diagrammatic representation in geometry

dialectica

Vol. 60, N° 4 (2006), pp. 369–382DOI: 10.1111/j.1746-8361.2006.01065.x

© 2006 The Author. Journal compilation © 2006 Editorial Board of

dialectica

Published by Blackwell Publishing Ltd., 9600 Garsington Road, Oxford, OX4 2DQ, UK and 350Main Street, Malden, MA 02148, USA

tion in GeometryDennis Potter

Diagrammatic Representation in Geometry

Dennis P

otter

†

A

BSTRACT

In this paper I offer a theory about the nature of diagrammatic representation in geometry. Onmy view, diagrammatic representaiton differs from pictorial representation in that neither theresemblance between the diagram and its object nor the experience of such a resemblance playsan essential role. Instead, the diagrammatic representation is arises from the role the componentsof the diagram play in a diagramatic practice that allows us to draws inferences based on themabout the ojbects they represent.

Introduction

In geometric practice, we employ both diagrammatic representational systems andlinguistic representational systems. The purpose of this paper is to offer a theory aboutthe nature of diagrammatic representation in geometry. It is very often assumed thatdiagrammatic representation in geometry is a type of pictorial representation. I arguefor a view according to which diagrammatic representation differs from pictorialrepresentation.

Before explicating my theory of diagrammatic representation I will examine sometheories of pictorial representation and how well they cohere with data about geometricdiagrams. The first of these theories is

the resemblance theory

. It claims that there isan objective resemblance between a picture and its object(s) that is recognized by thediagrammer. As a theory for geometric diagrams, this view faces several problems.One is the problem that diagrams are always imperfectly drawn. I call this

the problemof imperfection

.The second pictorial view of diagrammatic representation that I will consider is

the experiential theory

.

1

This theory claims that diagrams are like pictures in how theyrepresent. This theory also claims that pictures represent by virtue of being experi-enced as similar to the objects they represent. This won’t work in geometry since thereis nothing that it is like to experience an actual geometric object.

I will call the view that I advocate

the compositional pretence theory

. It is amodification of pretence theories of pictorial representation.

2

This theory claims that

1

See Hopkins 1998, 2003; Budd 1993 and Peacocke 1987.

2

See Walton 1990 and 2002.

†

Department of Philosophy and Humanities, Utah Valley State College, 800 WestUniversity Parkway, Orem, UT 84058, USA; Email: [email protected]

370 Dennis Potter


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a diagram represents by virtue of the diagrammer’s pretence in which she uses thecomponent parts of the diagram as props for the geometric parts of the diagram’sobject(s). I argue that this view solves the problems faced by the other theories. It alsoseems to entail that a geometric diagram is, in various ways, more like a map than apicture.

In this paper I am discussing diagrammatic representation only in Euclideangeometry. So, for the duration of this paper the reader should assume that by ‘diagram’I always mean ‘geometric diagram’.

Data to be explained

Diagrams and words represent differently. Words represent by convention alone. Or,in other words, what any particular string of symbols represents is

arbitrary

. Forexample, there is nothing about the shapes in the string ‘tree’ that makes it a morelikely candidate for representing trees than the string ‘dog’. The fact that ‘tree’represents trees is an historical accident.

Of course, diagrammatic representation also involves an element of convention.For example, we adopt the convention of using ‘hash marks’ to indicate congruenciesbetween lines and angles. However, diagrams do not represent by convention alone.Suppose I drew a triangular figure (see Figure 1) and then said, ‘Let this represent acircle’.

Figure 1

The right response to this would be: ‘That’s not a circle; it’s a triangle’. What thisexample shows is that there is a difference between diagrammatic representation andpurely conventional modes of representation. Clearly, we can stipulate that any givenshape will

linguistically

represent any type of object. Yet, it is equally clear that someshapes cannot

diagrammatically

represent some objects, at least for persons with ourperceptual faculties. In other words, any physical mark (or set of physical marks, ifthey are not visibly continuous)

has some limits

to what it can diagrammaticallyrepresent for us. Whereas linguistic representation is purely conventional, diagram-matic representation is not. In my view, this fact must be explained by any theory ofdiagrammatic representation. I will call this

the limitation datum

.

Diagrammatic Representation in Geometry 371


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Another way in which diagrammatic representation is different from linguisticrepresentation is that we can infer something about the represented object on the basisof the appearance of the diagram. If Figure 1 is supposed to represent a trianglediagrammatically, we can make some claims about what triangles are like on the basisof the features of the figure. On the other hand, if it is not supposed to representdiagrammatically, then we cannot make such claims.

3

For example, the linguistic string‘long’ has the property of being short, but the word ‘long’ stands for long things, notshort things. We cannot infer anything about the property represented by ‘long’ on thebasis of its properties

qua

string of symbols. So, a figure represents

diagrammatically

only if we can draw inferences about the object

on the basis of the structure of therepresentation

.Of course, we can only draw

some

inferences on the basis of the appearance ofthe diagram. If a figure of a triangle is drawn in black ink, we cannot infer that thetriangle is black. And if it is supposed to represent any triangle, then we cannot drawinferences on the basis of the fact that it appears to be acute. Coupled with the pointfrom the previous paragraph, the upshot is that we can draw inferences on the basisof some aspects of the diagram’s appearance but not on the basis of other aspects. Atheory of diagrammatic representation in Euclidean geometry should explain how thisline is drawn. I will call this

the inferential datum

.Another important datum concerning diagrammatic reasoning is that one and the

same physical mark (call it an

inscription

) can be several different diagrams. So, takeFigure 1 as an example again. It can be a diagram of a right triangle or of

any

triangle.Indeed, it could be a diagram of numerous different non-geometrical objects as well(e.g., a physical structure of some sort). The actual physical structure of the diagramunderdetermines what it is a diagram of. So, what a diagram depicts is (in part) up tous. I will call this

the flexibility datum

.Finally, Euclidean diagrammatic representation is consistent. This involves two

points. First, geometric diagrams are consistent in the types of marks used to constructthem. That is, the physical structure of each mark that is meant to count as a point isroughly similar to the physical structure of each other mark that is meant to count asa point as well, etc. Second, there are rules for extending a diagram of one type ofobject into a diagram of a new type of object. These latter relationships are importantfor the nature of diagrammatic inference. I will call these two points together

theconsistency datum

.

4

3

There is an exception to this. In proof theory for first-order predicate logic, inferencesare based precisely on the way the symbols look.

4

Here I use this term in the loose, more common sense way it is often used in ordinarydiscourse and not in philosophy or logic. Later I employ a more technical notion of consistencythat is not to be confused with this term.

372 Dennis Potter


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The resemblance theory

One pictorial theory of diagrammatic representation might say that diagrams representby having an objective structural resemblance to their objects

5

that is recognized bythe diagrammer.

6

The resemblance is a set of properties that they share.

7

The recognition of the objective resemblance between a diagram and its objectmight depend on one’s perspective. Consider the following line segment:

Figure 2

AB

appears to us to be more similar to a line segment than the ‘curve’ in the followingdiagram (

DE

).

Figure 3

However, this resemblance is one that might be seen only from a certain perspective.That is,

AB

is more similar to a line segment than

DE

only in how it appears to humansubjects under ‘normal conditions’ from a certain point of view

. When looking at

AB

under a microscope it might appear to be more like

DE

than a line segment – or, morelikely, it might not appear similar to either of them. Moreover, an inscription will lookdifferent from different points in space. The diagram of a square may appear to be a

5

A point of terminology: I will say that the diagram’s

objects

are those things that thediagram is meant to represent. So, instead of saying ‘The diagram resembles the objects that itrepresents’, I will say, ‘The diagram resembles its objects’.

6

The resemblance theory is not exactly the same as ‘the recognition theory’, discussedin Robert Hopkins (Hopkins, 2003, 653) and advocated by Dominic Lopes (Lopes, 1996),although there are some similarities. Lopes’ view, for example, claims that pictures representobjects by triggering our abilities to recognize those objects in the material world. I will nothave the space to deal with the recognition theory in this paper. However, it is not clear that itcould count as a theory of diagrammatic representation in geometry because we don’t recognizeactual Euclidean triangles in the material world.

7

Cf. (Barwise and Etchemendy, 22) and (Hammer 3ff). One may consider this claimto be constroversial. Instead, one may argue that the resemblance between a diagram and itsobjects is in virtue of a resemblance between their respective properties. But this is just to pushthe problem to the level of properties since properties will imperfectly resemble each otheras well.

A B

DE



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trapezoid when seen from an angle. So, a person’s ability to see a certain structuralresemblance between a diagrammatic inscription and the objects it represents dependson her visual point of view. This does not mean that the very

existence

of this structuralresemblance is relative to her point of view. Indeed, the structure exists whether shecan perceive it or not.

Given this, an advocate of the resemblance theory might claim that an inscriptioncan only be a diagram of a certain type of object

for S

if it shares a structure with thattype and

that structure is perceivable by S

.However, the resemblance theory has to deal with

the problem of imperfection

.Any resemblance between diagrammatic inscriptions and geometric objects is likelyto be imperfect. Many geometric properties are ones that cannot be reliably reproducedin a diagram. Take an arbitrary diagram of a triangle, for example. Its lines are notstraight. The sum of its angles is not 180 degrees. It is not even a triangle in the literalgeometric sense. This may sound odd since we are in the habit of saying that a giventriangular diagram is a triangle, is composed of straight lines, and has three angles.However, there are certain geometric properties that we cannot reliably reproduce.Nevertheless, even while noticing such gross imperfections in our diagrams, we oftenproceed to reason on the basis of them regardless. Sometimes we even make inferencesabout the properties of the represented objects even though the diagram does not havethese properties.

One way to respond to the problem of imperfection is to insist that although thereare many imperfections in diagrams there are other properties shared by the diagramand its objects. Instead of refuting the resemblance theory, the problem of imperfectionimplies flexibility in what counts as resemblance. The presence or absence of aresemblance between a diagram and its objects must not be sensitive to the presenceor absence of certain types of imperfections in the diagram.

For example, it might be that although the sides of the diagram of a triangle arenot straight, it shares relations of incidence with triangles. In particular, given thatoverlap of drawn lines is to be interpreted as a ‘point’, a diagram of a triangle willordinarily have at least three noncollinear points.

This view includes two claims. First, it is not that the diagram has the subjectiveproperty of

looking like

a triangle to

S

: there are properties that

objectively

inhere inboth the diagram and the triangle. We can take the inscription to be a diagram of atriangle because we perceive these properties. Second, and more strongly, this viewstates that there is a

general

account of the

type

of property that diagrams have incommon with their objects. So, the question we must answer now is ‘what is thisgeneral type of property?’

One proposal for the type of property shared between a diagram and its object(s)could be based on a distinction made by Kenneth Manders (Manders 1996, 392).

8

He

8

In a widely read but unpublished manuscript entitled, ‘The Euclidean Diagram’Manders refers to non-exact properties as ‘co-exact’. Manders does not really offer this

as

atheory of diagrammatic representation. He is actually concerned with examining the nature ofdiagrammatic inference in Euclidean practice. His approach is descriptive.

374 Dennis Potter


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proposes that there is a distinction between what he calls ‘exact’ and ‘non-exact’properties. Giving a characterization of exact properties he says,

Exact attributes are those which we, since Descartes’ time, could express by equa-tions. In traditional geometry, many of them were expressed or defined by equalitiesor proportionalities. Prominent examples include equality of lines (segments),angles, or other magnitudes, congruence of triangles or other figures, proportionalityof lines; that an angle is

right

(not that it is an angle), 4-point con-cyclicity relation-ships; the geometric character of lines or curves: that a line is

straight

, that a curveis a

circle

, an

ellipse

; that lines are

parallel

; that three lines or curves intersect in apoint (rather than intersecting pairwise in three distinct points); that a line is tangentto a curve (rather than intersecting it in two or more points close together).

Exact attributes are unstable under perturbation of a diagram, well beyond ourcontrol in drawing diagrams and judging them by sight. Thus, diagramming practiceby itself provides inadequate facilities to resolve discrepancies in response by par-ticipants as they draw and judge diagrams (Manders 1996, 392–393).

Exact properties are defined as those that can be

expressed by equations

. This tellsus two things about them: (i) exact properties are

not stable under perturbation

and(ii) they are

beyond our control in drawing diagrams and judging them by sight

. So,for an approach based on Manders’ distinction, the imperfections of the diagram wouldarise from the fact that it does not share

exact

properties with the objects it represents.If there are any properties that inhere in the diagram and its objects alike, then theymust be

non-exact

. Resemblance must be non-exact.Presumably non-exact properties are all those that cannot be expressed by an

equation. Manders says that the non-exact properties of a diagram are ones that‘express the

topology

of the diagram’ (Manders 1996, 393). He gives a list of exam-ples, but does not give a general characterization of what he means by topologicalproperties:

Non-exact attributes express recognition of regions (and their lower-dimensionalcounterparts, segments, and points) and their contiguities in the diagram. We mightsay they express the topology of the diagram. Prominent examples include non-empty delimited planar regions: triangles, squares, . . . (but not that the sides arestraight); circles (but not that they are circular rather than just elliptical or irregular);angles (must be less than two right angles, to delimit a region). In lower dimension,non-empty segments (but not the character of the curve of which they are segments);points as two-place (but not three-place or tangent) intersections of curves;

non-parallel

lines; non-tangencies. Further, contiguity and inclusion relations amongthese: point lies with region; point lies with segment; side opposed to vertex; line

divides region

into two parts; triangle

lies within

triangle;

alternate

angles, and soon (Manders 1996, 393).

This gives us an intuitive idea of the nature of topological properties. Another wayto think of topological properties is in terms of those properties that are preservedacross continuous transformations. And we have an idea of what these might be dueto our practical knowledge of how elastic bands can be transformed.



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This version of the resemblance theory, based on Manders’ ideas, has severalaspects that recommend it. First, it explains the limitation datum. There must be somenon-exact properties shared by the diagram and its objects. Second, it explains theflexiblity datum. Different geometric objects can share the same topology. So, differentgeometric objects can be represented by one given inscription. Third, it explains theconsistency datum. Diagrammatic representation would exhibit consistency acrossdiagrams and there would be consistent ways to extend diagrams into diagrams of newobjects, based on the former’s topological features.

One might also think that the resemblance theory

does

explain the inferentialdatum given the distinction between exact and non-exact properties. The theory entailsthat we can infer on the basis of some aspects of the appearance of the diagram andnot on the basis of other aspects. However, if diagrams represent only in terms of theirnon-exact properties, then we should only be able to draw non-exact inferences fromthem. Yet, many geometric inferences go beyond claims about non-exact properties.

9

Also, if we represent only the topological structure of the geometric object in thediagram, then we should say that we are investigating topology and not Euclideangeometry. This seems wrong. The use of diagrams to prove that the sum of the interiorangles of a triangle is equal to two right angles seems to be a paradigmatic case ofdiagrammatic inference. Yet this fact about the interior angles of a triangle is surelyan exact property of triangles and not a topological fact.

Another problem with this view is that transformations of a diagram that preserveits topological properties would go far beyond what we would count as being the samediagram. Indeed, imagine a typical diagram of a square that is made of elastic bands.Many of its transformations are not what we would allow to be diagrams of a square.In other words, this proposal is too permissive about what will count as a diagram ofan object.

Finally, it would be a remarkable coincidence if any particular type of mathematicalstructure were the structure of resemblance for all diagrammatic representation ingeometry. And this is true even if we focus only on geometric diagrams. There is asimple reason for this. One diagrammatic representation may have one set of conven-tions about what is to count as a line, point, etc., and another diagrammatic represen-tation may have another set of conventions. Each diagrammatic representation systemdepends on a different set of conventions. It would be quite surprising if there wereone common type of mathematical structure that underlies all such representations.

It might be that there is some one set of mathematical transformations that willcapture the structures at work in all geometric diagrammatic representation. However,if there were such a set then why wouldn’t it be merely an interesting empirical factabout the diagrammatic systems we use? And would it rule out our use of diagram-matic systems that do not fit within the specified set of mathematical transformations?

9 This point does not undermine the problem of imperfection. Indeed, it underlines it.The problem is that we can’t base inferences merely on what we can represent by perfectresemblance.

376 Dennis Potter

© 2006 The Author. Journal compilation © 2006 Editorial Board of dialectica

The experiential theory

Instead of arguing that a diagram represents by an objective resemblance that werecognize, one might argue that we experience it as resembling its objects. This is theview advocated by Christopher Peacocke (1987) and Malcolm Budd (Knowles andSkorupski 1993). This view appears to solve the problem of imperfection quite nicelybecause it does not require a strict congruence between the experienced picture andits object (Peacocke 1987, 386). The experience of resemblance can be in one respectin one case and in another respect in another case. A general account of actualresemblance is not needed on such a view.

However plausible the experiential theory is as a theory of pictorial depiction, itdoesn’t explain enough of the data concerning diagrammatic representation in geom-etry. First, it doesn’t explain the inferential datum because it doesn’t draw a distinctionbetween the properties of the inscription from which inferences can be drawn andthose from which inferences cannot be drawn. For example, any inscription drawn tobe a diagram of a triangle is likely to be experienced as an acute triangle, obtusetriangle, or right triangle.10 But if such a diagram is supposed to be a diagram of anytriangle in general and if we are trying to draw inferences about triangles in general,then the experiential account meets a specific problem. The problem is that we cannotbe sure that the experienced resemblance will not mislead us into drawing conclusionsthat only hold of a particular case of a triangle rather than the general case.

But there is a more pressing problem with the experiential account of pictorialrepresentation as applied to diagrammatic representation in geometry. This problemarises from the fact that we never visually perceive the objects of geometry. The pointis about the grammar of ‘looks like’ in the context of picturing as opposed to diagram-ming. I might say ‘the man in this picture looks like Grandpa’. And I might say that‘this looks like a triangle’. But surely we do not mean quite the same thing in thesecases. I have seen Grandpa. I have seen drawn ‘triangles’ in some loose sense ofthe word ‘triangle’. But I have not seen a Euclidean triangle. And it is the latter thatthe drawn triangle is supposed to represent. The drawn triangles are really not like theEuclidean triangle. There is nothing that it is like to see a Euclidean triangle.

Peacocke is right to point out that his account explains how we can depict a personthat we have not visually perceived (Peacocke 1987, 399). This is because we arevisually acquainted with some persons. But we don’t actually experience any Euclid-ean objects at all. So, it doesn’t make sense to say that we can experience somethingas a geometric object.

This last point is a very important point and I think it would be hard to overem-phasize it. Diagrams could not possibly be pictures in any normal sense since theycannot picture anything. They cannot look like anything. They are similar to so-called‘pictures’ of God as a man with a white beard sitting on a cloud or a throne. When

10 By this, I do not mean that there is something that it is like to experience an actualacute triangle. I only mean that it will be experienced in a way that is similar to how weexperience diagrams of acute triangles.



pressed, teachers of the religious doctrines of traditional theism will admit that theseare not really pictures of God, even if they are representations of God in some sense.

The compositional pretence theory

I believe that the correct view about diagrammatic representation is something likethe pretence theory of pictorial representation advocated by Kendall Walton (1990 and2002). Walton says that, ‘pictures are essentially props in visual games of make-believeof a certain kind’ (Walton 2002, 27). In this section I am going to argue for a particularadaptation of this theory that I call the compositional pretence theory. However, to beclear, I am not advocating it as a theory of pictorial representation. Instead, I claimthat it is a reasonable theory for diagrammatic representation. Moreover, I identifyways in which this theory entails that diagrams are more like maps than they are likepictures.

There are at least three types of pretence. First, in propositional pretence wepretend that something is the case. Children pretend that they are doctors or policeofficers. This involves feigning the belief that a proposition is true while knowing thatit is not. Second, in prop pretence we pretend that an object is something that it is not.A child might pretend that a stick is a sword. Or, in explaining a battle plan, we mightpretend that small plastic objects are tanks. Prop pretence entails propositional pre-tence since it involves pretending that something is true of a prop. Third, prop pretencemight or might not be compositional in its structure. Imagine a child pretending thata wooden structure is an airplane. Here the child also pretends that one part of it isthe fuselage, one part the wing, one part the cockpit, and so on. This is a case ofcompositional (prop) pretence. In the case of compositional pretence, parts of aphysical system are treated as props that stand in for parts of a represented system.

Given these distinctions about the nature of pretence, my view is that diagrammaticrepresentation involves compositional pretence. Prop pretence alone is not enoughbecause it does not explain how the structure of the inscription has anything to do withhow it becomes a diagram.

The notion of compositional pretence seems to fit well with the compositionalstructure of geometric objects. At a basic level we pretend that a certain type ofphysical mark is a certain type of basic geometric object(s). For example, I mightpretend that a drawn line is a geometric line. I call these basic diagrammatic propscomponents. The other components will be what we pretend to be points, curves,circles, lines, etc.

To a certain extent, the pretence involved in Euclidean diagramming is stipulative.However, this must be qualified in three ways. First, pretending that a drawn dot is apoint is not like adopting that type of mark as a name of a point. Instead, the drawndot is supposed to play a role in the diagram that its corresponding component playsin a geometric object. Second, the role it plays must be rule governed in some sense.There are certain things that one is allowed to do with it and other things that one isnot allowed to do with it. This normativity in the activity of geometric diagramming

378 Dennis Potter


entails that the component parts of a diagram can be related in certain ways but notin others. The upshot of this is that although there need be no particular type ofsimilarity between the diagram and the abstract geometric object it is supposed torepresent, there will be a similarity between the ways practitioners of geometry treatthe component parts that they take to represent the component parts of the abstractgeometric objects. To restate: the compositional theory does not require any particularstructural similarity between the diagram and its object, but only between the activitiesperformed on the objects that count as diagrams for those involved in geometric practice.

That said, it may be that structural similarities between geometric diagrams andtheir objects do in fact often exist in geometric practice. But the important element ofthe compositional theory is that they don’t really play an essential role in how thediagrams represent. They play only a contingent role. For example, perhaps it is mucheasier for me, and others with similar perceptual faculties to pretend that a certaininscription with such and such a structure is a triangle than it would be for us to pretendthe same of another inscription. If so, then this has to do with contingent features ofthe way our perceptual faculties are set up and not with the nature of diagrammaticrepresentation per se.11

Let’s see an example of how this normativity works. We may pretend that certaindrawn ‘lines’ are geometric lines (AB and CD in Figure 4), and overlaps of drawnlines are intersection points. If so, then anytime these lines overlap, our practice is thatwe must say that the overlap is an intersection point. That is, our geometric practicedoesn’t permit us to make the above stipulations and yet also claim that there is nointersection between AB and CD.

Figure 4

11 One may ask why we have similar imaginative natures with respect to what we canpretend of certain inscriptions. In other words, why are our faculties set up in such a way thateach of us is able to do/imagine the same things with (say) triangular inscriptions? It wouldseem that to explain this fact some kind of evolutionary story would be appropriate and thiswould be beyond the scope of this paper. In some way, this story would have to relate to therelated fact that our similar imaginative capacities enable us to use each other’s diagrams in theconstruction of edifices, for example.

A

B

C D



Similarly, our practice does not permit us to count AB and CD as parallel lines. Hereour choices about what count as lines and points of intersection aren’t consistent withthe inscription and our intention for it to represent two parallel lines. This is aconsistency that arises from the nature of our geometric diagrammatic practice andhow it is applied to this particular inscription combined with our particular choicesabout what to count as lines and intersections.12 This is the way we do things once wemake decisions about component parts. To put it generally: diagrammatic representa-tion in geometry is rule-governed and this normativity is part of what is essential abouthow diagrams represent. The similarity between diagrams is not their physical struc-ture, but the similar ways in which we impose rules on what we are allowed to dowith them and say about them.

Let’s examine how this theory deals with the data of diagrammatic representation.First, it seems that it explains the inferential datum. The shared practice of imposinga certain set of rules on a chosen set of component parts allows the geometer to drawsome inferences about the diagram. We can avoid worries about the problem thatdifferent diagrammers may use different inscriptions because it is the common practicethat matters and not the particular structure of the diagram itself.

It is not as easy for the compositional pretence theory to explain the limitationdatum. The problem with pretending is that it is so very flexible. Can’t we pretendthat anything is just about anything else? Here is where the compositional aspect ofthe compositional pretence theory does some work. It is my view that, in principle,any physical objects could be candidates for the component parts of a diagrammaticsystem. Perhaps, steins of beer could be points, glasses of wine could be lines, andshot glasses of bourbon could be circles. But we must be capable of imposing the rulesof our geometric practice on these objects in a way that we can do things with themthat are similar to what we do with the typical inscriptions that we use (drawn dots,drawn lines, and drawn ellipses). This is where our imaginative and/or physical skillsmay fail us with respect to some sets of potential component parts. So, according tothe compositional pretence theory, the limitation datum is as much about limitationson what we are able to do with certain objects (or what we can imagine that we mightbe able to do) as it is about the objects themselves. As a matter of contingent factconcerning the kind of perceptual beings we are, there will probably be a structuralsimilarity between the diagrams that I compose of triangles and that those that youcompose. But this is not what makes them diagrams of the same objects.

The compositional pretence theory easily explains the flexibility datum. Pretendingis a remarkably flexible activity. There are many different types of marks that we couldpretend to be geometric components. It is true that in actual geometric practice we

12 In principle, it is possible to imagine a diagrammatic practice that allows a certainkind of inconsistency. Perhaps Escher drawings would be an example. Geometric diagrammaticpractice needs to be consistent in the sense roughly sketched above because geometry needs tobe consistent in its implications. But this consistency arises from the nature of the rulesgoverning how we use the diagrammatic components.

380 Dennis Potter


typically employ one particular system of marks. But, as already mentioned, this isnot the only available system.

The compositional pretence theory explains the consistency datum as well. Indeed,the compositional nature of diagrammatic representation coupled with the rule-governed practices applied to the component parts entail that any diagram is alwaysembedded in a tacit system of diagrammatic representation (or better: a certain dia-grammatic practice). Once we have decided what we are going to use for points, lines,curves, and circles, the rest of the system of diagrammatic representation pretty muchfollows.

Where the compositional pretence theory really differs from the resemblancetheory is in its treatment of the problem of imperfection. The most important aspectof this solution is that the compositional pretence theory allows us to pretend that afigure has properties that it doesn’t really have. So, for example, although we cannotreliably draw straight lines, we can pretend that the ‘line’ we draw is straight. Anyimperfections in the diagram’s parts – qua representatives of the corresponding partsof the represented object(s) – are thereby ignored as irrelevant to the diagram. Pre-tending that a figure has properties that it does not have allows us to perceive relation-ships between those properties. So, for example, we may take the following to be aline with three points on it (we take the ‘dots’ to be the three points).

Figure 5

The line may not be precisely straight. But if we pretend that it is a straight lineand if we pretend that the dots are points on the line, then we can see that point B isbetween A and C.

Diagrams, pictures and maps

In certain ways, geometric diagramming is like map drawing. Both are activities thatare meant to fit into a certain practice with certain goals. Let’s see how this workswith maps. Suppose that there are three traffic lights on the road that runs between thefreeway exit and my house. If I draw a map with dots to represent traffic lights, a‘square’ for my house, and a ‘line’ for the freeway, then it may very well be asatisfactory map even if there are three dots that are only roughly between the line andthe square. It depends on whether you understand how I intended you to use it. Intrying to use the map to get to my house you deal with the components in wayscoordinated with the physical world in order to get some idea about where you arecurrently located. After you pass the second stoplight you might say, ‘we’re here’,pointing to a place on the map. By treating these items in this way one is able to findmy house, if the map is successful and if you understand how I intended it to be used,

A B C



i.e. the mapping practice that I have imposed on its configuration. This relationshipof ‘between-ness’ in the map may or may not be all that similar to the spatialrelationship between the freeway, traffic lights and my house. But the map is successfuljust in case you understand the rules that I intend you to follow in using it.

In fact, it is obvious that diagrammatic practice applied to theoretical geometry(as practiced mostly in educational facilities) is closely related to a practice that isused much more often by those in the construction business. A construction worker’sdiagramming is lose and imprecise (by comparison), but is embedded in a practicethat allows her or him to produce edifices. This practice is somewhat like mapping,except it is like mapping out things that don’t yet exist. Clearly, the above theory ofdiagrammatic representation has the added benefit of yielding a theory of appliedgeometry – i.e. if we take diagrammatic reasoning in theoretical geometry to be aserious part of that mathematical practice.

Arguing that the compositional pretence theory is a good explanation of howdiagrams are used in Euclidean geometry is not to say that it is the best theory ofpictorial representation. Indeed, it may be that in certain respects diagrams are quiteunlike pictures. For one, diagrams can represent abstract objects or mere idealizations,while pictures must represent something that can be visually perceived. Similarly, adiagram need not ‘look like’ the object it represents while a picture must look like itsobject in some respect. Finally, diagrammatic representation in geometry is systematicin the sense that the representational components are the same for different diagramsof the system. Pictorial representation need not be systematic.

These facts about the differences between geometric diagrams and pictures can besatisfactorily explained by the compositional pretence theory. Although a real geomet-ric triangle has no appearance associated with it, we can certainly pretend that aphysical inscription composed in such and such a way is a triangle. So, on thecompositional pretence theory, diagrams can represent abstract objects and they neednot look like those objects. Moreover, the compositional nature of diagrams accountfor how diagrammatic representation is systematic. It also accounts for how we candraw inferences in a systematic way on the basis of geometric diagrams.

If the compositional pretence theory of Euclidean diagrams is correct, then theyare more like maps than they are like pictures. Maps allow us to draw inferences aboutthe location of certain objects in a particular locale. Their use involves acts of pretencesuch as our saying, ‘right now, we are here’ while pointing to a spot on the map.Moreover, we don’t expect maps to look like the objects they represent (although theycan). Maps may be pictures (as when they are based on satellite photos); but usuallythey are not. There is one important way in which geometric diagrams are not likemaps. In the case of maps, the represented object is concrete or particular. A city mapdoes not represent any city, but rather a particular one. But geometric diagramsrepresent general classes of objects, rather than particular cases.

382 Dennis Potter


Conclusion

In this paper I have tried to say something about the nature of diagrammatic represen-tation in geometry. I have argued that diagrammatic representation needs to be under-stood in a way that distinguishes it from pictorial representation. Accordingly, I haverejected the resemblance and the experiential theories as explanations of diagrammaticrepresentation, even if they might be adequate for pictorial representation. Instead,following Walton, I have argued that the main intentional component in diagrammaticrepresentation is pretence. However, I have also argued that this pretence must becompositional in the sense that we not only pretend that the diagram is the objectrepresented but that some of its parts are the parts of the object represented. Thiscompositional component involves a normativity that arises from rules about how thecomponents can be used. Given this, the compositional nature of diagrammatic rep-resentation indicates some directions we might go in exploring diagrammatic infer-ence. Perhaps it will also help to adjudicate the question of whether diagrammaticinference in geometry can count as proof. But that’s a topic for another essay.

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