth and T is the temperature in degrees Kelvin at that point. With time come trajectories; with Newton we add in masses and forces. Now the mathematical statuary can be thought of as figures in, or functions defined on, higher dimensional spaces configuration spaces and phase spaces.... Trouble at the interface Of course we were idealizing! Who would think that there is an objective, sharp division between the geometric points inside and those outside a human body? But this idealizing fiction is the be-all and end-all of the seventeenthcentury geometric representation of nature, continued in the next several centuries of rational mechanics. 17 Yet it also leads inevitably to its own limits, where retrenchment from the idealization becomes imperative. Indeed, this process in which deliberate idealization brings us to its own limits, and thereby defines a new problematic for the scientist, imparts a new impetus for scientific progress. As illustration let us look back to geometric optics in its simplest form. 18 There light is treated as a set of rays, emanating from a source, propagating through transparent media according to three simple principles: the law of rectilinear propagation, that light rays propagating through a homogeneous transparent medium propagate in straight lines the law of reflection, which governs the rebound of light rays from reflecting surfaces

58 IMAGING, PICTURING, AND SCALING 43 the law of refraction, concerning the behavior of light rays as they traverse a sharp boundary between two different transparent media (e.g., air and glass). Hero of Alexandria established the law of reflection on the basis of a principle of economy in nature: that light will always follow the shortest path as it moves from surface to surface. 19 Proposition 1. If the light is unobstructed, it will travel in a straight line. 20 Proposition 2. (Law of reflection) If light is reflected from a surface (such as water or a mirror), the angle of reflection will equal the angle of incidence. By convention, the angle of incidence is taken to be the angle between the ray and the normal, i.e. the line perpendicular to the surface at the point of incidence; similarly for the angle of reflection. Hero s principle of economy, in which economy of action is identified as following the shortest path, is fine when the light is traveling through the same homogeneous medium all the way. When the ray travels through different media, say air and glass or air and water, it will follow the shortest path in each but change direction when it moves from one into another. This refraction depends on the density of the media. Suppose light strikes a water/air surface at an angle. Again we draw the normal, i.e. the perpendicular line at the point of incidence. The empirically ascertained findings were: The light entering the denser medium is refracted toward the normal; if entering from the denser medium into the less dense, it is refracted away from the normal, to the same extent. Ptolemy, in the second century AD treated this phenomenon systematically, but his work was lost to the Middle Ages until it became available, via Arab scholars, in the twelfth century. The Arab mathematician Alhazen also discussed refraction systematically and stated the above ca The quantitative description we are about to present was found in the seventeenth century. Let R be chosen so as to be as far from the point of incidence as Q. Draw perpendiculars from Q and R to the surface (assumed flat), to meet that surface in Q and R.

59 44 PART I: REPRESENTATION Proposition 3. The ratio Q P:PR is a constant, independent of the location of P, and depending only on the nature of the two media their Refractive Index. This Proposition is called Snell s (or Snel s) law after its discoverer in the seventeenth century. 21 Figure 2.1. Reflection and refraction But now let us look at a range of phenomena that lies just where reflection and refraction compete. (Today it is easy to see this illustrated on the internet by computer simulation. 22 ) Light is refracted if it strikes the surface at a shallow enough angle; but it is reflected, if it arrives at a sufficiently steep angle. What happens when a light source is moved so as to change the angle of incidence? Precisely where does the one phenomenon end, or the other begin? In the diagram, let QP equal PR, but consider various values for the angle QPQ. Let Q move downward toward the water surface; the angle of refraction away from the normal becomes larger, R moves up and to the left in our diagram. But noticing that the distance Q PisalwayslargerthanR P, what happens when Q and Q coincide? What happens then as the source moves still further? The only answer within this theory is that the light is at some point (at the critical angle) no longer refracted but completely reflected,

60 IMAGING, PICTURING, AND SCALING 45 and that when that happens there is suddenly a big jump to a distance below the surface a singularity, a discontinuity in nature! Perhaps even a contradiction in the theory given the traditional principle accepted, at the time when Willebrord Snel formulated his law, that nature makes no leaps. Is there really such an enormous discontinuity in nature at this point? At the level of observation open to a swimmer or fish, this phenomenon can be found. But extrapolation to what happens in the small is not valid; this is just the point where the model gives out, where the idealization reaches its limit of admissibility. The phenomena for which geometric optics works do include the more easily observable ones studied early in the history of optics. But there is no infinitely thin precise demarcation between water and air, and in any case, ignoring the wave character of light will only yield adequate results even for the observable phenomena in a limited range. Infinitely perfectible idealization? The assumption of continuity in all natural processes is no longer in force. That does not nullify the lesson illustrated by the above trouble at the interface, however. Agreed, we cannot demonstrate that in principle, asa matter of logic, mathematical modeling must inevitably be a distortion of what is modeled, although models actually constructed cannot have the perfection reachable in principle. But on the other hand, the conviction that perfect modeling is possible in principle what Paul Teller calls the perfect model model does not have an a priori justification either! 23 One conviction which supports the perfect model model is that however vague our ordinary language is, there is absolutely no vagueness in mathematics. This support too loses its plausibility, however, if we look not just to pure mathematics but to assertions made by way of application. There the fascinatingly creative changes in mathematics itself belie the idea. Since the scientific description of the world is couched throughout in mathematical language, we can put it this way: the scientific image itself harbors vagueness and ambiguity, at each historical moment of its development but this only comes to light in retrospect. Consider this beer glass on the table: each has a shape. What that shape is, precisely, we do not know. In the heady early days of the mathematization of our world picture it could be assumed that this shape is [described by] a precise function of the spatial coordinates. The edge of this table could be thought of as a straight line, hence [described by] a function

61 46 PART I: REPRESENTATION of form y = ax + b. But of course, the edge of a table is not perfectly straight.... If eventually the table, the beer glass, and their environment are re-conceived as assemblies of classical particles, they still occupy precise regions of space. These regions can be similarly represented by functions on the spatial coordinates. It may be a bit arbitrary exactly which particle assembly is the glass at any given time but upon any such arbitrary, admissible choice, the table and glass have a definite shape. 24 So now these objects can be represented by a mathematical model in the same way but more accurately than before though now definitely only in principle, not in practice! The shape is accurately modeled by a suitable mathematical function; what that function is, we do not know, but there must be such a function. We are speaking here of the continuum of classical mathematics which has equal use for the representation of each primary quality: length, duration, shape, size, number, mass, velocity, what have you. The equation of the primary quality shape with geometric shape is in effect the assertion that a certain representation is completely adequate. But now we must ask: what exactly is this representation? Not only the question as to what shape the glass has, but that question is continually answered differently. In the nineteenth century, mathematics developed to the point where it was sensible to ask: is this shape an analytic function? 25 Or is it only smooth, i.e. infinitely differentiable? There is no question but that, as a reconstruction of the world picture of Galileo, Descartes, and Newton, we can choose either option. They had not said that every physical magnitude in nature is an analytic function, but they had not conceived of any alternative. Nothing would have been lost from the subject as developed at that time if we thought of the functions then discussed in this way, the functions describing the primary qualities of real physical things, as all analytic nor if we thought of them as not necessarily analytic. Nor is there any kind of experimental evidence to cite in favor of one or the other option. The description was open, indeterminate in that respect. To see how far such nineteenth-century questions are beyond those that arose in the seventeenth century, reflect on what Descartes created when he created analytic geometry. When Pascal took issue with Descartes, it was because he felt the need for the existence of points given only as limits of infinite sequences, while Descartes was willing, within mathematics, to countenance only finitary constructions. 26 By the time the mathematicians

62 IMAGING, PICTURING, AND SCALING 47 could ask whether all functions are analytic or even continuous, and could contemplate negative answers thereto, that controversy was long past. To go even further, after the development of measure theory Birkhoff and von Neumann pointed out that when classical mechanics solves problems about systems with given precise configurations, we can construe it as using conveniently simplified descriptions. More realistic, they suggested, would be the description that results if we transform the precise descriptions by identifying regions that differ only by sets of measure zero. 27 Their reasons for thinking of that as more realistic may or may not be cogent, but it suffices here to note the conceptual possibility. That is, after the time of Lebesgue we can look back to the older description of nature and we have the new option for how to conceive mathematically of the shapes of things. 28 You will realize that I am simply giving examples of how, in many ways, we must in retrospect look upon the scientific image inherited from the older generation as open, vague, ambiguous in the light of our new understanding (that is: in the light of alternatives not previously conceived). What is the shape of this beer glass really? What was it in the Galilean, Cartesian, Newtonian scientific image? In each case the presupposition that it was one item in a certain class gives way to (i) the conditional that this was so if that shape was correctly represented by some item in that class, and (ii) the realization that there are other candidates. As it is for shape, so it is for each primary quality, represented by the mathematics of the continuum. Indeed, we need to cast our net more widely still, if we want to find all the ways in which we could now understand the scientific image fashioned in the seventeenth century. There is no such thing as the classical continuum, if that is meant to be the continuum on which the classical (= modern, 17th century) scientific image was erected originally. Cantor, Brouwer, and Weyl had equal right to regard it as erected on their continua, which are very different. Of course, today we will use the classical continuum to refer to the subject of real number theory as it now exists in main stream mathematics. That is the politics of linguistic usage. But even in what we now call classical mathematics, recall that we have the option of saying that quotient constructions are more accurate (and the simple use of real numbers merely a convenient artifice), as Birkhoff and von Neumann suggested. What would you like the shape of the beer glass to be? 29

63 48 PART I: REPRESENTATION So, what is the shape of the beer glass in the scientific image? What is meant by the assertion that its shape is one of the surfaces in the mathematical representation of nature? The openness of scientific description here come to light is irremediable. Of course, every time we outline a range of alternatives for ourselves, we can ascend our private throne are we not all kings and pontiffs in realms of the mind? and assert that one of these alternatives is the one true story of the world. When the range of alternatives is refined by new conceptual developments or simply by having our attention drawn upward by logical reflection we can choose a new option and make yet another declaration ex cathedra. Arbitrary perhaps, but as definite as can be, by choice. What we cannot pretend is to be non-arbitrary, or to close our text once and for all. Yet the form of understanding is always one of presumed objectivity and univocity. The scientific image is as replete with uncashed and ultimately uncashable promissory notes as the manifest image. Any practical context brings its own standards of appropriate precision, so it is neither proper nor practical to keep this open-endedness constantly salient, but to acknowledge that is not to deny it. Distortion by statistical abstraction (Simpson s paradox) Abstraction just removes some factors or parameters, and leaves the relations among the remainder intact, isn t that so? Just think of a set of premises: remove some, and the implications of the remainder, taken by itself, is precisely what it was before the removal. Think of a color photo; remove the color so that it becomes a photo in black and white: all features that are independent of color, such as relative size and angles, remain the same. So why ever think that abstraction can distort? Simpson s paradox in statistics gives the lie to this rhetorical question. Here is an example: the civil servants in a given city claimed that lighting and ventilation were seriously affecting their well-being and productivity. The city hired a statistician who showed conclusively by means of sampling that the productivity among workers in ill-lit and ill-ventilated spaces was no less than that among workers in general (or in better lit, better ventilated spaces) the productivity level was the same in both groups. So the complaint was concluded to be baseless. Eventually a new study was done, and the second statistician broke the data down by looking separately at women and at men. She showed clearly

64 IMAGING, PICTURING, AND SCALING 49 that among women, the productivity was less for workers in ill-lit and ill-ventilated spaces than elsewhere. She also showed that among men, the productivity was less for workers in ill-lit and ill-ventilated spaces than elsewhere! So relevance of working conditions did not show up until there was a subdivision by this third factor (gender). How is this possible? That is precisely Simpson s paradox: correlations can be washed out, or on the other hand brought to light, by averaging in different ways. Here is the solution to the puzzle: under all conditions the women were more productive than men working under the same conditions, but the women were predominantly working in poor conditions. 30 The first statistician abstracted from gender, and by this very means produced a picture which was misleading, and even conveyed a falsehood. The appearance for him, that is, the outcome in his measurement set-up, has to be assessed as how it looks in that set-up or relative to that set-up. His abstracting from gender would have been fine if that factor had been irrelevant to level of productivity under various conditions. On the other hand, we say now that it was relevant only because at a lesser level of abstraction in this respect, the correlation is different. (Mind you, there is no guarantee that subdividing further, by other features besides gender, won t undo the correlation again!) In this particular case we can say that by abstraction he produced a distorted picture of the reality, a picture in which lighting and ventilation were irrelevant to productivity. Abstraction is harmless only under very strict conditions of pertinence. Scale models and virtuous distortion The nearest to three-dimensional pictorial representation in use in science is surely the scale model 31 can it be conceived of in the terms proper to picturing? As we will see, scaling is not simple reduction or increase in size in all dimensions. In that sense, useful scaling trades not just on the obvious resemblances in shape but on distortion, both resemblance and non-resemblance being selective in a way dictated by the purpose at hand. The scaling cannot be proportionate in all respects. The pertinent question is whether there will be a sufficiently accurate resemblance in all relevant respects for the purpose at hand.

65 50 PART I: REPRESENTATION A presumption that this is always possible, and that the relevance will be transparently perceivable, has had a strong grip on the physical imagination. The idea of testing a hypothesis at a different scale tends to be immediately convincing. Think for example of the experiment proposed by Galileo concerning buoyancy. The conventional wisdom, which he disputed, was that a ship will ride higher in the water in the open sea than in port, due to the amount of water below it. This is difficult to test directly because of the choppy waves on the high seas. So Galileo proposed to place a small vessel in a shallow tank and load it with lead pellets until the addition of just one more pellet would make it sink and then to repeat this procedure in a much larger body of (quiet) water. 32 Doesn t this proposal design a definitive test of the hypothesis? 33 To take a more critical attitude, we must recognize that for any concept there are boundary cases where guidance by our usage so far dwindles and eventually gives out. Scale modeling displays the characteristics of picturing, by relying on selective resemblance to achieve its aim, but in a way that is subject to inevitable occlusion or distortion. Scaling as picturing A scale model represents, and yields information about what it is a model of, by selective resemblance. Are there such necessary limitations in that case, analogous to occlusion, marginal distortion; is scaling explicitly non-committal? Consider a scale model of an airplane. It has the same shape overall, but with the proportions reduced by a multiplicative factor, say Willit fly? Not necessarily. If it is to fly, to mention just one factor, it must have something to propel it; but its size limits necessarily what that can be. For example, the relation to air resistance will be quite different at this scale: the air, after all, has not been similarly scaled down in any way! There are other reasons, as we will see below the same shape is a deceptively simple concept. 34 Scale models can be produced for the sheer aesthetic pleasure of it, but more typically they serve in studies meant to design the very things of which they are meant to be the scaled down versions. This use and its subtleties were brought out clearly in the Second Day of Galileo s Two New Sciences. 35 His calculations involved an error, but his principles were

66 IMAGING, PICTURING, AND SCALING 51 correct. In modern terms we can summarize his conclusions easily for a cylindrical beam with constant density. Its strength decreases with its cross-sectional area, which is proportional to the square of its radius. But the mass is proportional to its volume, that is, to the cube of its radius. So the strength to mass ratio of such beams with the same density becomes N times less when the beam s size is increased by a factor of N. Beyond a certain point, the mass can no longer be supported, and the structure collapses under its own weight. As Galileo observes, large ships taken out of water are in danger of breaking for just that reason, and he gives examples for optimal bodily structure:... nor can nature produce trees of extraordinary size because the branches would break down under their own weight; so also it would be impossible to build up the bony structures of men, horses, or other animals so as to hold together and perform their normal functions if these animals were to be increased enormously in height... We can observe conversely that if a reduced structure is to remain feasibly like its original, some other features besides its size must be scaled as well, and not proportionately but appropriately. 36 Principles of Similitude and Approximation Roughly speaking, a scale model of X is an object which is structurally similar to X but suitably smaller. Similar and structurally have their usual context dependence as much as does smaller in any particular case, the goal implicit in suitable will determine the contextual parameters for each. And still roughly speaking: there are two assumptions in force when conclusions about the target are drawn from characteristics of a scale model. The first is that structurally similar objects will display the same behavior in structurally similar circumstances. This was glamorized by Richard Tolman in 1914 as his Principle of Similitude, more accurately called a principle of dimensional homogeneity, rather poetically expressed on a cosmic scale: The fundamental entities out of which the physical universe is constructed are of such a nature that from them a miniature universe could be constructed exactly similar in every respect to the present universe. (Tolman 1914, 1915)

67 52 PART I: REPRESENTATION So phrased it is a thesis in ontology; on the methodological side it could perhaps correspond to something like All laws of physics are to be, and all measurable effects are to be conceived of as, invariant under scale transformations of any kind. Amazingly this principle, which occasioned a good deal of response in the literature at the time, appears here in this evangelical form decades after the advent of Planck s quantum, also, almost ten years after Einstein s study of the photoelectric effect, and several years after Bohr s model of the atom. By this time it is certainly surprising to see the conviction that scale is essentially irrelevant to physical modeling. But Tolman is trying to capture a correct principle of scale invariance, though one that needs considerably more sophisticated formulation. If we doubt Tolman s principle, then inferences from scale models are just inferences from false assumptions. But there are useful fictions! The second principle in force is a Principle of Approximation. The centrality of this idea in applied science was highlighted by Reichenbach. 37 Think of how Newton proceeds to deduce the laws of motion for our solar system. Keeping his basic laws of mechanics as foundation, he adds the law of gravitation to describe this universe, then he adds that there is one sun and six planets to describe our solar system, and finally he adds that there is one moon to describe the more immediate gravitational forces on our planet Earth. Newton demonstrates that from a very idealized, simplified description of the solar system, something approximating the known phenomena follows. Well, so what? What s the point of deriving true conclusions from a false premise? This is the precise point where we notice this deep assumption at work: if certain conditions follow from the ideal case, then approximately those conditions will follow from an approximation to the ideal case. Mutatis mutandis, this assumption is in force when we have recourse to any model that we do not presume to be more than approximately similar to what it represents, but especially in the case of laboratory simulations. However, we cannot take this blithely as a context-independent methodological principle, given its phrasing with this sort of generality. The approximation to the ideal case, just as well as the similarity of a scale model to its original, is similarity in certain respects, with other aspects ignored as irrelevant for all practical purposes which means, for the purposes at hand

68 IMAGING, PICTURING, AND SCALING 53 in the context of application. Whether that ignoring will be vindicated is an empirical question; we can t very well decide it on principle. In fact, Reichenbach shows by illustration in statistical mechanics that even small departures in approximation can have widely divergent consequences a point now popularly familiar from Chaos theory. Fine, but the study of scale models, and in general studies that seem to be inspired by these rough and ready principles of similarity and approximation, are often useful, practical, and truly vindicated. So what are the facts of the matter, the real constraints on such modeling? Dimensions and invariance We can glean these from the critiques of Tolman published in a seminal paper by E. Buckingham and a critical article by Percy W. Bridgman, the physicist famous for originating operationalism as a philosophy of science. 38 The applications that Tolman outlined for his principle, both authors argue to be derivable in dimensional analysis, a technique whose development started with Fourier. I ll explain some of the basic ideas and then display the example of the screw propeller, which Buckingham analyzed in detail. Fourier had extended the geometrical notion of dimension to the now familiar general concept of physical dimensions, so that not just length, area, and volume but also mass, force, temperature, charge and the like are included: [E]very undetermined magnitude or constant has one dimension proper to itself, and... the terms of one and the same equation could not be compared, if they had not the same exponent of dimension. We have introduced this consideration into the theory of heat, in order to make our definitions more exact, and to serve to verify the analysis; it is derived from primary notions on quantities; for which reason, in geometry and mechanics, it is the equivalent of the fundamental lemmas which the Greeks have left us without proof. In the analytical theory of heat, every equation (E) expresses a necessary relation between the existing magnitudes [length] x, [time]t, [temperature] v, [capacity for heat] c, [surface conducibility] h, [specific conducibility] K. This relation depends in no respect on the choice of the unit of length, which from its very nature is contingent, that is to say, if we took a different unit to measure the linear dimensions, the equation (E) would still be the same. 39 This same passage, which introduces the general conception, also introduces the idea of dimensional homogeneity and the importance of invariance

69 54 PART I: REPRESENTATION under scale transformations for the fundamental equations of physical theory. 40 An equation must be dimensionally homogeneous to make sense: the dimension of the quantity on the left must be the same as that on the right. That is just the common place that you can t add apples and oranges except in the sense that you can take them both as fruits and count them that way. In more complicated cases, this homogeneity has to be checked. Take the equation of the distance covered s calculated from time t, velocity v, and acceleration a: s = vt + (1/2)at 2 Does that make sense? Here distance has the dimension of length, call it L; velocity has the dimension of length divided by time (T) and acceleration the dimension that has velocity divided by time. We must first check that on the right-hand side we are not trying to add apples and oranges, but rather things of the same sort. We check this by replacing each of the parameters by its dimension alone, while multiplying replacements of the terms with each other: (L/T)T (L/T)(1/T). T.T and then treat those dimensions algebraically the same as numbers. The result, after cancelling the Ts against each other, is L in both cases. Thus we are adding two like quantities, and the quantity denoted by the right side of the equation has dimension L. But the dimension of s is also L, so we have the required match. Without detailed scrutiny this may look like a calculation by rules of thumb far removed from rigor, but I will leave the detailed justification of dimensional analysis techniques to other sources. The second requirement, recall, is that of invariance under scale transformations. To achieve invariance under such transformations rewriting equations stated in terms of certain quantities in terms of others is precisely served in dimensional analysis by the search for dimensionless quantities as sole constituents for the fundamental equations. 41 As illustration we can begin with the familiar cgs system of units in mechanics: centimeter for length, gram for mass, and second for time. A different scale belonging to the same class of systems of units is one defined by multiplying each unit by a positive number. This is the form of a scale

70 IMAGING, PICTURING, AND SCALING 55 transformation. If we choose the numbers 100, 1000, 1 for this role, we define the MKS system, with the units meter, kilogram, second thus producing another system belonging to the same class of systems of units. The basic invariance requirement is now that to be significant, an equation must have the same form regardless of which member of the class of systems of units is chosen. This requirement was obviously respected well before Fourier, let alone before the subject of dimensionless analysis matured. Newton s famous F ma does not depend on its validity on a particular choice of units, and would not be famous if it did. 42 A dimensionless number more accurately speaking, a dimensionless parameter of the class is a quantity that has the same value in every system of units in the class. That is, it is an invariant of the set of admissible transformations, which are precisely the scale transformations. The Hauptsatz of dimensionless analysis, prominent in Buckingham s article, says in part that it is always possible to shift to a dimensionless representation. 43 The screw propeller Susan Sterrett points out the relevance of how the Wright brothers and their colleagues in the field were frustrated when they tried to extrapolate the behavior of children s flying toys to a larger scale. Making the object the same but larger ruined its capacity to fly why? Wasn t the toy a scale model for their construction? The fact is that here, as much as in the examples the Sophist pointed to, what counts as pertinent resemblance is not at all obvious in the way that looking alike is obvious. Buckingham, partly in service of his critique of Tolman, analyzed the case in detail. In the studies of the screw propeller, which had of course been started for ships but were then crucial to the development of the airplane, both rough and ready principles can be seen in a rigorous form. The thrust F of the propeller of given shape and immersion is taken to depend only on the diameter D, the speed of advance S, the number N of revolutions per unit time, the density and viscosity of the liquid, and the acceleration due to gravity. So suppose that a smaller propeller is meant to be a good scale model of a large one with respect to thrust in contrast with some other effects, here regarded as ignorable side effects (noise, shape of the wake,...). Then the equation which expresses F in terms of those other quantities must be the same for both cases, provided that the ratios that specify the shape and

71 56 PART I: REPRESENTATION immersion of the propeller stay the same. Any set of kinds of quantity that furnish the basic units for this dynamics can be changed in any ratios whatsoever without affecting this. How does this serve to guide a practical study? The obvious thing to do is to make the smaller propeller geometrically similar to the original, to immerse it similarly, and to construct the propellers so that the angle of attack of the blades on the water is the same. Is that enough to ensure that we can get information about the thrust of the original large propeller from the behavior of its scale model? It suffices only if one can completely control similarity in the effects of gravity, density, and viscosity. It is easy to see that for extremely small propellers those effects will be significant, and for larger but still small ones the difference is after all only a matter of degree. So in practice... one has to resort to some approximation. And here special conditions can be experimentally investigated to see what can and what cannot be ignored at various scales. For example, the pertinent mechanical behavior in very turbulent motion does not vary much with the viscosity of the fluid. And similarity in the effects of gravity will be approached when the ratio between the two speeds of advance approximates the ratio of the squares of the two diameters. Deep immersion will also prevent significant effects due to disturbance of the liquid surface. (Notice though that these are all matters of degree, and the purpose at hand may require a specific level of accuracy.) With all of this supposed under sufficient control, the ratio of the thrust of the small propeller to that of the large one will be (DS/D S ) 2, where the primes indicate the diameter and speed of advance of the large propeller, and not (DS/D S ), that is, not the ratio by which this product was altered in construction. Conclusion about imaging and scaling Imaging, recall, is representation that is effected through resemblance. Our discussion of mathematical statuary ended with the conclusion that we can see that mathematical representation of nature so far always involved some features that, in retrospect, with hindsight, we saw as necessary failures in resemblance. That does not imply that mathematical representations of the sort now available are also thus necessarily deficient, though some humility

72 IMAGING, PICTURING, AND SCALING 57 in even that respect is appropriate. But in application, the practice of actual construction of models of situations, the idea of perfect modeling is so far from realistic that it can certainly not be maintained. The resemblance of even so obvious an example of a scale model to its original is, as we have just seen, not nearly as simple as may strike the eye at first glance. True, a scale model of a vessel or propeller under study in a naval or aeronautics laboratory will look like a real one. But for it to have any use at all for the purpose at hand, there must be a delicately achieved pertinent similarity (to a pertinent degree of approximation) between the situation of these propellers revolving similarly when similarly immersed. This may well, and typically does, come at the price of dissimilarity: as Galileo already appreciated, the scaling must be different for different parameters. The selective in selective resemblance is a delicate, highly nuanced, contextually sensitive qualification and this point is general: it pertains to all pictorial representation.

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74 3 Pictorial Perspective and the Indexical Imagery, as defined above, is pictorial exactly if it bears hallmarks of perspectivity. In the notion of perspective, as so often, we have a cluster concept, with multiple criterial hallmarks. There is no defining common set of characteristics, only family resemblances among the instances. Whether or not something is aptly called perspectival depends on whether some appropriate subset of these hallmarks are present, but what amounts to appropriate we cannot delimit precisely either. The hallmarks listed above were occlusion, marginal distortion, texturefading (grain), angle, and with special importance, explicit non-commitment and the horizon of alternatives. These are all characteristics that relate to the content of the representation. But there is another notion closely connected to perspective which does not appear here. This does not pertain simply to content, but to how we relate to it; it comes to light when we very naturally think that for the painter or photographer, a picture is showing how the pictured scene looks from here. The painter s eye is located with respect to the content of the painting in a way that he himself can express with this is how it looks to me from here. The viewer may naturally say that this is how the scene looks from there. If I say, for example, that the photo shows the town as seen from the top of the church tower, that indicates something like that is how it would look to me if I were on top of the church tower. These are indexical statements, with the words here and there playing the context-sensitive role. For a critic describing a painting this may not be relevant. While s/he may refer to the painter s view, no special interest may attach either the painter s subjective situation or what the critic s own would be. But that

75 60 PART I: REPRESENTATION changes when we turn to representations subject to different norms and use. If we look to the painting or photograph to help us get around in the town, that does require us to locate ourselves with respect to the view presented by that picture. Since scientific representations are typically produced so as to serve some such practical end, at least in principle, this connection between perspective and the indexical becomes important there. The connection shows up in the sciences, for example, when talk of frames of reference is conducted in terms of observers (whose frames they are, so to speak). We need to look closely into both the character of perspective and the role of indexical judgments (such as self-attributions and self-locations) to see whether that is just an irrelevant heuristic or whether it brings to fore a fundamental connection between perspective, measurement, and theoretical representation. Pictorial Perspective and the Art of Measuring la pittura è una specie de natural filosofia, perché l imita la quantità equalità, la forma e virtù delle cose naturali. 1 The histories of perspective in painting, measurement, geometry and technology are thoroughly entangled. 2 Geometry is so-called because it began as the art of earth-measuring, and in Dutch its name is still the art of measuring (meetkunde). But as we ll see, a famous treatise on perspective was called by that name as well. Examining some of this history it will become quite clear that a picture in the modern perspectival style is essentially the outcome of a measuring procedure. Conversely, a measurement outcome is, in paradigm cases at least, a pictorial representation of the object measured. This paradigm example I will extrapolate subsequently: measurement falls squarely under the heading of representation, and measurement outcomes are at a certain stage to be conceived of as trading on selective resemblances in just the way that perspectival picturing does. Astrolabe and triangle How do you measure the width of a river while remaining on the shore? Or the height of a tower while remaining on the ground?

76 PICTORIAL PERSPECTIVE AND THE INDEXICAL 61 The first clue to the answer is of course that the two problems are essentially the same, related by a simple rotation from horizontal to vertical. The second clue is also geometrical: in a right triangle, the ratio of height to base is determined by the angle of sight along the hypotenuse. At this point today s student reaches for trigonometry, but these practical problems were solved long before that was available. By the end of the Roman Empire in the fourth century much of Greek mathematics was lost to the West, not to return there from Arabic sources until almost 800 years later. During these centuries practical surveying, architecture, and the scholarly study of practical geometry did continue, however. The practical techniques of the Roman surveyors survived, were preserved, collated, and taught among both artisans and cleric-scholars. 3 A representative text, the practical geometry manual of Hugh of St. Victor, in the century just before Euclid s Elements became available again, is divided into three parts: altimetry, planimetry, and cosmimetry (measurement relating to the earth, to the sun, and to other aspects of the cosmos). In retrospect we see the methods there presented as justifiable within geometry and geometric optics, but what is taught there is simply the practical technology of measurement. The instruments designed for this use cross staff, quadrant, and even the astrolabe introduced into the West about a century before this manual s date consist basically of a ruler with a sighting device (alidade) at the center, and part of a circle on which degrees are marked. These had significant use in navigation, but let us here concentrate on land-measurement. 4 The surveyor measuring the height of a tower, for example, adjusts the alidade until he can see the top through the two apertures. The angle thus formed determines the ratio of the height of the tower to the distance from the tower. Determines how? Though most of the mathematic theory was lacking, the astrolabe can be manually calibrated on relatively small similar triangles. This presumes understanding of geometric similarity; sufficiently much of Euclidean geometry was retained to understand this. The distance from the tower may itself not be measurable directly if it is far away, so Hugh s manual gives several forms of two station methods to use. Suppose that the astrolabe sighting is done at two points P and Q, at an unknown distance from the tower. Measure the distance between P and Q, and a few practical steps, starting with the two alidade readings, will yield the height of the tower:

77 62 PART I: REPRESENTATION Figure 3.1. Perspective Altimetry Let h be the height of the tower. The direct measurement by astrolabe at PgivesthefirstratioA= h/pt. The reading at Q gives the second ratio B = h/qt. Only the distance PQ is measured directly. From these three data, the height h can be calculated directly. 5 We will soon see this same configuration again. Alberti s De Pictura When Alberti wrote his monograph on the technique of perspectival drawing in 1435 a great deal of Greek mathematics and geometric optics had been assimilated in the years since Hugh s manual of practical geometry. But his way of writing was not so different from Hugh s, because Alberti was also a surveyor applying those practical arts as well as thinking about the theory behind them. 6 Since the practical geometry manuals focused on geometric figures created by physical objects and lines of sight, they were an obvious source for his study of perspective. His great innovation was to think of the visual cone or pyramid cut by the picture plane: [Painters] should understand that, when they draw lines around a surface, and fill the parts they have drawn with colours, their sole object is the representation on this one surface of many different surfaces, just as though this surface which they colour were so transparent and like glass, that the visual pyramid passed right through it.... (Alberti 1991: Book I, 48)

78 PICTORIAL PERSPECTIVE AND THE INDEXICAL 63 Art historian S. Y. Edgerton refers to Alberti s invention as Windows A painter drawing from life is as it were drawing on a window through which he is seeing his subject. In fact of course, the plane on which he draws (the canvas) is not the plane (imaginary window pane) cutting his visual pyramid. But he will succeed in accurately rendering his subject if what he produces is precisely what it would be if he did draw on that window plane. There was an elementary exercise for this skill whose sign we see in many paintings of that era: the checkerboard floor or pavement. Francesco Rosselli s Supplice de Savonarola (c. 1498), for example, depicts the central square in Florence precisely with such a checkerboard pavement seen in one-point perspective. The following illustrations show respectively how the painter is imagined to see and paint, and what it is that he sees as it will appear on the picture plane: Figure 3.2. Window and Checkerboard Notice how the left-hand diagram is really just the one above of the two station altimetry, flipped horizontally, but with the picture plane and some other sightlines added. The painter s eye corresponds in the geometry to the top of the tower of the earlier illustration. So the point of view is the opposite, as it were, but the geometry involved is the same. Alberti s concern was with technology. He began his development of perspectival drawing techniques by making a box with a small eye-hole in one side. 7 In the box there was a checker board laid horizontally on the bottom. Let us call what the checkerboard looked like, if viewed through the peephole, the checkerboard appearance. (That is to say, the box floor s appearance in observation or measurement made through the

79 64 PART I: REPRESENTATION peephole.) Studying this set-up in various ways he could make a drawing which, if placed upright in the box at a certain point, presented the same checkerboard appearance to the eye. In fact, viewers could not distinguish between the two when looking through the peephole. 8 Looking back at the second illustration you can see that parallel lines orthogonal to the picture plane converge to a point on the horizon, while lines in the other two orthogonal directions remain in place. So the lines between the tiles that are parallel to the picture plane remain parallel to each other; similarly the up-down lines remain vertical. This is one-point linear perspective. In two-point perspective parallelism is preserved in only one of the three directions; but this is hard to find in the history of painting till much later. We can think of one-point perspective as the style of representation that captures what is seen in the Alberti experimental situation the unmoving single eye at the peephole. We can equally think of it as the style in which of three orthogonal directions in space, two are preserved in effect in a grid of parallel lines, while in the third direction all straight lines converge to a point. Masaccio s Trinity, Botticelli s Episodes in the Life of Lucretia and Episodes in the Life of Virginia, as well as the philosophers favorite, Raphael s School of Athens, can be mentioned as examples of this style of visual representation. The Art of Measuring: mechanization of perspective Alberti s technique was applied by contemporary painters, though not nearly as rigorously as either Alberti or Vasari would have it. On the other hand, it was of use also to architecture, technical drawing, and machine design. Alberti s geometric and practical studies of perspective include ways to mechanize the process of perspectival drawing: So attention should be devoted to circumscription; and to do this well, I believe nothing more convenient can be found than the veil... whose usage I was the first to discover. It is like this: a veil loosely woven of fine thread... divided up by thicker threads into as many parallel square sections as you like, and stretched on a frame. I set this up between the eye and the object to be represented, so that the visual pyramid passes through the loose weave of the veil. (Alberti 1991, Book II, 65) His veil was precisely the painterly window we have been discussing, but now realized as a practical technological artifact. The veil with its grid

80 PICTORIAL PERSPECTIVE AND THE INDEXICAL 65 is a measuring instrument, designed to measure not such simple quantities as length or weight but as I shall discuss further below cross ratios, projective structure. This is explicitly recognized in Albrecht Dürer s treatise, where the technique is presented in a part entitled Unterweysung der Messung Teaching of Measurement, generally translated as Art of Measurement. The mathematically precise and practical character of this way of rendering the appearances implied its possible mechanization. 9 Thebasiswasineffecta very careful and systematic form of measurement, in which certain geometric features are faithfully captured on the picture plane. This way of understanding the episode is supported by a look at the machines that Dürer designed to produce perspectivally correct drawings, which show how far this can go. 10 In his most advanced artifact even the human element consists of fully determined mechanical motions. Measurement is precisely what this was, as basis for pictorial representation. The content of such a visual perspective is the content of a complex, technically advanced measurement outcome. Figure 3.3. Dürer, the Draughtsmen of the Lute

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