Mathematical Discourse: Language, Symbolism And Visual Images

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Mathematical Discourse Language, Symbolism and Visual Images

Kay L. O'Halloran

continuum LONDON

•

NEW YORK

Continuum The Tower Building, 11 York Road, London SE1 7NX

15 East 26th Street, New York, NY 10010

© Kay L. O'Halloran 2005 All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0-8264-6857-8 (hardback) Library of Congress Cataloging-in-Publication Data A catalogue record for this book is available from the Library of Congress Typeset by RefineCatch Ltd, Bungay, Suffolk Printed and bound in Great Britain by Cromwell Press Ltd, Trowbridge, Wilts

Contents Acknowledgements

viii

Copyright Permission Acknowledgements

ix

1

Mathematics as a Multisemiotic Discourse 1.1 The Creation of Order 1.2 Halliday's Social Semiotic Approach 1.3 Mathematics as Multisemiotic 1.4 Implications of a Multisemiotic View 1.5 Tracing the Semiotics of Mathematics 1.6 Systemic Functional Research in Multimodality

1 1 6 10 13 17 19

2

Evolution of the Semiotics of Mathematics 2.1 Historical Development of Mathematical Discourse 2.2 Early Printed Mathematics Books 2.3 Mathematics in the Early Renaissance 2.4 Beginnings of Modern Mathematics: Descartes and Newton 2.5 Descartes' Philosophy and Semiotic Representations 2.6 A New World Order

22 22 24 33 38 46 57

3

Systemic Functional Linguistics (SFL) and Mathematical Language 3.1 The Systemic Functional Model of Language 3.2 Interpersonal Meaning in Mathematics 3.3 Mathematics and the Language of Experience 3.4 The Construction of Logical Meaning 3.5 The Textual Organization of Language 3.6 Grammatical Metaphor and Mathematical Language 3.7 Language, Context and Ideology

60 60 67 75 78 81 83 88

4

The Grammar of Mathematical Symbolism 4.1 Mathematical Symbolism 4.2 Language-Based Approach to Mathematical Symbolism 4.3 SF Framework for Mathematical Symbolism 4.4 Contraction and Expansion of Experiential Meaning 4.5 Contraction of Interpersonal Meaning 4.6 A Resource for Logical Reasoning 4.7 Specification of Textual Meaning 4.8 Discourse, Grammar and Display 4.9 Concluding Comments

94 94 96 97 103 114 118 121 125 128

VI 5

CONTENTS The Grammar of Mathematical Visual Images

129

5.1 5.2 5.3 5.4 5.5 5.6

129 133 139 142 145

The Role of Visualization in Mathematics SF Framework for Mathematical Visual Images Interpersonally Orientating the Viewer Visual Construction of Experiential Meaning Reasoning through Mathematical Visual Images Compositional Meaning and Conventionalized Styles of Organization 5.7 Computer Graphics and the New Image of Mathematics 6

7

146 148

Intersemiosis: Meaning Across Language, Visual Images and Symbolism

158

6.1 6.2 6.3 6.4 6.5 6.6

158 163 171 177 179 184

The Semantic Circuit in Mathematics Intersemiosis: Mechanisms, Systems and Semantics Analysing Intersemiosis in Mathematical Texts Intersemiotic Re-Contexualization in Newton's Writings Semiotic Metaphor and Metaphorical Expansions of Meaning Reconceptualizing Grammatical Metaphor

Mathematical Constructions of Reality

189

7.1 Multisemiotic Analysis of a Contemporary Mathematics Problem 7.2 Educational Implications of a Multisemiotic Approach to Mathematics 7.3 Pedagogical Discourse in Mathematics Classrooms 7.4 The Nature and Use of Mathematical Constructions

189 199 205 208

References

211

Index

223

In memory of my father, Jim O'Halloran. For my brother Greg and his family.

Acknowledgements

This study of mathematical discourse is based on Michael Halliday's systemic functional model of language and Jim Martin's extensive contributions to systemic theory. Michael O'Toole's application of systemic functional theory to displayed art provides the inspiration for the models for mathematical symbolism and visual image presented here. Jay Lemke pioneered the application of systemic functional theory to science and mathematics as multisemiotic discourses. This work would not be possible without these founding contributions. My special thanks to the director and librarians from the John Rylands University Library of Manchester (JRULM) for making so readily available the mathematics manuscripts in the Mathematical Printed Collection. I thank Linda Thompson (Director of the Language and Literacy Studies Research Group, Faculty of Education, University of Manchester) for supporting my visit to JRULM. My special thanks to Philip J. Davis (Emeritus Professor, Applied Mathematics Division, Brown University) for his interest in this project. Our lively correspondence has contributed to the contents of this book. I thank Michael O'Toole and Frances Christie for their friendship and support, and I thank my past and present friends and colleagues at the National University of Singapore - most notably Joe Foley, Chris Stroud, Linda Thompson, Desmond Allison, Ed McDonald, Umberto Ansaldo and Lisa Lim.

Copyright Permission Acknowledgements

The author is grateful to the following organizations for the right to reproduce the images which appear in this book. Every effort has been made to contact copyright holders of material produced in this book. The publishers apologize for any omissions and will be pleased to rectify them at the earliest opportunity. Chapter 1 Plate 1.1(1)

Plate 1.3(1)

Photographs from Beevor (2002: Chapter 24) The photographs are reprinted with kind permission from: Photograph 43: Bildarchiv PreuBischer Kulturbesitz, Berlin Photograph 44: Ullstein Bild, Berlin Photograph 45: Jiirgen Stumpff/Bildarchiv PreuBischer Kulturbesitz, Berlin Language, Visual Images and Symbolism (Kockelkoren et al, 2003: 173) Reprinted with kind permission from Elsevier

Chapter 2 Plate 2.2(1) The Treviso Arithmetic (reproduced from Swetz, 1987: 140) Reprinted by permission from Open Court Publishing Company, a division of Carus Publishing Company, Peru, IL from Capitalism and Arithmetic by F. Swetz, © 1987 by Open Court Publishing Company The following have been reprinted by courtesy of the director and librarian, the John Rylands University Library of Manchester: Plate 2.2(2) Plate 2.2(3)

The Hindu-Arabic system versus counters and lines (Reisch, 1535: 267) Printing counter and line calculations (Reisch, 1535: 326)

X

COPYRIGHT PERMISSION ACKNOWLEDGEMENTS

Plate 2.2(4a) Euclid's Elements: Venice 1482, Erhard Ratdolt (ThomasStanford, 1926: Illustration II) Plate 2.2(4b) Euclid's Elements: Venice 1505, J. Tacuinus (ThomasStanford, 1926: Illustration IV) Plate 2.2(6) QuadraturaParaboles (Archimedes, 1615: 437) Plate 2.3(2) Hitting a target (Tartaglia, 1546: 7) Plate 2.3(3) Arithmetic calculations to hit a target (Tartaglia, 1546: 106) Plate 2.3(4) Positioning a target (Galileo, 1638: 67) Plate 2.4(la) Removing the human body (Descartes, 1682: 111) Plate 2.4(2a) Movement in space and time: the stone (Descartes, 1682: 217) Plate 2.4(2b) Movement in space and time: the model (Descartes, 1682:

217) Plate 2.4(4a) Plate 2.4(4b) Plate 2.4(5a) Plate 2.4(5b) Plate 2.5(1)

Context, circles and lines (Descartes, 1682: 226) Circles and lines (Descartes, 1682: 228) Descartes' semiotic compass (1683: 54) (Book Two) Drawing the curves (Descartes, 1683: 20) (Book Two) Illustration from Newton's (1729) The Mathematical Principles of Natural Philosophy

My thanks to the Bodleian Library, Oxford for access to the microfilm of the following manuscript held by the British Library, London: Plate 2.4(8a) Newton's (1736: 80-81) Method of Fluxions and Infinite Series Plate 2.4(8b) Newton's (1736: 100) Method of Fluxions and Infinite Series The following have been reprinted by courtesy of Dover Publications: Plate Plate Plate Plate Plate

2.2(5) 2.4(lb) 2.4(6a) 2.4(6b) 2.4(7)

Translation of Euclid (reproduced from Euclid, 1956: 283) Removing the human body: Newton (1952: 9) Descartes' description of curves (1954: 234) Descartes' use of symbolism (1954: 186) Newton's algebraic notes on Euclid (reproduced from Cajori, 1993: 209) Latin edition (1655) of Barrow's Euclid (Taken from Isaac Newton: A Memorial Volume [ed. WJ. Greenstreet: London, 1927], p. 168)

Chapter 4 Plate 4.3(1)

Mathematical Symbolic Text (Stewart, 1999: 139) From Calculus: Combined Single and Multivariable 4th edition by Stewart. © 1999. Reprinted with permission of Brooks/ Cole, a division of Thompson Learning: www.thompsonrights.com. Fax: 800 730-2215


xi

The following are reprinted with permission from Elsevier: Plate 4.5(1) Plate 4.7(1)

Mathematical Symbolic Text (Wei and Winter, 2003: 159) Textual Organization of Mathematical Symbolism (Clerc, 2003: 117)

Chapter 5 Plate 5.2(1)

Plate 5.7(1)

Interpretation of the Derivative as the Slope of a Tangent (Stewart, 1999: 130) From Calculus: Combined Single and Multivariable 4th edition by Stewart. © 1999. Reprinted with permission of Brooks/ Cole, a division of Thompson Learning: www.thompsonrights.com. Fax: 800 730-2215 Evolving Images of Computer Graphics (a) Figure 5 Stills from a computer-made movie: wrapping a rectangle to form a torus (Courtesy T. Banchoff and C. M. Strauss) (Davis, 1974: 126) Reprinted by kind permission of T. Banchoff and C. M. Strauss through Philip J. Davis (Emeritus Professor, Applied Mathematics Division, Brown University Providence, RI, USA) (b) MATLAB graphics, circa 1985 (courtesy of Philip J. Davis) Reprinted with kind permission Philip J. Davis (Emeritus Professor, Applied Mathematics Division, Brown University Providence, RI, USA) (d) Graphical and Diagrammatic Display of Patterns (Berge et al., 2003: 194) Reprinted with kind permission from Elsevier

Chapter 6 Plate 6.31

Plate 6.32

Newton's (1736: 46) Procedure for Drawing Tangents Reprinted from microfilm in the Bodleian Library, Oxford. Reproduced with permission from the British Library (London) which holds the original manuscript. The Derivate as the Instantaneous Rate of Change (Stewart, 1999: 132) From Calculus: Combined Single and Multivariable 4th edition by Stewart. © 1999. Reprinted with permission of Brooks/ Cole, a division of Thompson Learning: www.thompsonrights.com. Fax: 800 730-2215

xii


Chapter 7 Plate 7.1 (1)

Mathematics Example 2.24 (Burgmeier et al, 1990: 76-77) From Burgmeier, J. W., Boisen, M. B. and Larsen, M. D. (1990) Brief Calculus with Applications. New York: McGraw-Hill. © 1990. Reprinted with kind permission from The McGraw-Hill Publishing Company, New York

1 Mathematics as a Multisemiotic Discourse

1.1 The Creation of Order Success is right. What does not succeed is wrong. It was, for example, wrong to persecute the Jews before the war since that set the Anglo-Americans against Germany. It would have been right to postpone the anti-Jewish campaign and begin it after Germany had won the war. It was wrong to bomb England in 1940. If they had refrained, Great Britain, so they believe, would have joined Hitler in the war against Russia. It was wrong to treat Russian and Polish [prisoners of war] like cattle since now they will treat Germans in the same way. It was wrong to declare war against the USA and Russia because they were together stronger than Germany.

In this extract from Berlin: The Downfall 1945, Beevor (2002:429) summarizes the views of over three hundred pro-Nazi generals after Germany's defeat in the Second World War, based on a report of interviews by the Supreme Headquarters Allied Expeditionary Force in Europe (SHAEF). The German generals are seen to possess a view of events; one they envisaged would have worked towards victory rather than defeat. Their guiding principle, as expressed by Beevor (2002: 429), is 'Success is right. What does not succeed is wrong.' Many millions participated in the enactment of those views, and the familiar question arises as to how this could be possible. How could so many people be persuaded to take part in the events which unfolded during the course of the Second World War? There have been a variety of responses to this question. Goldhagen (1996), for instance, suggests that most of the ordinary Germans involved in the holocaust were 'willing executioners' who actually believed in the events that took place. No doubt a variety of means were used incrementally over a long period of time in order to mobilize the population in the war effort. In the past century, such massive mobilizations have not been confined to Germany. Weitz (2003), for example, documents the unprecedented programmes of genocide which have taken place in the twentieth century, including Stalin's Soviet Union, Cambodia under the Khmer Rouge and the former Yugoslavia. In these cases and many others, significant portions of the population take part in the war effort. But how can so many people be convinced of the necessity of such programmes, the impact of which lasts for generations?

2

MATHEMATICAL DISCOURSE

In attempting to answer such a question, it is worthwhile to consider a simple reformulation of the German generals' guiding principles 'Success is right. What does not succeed is wrong' (Beevor, 2002: 429). That is, if the phrase for the Nazi party is inserted, the statement becomes 'Success [for the Nazi party] is right. What does not succeed [for the Nazi party] is wrong.' Such a reformulation introduces in unequivocal terms the basis upon which the guiding principles are constructed. The simple inclusion of the beneficiary 'the Nazi party' makes clear the premise underlying the linguistic statement, and the specific interests which are being served. Such an inclusion also provides room for argumentation and negation, whereas the finality accompanying the original cliched statement 'Success is right' is much more difficult to counteract. In a similar manner, the import of linguistic choices may be seen in George W. Bush's statement to the world after 11 September 2001 attacks on the United States: 'Either you are with us, or you are with the terrorists' (CNN.com/US 20 September 2001). Expressed in simple terms of a relational set of circumstances, the dichotomy is based on pro-American interests ('with us') versus anti-American interests ('against us'). Such a simple division of the world into two opposing sets of relations leaves few options for a negotiated peace settlement along other possible lines of interest. Language functions in this way to structure the world largely in terms of categories, the nature of which depends upon the choices which are made. The value of using language and other systems of meaning to create a world view conducive to the war effort was well recognized in Nazi Germany. These strategies included the use of the media for news reports and documentaries (involving language, visual images and music), political speeches and rallies (for example, language, visual images, embodied action, music, and architectural features of the platform and seating arrangements), and particular styles of dress and the distinctive salute of the Nazi party (for example, the uniforms, insignias, actions and gestures). These strategies have direct parallels in existence today, where choices from the different resources combine to create particular meanings to the exclusion of others. However, the contexts which give rise to the ordering of reality are not confined to those which are specifically designed for mass consumption in the form of 'propaganda' programmes. Order is maintained, negotiated and challenged in every situation which involves choices from language, visual images, gesture, styles of dress and so forth. Page (2001: 10) comments: 'There is a privilege in being raised in a time of peace. A luxury that your life is not under immediate threat. War becomes something labelled as heroic, often patriotic, nationalistic. There is a cause, it is just and right, and it somehow excuses all the pain and all the loss.' The use of language and other sign systems for the structuring of thought and reality in the ways described by Page is the subject of this study. This approach is not intended to downplay strategies of physical and mental coercion and abuse. However, violence commences somewhere, and in many cases, for ordinary citizens at least, the starting point is the ordering

M A T H E M A T I C S AS A M U L T I S E M I O T I C D I S C O U R S E

3

of reality along certain lines through semiosis; that is, acts of meaning through choices from language and other sign systems. The major aim of this study is to introduce a theory and approach for examining the nature and impact of semiosis in contexts which span the supposedly inane to the discourses of immense influence, which include the subject matter of this investigation; namely, mathematics and science. War is chosen as the topic to introduce this approach. The role of language for structuring thought and reality is well recognized today within a wide range of disciplines which include sociolinguistics, critical discourse theory, communication studies, psychology and sociology (for example, Berger and Luckmann, 1991; Bourdieu, 1991; Fairclough, 1989; Gumperz, 1982; Halliday, 1978; Herman and Chomsky, 1988; Vygotsky, 1986). In addition, the functions of visual images are increasingly taken into account (for example, Barthes, 1972; Lynch and Woolgar, 1990; Mirzoeff, 1998; van Leeuwen andjewitt, 2001). This is especially important in the electronic age where the ease with which pictorial representations may be reproduced is expanding. Beevor (2002), for example, includes visual images in the form of black and white photographs and maps to depict the advance of the Red Army and the final collapse of the Third Reich. Berlin: The Downfall 1945 is a text or discourse constructed through choices from the English language, photographs and maps. These choices work together to create Beevor's account of the horror of the final months of the Second World War in Germany. In what follows, the types of meanings afforded by Beevor's (2002) photographs are investigated and compared to meanings which are made using language. Photographs 43-45 displayed in Plate 1.1(1) appear in Chapter 24 in Beevor (2002: 354—369). These photographs appear among a group of inserted photographs which are numbered 30-49. As seen in Plate 1.1(1), Photograph 43 is a picture of a German teenage conscript at the end of the war, Photograph 44 shows a Russian female medical assistant attending to a wounded Russian soldier, and the official signing of the final surrender by General Stumpff, Field Marshal Keitel and Admiral von Friedburg in May 1945 is shown in Photograph 45. In Beevor (2002), these pictures are preceded by photographs of Russians engaged in street fighting in Berlin, scenes outside the Reich Chancellery, convoys of Russian-controlled armed forces, German soldiers surrendering in Berlin, Russian soldiers washing and civilians cooking in the streets of Berlin, victory celebrations between delegates from the Red Army and the US Army, and German civilians escaping across the Elbe River to American territory. Immediately following Plate 1.1(1), there are further photographs of soldiers in the streets of Berlin, the Russian victory parade, and a full-page photograph of Red Army officials visiting the battleground inside the Reichstag. The photographs displayed in Plate 1.1(1) have contextual meaning within this sequence of photographs. Beevor's (2002) linguistic account of the fall of Berlin similarly unfolds as a staged text consisting of sentences, paragraphs, pages

4

MATHEMATICAL DISCOURSE 44

43

45

Plate 1.1(1) Photographs from Beevor (2002: Chapter 24) and chapters which have contextual meaning within the sequence of the narrative. However, there are differences in the types of meaning afforded by Beevor's linguistic and photographic account of the fall of Berlin. These


5

differences relate to the meaning potential of language and visual images. This point is developed below. From existing photographs of the fighting and aftermath in Berlin in 1945, a selection of photographs has been chosen to be included in Beevor (2002). In turn, each photograph in the sequence represents a set of choices made by the photographer, which, in the case of war, most likely happen more by chance rather than design. The photographer captures an instance of time according to the camera angle, the camera distance, the perspective and light conditions, for example. Certain scenes are frozen within the frame, and within those frames human figures are engaged in some form of action in a setting. Further to this, the photographs are developed and reproduced under certain conditions which include choices in terms of paper quality, darkroom techniques, and the possibility for various forms of editing, including cropping and erasure. Putting aside the materiality of the medium and the production process, following O'Toole's (1994) framework for the analysis of paintings, each photograph represents choices at the rank of the whole frame or the Work (in terms of the setting, actions and circumstance), the Episodes in each frame (the activities which are captured), the Figures (the individual people and other objects) and their Members (in terms of body parts and parts of the objects). The impact of these choices in the photographs displayed in Plate 1.1(1) merit close attention. The settings, physical actions, gestures, facial expressions and the nature of the averted gazes of the human figures in the photographs are juxtaposed in what is a grotesque opposition between the devastation faced by those involved in the fighting (Photographs 43-44) and the well-fed and well-attired defiance of those taking part in the official surrender (Photograph 45). This opposition is marked at each rank of the Work, Episode, Figure and Member. For example, the contrast between the physical and emotional state of the soldiers, the medical attendant and the German generals becomes evident in a glance. The quality, style and condition of their respective uniforms at the rank of Figure and Member are similarly diametrically opposed. Compositionally, even the grainy quality of the street scene where the Russian medical assistant attends to the injuries suffered by a soldier (Photograph 44) is placed in stark opposition to the smooth textual quality of the photograph of the official German surrender (Photograph 45). The situational contexts, actions, experiences and the emotional and physical states of the participants in the fall of Berlin according to circumstance, nationality, age, gender and position are thus constructed by the photographs. Even if Beevor had the space to describe these dimensions, the meanings of these black and white photographs are impossible to exactly reproduce in narrative form. A linguistic description cannot make the same meanings as Photographs 43-45. The scenes, the interplay of Episodes, the actions and events, the mood of the Figures realized through their embodied actions and appearance cannot be captured using words.

6

MATHEMATICAL D I S C O U R S E

In the same manner, 'Success [for the Nazi party] is right. What does not succeed [for the Nazi party] is wrong' cannot be captured pictorially. Different resources such as language and visual images have different potentials to create meaning. In simplest terms, language tends to order the world in terms of categorical-type distinctions, while visual images such as photographs create order in a manner which to varying degrees accords with our dynamic perceptual experience of the world. The two types of meanings afforded by language and visual images combine in Beevor's account of the fall of Berlin and the collapse of the Third Reich. The semantic realm explored in this study is not war, rather it is the world offered by mathematics, the discourse which underlies the scientific view of the world. This world came into being largely through the development and refinement of a new sign system, namely mathematical symbolism, which was designed to function in co-operation with language and specialized forms of visual images. The mathematical and the scientific ways of ordering the world permeate our everyday existence, and thus the aim of this study is to understand the nature and the implications of such a view. Before moving to the field of mathematics, Michael Halliday's social-semiotic approach which informs this study is introduced. 1.2 Halliday's Social Semiotic Approach

We impose order on the world, and that order is expressed semiotically through choices from a variety of sign systems. These semiotic resources, or sign systems, include language, paintings and other forms of visual images, music, embodied systems of meaning such as gesture, action and stance, and three-dimensional man-made items and objects such as clothes, sculptures and buildings. A culture may be understood as typical configurations of choices from a variety of semiotic resources. The lecture, the pop song, the political speech, the news report and the textbook are to a large extent predictable configurations of semiotic choices. In a general sense, this understanding of semiotics pertains to 'the specificity of human semiosis' (Cobley, 2001: 260) where 'Semiosis is the name given to the action of signs. Semiotics might therefore be understood as the study of semiosis or even as a "metasemiosis", producing "signs about signs" '(Cobley, 2001: 259). As Cobley (2001: 259) claims, 'Behind this simple definition [of semiotics] lies a universe of complexity.' Noth (1990) describes the diversity in theoretical and applied approaches to study of semiotics and Chandler (2002: 207) sees semiotics as 'a relatively loosely defined critical practice rather than a unified, fullyfledged analytical method or theory'. There are many schools and branches of theoretical and applied semiotics, with various definitions and meanings. Noth (1990), for instance, categorizes semiotics as being concerned with the study of language and language-based codes, text (for example, rhetoric and stylistics, poetry, theatre and drama, narrative, myth, ideology and theology), non-verbal communication, aesthetics and visual


7

communication. Noth (1990: 5-6) provides alternative subdivisions which include the semiotics of culture, multimedia communication, popular culture, anthropology, ethnosemiotics, and other topics such as psychosemiotics, socio-semiotics and semiotic sociology, together with the semiotics of disciplines such as mathematics, psychiatry, history and so forth. Michael Halliday's (1978, 1994, 2004) social-semiotic theory of language known as Systemic Functional Linguistics (SFL) is located within the theoretical realm of what Noth (1990: 6) terms 'socio-semiotics'. Halliday is concerned with the social interpretation of the meaning of language, and this view is extended to include other semiotic resources such as the maps and photographs found in Beevor (2002) and the mathematical symbolism and diagrams found in the discourse of mathematics. While the basic tenets of the Hallidayan approach to language are introduced below, more comprehensive accounts may be found elsewhere (for example, Bloor and Bloor, 1995; Eggins, 1994; Martin, 1992; Martin and Rose, 2003; Thompson, 1996). Halliday (1978, 1994) sees language as a tool, where the means through which language is used to achieve the desired results are located within the grammar. The grammar is theorized according to the functions language is required to serve. Halliday (1994) identifies the 'metafunctions of language' as (i) the experiential - the construction of our experience of the world, (ii) the logical - the construction of logical relations in that world, (iii) the interpersonal - the enactment of social relations, and (iv) the textual - the means for organizing the message. The grammatical systems through which these four metafunctions of language are realized are described in Chapter 3. From the Hallidayan perspective, meaning is thus made through choices from the metafunctionally based grammatical systems. The meaning of a choice (the sign or the syntagm) is understood in relation to the other possible choices within the system networks (the paradigmatic options). Halliday uses the term 'social semiotic' to explain that the meanings of the signs (the semiotic choices) depend on the context of use (the social). The meanings arising from choices from the system networks are negotiated within the social and cultural context in which those choices are made. For example, a linguistic statement such as 'Success is right' does not exist as an abstract independent entity. Rather, the statement means within a context of use, in this case in Beevor's (2002) account of the fall of Berlin. In the same fashion, contexts are established semiotically. For example, the fall of Berlin is constructed by Beevor (2002) and other historians through choices from the semiotic resources of language, maps and photographs. Similarly, the academic lecture is a typical configuration of semiotic choices from the resources of language, visual images, dress, gesture, objects, architecture, seating, lighting and so forth. The configuration of the academic lecture is recognizable by members of a culture, even though the form varies according to discipline and institution.

8


In order to communicate, members of a culture, or groups within that culture, must possess some sense of shared contextual meaning. Being part of a culture means learning, using and experimenting with the meaning potential of the semiotic systems to create, maintain and negotiate the reality which is socially constructed. Semiotic activity is also used for acts of resistance, which may materialize, for example, in the form of email messages or websites where 'standard' linguistic practices are subverted from the point of view of grammar, lexical choice, text colour and graphics. The dynamic nature of the electronic medium is such that the distinction between the spoken and written modes becomes increasingly blurred with the variations in genre configurations, language choices and graphical representations. However, these new practices eventually become in themselves standardized in much the same way that video texts in the music industry become predictable. The resistance which some discourses initially appear to offer (for example, in the music and film industry, sport and the internet) typically become absorbed into mainstream culture, often in the form of re-packaged commercial products. The contextual values attached to different choices or combinations of choices from semiotic resources are socially and culturally determined. Members of a culture recognize and maintain or resist those values. Companies such as McDonalds, Nike and Coca Cola, for example, invest large amounts of money in advertising to ensure that their brands and accompanying icons maintain 'the right' social value among the other products on offer. In this way they seek to create and maintain a market for groups of consumers. In one study, Cheong (2004) found that apart from the interpersonally salient component of an advertisement designed to attract the attention of the reader (in many cases a visual image), the only obligatory item in a print advertisement is the company logo. Presumably if the logo was missing, the intertextual relations with other texts in the advertising campaign would ensure that the brand is easily identifiable. Advertising as such means creating an image so that the product or service is viewed as desirable by groups of members of a community. Buying the product thus means acquiring the social and cultural connotative value of that product (Barthes, 1972, 1974). Human life is negotiated through semiotic exchange within the realms of situational and cultural contexts. Certain combinations of selections function more prominently to structure reality to the exclusion of others. Studies in Systemic Functional Linguistics (SFL) attempt to document and account for the typical linguistic patterns in different types of social interaction or genres; for example, casual conversation (Eggins and Slade, 1997), service encounters (Ventola, 1987), pedagogical discourse (Christie, 1999; Christie and Martin, 1997; O'Halloran, 2000, forthcoming b; Unsworth, 2000) and scientific writing (Martin, 1993b; Martin and Veel, 1998). Other studies of language look at typical patterns along contextual parameters such as gender (for example, Tannen, 1995) and sexuality (for example, Cameron and Kulick, 2003). Forensic linguistics, on the other


9

hand, is concerned with identifying typical language patterns of the individual (for example, Coulthard, 1993). Bourdieu's (1991) notion of symbolic and cultural capital of the 'habitus', which is the set of acquired dispositions of an individual or group of individuals, may be conceptualized as semiotic capital; that is, the ability to construct, interpret and reconstruct the world in contextually specific ways. However, the ability to make appropriate meanings in a range of contexts through the use of semiotic resources is unevenly distributed across sections of any community or culture. The reason for this unequal distribution of semiotic capital is related to the educational, economic, social and cultural background of individuals and groups within any community. For example, Bernstein (1977,1990) identifies the disadvantages students from lower social class backgrounds face in participating in the linguistic practices rewarded in educational institutions. In a sense, being 'educated' means being able to participate in certain types of 'valued' semiotic exchange; for instance, the discourses of medicine, science, business, law, music and art. Certain groups within a society, typically those with wealth and connections, are relatively well placed within the semantic domains which are rewarded (usually by members of that same group). Other groups to varying degrees are marginalized. Increasingly the market-driven practices adopted in schools and universities, such as making entrance dependent on money rather than merit, function to reinforce these divisions of inclusion and exclusion. Participation in everyday discourse includes semiotic exchange in terms of performative action; that is, selections in the form of gesture, stance, proxemics and dress. Whether delivering a conference paper or giving a political speech, the speaker needs to talk the talk (using appropriate linguistic and phonological choices), walk the walk (in terms of non-verbal behaviour and action), and increasingly look the look (in terms of clothing, hairstyle, make-up, body size, body shape, height, and skin and hair colour, for example) according to the parameters established as desirable in that culture. More generally men and women are urged to identify their 'unique selling point' (USP), be it the talk, the walk or the look. Increasingly acts of meaning inscribed on and through the human body (for example, physical appearance which is increasingly the product of medical procedures and other forms of practices involving drugs, chemicals and so forth) often outweigh the import of other acts of meaning (the talk). In the electronic medium, the performative action and physical creation of identity becomes a textual act. One is no longer constrained by semiosis emanating from the body and the immediate material context. Multiple identities can be established according to the limits of the electronic medium and platforms that are offered, and the user's ability to make use of different semiotic resources, including language, visual images, music and so forth. Semiotic capital comes into play in new ways through computer technology. The social-semiotic construction of reality (Berger and Luckmann, 1991)

10


is determined as much by what is included as to what is excluded. As seen in the example of the German generals' 'guiding principles', (i) there are limits to what options are selected, and (ii) there are limits to what can be selected from the existing systems. In the first case, semiotic selections function as meaning through choice, and so some options (for example, 'success') are chosen to the exclusion of others (for example, 'justice' or 'freedom'), while other possible options are left out (for example, 'for the Nazi party'). In the second case, although systems are dynamic and constantly changing with each contextual instantiation, there are nonetheless at any one time a limited number of options available. We are contained within particular semantic domains according to the limitations of the systems which are available. These systems, however, constantly evolve so that meaning making is a dynamic practice in which change is possible. Realms of meaning do not exist until they become semiotic choices; for example, the concepts of women's rights, gender and Freud's (for example, 1952, 1954) concept of psychoanalysis are comparatively new linguistic choices. Although perhaps pre-existing as disparate practices, the introduction of these options in language led to radically new ways of conceptualizing women, women's roles and what has become the inner psychosexual self. Similarly, the scientific revolution in the seventeenth century introduced radically new ways of conceptualizing the physical world. The basis for this scientific re-ordering of reality was the development of mathematics which offered new resources in the form of the symbolism and visual display. These semiotic resources combine in significant ways with language to create a new world order. The nature of that order is investigated in this study. 1.3 Mathematics as Multisemiotic

Mathematics and science are considered as 'multisemiotic' constructions; that is, discourses formed through choices from the functional sign systems of language, mathematical symbolism and visual display. These discourses are commonly constituted as written texts, although mathematical and scientific practices are not confined to these forms of semiotic activity. There are many different 'multimodal' genres constituting mathematical and scientific practices; for example, lectures, conference papers, software programs and laboratory investigations. In addition to the written mode, these types of semiotic activity involve spoken discourse, physical action and gesture in environments, which include digital media and day-to-day three-dimensional material reality. The major line of enquiry in this study, however, is directed towards multisemiosis in printed discourses of mathematics, largely because modern mathematical symbolism is a semiotic resource which developed in written format. In order to develop theoretical frameworks for mathematical symbolism and visual display, the print medium has been chosen for investigation. In addition, the effects of computer technology on the nature of mathematical discourse are also


11

considered in this study. With the exception of the systemic functional (SF) approach to mathematics (Lemke, 1998b; O'Halloran, 1996, 1999b, 2000, 2003a, 2003b, forthcoming a; Veel, 1999), few studies exist in the field of the semiotics of mathematics (for example, Anderson et al, 2003; Rotman, 1987,1988,1993,2000). Mathematical discourse involves language, mathematical symbolism and visual images as displayed in Plate 1.3(1), a page reproduced from Physica D, a journal for research in dynamical systems theory. Plate 1.3(1) contains equations (11), (12) and (13), which are mathematical symbolic statements spatially separated from the main body of the linguistic text. Symbolic statements and elements are also embedded within the linguistic text. For example, symbolic elements function as elements within the linguistic statements in the text located between equations (11) and (13). In addition, there are visual images in the form of mathematical graphs in the three panels labelled Fig. 2 in Plate 1.3(1). Mathematical written discourse may also contain tables which are forms of textual organization where the reader may access information quickly and efficiently (Baldry, 2000a; Lemke, 1998b). As seen in Plate 1.3(1), mathematical printed texts are typically organized in very specific ways which simultaneously permit segregation and integration of the three semiotic resources. An SF approach to mathematics as social-multisemiotic discourse means that each of the three semiotic resources - language, visual images and mathematical symbolism - is perceived to be organized according to unique discourse and grammatical systems through which meaning is realized. That is, each semiotic resource is considered to be a functional sign system which is organized grammatically. Mathematical texts such as those displayed in Plate 1.3(1) represent specific semiotic choices from the available grammatical systems in each of the three resources. As seen in the graphs and linguistic and symbolic components of the mathematics text in Plate 1.3(1), choices from the three semiotic resources function integratively. That is, the linguistic text and the graphs contain symbolic elements and the symbolic text contains linguistic elements. This feature of mathematical discourse means that the grammars of each resource must be considered in relation to each other. The similarities and differences in the organizing principles of the three semiotic resources are considered intra-semiotically in terms of the grammars and functions of each resource. In addition, mathematical discourse is considered inter-semiotically; that is, in terms of the meaning which arises from the relations and shifts between the three semiotic resources. Royce (1998a, 1998b, 1999) refers to intersemiotic semantic relations between linguistic and visual components of a text as 'intersemiotic complementarity', and ledema (2003: 30) calls the process of semiotic shift as 'resemioticization', which he defines as 'the analytical means for . . . tracing how semiotics are translated from one into another as social processes unfold'. In mathematics, intersemiotic shifts take place on a macro-scale across stretches of text, and they also take place on a micro-scale within stretches

12

MATHEMATICAL DISCOURSE J. Kockelkoren et al./Physica D 174 (2003) 16X-175

of the wavelength Xc of the patterns is at criticality about 13% off from the theoretical value; however, we are not interested here in the absolute value, but in the relative variation of X C /X. The difficulty of comparing theory and experiment on the variation of the wavelength is that the only theoretically sharply defined quantity is the wavelength

173

sufficiently far behind the front, A.^, and that one has to go beyond the lowest order Ginzburg-Landau treatment to be able to study the pattern wavelength left behind. For example, if we use a Swift-Hohenberg equation for a system with critical wavenumber kc and bare correlation length £o.

(11) then a node counting argument [4,6] yields for the asymptotic wavelength A.as far behind the front [6]:

(12) In the Rayleigh-Benard experiments, kc « 2.75/d, where d is the cell height; the theoretical value is £o = 0.385rf, so if our conjecture that the value is some 15% larger is correct, we get £Q ^ 0.4<W. This then gives

(13) As we stressed already above Xas is the wavelength far behind the front; for a propagating pulled front, there is another important quantity which one can calculate analytically, the local wavelength A* measured in the leading edge of the front. For the Swift-Hohenberg

Fig. 2. Top panel: shadowgraph trace of a propagating front in the experiments of FS for f = 0.012 [16]. The time difference between successive traces is 0.42fv, where f v is the vertical diffusion time in the experiments, and the distances are measured in units d (the cell height) (from [9]). Middle panel: similar data obtained from numerical integration of the Swift-Hohenberg equation also at e = 0.012 starting with a localized initial condition. The time difference between successive traces corresponds to 0.42/v. Bottom panel: velocity versus time in the experiment, as obtained by interpolating the maxima of the traces in the top panel, as explained in the text. The dashed line shows the analytical result (8) and the dotted curve the result of the amplitude equation simulation of Fig. 1 with nln" = 1.2. Note that the curves are not fitted, only the absolute scale is affected by adjusting £o

Plate 1.3(1) Language, visual images and symbolism (Kockelkoren et al, 2003: 173) of text. The potential of intersemiotic processes to produce metaphorical construals is formulated through the notion of 'semiotic metaphor'. Through close examination of the meaning realized within and across the three semiotic resources, the functions and the semantic realm of


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mathematics as a discourse are tentatively formulated in this study. It must be stressed that this is not an account of the entire field of mathematics. Rather it is an account of the semiotic processes and the discourse and grammatical strategies through which mathematics operates to structure the world. From this position, the semantic realm with which mathematics is concerned may be appreciated. This is in part achieved through a comparison of the functions of mathematics with those of language. However, mathematics evolved as a discourse capable of creating a world view which extends beyond that possible using linguistic resources alone. The result of that re-ordering in what is viewed as the scientific revolution is also considered in this study. The implications of viewing mathematical discourse as a multisemiotic construction are considered below. 1.4 Implications of a Multisemiotic View The multisemiotic approach, where language, visual images and mathematical symbolism are considered as semiotic resources (O'Halloran, 1996), originally stems from O'Toole's (1994, 1995, 1999) extensions of Halliday's (1978, 1994) SF approach to displayed art, and Lemke's (1998b, 2000, 2003) early work in mathematical and scientific discourse. The SF approach to mathematics is welcomed by Rotman (2000: 42) who explains that such an approach offers 'a linguistic/semiotic framework well grounded in natural language that . . . [is] abstract enough to include the making of meaning in mathematics'. Halliday's (1994) Systemic Functional Grammar (SFG) includes documentation of the metafunctionally based systems which are the grammatical resources through which meaning is made. Halliday's account of the abstract language systems includes statements of how these choices are realized in text. SFG is essentially a 'natural' grammar as it explains how language is organized to fulfil the metafunctions of language: the experiential, logical, interpersonal and textual. Halliday's (1994) model of language described in Chapter 3 provides the basis for the Systemic Functional Grammars (SFG) presented for mathematical symbolism and visual images in Chapters 4 and 5 respectively. These grammars and a framework with systems for intersemiosis are used for discourse analyses of mathematical texts in Chapters 6 and 7. The discussion includes an account of the educational implications of a multisemiotic view of mathematics and the nature of pedagogical discourses in mathematics classrooms. The SFGs for mathematical symbolism and visual images are inspired by O'Toole's (1994, 1995) systemic frameworks for the analysis of semiosis in paintings, architecture and sculpture. O'Toole (1994) demonstrates how the SF frameworks may be used so that the viewer can learn to engage directly with instantiations of displayed art rather than depending on the 'knowledge' handed down by art historians and other accredited experts. Bourdieu (1989) further explains that aesthetics and art appreciation are discourses which function covertly to maintain existing social class

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distinctions. In this view, 'taste' is a social and cultural product through which group and individual identities are indexed and, as with all symbolic investments, different values are placed on those indices. Needless to say, the highest values are accorded to those who constitute the powerful in society. the main reason for this close [semiotic] engagement with the details before our eyes is that it enables everyone to sharpen their perceptions and join in the discussion as soon as they begin to recognize the systems at work in the painting. And everyone can say something new and insightful about the work in front of them. Art history, on the other hand, requires a long apprenticeship . . . before they are expected to be able to contribute any new information to a discussion of the work in question. And what kind of information might this be? . . . Don't they in fact 'mystify' the painting and make us feel we have nothing to contribute? . . . the result is to build an insurmountable wall around this precious property. (O'Toole, 1994: 171)

Following O'Toole's (1994) example, rather than producing a discursive commentary about the nature of mathematics and its intellectual achievements, the intention behind the SF approach in this study is active engagement with mathematical text in order to understand the strategies through which the presented reality is structured, the content of that reality and the nature of the social relations which are subsequently established. The result is an appreciation and understanding of the functions of mathematical discourse and the strategies through which this is achieved. This is essentially a new approach to mathematics for practising mathematicians, and teachers and students of mathematics. This approach also offers insights for outsiders who typically possess a limited understanding and knowledge of mathematics. The implications of an SF approach to mathematics as a multisemiotic discourse are outlined below in relation to the key ideas and formulations developed in this study. These ideas are revisited in Chapter 7 after the theory and approach have been developed in Chapters 2-6. Mathematical and Scientific Language

The view of mathematics as multisemiotic has implications about the ways mathematical and scientific language are understood. Traditionally, the nature of scientific language has been viewed in isolation rather than as a semiotic resource which has been shaped through the use of mathematical symbolism and visual display. Scientific language developed in certain ways as a response to the functions which were fulfilled symbolically and visually. On a more global scale, our entire linguistic repertoire has been shaped by the use of other semiotic resources, with the result that many of our contemporary linguistic constructions are metaphorical in nature. For example, certain views become common sense under the guise of metaphorical labels such as 'economic rationalism', 'entrepreneurship' and

MATHEMATICS AS A M U L T I S E M I O T I C D I S C O U R S E

15

'freedom and democracy'. Despite their grammatical instantiation as nouns, these are not concrete or material objects. On the contrary, rather complex and dynamic sets of practices are subsumed under such labels. An understanding of the functions of mathematical symbolism, visual images and other semiotic resources permits a re-evaluation of the role of language in constructing such a naturalized view of the world. As with the vested interest behind the guiding principle, 'Success is right', metaphorical terms need to be critically understood in a historical and contextual manner in order to appreciate the premises behind their construction. The Grammar of Mathematical Symbolism

An SF framework for mathematical symbolism is presented so that the grammatical strategies through which meaning is encoded symbolically can be documented. This is significant because the grammatical strategies for organizing meaning in symbolic statements differ from those found in language. While members of a culture are capable of using language as a functional resource in various ways, typically the use of mathematical symbolism is restricted to certain groups. One reason for this limited access is that the grammar of mathematical symbolism is not generally well understood. It is important to demonstrate how mathematical meaning is organized, and how the unique grammatical strategies specifically developed in mathematical symbolism so that this semiotic could be used for the solution of mathematics problems. The underlying premise is that mathematical symbolism developed as a semiotic resource with a grammar which had the capacity to solve problems in a manner that is not possible with other semiotic resources. The SFG of mathematical symbolism presented in Chapter 4 explains how this functionality is achieved. Grammar of Visual Images in Mathematics

Visual images in mathematics are specialized types of visual representation, most typically in the form of abstract graphs, statistical graphs and diagrams. The systemic functional framework for abstract graphs is used to explain how the systems are organized to make very specific meanings which provide a link between the linguistic description of a problem and the symbolic solution. Once again, the functions fulfilled by mathematical visual images are different to those achieved linguistically and symbolically. The systems through which the functions of abstract graphs are achieved are discussed in Chapter 5. This discussion includes insights into the changing roles of visual images in mathematics due to the impact of computer technology. Visualization is undergoing a rapid resurgence due to the increasing sophistication of computer graphics which display numerical solutions generated by the computer. The new ways of manipulating and viewing data through computers are discussed in Chapter 5.

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Intrasemiosis and Intersemiosis

While the three semiotic resources in mathematics fulfil individual functions which are not replicable across the other resources (Lemke, 1998b, 2003; O'Halloran, 1996), the success of mathematics depends on utilizing and combining the unique meaning potentials of language, symbolism and visual display in such a way that the semantic expansion is greater than the sum of meanings derived from each of the three resources. Lemke (1998b) refers to this expansion of meaning as the multiplicative aspect of multisemiosis. Mathematical discourse thus depends on intrasemiotic activity, or semiosis through choices from the grammatical systems within each resource, and intersemiotic activity, or semiosis through grammatical systems which function across the three resources. Intersemiosis involves reconstrual of particular elements in a second or third resource through intersemiotic shifts or 'code-switching'. Intrasemiosis, or meaning within one semiotic resource, is important because the types of meaning made by each semiotic are fundamentally different. Intersemiosis, however, is equally important because not only is the new meaning potential of another resource accessed, but also metaphorical expressions can arise with such shifts. This important process, which may arise in any multisemiotic discourse, is developed in this study through the notion of semiotic metaphor. The functions of mathematics are therefore achieved through intrasemiosis and intersemiosis; that is, meaning through each semiotic resource, and meaning across the three semiotic resources where metaphor plays an important role in the expansion of meaning. Intersemiotic Mechanisms, Systems and Semiotic Metaphor

Intersemiotic mechanisms provide a description of the ways in which intersemiosis takes place across language, visual images and mathematical symbolism. The intersemiotic mechanisms take place through metafunctionally based systems which are documented in Chapter 6. Semiotic metaphor refers to the phenomenon of metaphorical construals which arise from such shifts across semiotic resources. This process means that expansions in meaning can occur when a functional element is reconstrued in a different resource. For instance, an action realized through a verb in language (for example, 'measuring') may be reconstrued as an entity in a second semiotic resource (for example, a visual line segment or a symbolical distance). Such reconstruals permit expansions of meaning on a scale which is not possible within a single semiotic resource. As explained in Chapter 6, one of the key elements in the success of mathematics is the metaphorical reformulation of elements across the three semiotic resources. Mathematics Education

The view of mathematics as a multisemiotic discourse is significant in a pedagogical context as often teachers and students do not seem to be aware


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of the grammatical systems for mathematical symbolism and visual display, and the types of metaphorical construals which take place in mathematics texts and in the classroom. The ways in which a social-semiotic perspective can inform mathematics teaching and learning are described in Chapter 7. This discussion is based on the functions of language, visual images and the symbolism, their respective grammatical systems and the nature of the intersemiotic activity. Chapter 7 includes a discussion of the nature of pedagogical discourse in mathematics. 1.5 Tracing the Semiotics of Mathematics

In order to introduce the types of meaning found in modern mathematics, a historical perspective is adopted in Chapter 2 to examine the semiotic unfolding of mathematics from the period of the early Renaissance to modern contemporary mathematics. The nature of the projects of early modern mathematics, as exemplified by Descartes and Newton, is seen to lead to the creation of a mathematical and scientific reality which is located within a limited semantic domain. However, at the same time, the semantic expansions afforded by the visual images and mathematical symbolism permitted expansions in the form of scientific description, prediction and prescription. Contemporary thought in mathematics, for example, chaos and dynamical systems theory, also reveals the changes in mathematical theorizations of reality. Significantly, the mathematical practices advocated by Descartes and Newton have been re-inscripted into new contexts in contemporary times. The beginnings of modern mathematics and science developed in what was originally conceived as a transcendental realm which necessitated the existence of God, as seen in the discussion of Cartesian and Newtonian philosophy in Chapter 2. The re-inscription of the supposedly 'value free' discourses of mathematics and science as universal truth into new realms of human endeavour such as the social sciences, education, business, economics and politics is questioned from the relatively fresh perspective of the socio-semiotics of mathematics in Chapter 7. This discussion also contributes to an appreciation of the metaphorical nature of our semiotic constructions and the limitations of the contexts in which mathematics may be usefully applied. Mathematics is thus first viewed in a historical context so the functions for which mathematics was originally designed and the context of that development may be appreciated. From this point, the new contexts in which mathematics is re-inscribed are critically examined. Although mathematics has expanded into new fields, the semiotic resources nonetheless essentially remain linguistic, visual and symbolic. Computation is considered a symbolic undertaking which is instantiated in an electronic medium. An understanding of the scientific view of the world made possible through mathematics is an overriding theme because such a view is vital for an understanding of contemporary Western culture which

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materializes as a technological project shaped by the discourse of mathematics and science. Looking back, the rationalist project of the eighteenth century and the consequent mathematical, scientific and technological achievements of the modern period appeared to hold much promise for the world. As Horkheimer and Ardorno (1972) claim, the much-touted aim of progress was the improvement of the human condition accompanied by freedom, equality and justice. In retrospect, however, such progress seems to have been made for the advantage of the relatively privileged few. In addition to providing the infrastructure for unequally distributed goods and services such as healthcare and education, advances in mathematical and scientific knowledge appear to have primarily provided the means for technological development which is directed and controlled by military, business and political interests. As Davis (2000: 291) claims: 'Through advanced science and technology, warfare utilizes many mathematical ideas and techniques. The creation of vast numbers of new mathematical theories over the past fifty years was due in considerable measure to the pressures and the financial support of the military.' The self-evident deliverables of the scientific project were underscored in the aftermath of the Second World War and, in a more recent case, the US-led war in Iraq in 2003 where the destructive power of military technological innovation was widely televised. As Horkheimer (1972: 3) claims: Tn the most general sense of progressive thought, the Enlightenment has always aimed at liberating men from fear and establishing their sovereignty. Yet the fully enlightened earth radiates disaster triumphant.' Today the extent to which the military, business and the political institutions can be differentiated as separate functioning bodies becomes increasingly difficult to ascertain. One could include universities on the list of institutions which increasingly function pragmatically along the lines of business-orientated commercial interests. The soundness of reason depends on the explicit or implicit premises upon which that reasoning is based. The view that mathematical and scientific reasoning is constructed to order the world along certain principles which change is not new (for example, Derrida, 1978; Foucault, 1970,1972; Kuhn, 1970). However, the approach adopted in this study is to understand the systems and strategies through which that ordering takes place. In this way, the functions of these discourses may be understood, and through such awareness we can understand our own positions and explore possibilities other than those directly offered. This is an exploration of the world view offered by mathematics and science, a view which dominates our everyday thinking. It is also a critique of that world view which is so often misunderstood as universal truth. The path is developed through an excursion through early printed mathematical texts to understand the context behind modern mathematics. SFGs are used to critically interpret the nature of meanings made in contemporary mathematics. Through an understanding of the discourse,

MATHEMATICS AS A MULTISEMIOTIC DISCOURSE

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we may start to count the gains and costs of the mathematical and scientific view of the world. In the view of Davis (2000: 291): The mathematical spirit both solves problems and creates other problems. What is the mathematical spirit? It is the spirit of abstraction, of objectification, of generalization, of rational or 'logical' deduction, of universal quantization, of computational recipes. It claims universality and indubitability. I have the conviction . . . that this spirit is now . . . pushing us too hard, pushing us to the edge of dehumanization.

The ways in which 'mathematics is pushing us too hard' are investigated through an understanding of mathematics as a multisemiotic resource. Only then can we begin to appreciate the ways in which this discourse and scientific order function to shape our view of ourselves, and our relations to others and the world around us. 1.6 Systemic Functional Research in Multimodality

This study of mathematics represents part of a growing movement in SFL (see ledema, 2003) where language is conceptualized as one resource which functions alongside other semiotic resources. This research field is commonly called 'multimodality', or the study of 'multimodal discourse' (for example, Baldry, 2000b; Baldry and Thibault, forthcoming a; Kress, 2000, 2003; Kress et al., 2001; Kress and van Leeuwen, 1996, 2001; Levine and Scollon, 2004; O'Halloran, 2004a; Unsworth, 2001; Ventola et al., forthcoming). Apart from the research in mathematics (Lemke, 2003; O'Halloran, 1996, 1999b, 2003b, forthcoming a), studies have been completed in a wide range of fields including science (Baldry, 2000a; Kress et al, 2001; Lemke, 1998b, 2000, 2002), biology (Guo, 2004b; Thibault, 2001), multiliteracy (Lemke, 1998a; Unsworth, 2001), film and television (ledema, 2001; O'Halloran, 2004b; Thibault, 2000), music (Callaghan and McDonald, 2002), museum exhibitions (Pang, 2004), shopping displays (Ravelli, 2000), TESOL (Royce, 2002), hypertext and the electronic medium (Jewitt, 2002; Kok, 2004; Lemke, 2002) and advertising (for example, Cheong, 2004). Research in the field of multimodality also includes the development by Anthony Baldry et al (Baldry, 2004, forthcoming; Baldry and Thibault, 2001, forthcoming a, forthcoming b) of an on-line multimodal concordancer, the Multimodal Corpus Authoring (MCA) system, which is web-based software for the analysis of phase and transitions in dynamic texts such as television advertisements, film and web pages. There have been attempts to construct grammatical frameworks for different semiotic resources (see Kress and van Leeuwen, 1996, 2002; O'Halloran, 2004a). However, with the exception of Thibault's (2001) approach to the theory and practice of multimodal transcription and Baldry and Thibault's notion of phase for the analysis of dynamic texts (Baldry, 2004, forthcoming; Baldry and Thibault, 2001, forthcoming b), few comprehensive theoretical and practical approaches have been developed in the field of multimodality. Consequently, a meta-language for

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MATHEMATICAL

DISCOURSE

an overarching model for the theory and practice of multimodal discourse analysis remains at a preliminary stage. Partly as a consequence of this lack of a meta-theory, there exist problems of terminology in studies of multimodality, as noted by ledema (2003: 50). For example, there is confusion over the use of the terms 'mode' versus 'semiotic', and, consequently, 'multimodal' versus 'multisemiotic'. Given that this field represents a relatively new area of research, this is to be expected as the much needed frameworks undergo development. As an example of mixed terminology, Kress and van Leeuwen (2001: 21-22) define 'mode' as the 'semiotic resources which allow the simultaneous realization of discourses and types of (inter) action. Designs then use these resources, combining semiotic modes, and selecting from the options which they make available according to the interests of a particular communication situation.' From this position, Kress and van Leeuwen (2001: 22) see Narrative, for example, as a mode. In this study, however, the term 'semiotic' is used to refer to semiotic resources such as language, visual images and mathematical symbolism. These semiotic resources have unique grammatical systems through which they are organized. Any discourse that involves more than one semiotic resource is therefore termed 'multisemiotic' rather than 'multimodal'. The use of the term 'multimodal' is explained below. The term 'mode' in SFL, following Halliday and Hasan (1985), typically means the role language is playing (spoken or written) in an interaction. This sense, adopted in this study, is concerned with the nature of the action of semiosis; that is, whether it is auditory, visual or tactile, for example. It follows that different semiotic resources are constrained in terms of possible modes through which the semiotic activity can take place. For example, language may be instantiated orally or visually, but visual images are instantiated through the visual mode in different media such as print, electronic media and three-dimensional space. On the other hand, Kress and van Leeuwen (2001: 22) use the term 'medium' to refer to the '[m]aterial resources used in the production of semiotic products and events, including both the tools and the materials used (for example, the musical instrument and air; the chisel and the block of wood. They are usually specially produced for this purpose, not only in culture (ink, paint, cameras, computers), but also in nature.' In order to maintain existing systemic terminology, in this study the term mode is used to refer to the channel (auditory, visual or tactile, for example) through which semiotic activity takes place, medium for the material resources of the channel, and genre for text types such as the Narrative (which is realized through language in the spoken or written form). The term multisemiotic is used for texts which are constructed from more than one semiotic resource and multimodality is used for discourses which involve more than one mode of semiosis. A radio play featuring speech, music and diegetic sound is therefore multisemiotic rather than multimodal as it involves multiple semiotic resources realized through the auditory mode of


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sound through the medium of the radio. However, a website which contains written linguistic text and a music video clip is multisemiotic (involving language, visual images, music) and multimodal (visual and auditory). The practices adopted here do not attempt to solve the problems of mixed usages of terminology, rather they seek to clarify the use of the terms adopted in this study. In this respect, mathematics is referred to as multisemiotic as it consists of three semiotic resources, language, visual images and mathematical symbolism. Mathematics is considered to be primarily a written discourse produced in printed and electronic media. There are also multimodal genres in the field of mathematics, such as the academic lecture, which involves spoken discourse and other semiotic resources. The multimodal nature of mathematical pedagogical discourse is discussed in Chapter 7.

2 Evolution of the Semiotics of Mathematics

2.1 Historical Development of Mathematical Discourse

A historical view of the changing nature of multisemiosis in mathematical discourse from the early Renaissance to the present is a useful method for introducing in general terms the development of the semantic realm of mathematics (O'Halloran, 2003b, forthcoming a). Such an examination of visual images, symbolism and language in mathematical texts demonstrates how particular dimensions of meaning are incorporated to the exclusion of others. This excursion includes a discussion of the first known printed mathematics book, the Treviso Arithmetic 1478, and an examination of early mathematical and scientific printed texts from the sixteenth to the eighteenth centuries. In particular, Descartes' shift of emphasis from perception to what he called 'the intellect' and Newton's reformulation of nature in mathematical terms are investigated. Descartes (1596-1650) and Newton (1642-1727) are seen to provide important points of departure in the seventeenth century for what was to become the contemporary mathematical and scientific project. In the first case, Descartes successfully used mathematical symbolism to describe and differentiate between curves. It appears that this success with symbolic and visual semiotic tools was incorporated into an approach upon which Descartes could base his philosophical method, a method aimed at securing 'true knowledge'. This method involved dispensing with the 'secondary' qualities of matter, such as colour, odour and taste perceived by the bodily senses, and accepting only 'primary' qualities which could be dealt with through 'the mind'. Newton developed Descartes' mathematical semiotic tools to provide a symbolic description of physical reality. In doing so, Newton in fact re-admitted sense experience to the philosophical and scientific realm in such a way that made the invisible (for example, force and attraction) visible through mathematicized symbolic description (Barry, 1996). The scientific age and the use of experimentation and technology began in earnest with Newton. In this movement, much effort was expended in developing mathematical symbolism as a semiotic resource with a grammar that could directly interact with the grammars of graphs,

E V O L U T I O N OF THE S E M I O T I C S OF M A T H E M A T I C S

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diagrams and language. In this chapter, some necessarily fragmented glimpses of these early events are traced in order to introduce the nature of meanings found in contemporary mathematics. To make this undertaking more accessible, the historical investigation into the semiotic realm of mathematics and science takes the form of a discussion of the illustrations and diagrams which appear in early printed mathematical texts. In addition, observations concerning the different forms of symbolism in these texts are made. As Cajori (1993) explains, the history of the development of mathematical symbolism is complex and involves rivalry among mathematicians. Cajori's (1993) detailed account of the history of mathematical notation reveals that the majority of forms of mathematical symbolism became obsolete with only a few forms surviving to the present day. These developments are not included in the account presented here, nor is it possible to include a discussion of the origins of algebra documented by mathematical historians (for example, Klein, 1968). The broader view presented in this study is that algebra developed in three stages (see Joseph, 1991; Swetz, 1987). First there existed rhetorical algebra which involved linguistic descriptions and solutions to problems. The second stage was syncopated algebra where quantities and operations which were used frequently were symbolized. The last stage of the development was symbolic algebra where the mathematical symbolism developed as a semiotic resource in its own right. Rather than providing a complete description of the three stages, the changing nature of multisemiosis in mathematics as the symbolism developed is explored. This discussion of the history of mathematics differs from most accounts in that the view is essentially semiotic. In other words, out of the possible options within the different sign systems for language, visual images and symbolism, it may be seen that only certain ranges of choice were incorporated in mathematics during different time periods. The shifting nature of those choices becomes evident as the printed classical mathematical texts of antiquity and early practical arithmetic books were replaced with new forms of semiosis in the mathematics of the early Renaissance. Descartes and Newton reformulated the mathematical realm in what marks the beginning of modern science during the seventeenth century. However, today that mathematical realm functions in different contexts from that which supported the original mathematical formulations. The implications of this re-contextualization of mathematical and scientific formulations in fields which include the arts, social sciences and humanities are addressed in Chapter 7. The reasons for the changes in the nature of mathematics are traced to the cultural, intellectual and economic climate of the different time periods, the functions which the mathematics was designed to serve and the available technology. This remains true today where economic, commercial, political and private interests combine with advances in computer technology to determine the type of mathematics developed and the nature of scientific projects which are undertaken. As Wilder (1986) claims,

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mathematics is a cultural practice, and so, like other forms of discourse, it is politically motivated. Consequently, following Koestler (1959) and later philosophers of science such as Kuhn (1970), the development of mathematics and science has not been orderly: 'The progress of science [and mathematics] is generally regarded as a clean, rational advance along a straight descending line; in fact it has followed a zigzag course, more bewildering than the evolution of political thought' (Koestler, 1959: 11). In the following discussion, views of the changing nature of the semiotics of mathematics are oudined in relation to the cultural and situational contexts which gave rise to those discourses. In explaining the effectiveness of mathematics, Hamming (1980) claims that what is seen is what is looked for, that the kind of mathematics used is selected from a range of possible choices and that in this process very few problems are answered. Following this line of argument, mathematics is seen to deal with a limited semantic field in limited ways, but in doing so has the potential to solve problems which would be impossible to solve using other semiotic resources. Seen in this light, the breakthrough which led to the scientific revolution was a new way of conceptualizing the world using new forms of semiosis. This is basically the position developed in the following discussion of the evolution of modern mathematics. 2.2 Early Printed Mathematics Books

The first known printed Western mathematical book is the Treviso Arithmetic 1478. Partially translated from Italian into English by David Eugene Smith in the 1920s, the first complete translation appears in Frank Swetz's (1987) Capitalism and Arithmetic. The author of the original manuscript is unknown and the title arises from the date and place of publication, the Italian town of Treviso. The book is concerned with practical arithmetic for calculations in trade and commerce. Swetz (1987) explains that the content is typical of the earliest known mathematics books in Europe. The majority of the books were written by masters in reckoning schools and guilds which flourished in Italy during the fifteenth century. These institutions were popular places to learn the mathematics necessary for the expanding merchant trade. As seen in Plate 2.2(1), the Hindu-Arabic system is used for the calculations in the Treviso Arithmetic, and there are drawings to demonstrate how the calculations are performed. Although Swetz (1987) is an English translation of an Italian text, it can be seen that the original text is constructed semiotically through the use of language, numerical symbols and particular forms of drawings. The style is rhetorical algebra where unknown quantities are realized as entities such as 'profit' rather than symbolic quantities such as 'P'. According to Swetz (1987), in the late fifteenth century some Italian writers were starting to use syncopated algebra with forms of abbreviations for recurring terms and mathematical operations.


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Thus the problem is solved, and the answer is that there falls to Piero as profit 138 ducats, 21 grossi, n pizoli and Plate 2.2(1) The Treviso Arithmetic (reproduced from Swetz, 1987: 140) At the time of the Treviso Arithmetic, there were controversies over the best method for performing arithmetical calculations as pictured in Plate 2.2(2). The controversy concerned the abacists, who manually used counters and ruled lines to perform the calculations and recorded the result in Roman numerals, and the algorists, who used the Hindu-Arabic numeration system and algorithms to calculate and record. Computation at this time centred around prestigious counting tables, and the proposed shift to algorists' Hindu-Arabic system represented a threat to those who had vested interests in maintaining the tables. Despite the obvious benefits of the new system, the shift to the Hindu-Arabic numeration system, first introduced in Europe as early as AD 1000 was slow because of the resistance exerted by those who controlled the tables. By the fifteenth century Italy, however, was ahead of other European countries in using the

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Plate 2.2(2) The Hindu-Arabic system versus counters and lines (Reisch, 1535: 267) Hindu-Arabic system as the means for performing arithmetic calculations. The Treviso Arithmetic demonstrates how the nature of mathematics is influenced by cultural and economic concerns of the time (in this case, merchant trade and commerce) and pressure from special interest groups (for example, those supporting the Hindu-Arabic number system). In addition, technology plays an important role as seen below in the discussion of the


27

impact of the printing press on mathematics. More generally, the nature of the development of mathematics is a convergence of these factors. The use of printing press technology explains the increased popularity of the Hindu-Arabic system which lent itself to this type of reproduction (for example, Dantzig, 1954; Eisenstein, 1979; Swetz, 1987). The calculations completed through lines and counters on the counting tables were clumsy to reproduce and required special printing techniques. As may be seen in Plate 2.2(3), the diagrams on the left-hand side (which include small pictures of hands) demonstrate how the calculations are performed using the lines and counters. As mathematical texts increasingly appeared in print, this form of representation could not compete with the more efficient semiotic form of the Hindu-Arabic system of computation. The expansion of the use of the Hindu-Arabic system is significant for two reasons. First, as commercial mathematics increasingly became semiotic instantiations in the written mode, the algorithms for the calculations became more widely disseminated and commercial arithmetic moved from the hands and counters of the few to a wide audience. Second, Swetz (1987: 32) explains that the increased focus, attention and recording of the mathematical techniques in the Hindu-Arabic practical arithmetical texts in effect paved the way for the development of symbolic algebra. The printed text permitted close examination and development of arithmetical algorithms, and the standardization of mathematical procedures, techniques and symbols which led to the range of mathematical notations documented by Cajori (1993). Under the economic and intellectual impetus of this time, not only were mathematical techniques being more widely learned but they were, in many cases, new techniques based on the use of Hindu-Arabic numerals and their accompanying algorithms. From this period onward, computation involving numbers would be more easily executed and efficiently recorded. The visual stimulus of a mathematical process written out allowed for a re-examination and questioning of the process; patterns could be noted and mathematical structure discerned. Printing also forced a standardization of mathematical terms, symbols, and concepts. The way was now opened for even greater computational advances and the movement from a rhetorical algebra to a symbolic one. (Swetz, 1987: 284)

Swetz (1987) explains that at the time the Treviso Arithmetic was published, the Classicist mathematicians demanded a printed edition of Euclid. However, it appears that this priority was placed second to arithmetic where 'Practical necessity was the motivating force in this printing decision' (Swetz, 1987: 25). 'Perhaps the typographical problems inherent in setting type for geometrical figures were responsible for the delay, but more likely it was due to the economic and intellectual demands of the marketplace' (ibid.). In Thomas-Stanford's (1926: 3) view, the early Venetian printing presses published few mathematics books due to the problems of printing the diagrams. Two examples of early printed editions of Euclid's Elements are displayed in Plates 2.2(4a-b). Thomas-Stanford (1926: 3) states that the

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Plate 2.2(3) Printing counter and line calculations (Reisch, 1535: 326) first printed edition, which appeared in 1482, was 'an epoch-making' book in many respects: 'It was [one of] the first attempt[s] - and a highly successful one - to produce a long mathematical book illustrated by diagrams.' In this version of Euclid displayed in Plate 2.2(4a), the running linguistic text in the style of rhetorical algebra is ornately decorated and the diagrams are neatly offset to one side. Plate 2.2(4b) shows the richness of the border patterns and text which appears in the early editions of Euclid. In the

E V O L U T I O N OF THE S E M I O T I C S OF MATHEMATICS

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Plate 2.2(4a) Euclid's Elements: Venice 1482, Erhard Ratdolt (ThomasStanford, 1926: Illustration II) original version of Plate 2.2(4b) reproduced in Thomas-Stanford (1926: Illustration IV), parts of the text and the decoration are coloured red. Thomas-Stanford (1926: 4) observes that' [it] would almost seem that at Venice especially the printers sought by a refinement of ornamentation to relieve the austerity of the subject-matter', the nature of which may be appreciated from the modern translation of Euclid which is displayed in Plate 2.2(5). The mathematics appears as objective statements which are accompanied by perfect geometrical shapes. Apart from the statement, 'I say that. . .' in Euclid's discourse, the author is absent. In Euclid's writings

30


Plate 2.2(4b) Euclid's Elements: Venice 1505, J. Tacuinus (ThomasStanford, 1926: Illustration IV) displayed in Plates 2.2(4a-b) and 2.2(5), there is also a noticeable lack of symbolism in the text, which appears only in the form of a, b, c and d and A, B, C and D to refer to the points, sides, angles and triangles in the


31

PROPOSITION 18. In any triangle the greater side subtends the greater angle. For let ABC be a triangle having the side AC greater than AB; I say that the angle ABC is also greater than the angle BCA. For, since AC is greater than AB, let AD be made equal toAB [i. 3], and let BD be joined. Then, since the angle ADB is an exterior angle of the triangle BCD, it is greater than the interior and opposite angle DCB. [i. 16] But the angle ADB is equal

to the angle ABD, since the side AB is equal to AD; therefore the angle ABD is also greater than the angle ACB; therefore the angle ABC is much greater than the angle ACS. Therefore etc. Q. E. D. Plate 2.2(5) Translation of Euclid (reproduced from Euclid, 1956: 283) mathematical diagrams. As we shall soon see, Newton rewrote Euclid's geometry in symbolic form. Needless to say, the quality of the production of the mathematics printed texts was not always consistent. For example, as displayed in Plate 2.2(6), the translated version of Archimedes (1615: 437) printed in Paris contains mathematical diagrams which are crude especially with respect to line width and horizontal alignment on the page. In addition, typesetting lines separate the Greek and Latin versions of the text, and the headings and margins. Presumably early printing presses possessed to different degrees the technology, expertise and finance to produce printed mathematical books. While the absence of mathematical symbolism in Archimedes' (287-212 BC) rhetorical-style text is not surprising, the textual layout includes spatial separation of the text and diagrams, which is a feature of contemporary mathematical texts. Febvre and Martin (1976: 259) claim that 'printing does not seem to have played much part in developing scientific theory at the start'. This view is based on an observation that some influential works in arithmetic and

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Plate 2.2(6) Quadratura Paraboles (Archimedes, 1615: 437)


33

algebra, such as Nicolas Chuquet's (1484) The Triparty, remained in manuscript form in the late fifteenth century. However, early printing presses must have taken time to become established and to be economically viable, the printers must have made careful choices as to what they published. Given the economic demands of the time and the pressure from certain established circles, it is not surprising that the first books in the late fifteenth century were concerned with practical arithmetic and classical texts rather than new works in algebra. When the printed texts on algebra did appear, they were influential. Whitrow (1988), for example, states that the key to the mathematical revolution in the sixteenth century was the beginnings of the development of algebra, and the first book on the subject was Luca Pacioli's Summa de Arithmetica (1494). Whitrow (1988: 267) notes: '[this book] was extremely influential, presumably because it was printed'. The printing of mathematical texts had an immense impact on the mathematics that was subsequently developed, for as Eisenstein (1979: 467) claims '[cjounting on one's fingers or even using an abacus did not encourage the invention of Cartesian coordinates'. Eisenstein (1979) further explains that Newton mastered the classical works of the ancients and contemporary mathematicians such as Descartes from the books he obtained from libraries and book fairs. Newton was self-taught, and this differed greatly from previous practices where learning took place in an oral tradition which involved the elder masters. Likewise, Leibniz had read most of the important mathematical texts of his time before he was twenty years of age (Smith, 1951). Mathematics became widely accessible, and in some sense standardized, through the medium of the printing press. Before moving beyond the times of the Treviso Arithmetic, it is important to note that commercial arithmetic was not the only concern of this time simply for the reason that commerce and trade do not only involve counting. As Swetz (1987: 25) explains, the reckoning masters were the forerunners of applied mathematicians, and their concerns spanned commercial arithmetic to land surveying, construction of calendars, and cask gauging. As trade and colonialization expanded, there was a need to refine navigational techniques and increase military strength. As becomes evident in Section 2.3, mathematical and scientific descriptions at the beginning of the Renaissance included the study of warfare. At this time, mathematics became a recognized profession which was freed from the mysticism of the Middle Ages, and it developed into a discipline that ranked alongside or above other more established fields of study. In this climate, studies in mathematics expanded rapidly into new ways of thinking about old ideas, and new ways of thinking about new ideas. 2.3 Mathematics in the Early Renaissance

The new movements in sixteenth-century Europe were fuelled through the decline of feudalism and the growth of cities and towns in which the

34


wealth created through manufacture, commerce and trade meant the introduction of a new power base. In the climate of relative social stability in the West, a re-examination of ideas occurred in what is generally termed the Renaissance. Swetz (1987: 5) explains that 'Intellectual humanism was patronised by capitalism and secularism, which broadened man's horizons of inquiry and innovation.' The arts flourished and the nature of mathematical and scientific thinking also changed. The nature of the change in mathematics is explored below through the examination of several printed mathematical texts of that time. Niccolo Fontana, known as Niccolo Tartaglia, was a pupil of the Italian reckoning schools and later became a prominent mathematician in his own right. Tartaglia wrote La Nova Scientia (The New Science) in 1537, and the frontispiece to this book, displayed in Plate 2.3(1), contains a picture of a tower with different academic fields represented by human figures. The new place of mathematics in what marks the beginning of the Renaissance is made clear in this scene. After being admitted by Euclid and passing by the two firing cannons with their attendants, the two female figures standing by Tartaglia to greet the visitor are labelled Arithmetic and Geometry. The other female figures are labelled Astronomy, Music, Poetry, and Astrology. The figures of Plato and Aristotle stand at the entrance to the second level and the female figure of Philosophy is located at the top level. The banner that Plato holds reads 'hue geometriae expers ingrediatur' or 'Nobody enters who is not expert of mathematics' (translated by David Pingree in Davis, 2000: 293). As Davis (2000: 293) explains, 'What we have here is the hierarchy of knowledge as set out by St. Thomas Aquinas . . . that lacks its topmost thomist level: theology!' The message is that those seeking wisdom must know mathematics, and it appears that an integral part of that knowledge is somehow associated with cannons, a topic which is further investigated below. In the new spirit of the early Renaissance, mathematical discourse began to appear in a very different form from the earlier classics which were still in place as the authorative texts. The circumstantial context of the mathematical problem was often made explicit, and, in addition, the human realm was depicted. For instance, the concept of volume is illustrated through spears which pierce the body of a naked man standing on a mound of earth in Reisch (1535: 424). This conception of volume differs dramatically from that found in Euclid, for example. Tartaglia's (1546: 7) research into the trajectory of cannonballs displayed in Plate 2.3(2) clearly shows the circumstantial context of the problem including the target which is to be hit. The accompanying mathematical symbolic text includes extended arithmetical calculations as seen in Plate 2.3(3). The arithmetic is difficult to read as it is embedded within the linguistic text, and the notation does not include the shorthand forms which are found today; for example, 1020 for 100,000,000,000,000,000,000. This number appears as '100000000000000000000' in Tartaglia (1546). Apart from Tartaglia, military concerns in the form of hitting targets are reflected in other studies;


35

Plate 2.3(1) Tartaglia's (1537) La Nova Scientia Frontispiece for example, Galileo (1638: 67) who attempts to calculate the angle of elevation of a building on a hill, this time from different positions on the ground as seen in Plate 2.3(4). Many mathematical texts in the early Renaissance appear to involve human figures participating in some form of physical or perceptual activity

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Plate 2.3(2) Hitting a target (Tartaglia, 1546: 7) where the circumstantial context of the exercise is included. For example, the context surrounding Tartaglia's (1537) concern with predicting the path of a cannonball is explicit in Plate 2.3(5). A man is engaged in firing a cannon to hit a target which appears to be the building on the other side of the river or lake. There is a human actor engaged in a material activity which is presented as the problem to be solved. The problem is approached through geometrical constructions involving lines and a circle (which was later proved to be incorrect through the work of Galileo). While Tartaglia later regretted his work on cannon fire as a contribution to the art of warfare (see Davis, 2000: 293), these texts were nonetheless explicit as to the purpose of the mathematical exercise. Contemporary textbooks in mathematics also have images where the context of the problem is visualized; for example, introductory exercises pose problems in order to introduce the mathematical theory which is to be developed, and practice exercises apply that theory. However, reasons for the theory such as warfare are not typically depicted, at least in the public eye. Rather, the mathematical theory is developed in the abstract, and retrospectively shown to have applications, or alternatively, a suitable example of a problem is posed so that the reader can appreciate the usefulness of the mathematics which is subsequently developed. Mathematical theory is, however, often presented non-contextually in contemporary discourse. However, in the


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Plate 2.3(3) Arithmetic calculations to hit a target (Tartaglia, 1546: 106)

38


Plate 2.3(4) Positioning a target (Galileo, 1638: 67) Renaissance texts viewed here, the realm of human activity is an integral part of the mathematics texts. The semiotic visual rendition of the mathematics problem permits a re-organization of perceptual reality. For example, Tartaglia's (1537) drawing of the men firing the cannon in Plate 2.3(5) demonstrates how the visualization of the problem allows new objects or entities to be introduced semiotically. That is, the line segments, circles and arcs and the resultant triangles are entities which only exist in the semiotic construal of the material context of the problem. Significantly, these new entities become the focus of attention for Descartes who attempts to construct different curves and in doing so discovers that they may be described algebraically. From this point, Descartes discards the human realm of sense perception. In what follows, an examination of Descartes' mathematical and philosophical writings reveals that mathematical symbolism developed as a semiotic system to form 'a semantic circuit' with the visual images and language. That is, the visual images of the curves, the symbolic description of those curves and the use of language function hand-in-hand to create a new version of reality. 2.4 Beginnings of Modern Mathematics: Descartes and Newton

There was a shift in the nature of the semiotic construction of mathematical problems in the seventeenth to eighteenth centuries where the circles, curves and line segments increasingly became the major focus of attention rather than the depiction of the material context of the problem that featured so prominently in the early Renaissance texts. For example, the human body gradually disappears, or alternatively is replaced by a part of


Plate 2.3(5) Predicting the path of cannon fire (Tartaglia, 1537)

39

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the body, as we may see in the case of Descartes' (1682: 111) eyes in Plate 2.4(la) and Newton's drawings of the path of light in Plate 2.4(lb). While the mental process of perception is still construed, the body part of 'the eye' now acts as the sensor. In these new diagrams, the focus shifts away from the human actor and the context of the problem to the semiotic entities of lines, curves and triangles. With the decline of the human agent, the line segments, circles and arcs and their accompanying spatial and temporal relations take centre stage. In Newton's diagram in Plate 2.4(lb),

Plate 2.4(la) Removing the human body: Descartes (1682: 111)

Plate 2.4(lb) Removing the human body: Newton (1952: 9)


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for example, the dynamic process of the refraction of light is constructed through intersecting line segments through a lens. The drawing of the eye contextualizes the scene, but this eye possesses secondary importance compared to the lens and the path of light depicted by the line segments. This secondary importance is signalled by the size and position of the eye compared to size, centrality and labelling of the line segments and the lens. The human figure disappears in visual representations of material actions as well as acts of perception. For example, in Descartes' (1985c: 259) drawing in Plate 2.4(2a), a hand rather than a complete human figure is drawn swinging a stone from point A to point F. The situational context of the problem is absent and the stone is not swung for any conceivable purpose other than to trace the movement of the stone. Descartes labels the path A, B and F at different points which adds a temporal dimension to the visual semiotic representation. Descartes is concerned with spatial and temporal dimensions of the path in the visual semiotic construction of the problem. Although the path of the cannonball is drawn in Plate 2.3(5), Tartaglia does not attempt to mark so explicitly the unfolding temporal dynamics at particular points of time, but rather the more dynamic aspect

Plate 2.4(2a) Movement in space and time: the stone (Descartes, 1682: 217)

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of his drawing appears in the firing of the cannon. Descartes' attention, however, lies with the path of the stone at different times which are explicitly drawn, labelled and marked spatially with line segments. Descartes draws a model of a compass to trace the movement of the stone in Plate 2.4(2b). This drawing depicts a material compass that swings on a pivot at point E. As we shall soon see, the semiotic compass plays a major role in the development of Descartes' geometry. Newton's (1953: 31) construction of the path of two swinging pendulums

Plate 2.4(2b) Movement in space and time: the model (Descartes, 1682: 217)

Plate 2.4(3) Movement in space and time: the pendulum (Newton, 1953: 31)


43

is displayed in Plate 2.4(3). In this diagram the human figure and the context have completely disappeared, and the material everyday object such as a stone is replaced with a pendulum, which is a piece of scientific equipment like the lens in Plate 2.4(lb). Newton marks the relative position of the pendulums in more detail than Descartes' path of the stone in Plate 2.4(2a). One can see the development of the semiotic construction of the prediction of the path of objects as a continuous mapping of spatial and temporal dimensions. The shift from the realm of the material, everyday world of human action and perception to de-contextualized visual images in the beginnings of modern science is apparent in Descartes' drawings in Plates 2.4(4a-b). These illustrations display the path of a ball through water. In Plate 2.4(4a), a man is shown hitting the ball downwards into what looks like a lake or a pond. We have, in a manner similar to Tartaglia's drawings, features of the context of the situation which include a complete human figure involved in some material action in a setting. The path of the ball is constructed as a series of line segments in relation to a circle. However, in Plate 2.4(4b) Descartes shifts his attention to the line segments and curves. Once again, the human figure and the context are eliminated and the major participants are the new semiotic entities of line segments and circles which are situated in specific relations to each other. The appearance of de-contextualized visual images where the major processes are spatial, temporal and relational with entities in the form of

Plate 2.4(4a) Context, circles and lines (Descartes, 1682: 226)

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Plate 2.4(4b) Circles and lines (Descartes, 1682: 228) the line segments, circles and curves requires explanation. One reason is the growing significance of the role of the mathematical symbolism. Descartes discovered that curves could be described using mathematical symbolism, and therefore he moved (a) from the semiotic construction of the material context (b) to the semiotic construction of the lines and curves (c) which were described symbolically (d) to solve the problem. Newton and other mathematicians used this path to lay the foundations of modern science. This path is explored in the remainder of this chapter. Descartes was interested in constructing curves using a semiotically grounded compass which was conceived from the material compass and ruler used by the Greeks. From the material actions depicted in Plate 2.4(5a), Descartes devised a method of semiotically constructing curves based on proportionality as displayed in Plate 2.4(5b). For Descartes, 'This new [semiotic] instrument does not have to be physically applied; it is enough to be able to visualise it and use it as a computing device. In other words, pen and paper is all that is required, since the nature of the curve is revealed in its tracing' (Shea, 1991: 45). The shifts in the nature of the semiotic construals of Descartes' geometry seem to occur in stages. In the initial stages, the semiotic rendition included drawing the material action of tracing the curve as displayed in Plate 2.4(5a). However, this material drawing of the curve (which includes the actions of the hand) developed into a semiotic rendition of the material compass to trace curves as displayed in Figure 2.4(5b). Descartes' main concern was the proportional relations which he mapped visually and spatially as curves using his semiotic compass. Descartes discovered that mathematical symbolism could be used to differentiate between the curves he constructed. Although this was a major


45

Plate 2.4(5a) Descartes' semiotic compass (1683: 54) (Book Two)

Plate 2.4(5b) Drawing the curves (Descartes, 1683: 20) (Book Two) breakthrough, Descartes' interest remained in the construction of the curves and he did not realize, as later mathematicians such Newton and Leibniz did, that the symbolism provided a complete description of the curves rather than a means for construction. While the method amounted to an algebraization of ruler-and-compass constructions (Davis and Hersh, 1986), Descartes nonetheless 'simplified algebraic notation and

46


set geometry on a new course by his discovery that algebraic equations were useful not only in classifying geometrical curves, but in actually devising the simplest possible construction' (Shea, 1991: 67). Davis and Hersh (1986: 5) comment: 'In its current form, Cartesian geometry is due as much to Descartes' own contemporaries and successors as to himself.' Despite this, Descartes' increasing reliance on the symbolism is evident in his geometry as displayed in Plates 2.4(6a-b). The symbolism features as an integral part of Descartes' geometry, one that now depends on language, visual images and mathematical symbolism. The significance of Descartes' algebraic descriptions for curves cannot be underestimated because this is the point from which modern mathematics and science developed as an integrated multisemiotic discourse in which a central role was assigned to the symbolism. For example, Newton's reliance on mathematical symbolic descriptions is seen in Plates 2.4(7) and 2.4(8a-b). Newton rewrites Euclid's geometry in algebraic terms in Plate 2.4(7), and in printed versions of Newton's work in Plates 2.4(8a-b) the reliance on algebraic symbolism as a method for reasoning is evident. Newton proceeds symbolically step-by-step in Plate 2.4(8a), and efficiently organizes these symbolic descriptions into table format in Plate 2.4(8b). Newton's semiotic is the symbolism which functions in conjunction with language and the mathematical graphs. The implications and circumstances surrounding Descartes' move to the symbolic are worthy of further investigation, not only because this provided Newton with the tools to rewrite nature, but also because it appears that the newly de-contextualized and algebraicized geometry provided Descartes with the foundations for his influential philosophy. Descartes' method is concerned with establishing what is 'true knowledge' through the path of intelligibility rather than sensory experience. This method appears to operate within the boundaries of Descartes' new form of symbolic semiosis, which offers much more at the price of admitting substantially less. The programme of objectivity and truth in the mathematical descriptions inherited from Descartes exists today. 2.5 Descartes' Philosophy and Semiotic Representations

While Descartes did not fully utilize the potential of the mathematical symbolic descriptions, he certainly appreciated the power of this form of semiosis. Descartes repeatedly insists, for instance, that language is inadequate for his purpose of achieving certainty of knowledge beyond the commonsense kind. Descartes' distrust of the linguistic semiotic is openly expressed in the Second Meditation in 'Meditations on First Philosophy' (Descartes, 1952, 1985b) where he attempts to describe what can be known with certainty through a discussion of a ball of wax. Descartes explains that knowledge achieved through the senses (for example, colour, flavour, smell, shape and size) is unreliable because these properties change as the wax is heated. Descartes concludes that mental facilities alone permit


*to

LA GEOMETRIE.

Plate 2.4(6a) Descartes' description of curves (1954: 234)

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48


Plate 2.4(6b) Descartes' use of symbolism (1954: 186)


Plate 2.4(7) Newton's algebraic notes on Euclid

49

50


80

The Method ..of FLUXIONS,

llli To determine -a Conic SeSiimt, at any.Point of which, the Curvntitre and Pofitian of the Tangent, (inrefpeftofthe AxhJ) may be like to the Curvature and' Pofition of the. tangent, at a Point aj/igiid of any other Curve. "21. The ufe of which Problem is. this* that inftead of Ellipfes of the fecond kind, whole Properties- of refradling Light are explain'd by DCS Cartes in his Geometry, Conic Sections may be fubftituted, which Hull perform the fame | thing, very nearly, as to their Reirasftions. And the fame may be underftooci of other Curves.

P R O B. VII. 20 find as .many Curves as you pleafe^ wbofe Areas may be exhibited by finite. Equations.

EXAMPLES.

Plate 2.4 (8a) Newton's (1736: 80-81) Method of Fluxions and Infinite Series


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10. And thus from the Areas, however they may be feign'd, you may always determine the Ordinates to which they belong. P R O B.

VIII.

70 find as many Curves as you pleafe^ wbofe Areas Jhall have a relation to the Area of any given Curve> ajjlgnable by finite Equations. i. Let FDH be a given Curve, and GEI the Curve required, and conceive their Ordinates DB and EC to move, at right Angles upon

their Abfcifles or Bafes AB and AC. Then the Increments or Fluxions o( the Areas which they defcribe, will be as thofe Ordinates drawn

M Plate 2.4(8a) - cont

into

52 1OO


fhe Method < ? / F L U X I O N S ,

Plate 2.4(8b) Newton's (1736: 100) Method of Fluxions and Infinite Series

EVOLUTION OF THE SEMIOTICS OF MATHEMATICS 53

examination of the reliable essence of matter which he conceptualizes as motion and extension in the form of length, breadth and depth. Two translations where Descartes explicitly criticizes the use of language are reproduced below: the first is an English translation of Descartes' original 1642 Latin text and the second is a translation of the 1647 French version of that Latin text. But as I reach this conclusion I am amazed at how [weak and] prone to error my mind is. For although I am thinking about these matters within myself, silently and without speaking, nonetheless the actual words bring me up short, and I am almost tricked by ordinary ways of talking. (Descartes, 1985b: 21) I am indeed amazed when I consider how weak my mind is and how prone to error. For although I can, dispensing with words, [directly] apprehend all this in myself, none the less words have a hampering hold upon me, and the accepted usages of ordinary speech tend to mislead me. (Descartes, 1952: 209)

The two translations express Descartes' criticism of the use of language to describe knowledge which he claims is certain. His intent is clear: 'But aiming as I do at knowledge superior to the common, I should be ashamed to draw grounds for doubt from the forms and terms of ordinary speech' (ibid.: 210). The semiotic Descartes installs as the one most appropriate for his purposes is mathematical symbolism. In this process, the accompanying geometrical curves are only to be used as an aid to thought. Descartes describes his symbolic expressions which are to replace the linguistic descriptions: Whatever, therefore, is to be regarded as an item . . . we shall designate by a unique sign, which can be freely chosen. For convenience sake, we employ the letters, a, b and c, etc., to express magnitudes already known, and A, B, C, etc. for unknown magnitudes. To them we shall often prefix the signs 1, 2, 3, etc., to indicate their numerical quantity, and shall also append them to indicate the number of relations which are to be recognised in them. Thus if I write 2a\ that will be as if I should write the double of the magnitude signified by the letter a, which contains three relations. By this device not only do we obtain a great economy in words, but also, what is more important, we present the terms of the difficulty so plain and unencumbered that, while omitting nothing which is needed, there is also nothing superfluous, nothing which engages our mental powers to no purpose . . . Rule XVI (ibid.: 101)

Descartes (1952: 101-102) claims that linguistic descriptions such as 'the square' or 'the cube' are confusing, and that they should be abandoned. The first is entitled the root, the second the square, the third the cube, the fourth the biquadratic, etc. These terms have, I confess, long misled me. For, after the line and square, nothing it seemed to me allowed of being more clearly exhibited to the imagination than the cube and other shapes; and with their aid I solved not a few difficulties. But at last after many trials I came to realise that by this way of conceiving things I had discovered nothing which I could not have learnt much more easily and

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distinctly without it, and that all such denominations should be entirely abandoned, as being likely to cause confusion in our thinking.

The underlying reason for Descartes' claim that the linguistic descriptions cause 'confusion in our thinking' is that the linguistic version means something quite different to what is generally considered to be the symbolic equivalent. As explained in the discussion of language and photographs in Chapter 1, different semiotic resources have the potential to mean different things. In Descartes' example, the linguistic term 'the cube' is semantically a fixed entity through its grammatical instantiation as a noun. It is an object, a thing, a participant or entity. On the other hand, the symbolic ax1 is not a fixed object, rather it is a complex of interactive participants which combine through the process of multiplication as seen in the expanded form ax xx x x x. The symbolic expression is not a stable fixed entity like the linguistic nominal group, rather it is a dynamic complex which may be reconfigured in different ways. This type of symbolic expression offers countless alternatives to describe different curves; for example x3, 2X3, Sx3 and so forth. Given that Descartes' aim is to construct and differentiate between curves, it is not surprising that the symbolic descriptions are preferred. Descartes' (1985a: 9-78, 111-151) 'Rules for the Direction of the Mind' and 'Discourse on the Method for Rightly Conducting One's Reason and Seeking the Truth in the Sciences' explain his method for securing true knowledge. Basically the method entails finding the simplest parts which are known to be true, and ordering and enumerating these parts to understand the more complex whole. The results should be checked to make sure that there are no errors. Shea (1991: 131) explains more fully the method:' (a) nothing is to be assented to unless evidently known to be true; (b) every subject-matter is to be divided into the smallest possible parts, and each dealt with separately; (c) each part is to be considered in the right order, the simplest first; and (d) no part is to be omitted in reviewing the whole'. This is Descartes' method. Construct the problem out of the simplest elements possible, and rearrange those elements to solve the problem of the more complex. The symbolic descriptions fulfil Descartes' criterion for the right philosophical method upon which to proceed to secure true knowledge because the method appears to be based upon his success in algebraicized geometry. The relationship between Descartes' philosophical method and his algebraicized geometry is quite apparent. First, the (algebraic) elements are broken down into their simplest components to understand the more complex (symbolic) configuration. Second, the (symbolic) statements do not include any superfluous information which may function as a distraction. Third, the (symbolic) expressions may be checked to ensure there is no error. Descartes' procedures in geometry match his method for securing true knowledge. There are several important implications of Descartes' mathematics and philosophy which shaped the course of modern mathematics and science.


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Descartes' philosophy rested on mathematical formulations and procedures which revolutionized the nature of semiosis through the prominence accorded to the symbolic and the secondary position accorded to the linguistic. Language was considered inadequate for the knowledge to which Descartes aspired. Language belonged to the common-sense world of perceptions and sensory input which was deemed unreliable. The famous Cartesian mind and body duality was attached to the semiotic forms which were used to represent each realm. Symbolic and specialized mathematical forms of visual semiosis were located in the realm of the mind which was, according to Descartes, the proper site for securing knowledge. Language belonged to the realm of the everyday and to the body and its sensory apparatus. This remarkable shift in emphasis resulted in a sharp dichotomy between the different forms of semiosis, and this included a difference in the values attached to each. The differing values accorded to the sciences and the arts and social sciences continue today. The focus of concern in mathematics became curves or patterns which were exactly describable through the symbolism. The types of processes in these visual representations are spatial and temporal relations, and relative proportional rates of change. The major visual participants are lines, line segments, circles, arcs and curves and geometrical shapes which are the visual representations of the relations. Those relations are described symbolically through mathematical processes such as multiplication, addition, subtraction and division between the symbolic participants. The continuous nature of these relations is depicted graphically and described exactly through the symbolism. Human actors participating in material, affective, perceptive, behavioural or verbal processes in the context of the everyday were removed from the realm of mathematics and science. The new semiotic tools are designed to work within particular semantic fields, and these do not include the human realm of the material, the emotive, and the sensory which were considered superfluous. The human realm was put aside in this major re-evaluation of knowledge. The mystical claims of the Middle Ages were replaced with a new type of knowledge and a new basis for legitimizing truth. In addition, the claims upheld on the basis of mathematical descriptions were backed by experimental evidence. With the material success achieved through the mathematicization of nature, gradually the God which was central to Descartes and Newton's formulations (see Plate 2.5(1)) was removed from modern science. The significance of this re-contextualization in the modern mathematical view of the world is explored in Chapter 7. As Koestler (1959: 11) comments 'all cosmological systems [visions of the universe] reflect the unconscious prejudices, the philosophical or even political biases of their authors; and from physics to physiology, no branch of Science, ancient or modern, can boast freedom from metaphysical bias of one kind or another'. The question remains as to the impact of our adoption of the mathematical in a present-day context which differs from the cosmology within which it developed.

Plate 2.5(1) Illustration from Newton's (1729) The Mathematical Principles of Natural Philosophy


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Mathematical symbolism originated from rhetorical algebra in the form of linguistic descriptions and commands which explained the method through which to proceed. Syncopated algebra saw abbreviations for recurring participants, but basically the linguistic grammar still provided the basis for these discourses. However, through the work of Descartes, Newton and other mathematicians, especially Leibniz who 'made a prolonged study of matters of notation' (Book Two, Cajori, 1993: 180), the symbolism developed as the semiotic tool which was central to the mathematics which subsequently developed. The symbolic grammar was based on economy through the need to describe in the simplest and most condensed form that which also needed to be rearranged to explain the more complex. This meant the development of new systems in the grammar of mathematical symbolism which did not exist in language. There was to be no confusion, no room for error and no superfluous information in the new forms of reasoning provided by the mathematical symbolism. This grammar of mathematical symbolism is the focus of Chapter 4. From this point, Newton started a trend which could only be called a new world order. 2.6 A New World Order The new approach advocated by Descartes proved to be significant because Newton and others created a movement which involved a new representation of the physical world using new semiotic tools. In this movement, matter and perceptual data were re-admitted by Newton, but in a new mathematicized form (Barry, 1996). As Sweet Stayer (1988: 3) claims, while Newton explained the motion of bodies through his calculus and completed research in the fields of optics, tides, thin films and gravitation, The Mathematical Principles of Natural Philosophy was the culmination of his work, and it 'profoundly changed the perspective with which we view the world'. Newton's new semiotic constructions explained the visible world through invisible properties which were made 'real' or 'concrete' through mathematical symbolic description. One key to this success was that the mathematical symbolism, the visual images and language worked together. Descartes discarded sense data and developed a method with a form of semiosis which could describe exactly relations. These entities could be visualized and they could be described exactly in a symbolic form which allowed the rearrangement of those relations to solve problems and conceptualize the more complex phenomena. Newton dispensed with Descartes' position in that he accepted the world of perception, but at the same time he reconstructed that world using Descartes' semiotic tools. Newton used mathematical symbolism to create metaphorical entities which explained the everyday material world. Newton's new mathematical tools permitted exact description of that which was perceived in terms of the properties of matter. The physical world became the object of concern, and with this new engagement, the means for industrialization, colonialization and commerce rapidly increased. The reason for the status of mathematics is

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precisely the goals and objectives which have been fulfilled through this form of semiotic representation. Mathematics and science fulfil functions which have transformed the face of the world and life on earth. Newton instigated a movement which increased control over the physical world because he included experimentation as an integral part of his scientific method. Galileo first established the ideas of experimentation: 'Galileo's work laid the foundations of the modern scientific method which regards the collection of experimental evidence as the essential prelude to the formulation of scientific laws and theories' (Hooper, 1949: 201). One of Newton's major contributions was the use of technology and scientific equipment in a laboratory setting to test empirically his theories. The theories had to fit the empirical evidence, or at least are seen to fit. Science became a matter of description, prediction and prescription within the confines of the practices established by the laboratory and the semiotic tools which permitted those representations. Although matter may have been re-admitted by Newton, there were constraints on how the sensory phenomena could be viewed and described, and those constraints were established through his semiotic tools and the technology of his scientific equipment. As Descartes openly states, all 'superfluous' information was removed, and what remained was what was possible with the symbolism, visual images and language which formed the semantic circuit with which Newton constructed the new world order. New branches of mathematics have developed since Newton, and the idea of an ordered mechanical physical world has been largely abandoned with the development of the notion of chaos and dynamical systems theory. This new view of the world is based on the idea of non-linearity where it is assumed that the behaviour of physical systems is in fact indeterminate; that is, the behaviour of a system cannot be predicted exactly. Davies (1990) explains that the approach advocated by Descartes where systems are broken down into constituent components to understand the complex whole is reasonably successful because most physical systems behave in this linear format up to a certain point. This method of analysis is only partially successful however: 'On the other hand, they [all physical systems] turn out to be nonlinear at some level. When nonlinearity becomes important, it is no longer possible to proceed by analysis, because the whole is now greater than the sum of the parts' (ibid.: 16). When this point is reached, the constraints, boundary conditions and initial conditions of the system must be taken into account if the behaviour of the system is to be predicted with some degree of success. The new mathematics of non-linear dynamical systems theory is made possible through computer technology. As the computing ability increases, together with the potential for highly sophisticated dynamic graphical images, so the nature of the mathematics changes; that is, mathematics and science are intimately linked to the state of the art of computer technology which affords new possibilities in what has literally become a virtual world. Indeed, computer technology is such that visual images are now

EVOLUTION OF THE SEMIOTICS OF MATHEMATICS

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increasingly exploited as a semiotic resource that offers new possibilities for modelling the world. It could be that the superfluous information once discarded by Descartes can now be incorporated in a new more fully inclusive view of the world. In conclusion, mathematics and science offer a particular representation of the world - one that is limited by the semiotic tools and the technology employed in its construction. In order to appreciate the nature of that construction, texts and contexts must be analysed to understand the types of meanings that are made, and the means through which this is achieved. For this reason, the grammatical systems for mathematical symbolism and visual display are presented in Chapters 4—5. After discussing the unique grammars for each resource, the semantic circuit in mathematics involving the linguistic, the visual and the symbolic is discussed in Chapters 6-7. Discourse analyses of mathematical texts demonstrate that intersemiosis across the three resources is critical for the semantic expansions that take place in mathematics. The analyses represent a close engagement with mathematics as a multisemiotic discourse in order to appreciate the potential and the limitations of the meanings which are made.

3 Systemic Functional Linguistics (SFL) and Mathematical Language

3.1 The Systemic Functional Model of Language

The investigation of multisemiosis in mathematics is based on Michael Halliday's (1973, 1978, 1985, 1994) systemic functional (SF) approach to language which has been extended by Jim Martin and others to incorporate discourse systems (Martin, 1992; Martin and Rose, 2003), genre and ideology (for example, Christie, 1999; Christie and Martin, 1997; Hasan, 1996b; Martin, 1997). An outline of systemic functional (SF) theory and the accompanying grammatical and discourse systems of language is provided in order to explain the conceptual apparatus underlying this study of mathematics. The discussion is necessarily technical, but further explanations of SF theory are provided elsewhere (for example, Bloor and Bloor, 1995; Eggins, 1994; Eggins and Slade, 1997; Halliday, 1994; Halliday and Matthiessen, 1999; Martin etal, 1997; Matthiessen, 1995; Thompson, 1996) including a collection of Halliday's writings (Webster, 2002-). The description of Systemic Functional Linguistics (SFL) and the discussion of the nature of mathematical language in this chapter function to contextualize the systemic frameworks for mathematical symbolism and visual display developed in Chapters 4-5. This leads to an investigation of the meaning arising from the integrated use of language, mathematical symbolism and visual display in Chapters 6-7. The description of mathematical and scientific language in this chapter is general (for a more detailed analysis, see Halliday and Martin, 1993; Halliday and Matthiessen, 1999; Martin and Veel, 1998) as the major concern of this study is the extension of SF theory to mathematical symbolism and visual display in order to investigate the multisemiotic nature of mathematical discourse. In what follows, Halliday's SFL model and linguistic systems at the rank of clause and clause complex and Martin's discourse systems at the level of paragraph and text are used to examine the nature of mathematical language. It becomes apparent in this discussion that the study of mathematical and scientific language needs to take into account the meaning arising from the symbolism and visual display. Martin's discourse systems where meaning is made across stretches of text are therefore

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extended in Chapter 6 to include meaning arising from intersemiosis across linguistic, visual and symbolic components of the text. In addition, the description of grammatical metaphor in this chapter is further developed through an examination of semiotic metaphor in Chapters 6-7. In a similar fashion, the discussion of register, genre and ideology is revisited in relation to the multisemiotic nature of the discourse of mathematics in Chapter 7. Halliday 's SF Theory of Language The fundamental assumption behind Halliday's SF social-semiotic theory is that language is a resource for meaning through choice. Halliday (1994) comprehensively documents the grammatical systems through which language is used to achieve different functions. For, as Halliday explains, language has evolved to satisfy human needs and its grammatical organization is therefore functional with respect to those needs. Halliday (ibid.: xiii) states: 'A functional grammar is essentially a "natural" grammar, in the sense that everything in it can be explained, ultimately, by reference to how language is used.' Any instance of written or spoken language does not unfold haphazardly as an abstract artefact as formal linguists would lead us to believe, but rather all texts are constructed in some context of use. The choices in the text's patternings reflect the uses that language is serving in that particular instance. The underlying assumptions of SFL may be contrasted to the position adopted in formal linguistics where language is conceptualized as a system of rules. Descriptions in these traditions show which sentences are acceptable and explanations reveal 'why the line between in and out falls where it does in terms of an innate neurological speech organism' (Martin, 1992: 3). Rather than adopting an individual mentalist perspective, SFL views language as a resource consisting of a network of relationships. Descriptions show 'how these relationships are interrelated' and explanations reveal 'the connections between these relations and the use to which language is put' (ibid.: 3). SFL is thus orientated to choice, 'what speakers might and tend to do', as opposed to restriction, 'what speakers are neurologically required not to do' (ibid.: 3-4). The SFL approach is concerned with the analysis of how language is used to achieve certain goals through the description of lexicogrammatical (that is, lexical and grammatical) and discourse systems, and the analysis of the choices that have been made in any instance of language use. SFL discourse analysis is a critical interpretation of how language choices function to construct a particular view of reality, and the nature of social relations that are enacted in that construction. SFL evolved from Firthian linguistics and consequently is a type of system/structure theory where the key idea is meaning through choice from the available systems. Following Hjelmslev (1961), paradigmatic relations are mapped onto the available options in the system network (the range of

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choices) and syntagmatic relations are mapped on to actual choices (the process which takes the form of a chain of words). The concept of 'realization' relates system and process in that realization statements specify the systems in process and give the structural arrangement of the selected options. Halliday does not privilege either system or process: 'I prefer to think of these [system and process] as a single complex phenomenon: the "system" only exists as potential for the process, and the process is the actualisation of that potential. Since this is a language potential, the "process" takes the form of what we call a text' (Thibault, 1987: 603). The systems in Halliday's grammar of English are organized according to meaning. Halliday separates the two main types of meaning into the 'ideational' or the reflective, and the 'interpersonal' or the socially active. Halliday further separates ideational meaning into two components, the 'experiential' and the 'logical', which are respectively concerned with the construction of experience and logical relations in the world. Halliday also identifies the enabling function of language, the 'textual' component which organizes language choices into coherent message forms. These four types of meaning, the experiential, logical, interpersonal and textual, are called the metafunctions of language as they are manifestations of the general purposes of language: ' (i) to understand the environment (ideational), and (ii) to act on the others in it (interpersonal)' (Halliday, 1994: xiii). The SF approach means that although the grammatical classes such as nouns, verbs, adjectives and so forth still have a place, for example, in the descriptions of grammatical metaphor (Derewianka, 1995; Halliday and Martin, 1993; Martin et al., 1997; Simon-Vandenbergen et al, 2003), the elements of language are described by functional rather than word class labels. Language is conceived as an 'organic configuration of functions' and 'each part is interpreted as functional with respect to the whole' (Halliday, 1994: xiv). There is equal emphasis on the interpretation of the interpersonal metafunction as well as experiential, logical and textual meanings in SFL. As Poynton (1990) explains, the focus on social relations and the expression of personal attitudes and feelings has traditionally been marginalized in the majority of linguistic theories. The focus on system and referential meaning in linguistics, perpetuated with Chomsky's reformulation of Saussure's (1966) langue/parole (language system versus language use) distinction as competence/performance, was accompanied by an explicit emphasis on the cognitive domain (for example, Chomsky, 1965, 2000). Poynton (1990) explains that such cognitively orientated conceptions of language support dichotomies such as objective versus subjective, and reason versus emotion. As Poynton claims, the higher values accorded to objectivity and reason have obvious significance in the development of mathematics and science, and they have also been invoked in areas of social control. These issues are explored through an interpretation of the concepts of reason, objectivity and truth based on the analysis of a mathematics text in Chapter 7.


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Halliday (1994) is largely concerned with lexicogrammar at the ranks of word, word group/phrase, clause and the clause complex. On the other hand, Martin (1992; Martin and Rose, 2003) is concerned with metafunctionally based systems which operate across paragraphs and the whole text. Martin's work follows from Halliday and Hasan's (1976) systemic analysis of textual cohesion where the basic opposition is between structural (grammatical) and non-structural (cohesive) devices. Martin (1992: 1) organizes his divisions stratally 'as an opposition between grammar and semantics (between clause orientated and text orientated resources for meaning)'. Martin thus establishes a separate discourse semantics stratum to complement Halliday's lexicogrammar. Martin's proposals lead to a language plane with two strata,1 discourse semantics and lexicogrammar, and an expression plane which is concerned with phonology and graphology/ typography (Eggins, 1994: 81-82). The resulting SF model of language, which also includes the communication planes of register, genre and ideology, is displayed in Table 3.1 (1). Halliday's Lexicogrammatical Systems and Martin's Discourse Systems

The major systems in Halliday's lexicogrammar and Martin's (1992; Martin and Rose, 2003) metafunctionally based discourse systems are listed in Table 3.1(2). Following systemic conventions, the lexicogrammatical and discourse systems are capitalized. The major lexicogrammatical systems are MOOD for interpersonal meaning, THEME for textual meaning and Table 3.1(1) Language, Expression and Communication Planes IDEOLOGY GENRE REGISTER LANGUAGE CONTENT

Discourse semantics

Paragraph and text Lexicogrammar

Clause complex Clause Word group and phrase Word

EXPRESSION

Phonology Graphology/Typography

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Table 3.1(2) Metafunctional Organization of Halliday's (1994) Lexicogrammatical Systems and Martin's (1992; Martin and Rose, 2003) Discourse Systems Metafunction

Lexicogrammar

Discourse Systems

interpersonal

clause: MOOD; MODALIZATION; MODULATION; POLARITY; TAGGING; VOCATION; ELLIPSIS

NEGOTIATION (exchange rank including SPEECH FUNCTION at the move rank) Structure: Exchange Structure linking moves

word group: PERSON; ATTITUDE (attitudinal modifiers, intensifiers); COMMENT (comment adjuncts); LEXIS (expressive words, stylistic organization of vocabulary) textual

clause: THEME clause and word group: SUBSTITUTION; ELLIPSIS word group: DEIXIS (nominal)

APPRAISAL Structure: surges, flows and falls mapped through word groups, phrases and clauses in text

IDENTIFICATION (phoricity, reference) Structure: reference chains linking participants

logical

clause complex: LOGICO-SEMANTIC RELATIONS and INTERDEPENDENT

CONJUNCTION and CONTINUITY (based on classifications of LOGICO-SEMANTICS RELATIONS and semantic relations respectively) Structure: conjunctive reticula linking messages

experiential

clause: TRANSITIVITY; AGENCY

IDEATION (lexical relations) Structure: lexical strings and nuclear relations linking message parts

word group: TENSE; LEXIS (lexical 'content'); collocation

TRANSITIVITY for experiential meaning at the clause rank, which is the basic unit in which the semantic features are represented. The elements in the clause (word, word group/phrase) are explained by their functions in each of the metafunctionally based systems. At the rank of clause complex or sentence, the systems for logical meaning are LOGICO-SEMANTIC RELATIONS and INTERDEPENDENT. Halliday's systems are discussed in

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relation to the types of selections found in mathematical discourse in Sections 3.2-3.5. These systems are also considered in relation to the grammatical organization of mathematical symbolism in Chapter 4. The realization structures for Halliday's grammatical systems take different forms. Textual meanings arising from the system of THEME and interpersonal meanings through the system of MOOD are described by paradigmatic oppositions realized through syntagmatic structures of the functional categories. Experiential meanings from TRANSITIVITY choices, on the other hand, are represented as clusters of participant/process/ circumstance rather than sequences of functional elements in the clause. In addition, the patterns of realization vary across the metafunctions. Following Halliday, the realizations of the metafunctions in discourse take the forms of particle-like experiential meanings, irregular prosodic swells of interpersonal meanings (where the concept of volume comes into play) and regular periodic wave-like textual meanings. The realization of the logical metafunction at the rank of clause complex through the system of LOGICAL-SEMANTIC RELATIONS is somewhat different from the other grammatical systems as particular elements (for example, the structural conjunctions 'and' and 'or') may be selected more than once in a clause complex, while the other systems have multiple variables which may be selected only once in the clause. Logical meaning is described by the types of INTERDEPENDENCY relations (dependent or independent) and by LOGICO-SEMANTIC relations between clauses (relations of logical expansion or projection of speech and thought). Martin's (1992; Martin and Rose, 2003) discourse systems include NEGOTIATION, APPRAISAL, IDENTIFICATION, CONJUNCTION and CONTINUITY, and IDEATION. The metafunctional organization and structure of the discourse systems are included in Table 3.1(2). NEGOTIATION is orientated towards spoken discourse, but Martin and Rose (2003) also include the system of APPRAISAL for capturing graduations of attitude (affect, judgement, appreciation) and engagement in written (and spoken) discourse. 'Appraisal is concerned with evaluation: the kinds of attitudes that are negotiated in a text, the strength of those feelings involved and the ways in which values are sourced and readers aligned' (ibid.: 22). Martin's concept of discourse systems is useful for the analysis of stretches of text which involve language, visual images and mathematical symbolism. Martin's frameworks, however, need reworking as they are developed for the analysis of linguistic text (that is, intrasemiosis in language), rather than the analysis of meaning within and across different semiotic resources (that is, intrasemiosis in mathematical symbolism and visual images, and intersemiosis across the three semiotic resources). Discourse systems similar to those proposed by Martin are introduced in the SF frameworks for mathematical symbolism and visual images in Chapters 4 and 5, and the framework for intersemiosis in Chapter 6. The analysis of intra- and inter-semiosis in mathematical discourse includes the typography

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of the text at the expression stratum, the importance of which becomes evident in Chapters 4-6. This stratum has not typically been included in SFL analysis. Martin's discourse structures interact systematically with each other and the lexicogrammatical structures (giving rise to incidences of grammatical metaphor, for example) resulting in the 'texture' of a text. The ways in which the discourse systems co-operate with each other to make a text is not as well understood as the nature of interaction across the grammatical and discourse strata (see for example, Halliday, 1994; Hasan, 1984). Martin (1992:392) refers to the systematic interaction between discoursal and grammatical structures as modal responsibility, cohesive harmony and the method of development of the text. While these formulations are not specifically developed in this study, the grammatical density arising from the interactions between language, visual images and the symbolism across different strata becomes apparent in the analyses of the mathematics texts presented in Chapters 6-7. The texture of discourse in this case involves the dense patterns which emerge from the integrated use of language, mathematical symbolism and visual display (O'Halloran, 2000, 2004c). SFL Discourse Analysis

In SFL discourse analysis, clauses are marked by slashes / / . . . / / and the elements within each clause are analysed according to the metafunctionally based grammatical and discourse systems. Elements in the clause are analysed several times, and functional labels are attached according to choices made from each system. Clauses are also classified as major or minor, and complete or ellipsed. In the case of spoken discourse, abandoned clauses may also be tagged. Minor clauses are 'clauses with no mood or transitivity structure, typically functioning as calls, greetings and exclamations' (Halliday, 1994: 63). Eggins (1994: 172) explains that these minor clauses are 'typically brief, but their brevity is not the result of ellipsis'. The classification of clause type is useful for examining interpersonal patterns of domination and deference. Clauses are classified as to whether they contain rankshifted or embedded elements. Using Halliday's (1994: 63) notion of ranks (word, word group/phrase, clause and clause complex), rankshifting is the process whereby a clause or phrase functions at the lower rank of word or word group. That is, embedded clauses (indicated by sets of square brackets [ [ . . . ] ]) and phrases (indicated by the square brackets [ . . . ] ) function within the structure of a word or word group, thus shifting rank from clause and phrase to the lower rank of word/word group.2 Halliday (ibid.: 242) explains that embedded clausal and phrasal elements may function as a Postmodifier in nominal groups; for example, //the job [[I want]] was advertised//; and adverbial groups; for example, //she reacted more strongly [[than they expected]]//. Alternatively the rankshifted element may function as a Head of a nominal group; for example, //[[that so many


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staff are leaving]] is cause for concern//. As may be seen from these examples, rankshifting in language serves the important function of packing information into the clause. The concept of rankshift is particularly significant in the grammar of mathematical symbolism in Chapter 4 where the lexicogrammatical strategies for encoding meaning in the symbolism are seen to be different from those found in mathematical and scientific language. In the following discussion, the major grammatical and discourse systems are explained, and the nature of the selections found in mathematical and scientific language is discussed according to metafunction. The following linguistic extract from Stewart's (1999: 132) textbook Calculus is used to illustrate features of mathematical language: From Equation 3 we recognize this limit as being the derivative off at x,, that is /' (x,). This gives a second interpretation of the derivative: The derivative/' (a) is the instantaneous rate of change of y=f(x) with respect to xwhen x= a. The connection with the first interpretation is that if we sketch the curve y=f(x), then the instantaneous rate of change is the slope of the tangent to this curve at the point where x = a. This means that when the derivative is large (and therefore the curve is steep, as at the Point P in Figure 4), the ^values change rapidly. When the derivative is small, the curve is relatively flat and the ^values change slowly.

The multisemiotic text for this extract is reproduced in Plate 6.3(2) in Chapter 6 where the meanings arising from intersemiosis between the linguistic, symbolic and visual choices in the text are analysed. It becomes apparent in the analysis of Stewart (1999: 132) that mathematical and scientific language must necessarily take into account the visual and symbolic components of the text. 3.2 Interpersonal Meaning in Mathematics

The analysis of interpersonal meaning is concerned with the nature of the social relations which are enacted through linguistic choices from the systems listed in Table 3.1 (2). The description of the system of MOOD is given in Halliday (1994: 71-105). In essence, the MOOD system (where choices are made for Subject, Finite, Mood Adjuncts, Comment Adjuncts, Predicator, Complement and Circumstantial Adjuncts) is related to the SPEECH FUNCTION which is concerned with the giving/demanding information (statements and questions) and goods and services (commands and offers). The SFL analysis is concerned with choice: how do interactants negotiate the exchange of information and goods and services, and what does this reveal about their social relations? A one-to-one relationship between the grammatical classes of MOOD (declarative, interrogative, imperative and exclamative) and the SPEECH FUNCTION (statement, question, command and offer) does not exist, so the co-text and context are taken into consideration in the analysis. However, the congruent or unmarked case is that statements are realized

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through declarative Mood (SubjectAFinite, for example, 'this is . . . ' ) , questions are realized through interrogative Mood (FiniteASubject, for example, 'is this . . .?') and the WH-element (WH, for example, 'why . . .?'), commands through imperative Mood (Predicater, for example, 'solve . . .') and offers through modulated interrogative Mood (modulated FiniteASubject, for example, 'would you . . .'). Incongruent selections result in interpersonal metaphors where there is variation (for whatever reason) in the enactment of the social relations. The system networks for MOOD and SPEECH FUNCTION are displayed in Figure 3.2 (I), 3 together with the Exchange Structure which consists of sequences of moves. SPEECH FUNCTIONS include responses to the statements, questions, commands and offers and Martin's (1992: 66-76) 'dynamic moves' for spoken discourse, which are requests and responses for the tracking moves of Backchannel, Clarification, Check, Confirmation and Challenging moves. The SPEECH FUNCTIONS of Call and Greeting have also been included, together with moves of Reacting in Figure 3.2(1). These classifications frame the types of moves found in written mathematical texts. As displayed in figure 3.2(1), the SPEECH FUNCTION and MOOD systems relate to the Exchange Structure, which is Martin's (1992) discourse system of NEGOTIATION. The Exchange Structure is based on Halliday (1994) where the opposition is between information and goods and services moves. Following Berry (1981), moves are classified as primary knower (Kl) and primary actor (Al) moves, and secondary knower (K2) and secondary actor (A2) moves. Kl is the speaker/writer who has the information which is being negotiated, and Al is the participant who performs the action. The Exchange Structure consists of obligatory moves Kl and Al and three optional moves. These are delaying moves (dKl and dAl), secondary knower (K2) and secondary actor moves (A2). Berry's (1981) moves are developed from Halliday's (1994: 69) SPEECH FUNCTION classifications of 'initiating and responding' to the 'giving and demanding' of goods and services and information. While all the classifications presented in Figure 3.2(1) do not typically occur in written mathematical texts, it is useful to consider the selections afforded in dynamic spoken contexts to situate the type of discourse found in written mathematics. In order to incorporate Ventola's (1987, 1988) 'move complexes', the categories of initiation, request, response and closure moves are supplemented with 'K-Continuation' and 'A-Continuation', and 'K-x' and 'A-x' moves. These move-complexes are based on Halliday's (1994: 220) LOGICO-SEMANTIC RELATIONS of expansion (elaboration, extension and enhancement) and projection (locution and idea) and interdependency relations (see Section 3.4 on logical meaning). A 'continuation' move realizes a continuation of the same SPEECH FUNCTION which was established in the previous move even though the clause selects independently for MOOD. This move is realized through paratactic relations of interdependency between the two clauses. An 'x' move realizes a continuation of

Proposition Dynamic move Followup Reacting

Call Greeting Knowledge

EXCHANGE new

continuation exchange number

dK1 K1 K2 K-lnitiation K-Continuation K-x K-Response K-Request

SPEECH FUNCTION

Action

A-lnitiation A-Continuation A-x A-Response A-Request

declarative WH - interrogative MOOD

YN - interrogative imperative

Proposal Dynamic move Followup Reacting dA1 A1 A2

statement exclamative expletive question acknowledgeme contradiction answer disclaimer clarification challenge backchannel check confirmation

SPEECH FUNCTION

command offer compliance refusal acceptance rejection clarification challenge backchannel check confirmation

Figure 3.2(1) NEGOTIATION (Exchange Structure), SPEECH FUNCTION and MOOD

paralinguistic none

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the same SPEECH FUNCTION in hypotactically related finite and nonfinite clauses. In this way, it is possible to map each clause as an element in the unfolding Exchange Structure. SFL analysis involves analysing choices in the text for mood, speech function and negotiation (exchange structure). While this framework is used to discuss the linguistic selections in Stewart (1999: 132), it is apparent that discourse moves in written mathematics often involve shifts from one semiotic resource to another. Stewart (1999: 132) makes reference to 'Equation 3' and 'Point Pin Figure 4'. In a similar fashion, the command to solve a problem is typically undertaken symbolically. However, the SFL grammatical and discourse frameworks provide the starting point for the development of grammars for the symbolism and visual display and the theorization of intersemiotic shifts and transitions in mathematical discourse. In what follows, the nature of linguistic selections for interpersonal meaning in Stewart (1999: 132) are investigated. The selections from the systems of MOOD, SPEECH FUNCTION and NEGOTIATION function to establish unequal relations between the writer and the reader of the mathematics text. In the following chapters, it becomes apparent that the dominant position of the writer is reinforced across choices for mathematical symbolism and visual display. Similarly, unequal social relations are established between the teacher and the students in the context of the mathematics classroom. While mathematical pedagogical discourse is dominated by the teacher (Veel, 1999), the nature of those social relations in classrooms differs on the basis of gender and social class (O'Halloran, 1996, 2004c). The nature of the linguistic selections which reinforce the position of dominance of the author of the mathematics text is discussed below. Given the monologic format of the written discourse of mathematics, the writer assumes the speech roles. Foremost, the writer is the primary knower (Kl) who gives information in the form of statements through declarative Mood. These statements are typically complete, and so the writer provides detailed information. In the case of mathematics textbooks such as Stewart (1999), the author also assumes the role of secondary knower (K2) who asks the questions and, most typically, provides the answers (Kl). The writer takes the role of the one who commands (A2). In addition, as primary actor (Al), the writer checks that commands have been completed correctly by providing the solutions to problems. Mathematics lends itself to these types of social relations between the writer and the reader as mathematics is a written discourse. The Exchange Structure typically involves long sequences of moves as seen in Stewart (1999: 132). The extended exchanges contribute to the steady interpersonal rhythm of mathematical discourse, with its overarching aim of deriving results through long sequences of logical reasoning. Stewart (1999: 132) attempts to vary the interpersonal nature of the extended exchanges through various strategies, which include the use of 'we' as Subject. However, such selections as 'we' give rise to interpersonal

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metaphors where it is clear that the relations are being manipulated. For example, in the statement 'From Equation 3 we recognize this limit as being the derivative of/at xl that is,/' (xj)', the inclusive 'we' serves interpersonal rather than experiential meaning as the statement may be seen as a metaphorical variant for the command 'Recognize this limit as being the derivative of/at x1} that is,/' (xj)'. Alternatively, the process 'recognize' in the projecting clause may be seen to be metaphorical and unnecessary with respect to the more direct statement 'this limit is the derivative of/at x1; that is/' (xj)'. Although the writer attempts to vary the social relations with the reader through Subject choice and metaphorical expressions, the reader nonetheless remains the receiver of information, and the one whose answers and responses are checked against those provided by the author of the mathematics text. The degrees of probability and obligation associated with the linguistic statements, questions, commands and answers in mathematical discourse are similarly consistent. Halliday's (1994: 354—363) graduations in probability and usuality (MODALIZATION) and inclination, obligation and potentiality (MODULATION) are associated with propositions ('information') and proposals ('goods and services') respectively. The descriptive categories for MODALIZATION and MODULATION and the value and orientation of the selection are displayed in Figure 3.2(2).

probability MODALITY usuality obligation C-MODULATION potentiality inclination O-MODULATION potentiality

VALUE

-maximal high median low objective subjective

ORIENTATION

explicit implicit

Figure 3.2(2) MODALIZATION and MODULATION

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The statements are unmodalized in terms of graduations of probability and usuality in the extract from Stewart (1999: 132). For instance, the absence of modality (realized through the Finite selections such as 'might', 'could', or 'should') functions to make the mathematics statements appear as correct and factual. The lack of modalization is accompanied by maximal obligation in the commands. POLARITY is simply positive ('is') and negative ('is not'). This contributes to the steady interpersonal orientation of a discourse which possesses an unqualified level of certainty. As noted in Chapter 4, probability in mathematical discourse is typically expressed symbolically through relational clauses. For example, probability may be expressed through approximations such as x =» 0.5. The typical absence of selections from Halliday's (1994: 82-83) system of MOOD ADJUNCTS displayed in Figure 3.2(3) also functions to create an aura of factuality. For example, in Stewart (1999: 132) Mood Adjuncts indicating plays with probability (for example, 'possibly', 'perhaps' and 'certainly') are not selected. Instead a certain presumption arises from the unmodalized statements and unmodulated commands, the nature of the processes which are selected (see Section 3.3) and the long implication sequences in the Exchange Structure arising from selections for logical meaning (see Section 3.4). This is not to say that Mood Adjuncts are not selected in mathematical discourse. However, the nature of such adjuncts may replicate the high level of presumption and obviousness found in the pedagogical discourse of mathematics (O'Halloran, 1996, 2004c). The objective, rational and factual stance of mathematics is the product of the nature of the selections for interpersonal meaning as they combine with a limited range of process types and participants (see Section 3.3) with an emphasis towards logical meaning (see Section 3.4). Descartes' removal of the human realm in mathematics and science is apparent in modern mathematics. The types of interpersonal choices from language, mathematical symbolism and visual display function to simultaneously restrict and expand experiential and logical meaning in mathematics. In the context of the mathematics classroom, teachers introduce a variety of interpersonal strategies to maintain solidarity and group cohesiveness and to relieve the interpersonal stance of the subject matter (O'Halloran, 1996, 2004c). However, if one considers high school mathematics texts, books on mathematics and other generic forms of mathematical discourse, the probability usuality MOOD ADJUNCT

presumption inclination time degree

Figure 3.2(3) MOOD ADJUNCTS

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interpersonal stance of mathematical written discourse is largely consistent. This observation explains the inclusion of metaphorical forms of expression and non-generic choices such as cartoons, drawings and photographs which are inserted to disrupt the interpersonal orientation of mathematics texts. Furthermore, the issue of lexical choice for interpersonal meaning in mathematics is addressed below. Halliday (1961: 267) states that 'The grammarian's dream is ... to turn the whole of linguistic form into grammar, hoping to show that lexis can be denned as "most delicate grammar" '(see Hasan, 1996a). Two major areas of interest are lexis specific to the field of mathematics which is considered under experiential meaning (see Section 3.3), and lexical items which are interpersonally marked. The relevant notion is one of 'core vocabulary' where certain lexical items are more central than others in describing experiential or intersubjective reality. Carter describes tests for coreness which involve syntactic and semantic relations, and neutrality. From this perspective, the coreness of lexical items is the extent to which they are 'more tightly integrated than others into the language system; that is, they occupy places in a highly organized network of mostly structurally-semantic and syntactic interrelations' and are 'more discoursally neutral than others, that is, generally they function in pragmatic contexts of language use as unmarked and non-expressive' (Carter, 1998: 36). Expressive linguistic selections orientated towards interpersonal meaning are included in Martin and Rose's (2003: 54) system network for APPRAISAL, which is reproduced in Figure 3.2(4). APPRAISAL is concerned with evaluation: how the text functions to align the reader or speaker with the various propositions or proposals which are put forth. This includes lexical items and cases of amplification, special forms of address and so forth. Further research will see the development of the system of APPRAISAL so that different strategies for positive and negative evaluation monogloss ENGAGEMENT heterogloss

PROJECTION MODALITY CONCESSION

AFFECT APPRAISAL

ATTITUDE

JUDGEMENT APPRECIATION

FORCE GRADUATION FOCUS

Figure 3.2(4) APPRAISAL Systems Reproduced from Martin and Rose (2003: 54)

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may be uncovered. The three main appraisal systems given by Martin and Rose (2003: 54-55) are attitude, amplification (or graduation) and source (or engagement): Attitude comprises affect, judgement and appreciation: our three major regions of feeling. Amplification covers grading, including force and focus; force involves the choice to raise or lower the intensity of gradable items, focus the option of sharpening or softening an experiential boundary. Source covers resources that introduce additional voices into a discourse, via projection, modalization, or concession; the key choice is one voice (monogloss) or more than one voice (heterogloss).

The absence of lexical items orientated towards expressive or evaluative interpersonal meaning is apparent in the extract from Stewart (1999: 132). Lexical choice in mathematics is largely orientated towards experiential and logical meaning rather than interpersonal meaning. This does not mean, however, that the mathematics writer does not make evaluations. Appraisals of what is presented in mathematics exist in different genres in different forms. For example, the authors of research papers in mathematics presumably cast a favourable impression on the results which are established. However, such judgements presumably appear as factual rather than evaluative. For example, Mood Adjunct selections (for example, 'of course' and 'typically') which may combine with an explicit objective orientation towards modality ('it is certain' and 'it appears that') mean that evaluations are made through grammatical choices which do not necessarily include interpersonally expressive lexis (for example, 'that is excellent' or 'that is really ridiculous'). The linguistic strategies for evaluation in mathematics and scientific discourse require further research. In addition, perhaps APPRAISAL may be understood as a meta-system arising from the layering and juxtaposition of functional choices across experiential, interpersonal, textual and logical systems, rather than a discourse system in its own right. Further work is needed, however, to establish how such layers function to orientate the reader. The apparent lack of the need to explicitly evaluate contributes to the view of mathematics as a rational discourse of truth. However, as discussed in Chapter 2, mathematics dispensed with many realms of human activity. As the semiotic which provides the meta-discourse for that which is performed symbolically and visually, language choices in mathematics are circumscribed within certain semantic domains. The limited fields of meaning are considered in the discussion of experiential meaning in Section 3.3 and the packing of that information through grammatical metaphor in Section 3.6. However, the point is that the discourse of mathematics appears as factual and objective truth because of the types of interpersonal choices which are made using language, and the precise organization of those choices in the mathematics text (see Section 3.5 for textual meaning). This orientation is supported by the nature of experiential and logical choices (see Sections 3.3 and 3.4), and by the available options in the system networks for the grammar of visual images and the

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symbolism. This point is explored further in the chapters concerned with the symbolism, visual images and intersemiosis in mathematics. 3.3 Mathematics and the Language of Experience

Following Halliday (1994), experiential meaning at the rank of clause is realized through the system of TRANSITIVITY displayed in Figure 3.3(1). The construction of experience takes the form of choices for process, participants and circumstance. Halliday (1994) includes the ergative interpretation of experience in the form of AGENCY. The associated functional elements are the Medium, Agent, Beneficiary, Range and Circumstance. Halliday's (1994: 166) descriptive categories have been extended to include mathematical 'Operative' processes in mathematical discourse as displayed in Figure 3.3(1). This new process type initially appeared in mathematical symbolism in the form of mathematical processes such as addition, subtraction, multiplication, division, powers and roots, and other mathematical operations. The meanings of these processes in mathematical symbolism do not accord with existing processes categories. The linguistic versions of these process types have thus been categorized as Operative processes. The rationale and justification for the inclusion of Operative processes in the mathematical symbolism is found in Chapter 4. The stages through which mathematics became concerned with particular realms of meaning to the exclusion of others are discussed in Chapter 2. Mathematics dispensed with the human realm, and became concerned with dynamic relations which could be viewed visually and described symbolically. Relations took a visual form, and linguistic descriptions shifted to the symbolic formulations. As language functions as the meta-discourse for these descriptions and visual instantiations, the nature of experiential meaning in mathematics simultaneously expanded to incorporate the new meanings, and contracted to the limited semantic realms with which the visualizations and symbolic descriptions were concerned. The impact on the nature of mathematical language arising from the semantic expansions made possible through the symbolism and visual display may be seen in Stewart (1999: 132). This includes the relatively high incidence of relational processes and the metaphorical nature of the participants. These features of experiential meaning in mathematical language are considered below. The major process type found in mathematical language appears to be the relational process, which Halliday (1998: 193) explains is the favoured process type in science. It appears that as mathematical symbolism became concerned with the description of relations, the same shift occurred within language which was being used to describe and contexualize the visualizations and symbolic descriptions. Halliday (1993a, 1993b, 1998), Halliday and Matthiessen (1999) and Martin (1993a, 1993b) explain that the regrammaticization of experience which takes place through scientific language involves relational processes and entities in the form of grammatical

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Actor Goal Range Recipient Client

Material

Senser Phenomenon

Mental

Sayer Verbiage Receiver

Verbal

Intensive Attributive

Circumstantial

PROCESS

Possessive

Relational

Carrier Attributor Beneficiary Attribute

Intensive

PARTICIPANTS Identifying

Circumstantial Possessive

Existential

Existent Behaver

Behavioural

Behaviour

X1 X2 X3 Xs

Operator Operative

Participant

Extent

duration distance

Location

Manner CIRCUMSTANCE Cause

temporal spatial means quality comparison reason purpose behalf fcomitation

Accompaniment

u

Matter Role

Figure 3.3(1) The TRANSITIVITY System

addition

Identified Identifier

Token Value

(SFL)


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metaphors, in particular, nominalizations. Such metaphorical participants permit experiential meaning to be economically packaged within the nominal group structures which are aligned through relational clauses. The impact of this regrammaticization of experience is discussed in relation to grammatical metaphor in Section 3.6. The relatively high incidence of relational processes and metaphorical participants is found in mathematical language. For example, relational processes realized through 'is' and grammatical metaphors (bold) appear in Stewart (1999: 132): '//The connection with the first interpretation is [ [that if we sketch the curve y =f(x), \ \ then the instantaneous rate of change is the slope of the tangent to this curve at the point [[where x= a ]]]]'. The repacking of experiential content through relational processes and grammatical metaphor is reconsidered in Chapter 6 where the notion of grammatical metaphor is linked to semiotic metaphor. The reasons for the current forms of scientific and mathematical language include the impact of the functions which are fulfilled by mathematical symbolism and visual images. The basis of the discourse system of IDEATION for experiential meaning is lexis (Martin, 1992). The discourse units underlying the lexical items are lexical relations which are concerned with (i) taxonomic relations, (ii) nuclear relations and (iii) activity sequences. The discourse structures realizing lexical relations are called lexical strings which run through the text. In the case of taxonomy, the two types of lexical relations are superordination involving subclassification and composition involving part/whole relations. The types of taxonomic relations are summarized in Eggins (1994: 101-102). In mathematics, the taxonomies for mathematical terms are extended and precise; for example, triangles are defined according to the size of the angles and sides. This serves to order mathematical reality in exact ways, leading to condensation in mathematical texts; for example, the term 'isosceles triangle' incorporates a range of meanings. Mathematical taxonomies, however, are not explored here. The second category of lexical relations involves nuclear relations. 'Nuclear relations reflect the ways in which actions, people, place, things and qualities configure as activities in activity sequences' (Martin, 1992: 309). These relations have previously been handled in SFL under collocation. In the case of mathematics, nuclear relations stretch across linguistic, visual and the symbolic components of the mathematics text. Nuclear relations are realized through configurations of Halliday's functional categories of process, participant and circumstance in the system of TRANSITIVITY, and the corresponding systems in mathematical symbolism and visual display. The model of nuclearity adopted for language and the mathematical symbolism follows Martin (1992: 319). Centre PROCESS = Range: process

Nucleus + MEDIUM + Range: entity

Margin Periphery + AGENT x CIRCUMSTANCE + BENEFICIARY

78


The third category of lexical relations is expectancy and implication relations between activities in activity sequences: 'These relations are based on the way in which the nuclear configurations . . . are recurrently sequenced in a given field' (Martin, 1992: 321). The relations in mathematics are realized through conjunctive relations, with implication relations typically involving conditional and consequential type relations. As suggested by the extract from Stewart (1999: 132), given the emphasis on logical meaning and the derivation of results, implication chains involving the semiotic construction of mathematical knowledge through language, symbolism and visual images are extended and complex (O'Halloran, 1996, 2000). Halliday (1978) explains that the need to conceptualize abstract relations in mathematics using linguistic modes of expression causes grammatical problems. Apart from borrowing everyday linguistic terms, mathematical language is technical and often involves complex taxonomies of terms in nominalized forms. Halliday (1993b: 69-85) describes the difficulties in mathematical and scientific language which involve interlocking definitions, technical taxonomies, special expressions, lexical density, syntactical ambiguity, grammatical metaphor and semantic discontinuity. However, these problems cannot be viewed in isolation. Rather the difficulties with mathematical language must be viewed in connection with symbolic and visual descriptions. Further to this, the texture of mathematical discourse (linguistic, visual and symbolic) involves grammatical intricacy (like spoken discourse) and lexical density (like written discourse) which results in grammatical density (O'Halloran, 1996, 2004c). In other words, the language of mathematics is best investigated in relation to functions and grammar of mathematical symbolism and visual display to understand the functions of contemporary linguistic constructions in mathematics. 3.4 The Construction of Logical Meaning

Martin's (1992) discourse systems of CONJUNCTION and CONTINUITY are informed by Halliday's paradigm for clause complex relations in the form of INTERDEPENDENCY and LOGICO-SEMANTIC RELATIONS. Halliday's (1994: 221) description of clause complex relations is based on the system of TAXIS which is common to word, group, phrase and clause complexes alike. Halliday (1994) distinguishes hypotaxis as a dependent modifying relation and parataxis as an independent continuing relation. As illustrated in Figure 3.4(1), clause complexes are classified as paratactic and hypotactic. In addition, cohesive or intersentence logical relations are based on Halliday (1994: 220) and Martin (1992: 179). Halliday's (1994: 219-220) system of LOGICO-SEMANTIC RELATIONS is also concerned with EXPANSION and PROJECTION. The categories of EXPANSION describe the relations whereby a secondary clause expands the primary clause through Elaboration ('='), Extension ('+') and Enhancement ('x'). Secondary clauses realizing Elaboration ('that is' type relations) function to restate, specify, comment on or exemplify the

elaboration

elucidative

opposition clarification

additative

additative alternation

comparative •

similarity contrast

temporal

simultaneous successive

extension expansion enhancement paratactic independent

consequential

cohesive .none

locution projection idea implicit

dependent

hypotactic explicit internal external

Figure 3.4(1) LOGICO-SEMANTIC RELATIONS and INTERDEPENDENCE

purpose condition consequence concession manner

80


content of the primary clause. Secondary clauses realizing Extension ('and') add new elements to the primary clause by giving exceptions or offering alternatives. Secondary clauses realizing Enhancement ('so', 'yet', 'then') serve to qualify the primary clause with circumstantial features of time, place, cause or condition. Projection describes the situation whereby the secondary clause is projected through the primary clause as a Locution or Idea. Locution is the realization of the secondary clause as wording (") while Idea realizes the secondary clause as an idea ('). As illustrated in Figure 3.4(1), the type of INTERDEPENDENCE (paratactic or hypotactic) is cross-referenced with the type of LOGICO-SEMANTIC RELATION (expansion or projection). The discourse semantic systems of CONJUNCTION and CONTINUITY are modelled through covariate dependency structures called conjunctive reticula (Martin, 1992). The discourse system of CONTINUITY differs in that items are realized in the Rheme as opposed to textual Theme. Following Martin (1992), these systems are organized by listing the clauses down the page.4 Succeeding moves are shown to be dependent on preceding ones by dependency arrows pointing upwards towards the presumed message. Typically conjunctive relations are anaphoric but in the case of the forward relations, an arrow is placed at both ends of the dependency line. Implicit conjunctions are shown where they could have been made explicit in the discourse. The systems of CONJUNCTION and CONTINUITY may be used to describe logical relations in mathematics (O'Halloran, 2000: 378) which typically involve discourse moves across linguistic, symbolic and visual parts of the text. The step-by-step development of logical reasoning is an important function of symbolic mathematical discourse discussed in Chapter 4. The analysis of logical meaning in mathematics involves long and complex chains of reasoning which favour consequential-type relations (O'Halloran, 1996, 1999b, 2000). Typically these chains of reasoning (at least in the symbolic text) are primarily based on pre-established mathematical results. The significance of logical meaning in mathematical linguistic text is evident in the analysis of the extract from Stewart (1999: 132) displayed in Figure 3.4(2). There are complex nested structures of logical relations realized through structural conjunctions and conjunctive adjuncts, and there are also clause complex relations within rankshifted clause configurations. The analysis also reveals that logical meaning is realized metaphorically in the form of processes. That is, logical meaning is realized through the processes 'gives' and 'means' in the following clauses: 'This gives a second interpretation of the derivative', and 'This means [[that when the derivative is large «(and therefore the curve is steep, as at the Point P in Figure 4)» || the y-values change rapidly]]. Such processes are examples of grammatical metaphor (see Section 3.6) where logical meaning is encoded through process type. However, as evidenced in the short extract from Stewart (1999: 132), logical relations typically stretch across symbolic, visual and linguistic components of the text.

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From Equation 3 we recognize this limit as being the derivative of f at x 1 that is, f'(x1) This gives a second interpretation of the derivative: (that is) The derivative f'(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a. The connection with the first interpretation is [[that if we sketch the curve y = f(x), || then the instantaneous rate of change is the slope of the tangent to this curve at the point [[where x

= a]]]] This means [[that when the derivative is large «(and therefore the curve is steep, as at the Point P in Figure 4)» ||the y-values change rapidly]]. When the derivative is small the curve is relatively flat and the y-values change slowly Stewart[, 1999 #465: 132]

Figure 3.4(2) Logical Relations in Stewart (1999: 132) 3.5 The Textual Organization of Language At the lexicogrammatical stratum, textual meaning is realized as GivenA New through the system of THEME (Halliday, 1994) which is composed of two functional elements: the Theme and Rheme. Following Halliday (1994: 38), ' [t]he Theme is the element which serves as the point of departure for the message; it is that with which the clause is concerned. The remainder of the message, the part in which the Theme is developed, is called . . . the Rheme'. The system network for THEME is given in Figure 3.5(1). The Theme analysis, which is concerned with the organization of New information, permits the development of the text to be tracked at the rank of clause and clause complex. In addition to Theme, Martin and Rose (2003: 175-205) discuss thematic development in terms of phase and the whole text. That is, hyperThemes function to organize information at the rank of paragraph, and macroThemes provide the focus for the text. Thus the Conjunction - structural textual

Conjunctive Adjunct Continative Vocative

THEME interpersonal -

Modal Adjunct Finite

ideational

RHEME

Figure 3.5(1) The System of THEME

topic

82


organization of the text is investigated as regular periodic waves of increasing amplitudes at the ranks of clause, clause complex, paragraph and text. The THEME analysis for Stewart (1999: 132) (where Theme selections appear in bold) is given below: //From Equation 3 we recognize// //this limit as being derivative of/at x^// //that is, /'(*!)//

//This gives a second interpretation of the derivative:// //(that is) The derivative/' (a) is the instantaneous rate of change of y—f(x) with respect to x// //when x = a// //The connection with the first interpretation is [ [that if we sketch the curve y = f (x), 11 then the instantaneous rate of change is the slope of the tangent to this curve at the point [ [where x = a] ] ] ] // //This means [ [that when the derivative is large « (and therefore the curve is steep, as at the Point P in Figure 4)» 11 the y-values change rapidly]]// //When the derivative is small// //the curve is relatively flat// //and the y-values change slowly// The analysis demonstrates that the mathematical linguistic text is carefully organized to carry forth the argument. Marked Themes are selected ('From Equation 3', and 'When the derivative is small') to foreground important experiential content. Information is not only packaged into nominal group structures through grammatical metaphor, but clausal rankshift also appears to be a significant method of organizing experiential meaning. In addition, selections such as 'this' link the clause to previously established results. Martin (1992: 416) sees these types of selections as a case of textual grammatical metaphor (see Section 3.6). The linguistic text in Stewart (1999: 132) reflects grammatical intricacy as well as lexical density. More generally, these two types of complexity combine in mathematical discourse to give grammatical density (O'Halloran, 2000, 2004c) as discussed in Chapter 7. The discourse system of IDENTIFICATION is used to track participants where the basic opposition involves phoricity whereby information is recoverable from the text or context. That is, a participant is either newly presented ('addition'), or alternatively the identity of the presumed participant has to be retrieved in some way from the text or context (Halliday, 1994: 312-316). The means of retrieval are described by the types of phora (see the system network in Martin, 1992: 126). This includes 'bridging reference' where the referent has to be inferentially derived from the context rather than by direct reference, and 'multiple reference' which results in ambiguity. There is also 'generic' or 'specific' reference: 'Generic reference is selected when the whole of some experiential class of participants is at stake rather than a specific manifestation of that class' (Martin, 1992: 103).

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These classifications are used to track the participants in mathematical discourse in order to understand how reference functions in mathematics. The linguistic text in Stewart (1999: 132) illustrates that tracking participants in mathematics necessarily involves the linguistic, symbolic and visual components of the text. In addition, reference chains in mathematics are complex as they split and cojoin as mathematical participants are rearranged for the solution to problems and the mathematical relations which are described are visualized (O'Halloran, 1996, 2000). The complexity of tracking participants may be seen in Stewart where 'the derivative' is variously referred to as '/'(xj)' and '/'(a)' and participants are reconfigured in other ways, for example, 'at x^ and 'x= a\ Tracing participant reconfigurations across the three semiotic resources necessarily involves knowledge of the grammars of language, mathematical symbolism and visual display. 3.6 Grammatical Metaphor and Mathematical Language

Grammatical metaphor is an important concept for understanding the nature of scientific language (for example, Chen, 2001; Derewianka, 1995; Halliday, 1994; Halliday and Martin, 1993; Martin, 1992, 1997; Martin and Veel, 1998; O'Halloran, 2003b; Simon-Vandenbergen et al, 2003). This discussion forms the basis in Chapter 6 for the extension of the concept of grammatical metaphor to semiotic metaphor. In this formulation, the notion of grammatical metaphor is extended to take into account the types of meaning expansions which take place intersemiotically in multisemiotic texts. The nature of the systems and lexicogrammatical strategies for encoding meaning in language, visual images and symbolism are the product of the interaction of the three resources. Grammatical metaphor is a 'variation in the expression of a given meaning' which appears in a grammatical form although some lexical variation may occur as well (Halliday, 1994: 342). The typical or unmarked form is referred to as the congruent realization and the other forms which realize some transference of meaning as the metaphorical form. The presence of grammatical metaphor necessitates more than one level of interpretation, the metaphorical (or the transferred meaning) and the congruent. Martin (1993a: 237) states: 'the fact that we have to read the clause on more than one level is critical - the metaphor makes the clause mean what it does'. If, therefore, an expression can be unpacked grammatically to a congruent meaning, it is a case of grammatical metaphor. Halliday's categorization of the types of grammatical metaphor (see Table 1.9 in Martin, 1997: 32) is given in Table 3.6(1). The types of grammatical metaphor are organized metafunctionally according to rank in Table 3.6(1). There exist logical, experiential and interpersonal metaphors at ranks of clause complex, clause and word group. Grammatical metaphor involves the shifts to 'entity', 'quality', 'process' and 'circumstance' from congruent realizations listed in Table 3.6(1).

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Table 3.6(1) Halliday's Grammatical Metaphor (see Table 1.9 in Martin, 1997: 32) RANK AND METAFUNCTION

GRAMMATICAL METAPHOR

Clause complex: LOGICAL relator

entity

Examples: so if because

(nominal group) cause/proof condition reason

relator

quality

Examples: then so

(nominal group) subsequent/follow resulting

relator

process (clause)

Examples: then

follow cause complement

so and

circumstance (clause)

relator Examples: when

in times of/in . . . times under conditions of/under . . . conditions due to

if

therefore Clause: LOGICAL (internal relations) EXPERIENTIAL and INTERPERSONAL process

Examples: event auxiliary - tense - phase - modality

(nominal group) transformation transform will/going to try to can/could/may/will

process Examples: event

entity

prospect attempt possibility, potential, tendency quality

(nominal group) increasing poverty poverty is increasing

( S F L ) A N D MATHEMATICAL LANGUAGE Table 3.6(1) - cont process auxiliary - tense - phase - modality

quality was/used to begin to must/will [always] may

previous initial constant possible/permissable

process Examples: divide

circumstance

circumstance Examples: with to

entity (nominal group)

circumstance

quality (nominal group)

Examples: manner other other

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'h' [on V]

accompaniment destination

[decided] hastily [argued] for a long time cracked on the surface

circumstance (clause) Examples: be about be instead of

hasty decision lengthy [argument] surface [cracks] process (clause) concern replace

Word group: LOGICAL (internal relations), EXPERIENTIAL and INTERPERSONAL quality Examples: unstable entity Examples: the government [decided] the government couldn't decide

entity (nominal group) instability modifier [expansion] (nominal group) the government [decision] [a/the decision] of/by the government the government's [indecision] [the indecision of the government] governmental [indecision]

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Table 3.6(1) - cont Non-entity: LOGICAL (internal relations), EXPERIENTIAL and INTERPERSONAL entity (nominal group) the phenomenon o f . . .

process (clause) . . . occurs/ensues

The new case where a process is realized by circumstance (as illustrated by 'h on r' to mean 'h divided by r') which has appeared in mathematical classroom discourse (O'Halloran, 1996) is added to Halliday's categories in Table 3.6(1). The majority of cases of grammatical metaphor involve the process of nominalization whereby a grammatical class or structure realizing a process, circumstance, quality or conjunction is turned into another grammatical class, that of a nominal group realizing a participant. Following Halliday (1993a, 1993b, 1998) nominalization is conceived as 'the predominant semantic drift of grammatical metaphor in modern English' (Martin, 1992: 406), which has largely resulted from changes in the English language to realize a scientific view of the world. That is, 'a new variety of English' was created 'for a new kind of knowledge' (Halliday, 1993b: 81), one in which the main concern was to establish causal relations. As Halliday explains, the most effective way to construct logical arguments is to establish steps within a single clause, with the two parts 'what was established' and 'what follows from it' reified as two 'things' or participants realized through nominal group structures. These two participants are then connected with a process in a single clause. The strategy of recursive modification of the nominal group is also employed in scientific discourse. These two devices are typical of contemporary written discourse, and as Halliday and Martin (1993: 39) point out, nominalizations may serve important ideological functions because they are less negotiable than the congruent form: 'you can argue with a clause but you can't argue with a nominal group'. Cases of grammatical metaphor may be mapped through the system network as displayed in Figure 3.6(1). In addition to experientially based types of grammatical metaphor, interpersonal metaphors occur in conjunction with the systems of MODALIZATION and MODULATION (see Martin et al, 1997: 70). Following Halliday (1994: 354—363) metaphors of modalization and modulation are realized through the use of modal auxiliaries (modal Finites) with high, median and low values of probability and usuality, and obligation, inclination and potentiality respectively. MODALIZATION and MODULATION vary in orientation with respect to two criteria: first, objectivity and subjectivity; and

(SFL)


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experiential logical interpersonal textual relator

circumstance

circumstance

process

process

quality

quality entity -zero

entity nodifier process entity

Figure 3.6(1) System Network for Grammatical Metaphor second, implicitness and explicitness. A subjective explicit orientation is realized through a projecting clause; for example, 'I know this is correct'. An explicit objective orientation is realized through encoding of the objectivity; for example, 'it is certain this is correct'. Interpersonal metaphors are also realized through incongruence between MOOD and SPEECH FUNCTION selections (Halliday, 1994: 363-367). As explained in Section 3.2, the unmarked MOOD realizations of the SPEECH FUNCTIONS are statement realized by declarative (SubjectAFinite), question by interrogative (FiniteASubject and WH), command by imperative (Predicator), and Offer by modalized interrogative (modalized FiniteASubject). Mathematical discourse includes Modal and Mood metaphors (see Section 3.2). Martin (1992: 416) introduces textual grammatical metaphor which is orientated towards organizing the text as ' "material" social reality'. Martin gives four types of textual metaphor which contribute to this organization of text: (i) 'meta-message relations' as found in Francis' (1985) anaphoric nouns (for example, 'reason', 'example', 'point' and 'factor'); (ii) 'text reference' which identifies facts rather than participants (for example, 'this'); (iii) 'negotiating texture' which can, for example, exploit monologic text as dialogic (for example, 'let me begin by'); and (iv) internal conjunction which orchestrates text organization as opposed to field organization (for example, 'as a final point'). As Martin (1992: 416) points out, rather than being orientated towards logical meaning, these types of textual metaphors may be orientated towards the interpersonal. For example, That point is just silly' (Martin, 1992: 417) is a textual metaphor of the type 'meta-message relation' which is orientated to interpersonal meaning in the form of APPRAISAL. Derewianka (1995: 238) explains that the

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functioning of a nominal element to 'summarise or "distil" a figure or sequence of figures' does not necessarily mean that 'any instance of this type is inherently metaphorical'. According to Derewianka (1995: 238), what needs to be taken into account is a change in the level of generalization and abstraction. Nonetheless, Martin (1992: 395) sees grammatical metaphor as an important strategy for creating texture: 'The resources for weaving chains and strings through different grammatical functions . . . are important ones: but they provide only a very partial picture of the way in which meanings are packaged for grammatical realisation. The real gatekeeper is grammatical metaphor.' As evident in the discussion of experiential meaning in Stewart (1999: 132) in Section 3.3, mathematical discourse involves grammatical metaphor. The analysis of multisemiosis in mathematical and scientific texts enhances our understanding of the role and function of grammatical metaphor. Once the notion of semiotic metaphor is introduced in the form of metaphorical realizations which take place with intersemiotic shifts across semiotic resources, the semantic drift in language where grammatical metaphor developed intrasemiotically in language as a means of re-packaging information becomes understandable in the context of the functions and roles which are fulfilled symbolically and visually. This important point necessitates further discussion of the nature and functions of grammatical metaphor in Chapter 6. As well as grammatical and semiotic metaphors, lexical metaphors (Halliday, 1994: 340-342) may also be examined in mathematics, although this is not undertaken in this study. Lexical metaphors are metaphors in the more classical sense of the term where 'a particular lexeme is said to have a "literal" and a "transferred" meaning'(Derewianka, 1995: 109). In terms of distinguishing grammatical and lexical metaphors, both have 'a semantic category which can be realized congruently or metaphorically' but with grammatical metaphor, 'what is varied is not the lexis but the grammar' (ibid.). Although this field of study is worthy of investigation, the major concern here is shifts in meaning which arise grammatically in mathematics through the interactions between the semiotic resources of language, visual images and mathematical symbolism. 3.7 Language, Context and Ideology SFL views language as a social-semiotic, a system of meanings that construe the reality of a culture. This construction is described metafunctionally: the ideational metafunction construes 'natural reality'; the interpersonal metafunction construes 'intersubjective reality'; and the textual metafunction construes 'semiotic reality' (Halliday, 1978; Matthiessen, 1991). This intrinsic functional organization of language is modelled as interacting with the organization of social context in what Halliday (1985) terms as language's extrinsic functionality. In other words, language is viewed as construing the social context with the net result being the reality of a


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culture. Conversely, the social context impinges upon language use. Martin (1992) describes the relationship between language and social context as one of mutual engendering where instances of language use, collectively called texts, are social processes which are analysed as manifesting the culture they in part largely construct. These SFL formulations are extended to other semiotic resources (for example, Baldry, 2000b; Halliday, 1978; Kress and van Leeuwen, 1996; O'Halloran, 2004a; O'Toole, 1994; Ventola el al., forthcoming) which, in the context of this study, are language, mathematical symbolism and specialized forms of visual display. The analysis of text becomes 'the analysis of semantic choice in context' (Martin, 1992: 404) where context is conceived as consisting of the context of the situation and the context of culture. Context is viewed as a semiotic system manifested in whole or part through language and other semiotic resources. The levels of semiosis articulated by this process of realization are referred to as communication planes. The difference between language, mathematical symbolism and visual display on the one hand, and context on the other, is that the former have their own means of organizing expression (through typography/graphology and so forth) while context depends on other semiotic planes for realization (Ventola, 1987). The context of a text consists of two communication planes: register at the level of context of situation, and genre at the level of the context of culture. Register is constituted by contextual variables of field, tenor and mode which work together to achieve a text's goal. Field is concerned with experiential meaning (what is actually taking place), tenor with interpersonal meaning (the nature of the social relations) and mode with textual meanings (the role language is playing) (Halliday, 1978, 1985). The three register variables of field, tenor and mode can be viewed as working together to achieve a text's goals, 'where goals are defined in terms of the systems of social processes at the level of genre' (Martin, 1992: 502-503). Genre networks are formulated on the basis of similarities and differences between text structures which define text types. A culture consists of particular ways of meaning, which are described through genre, register and the integration of different forms of semiosis. In this study, the focus is directed towards the language and expression plane, rather than register and genre. However, a brief discussion of the register of mathematical language in terms of field, tenor and mode is included below. The written mode of mathematics means that semiosis in the form of language, visual images and the symbolism is constitutive of the mathematics which is developed, rather than contextual meaning arising from the immediate material setting. While mathematics does involve genres other than the written (such as the academic lecture, the conference paper and so forth), in essence modern mathematics developed as a written discourse. The textual organization of linguistic, symbolic and visual components and the compositional arrangement of those selections in mathematics are sophisticated. Mathematical discourse is concerned with particular realms of experiential content according to the field of mathematics (for

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example, elementary mathematics, calculus, pure mathematics and applied mathematics) and the genre. At this stage, the nature of the tenor relations which are established in mathematics is worthy of closer inspection as these relations orientate the reader towards the mathematics which is presented: 'Tenor refers to the negotiation of social relationships among participants' (Martin, 1992: 523). Tenor is the projection of interpersonal meaning realized through discourse semantics and lexicogrammatical systems in the language stratum. Tenor is mediated along the three dimensions of power (which Martin refers to as 'status'), contact and affect (Martin, 1992; Poynton, 1984, 1985, 1990) as displayed in Figure 3.7(1). Status refers to 'the relative position of the interlocutors in a culture's social hierarchy', contact is 'their degree of institutional involvement with each other' and affect includes 'what Halliday (1978) refers to as the "degree of emotional charge" in the relationship between participants' (Martin, 1992: 525). The principle of reciprocity of choice is significant in terms of the realization of status in spoken discourse. Patterns of dominance and deference in which the status of the writer/speaker is reflected take place through the kinds of linguistic choices which are made. Equal status is realized through selections of the same kinds of options for both interlocutors while unequal status is realized through non-reciprocal choices. As Martin (1992: 528) explains, there is 'a symbolic relationship between position in the social hierarchy and various linguistic systems, especially interpersonal ones'. The contact, or degree of involvement, is equivalent to what Hasan (1985) describes as 'social distance', the frequency and range of interaction. The principle of proliferation is used in which a high degree of contact means a wider range of options are available, while a low degree of contact means a smaller range of options. The basic realization principle of affect is amplification in which speakers can vary the 'volume' from normal writing/listening levels. Martin (1992: 529-535) lists features of interaction patterns, discourse semantics, lexis, grammar, and phonology which realize patterns of dominance and deference, involved and uninvolved contact, and dimensions of affect. The lexical and grammatical realizations of these tenor dimensions STATUS reciprocity

TENOR

CONTACT proliferation

equal unequal involved distant positive

AFFECT

marked negative

amplification

Figure 3.7(1) TENOR Dimensions adapted from Martin (1992: 526)


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are represented in a three-dimensional space in Figure 3.7(2). While many of these systems for spoken language do not operate in mathematics written texts (for example, swearing, slang and so forth), the three dimensional space is nonetheless useful as a means of framing the range of options which are available in mathematics. Such a framework is also useful for the analysis of pedagogical discourse in mathematics. The nature of the linguistic choices in written mathematics means that the discourse operates from an uncontested position of dominance. The linguistic choices are not reciprocal, there is minimal affect, and the contact is involved but distant. The nature of interpersonal relations is further discussed in relation to the symbolic and visual components of mathematics text in the following chapters. Martin (1992: 507) explains that ideology may be seen as 'the system of coding orientations constituting a culture'. Incorporating Foucault's (1970, 1972, 1980a, 1984, 1991) formulations of knowledge, power and discourse, SFL analysis is concerned with how texts relate to each other, and how one text relates to all texts that may have been. As texts are interpreted in a multidimensional intertextual semiotic space, this allows the selections which have been made to be effectively placed alongside all other possibilities, thus revealing the ideological positioning of the choices that have been made. Ideology has genre, and hence register and language as its expression plane. The ideological orientation of mathematics is discussed in relation to the concepts of abstraction, contextual independence, reason, objectivity and truth in Chapter 7. The adoption of a multisemiotic perspective of discourse facilitates a holistic understanding of text, context and culture. The inclusion of other forms of semiosis and the study of intrasemiotic and intersemiotic processes enhances the theoretical possibilities afforded by SFL. For example, the SFL framework presented in Table 3.1(1) is extended to incorporate other semiotic resources in the 'Integrative Multisemiotic Model' (IMM) in Lim (2002, 2004). Such a multisemiotic systemic functional model incorporates (i) the grammatical systems for other semiotic resources, (ii) intrasemiosis within the semiotic resources, (iii) intersemiotic mechanisms for meaning across semiotic resources, (iv) systems which operate on the Expression stratum (for example, Colour and Font Style and Size), and (v) the materiality and medium of the text (for example, print versus electronic medium). A multisemiotic approach reveals differences between the functions and systems of semiotic resources across different strata. For example, Lim's (2002: 37) division of metafunctionally based systems shows a separation of metafunctional boundaries with respect to the systems which operate at the grammatical stratum for language. However, the 'system-metafunction fidelity' (Lim, 2002, 2004), or the measure of dedication of a system to one particular metafunction, breaks down on the expression plane. These systems (for example, systems such as Font, Colour and so forth) do not have the clear metafunctional orientations which are found in grammatical systems. Choices from the system of Colour, for example, can function interpersonally to attract attention, textually for cohesive purposes, and

GRAMMAR Residue ellipsis polarity matched attitude concur comment invited vocation respectful person 2nd tagging checking agency: I/medium modalization low " modulation: inclination LEXIS euphemize tempered swearing covert GRAMMAR minor clauses Mood ellipsis Mood contraction vocation ' range of names nick-name

LEXIS attitudinal taboo swearing

POWER - Defer

GRAMMAR exclamative attitude comment minor expressive intensification repetition prosodic nm gp diminutives; mental affection manner degree

GRAMMAR major clauses no ellipsis no contraction no vocation single name full name CONTACT Distant/ uninvolved

AFFECT Positive POWER Dominate AFFECT Negative

CONTACT Intimate/ involved

LEXIS specialized technical slang general words

GRAMMAR exclamative attitude comment minor expressive intensification repetition prosodic nm gp diminutives; mental affection manner degree

LEXIS attitudinal taboo swearing

LEXIS core non-technical standard specific words

GRAMMAR no ellipsis polarity asserted attitude manifested ~ comment presented vocation familiar person 1st tagging invited agency: I/agent modalization high modulation: obligation LEXIS explicit bodily functions swearing overt

Figure 3.7(2) Lexicogrammatical Aspects of the Realization of TENOR adapted from Martin (1992: 529-535)


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experientially for representational meaning. In a similar fashion, semiotic resources have different grammatical systems, and such differences in the meaning potential have implications for the functions which are fulfilled by that resource, as seen in the discussion of the grammar of mathematical symbolism and visual images. The formulation of SF frameworks for mathematical symbolism and visual display and the analysis of intersemiotic processes reveal that the key to the success of mathematical discourse is the ability to create a semantic circuit across the linguistic, symbolic and visual components of the mathematics text through the specialized grammars of each resource. These semantic circuits give rise to metaphorical expressions in the form of semiotic metaphors. The analysis of the mathematical texts in Chapter 7 leads to a further discussion of the ideology and orientation of mathematics. Martin's (1992: 507) focus is situated within the dynamic view of ideology which is 'concerned with the redistribution of power - with semiotic evolution'. These concerns provide the impetus for this study. Notes 1 Martin's (1992: 14-21) arguments for stratification of the content plane include the following limitations of the lexicogrammar: semantic motifs cannot be generalized because of diverse structural realizations; the multiple levels of semantic layers resulting from grammatical metaphor cannot be fully accounted for; generalizations across structural and nonstructural textual relations such as those found in cohesion are not possible; and the semantic stratum is more abstract and the systems are composed of larger units which differ in structure from those found elsewhere. Martin (1992) and Martin and Rose (2003) are concerned with overcoming these limitations and capturing semantic interdependencies in the whole text which are otherwise only partially accounted for by the lexicogrammar. The type of structures are open ended in so far as the issue is not one of constituency, but rather interdependency. 2 Although word groups and phrases occupy the intermediate position on the rank scale, Halliday (1994: 180) distinguishes between the two: 'A PHRASE is different from a group in that, whereas a group is an expansion of a word, a phrase is a contraction of clause.' Halliday's (ibid.: 242) classification of rankshifting thus covers clausal and phrasal elements. 3 In SF system networks, the curly brackets mean 'select from each of the systems' (that is, 'select from this and this'), while the square brackets mean 'select only one of the options' (that is, 'select this or that'). 4 Typically in conjunctive reticula, external relations are modelled on the right-hand side, and the internal relations are modelled down the left with external additive relations positioned in the centre. This allows the conjunctive relations to be separated into those which function in a rhetorical (internal) sense compared to those which function in a more experiential (external) sense.

4 The Grammar of Mathematical Symbolism

4.1 Mathematical Symbolism

The historical perspective covered in Chapter 2 reveals how mathematical symbolism developed as a tool for reasoning through the discovery that curves could be described algebraically and the increasingly important aim of rewriting the physical world in mathematicized form. Mathematical descriptions eventually replaced metaphysical, theological and mechanical explanations of the universe (see for example, Barry, 1996; Kline, 1972, 1980; Wilder, 1981). Today, many fields of human endeavour are written in mathematicized or pseudo-scientific form. The scientific view of the world is not confined to the physical universe; rather it underlies our day-to-day conception of reality. Mathematical discourse succeeds through the interwoven grammars of language, mathematical symbolism and visual images, which means that shifts may be made seamlessly across these three resources. However, each semiotic resource has a particular contribution or function within mathematical discourse. Language is often used to introduce, contextualize and describe the mathematics problem. The next step is typically the visualization of the problem in graphical or diagrammatic form. Finally the problem is solved using mathematical symbolism through a variety of approaches which include the recognition of patterns, the use of analogy, an examination of different cases, working backwards from a solution to arrive at the original data, establishing sub-goals for complex problems, indirect reasoning in the form of proof by contradiction, mathematical induction (if Sk is true, and Sk+1 is true whenever Sk is true, then Sn is true for all n) and mathematical deduction using previously established results (Stewart, 1999: 59-60). The generalized solutions and mathematical models are used for predictive purposes. Before discussing intersemiotic processes which take place across language, the symbolism and visual images in Chapters 6-7, intrasemiosis within mathematical symbolism and visual display is explored in Chapters 4—5 respectively. The unique functions of each resource are discussed through SF frameworks and an investigation of choices from the systems which are found in symbolic and visual

THE G R A M M A R OF M A T H E M A T I C A L S Y M B O L I S M

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parts of the mathematical texts. In this way, the grammatical strategies for encoding meaning in each resource may be understood before proceeding to the complex problem of understanding how meaning is made intersemiotically across the three resources. The general nature of meanings afforded by language, mathematical symbolism and visual images is described by Lemke (1998b, 2003). That is, language is seen to be orientated towards making categorical-type distinctions (for example, Bateson, 1972; de Saussure, 1966; Messaris, 1994); that is, typological-type meanings. Mathematical symbolism, on the other hand, is seen to make meanings by degree in the form of continuous descriptions of patterns of co-variation; that is, topological-type meanings. For example, one can observe that the tiger population in an ecosystem is decreasing, and one may even comment: 'there are not many tigers around these days'. The linguistic statement makes a categorical type assessment of the situation regarding tigers: 'there are not' (Existential process with negative polarity) 'many tigers' (Existent) 'around' (circumstance-Location of place) 'these days' (circumstance-Location of time). However, through 'predator-prey' type mathematical models, the relationship between the number of tigers and the number of men, for example, can be specified in order to study the patterns of the interaction between the two species, and to predict the tiger population at any one time, including when they may be expected to become extinct. The mathematical model expresses the relationship between the number of men and the number of tigers as a continuous function over time. For example, if M represents the predator 'man', T represents the prey 'tiger', and t represents time, then such a model would take the form

(Stewart, 1999: 662). In

addition to describing patterns of variation over time, the symbolism has the potential to express the exact relations of parts to a whole. For example, a triangle with base b and height h is related to the area of the triangle A (A) through the symbolic statement As the nature of patterns of variation is not easily discernible from the symbolic statements, the graphs and diagrams are used to give more intuitive understanding of the relationships which are encoded symbolically (Lemke, 1998b). For example, the predator-prey mathematical model for the relationship between the population of tigers and the number of men can be displayed graphically to give a sense of the type of relationship encoded in the mathematical model. Visual images supersede language in terms of the ability to represent continuous spatial relations. However, mathematics visual patterns are often only partial descriptions over a limited domain, and they are limited in terms of their ability to be used for calculations. This shortfall is becoming less marked with the development of the power of computers to display and manipulate visual patterns, a theme which is explored in Chapter 5. The computer is revolutionizing the

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types of mathematics being developed due to the increased facility for performing numerical calculations and displaying the resulting visual patterns through computer graphics. The functions and grammar of mathematical symbolism are examined through the development of an SF framework. This framework is used to explore the nature of interpersonal, experiential, logical and textual meanings afforded by symbolism, and the strategies through which these meanings are encoded. A similar exploration of intrasemiosis in visual images takes place in Chapter 5. From this point, it is possible to examine how language, the symbolism and the visual images combine intersemiotically to create meaning in mathematical discourse. The examination of the grammars of mathematical symbolism and visual display on a separate basis is a somewhat artificial approach as historically these semiotic resources developed together in mathematical discourse. The key to the success of mathematics is that the three grammars function integratively. However, if the process of semiosis is 'frozen' in stages where meaning is made primarily within one resource rather than across the three resources, the contribution of that one resource may be appreciated. This appears to be a necessary preliminary first step to understand how the three semiotic resources function together. The functions and grammar of the mathematical symbolism and visual images are therefore first investigated individually. As will become evident, the grammatical strategies for encoding meaning in mathematical symbolism differ from those found in scientific language. This is not surprising as the symbolism was designed to fulfil different functions, and its grammar evolved accordingly. The nature of scientific language, with its propensity to pack experiential meaning into extended nominal group structures in the form of grammatical metaphors which are configured with relational processes (for example, Halliday and Martin, 1993; Martin and Veel, 1998), is the resultant product of the impact of the use of the symbolism and the visual display in mathematical and scientific discourse. From the discussion of intrasemiosis in mathematical symbolism and visual display, intersemiotic processes and their impact on scientific language are investigated in Chapter 6. 4.2 Language-Based Approach to Mathematical Symbolism

The language-based approach to the SF framework for mathematical symbolism adopted in this study is justified by the fact that the symbolism developed as a semiotic resource which evolved from language. The stages of the development of algebra, for example, have been characterized as rhetorical where instructions were in the form of linguistic commands, syncopated where recurring linguistic elements for participants and processes were symbolized, and symbolic where mathematical symbolism developed as a semiotic resource (see Chapter 2 and Joseph, 1991). The symbolism developed a functionality through new grammatical systems


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which permitted semantic expansions beyond that capable with language, but at the same time it depended upon employing certain linguistic elements and a range of grammatical strategies inherited from language. Furthermore, symbolic statements are typically embedded within linguistic text. Thus, despite the new functionality of mathematical symbolism, it nonetheless requires a surrounding linguistic co-text to contextualize the symbolic descriptions and procedures that take place. The dependence on the linguistic semiotic suggests that the symbolism did not develop a wellrounded functionality, which becomes evident in the discussion of the types of meaning which are possible using mathematical symbolism. The language-based approach permits the semantics of the mathematical symbolism to be understood and contextualized in relation to the types of meaning afforded by the linguistic semiotic. The unique relations between language and mathematical symbolism explain the nature of the mappings that may be made between the two semiotic resources. For example, there exist acceptable wordings in natural language for mathematical symbolic statements, although this is not an exact one-to-one correspondence. Mathematical statements are recoverable from linguistic statements, although in some cases this is problematic because the linguistic construals are metaphorical (see the discussion of semiotic metaphor). The unique relations between language and mathematical symbolism serve to highlight an important difference between these two resources and other semiotic resources such as art, sculpture and architecture where such accurate mappings do not exist. For instance, unlike a mathematical symbolic statement, a painting or a sculpture is not recoverable from any combination of words. After introducing the SF framework for mathematical symbolism, the types of semantic shift in the evolution of the symbolism are discussed in terms of the expansion and contraction of experiential meaning, the narrowing of interpersonal meaning, the development of selected types of logical meaning, and the refinement of textual meaning. These types of semantic shift mean that mathematical symbolism developed as a semiotic resource with a grammar through which meaning is unambiguously encoded in ways which involve maximal economy and condensation. The economical means of encoding meaning in the symbolism permit the easy rearrangement and manipulation of relations so that mathematical models can be constructed and problems can be solved. This perspective is developed in the following discussion of the grammar of mathematical symbolism. A summary of the major points concerning the grammar of mathematical symbolism appears in Section 4.8 4.3 SF Framework for Mathematical Symbolism

The SF model for mathematical symbolism displayed in Table 4.3(1) is based upon Halliday (1994) and Martin's (1992; Martin and Rose, 2003) systemic model for language. The communicative planes of ideology, genre

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Table 4.3(1) SF Model for Mathematical Symbolism MATHEMATICAL SYMBOLISM CONTENT

Discourse Semantics Inter-statemental relations Grammar Statements (or clause complex) Clause (// //) Expressions ([[ ]]) (rankshifted participants of the clause which are the result of mathematical operations) Components (the functional elements in expressions)

DISPLAY

Graphology and Typography

and register are applicable to the multisemiotic mathematical texts considered in Chapters 6-7. In the language plane, the content stratum for mathematical symbolism consists of discourse semantics and grammar strata with the ranks of statement (clause complex), clause, expression and component. The model parallels the discourse stratum and lexicogrammatical ranks of clause complex, clause, word group/phrase and word for language. The 'display plane' for mathematical symbolism corresponds to the 'expression plane' for language in the model. The term 'display plane' is used rather than 'expression plane' because a new grammatical rank of 'expression' is introduced for mathematical symbolism in Table 4.3(1). The need for the inclusion of the rank of expression in the grammar of mathematical symbolism will become apparent in Section 4.4. The SF framework for a grammar for mathematical symbolism is presented in Table 4.3(2). This framework provides a description of the major systems through which mathematical symbolism is organized as a semiotic resource for experiential, logical, interpersonal and textual meaning for the content and display planes. The discourse systems for mathematical symbolism parallel those found in language. However, as discourse moves often span linguistic, symbolic and visual components of the text, Martin's discourse systems are extended in Chapter 6 in the attempt to theorize intersemiosis between the three resources. In the model presented in Table 4.3(2), systems which operate at the level of the display plane are also included. It is recognized that options in the expression of the semiotic choices in the mathematics text (for example, Colour, Font Size and Style) function to create meaning (for example, Kress and van Leeuwen, 2002; Lim, 2004; O'Halloran, 2004a). Traditionally, the expression stratum in language has been under-theorized in SFL where the major concerns have been the language plane and the communicative planes of register, genre and ideology.

THE G R A M M A R OF MATHEMATICAL SYMBOLISM

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Table 4.3(2) Grammar and Discourse Systems for Mathematical Symbolism DISCOURSE SEMANTICS EXPERIENTIAL

LOGICAL

INTERPERSONAL

TEXTUAL

IDEATION • Activity Sequences consisting of Operative process and participant reconfigurations (progressive steps of simplification and solution) • Nuclear relations (participant and process) • Collocation (symbolic relations and strings through taxonomies, definitions, axioms and theorems)

CONJUNCTION and CONTINUITY (based on EXPANSION) • Sequential placement of statements (explicitly marked when the logical connection is non-sequential) • Extension of TAXIS into long implication sequences

NEGOTIATION Exchange Structure and SPEECH FUNCTION at the move rank • Consists of moves and move-complexes

IDENTIFICATION • Direct Repetition • Referential cohesion (based on definition, operational properties with explicit repetition of reference) • Positional notation (the sequential downward placement of statements and positional placement functional components)

Structure: Exchange Structure linking moves

Structure: conjunctive reticula

Structure: reference chains linking participants

Structure: strings for tracing activity sequence reconfigurations

INTER-STATEMENTAL RELATIONS EXPERIENTIAL

LOGICAL

INTERPERSONAL

TEXTUAL

• Positional notation to indicate continuations of Activity Sequences • Repetition of processes and participants in new configurations

EXPANSION • Conjunctions and cohesive conjunctions • Implicit and explicit conjunctions (external symbolic and linguistic conjunctive devices; internal substitution and operative properties) • Apposition • Parenthesis • Labelling

SPEECH FUNCTION (statements and limited forms of command) • Intricacy of symbolic representation • Abstractness (nature of participants, processes) • Discursive links (using verbal code of main text within the mathematical array) • Labelling

• Positional notation (the sequential downward placement of statements and positional placement functional components) • Dependent clauses (thematic or spatially marked) • Ellipsis (marked by spatial position) • Discursive links (using verbal code of main text within the mathematical array) • Labelling

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Table 4.3(2) - cont STATEMENTS // EXPERIENTIAL

LOGICAL

• Rhetorical 'temporal' TRANSITIVITY conjunctive relations • Processes (Operative, realized through relational and Rule of Order of existential) operations and use of • Participants are brackets rankshifted configurations of Operative processes • Circumstantial features (minor clauses, dependent clauses or fused within participant structure) • Ellipsis of Operative processes • Rule of Order of operations

INTERPERSONAL

TEXTUAL

MOOD (with one symbol for the Finite and Predicator) • MODALITY (consistently high, implicit objective orientation) POLARITY (presence or absence of a slash through the process symbol) • Intricacy (embedded processes) • Abstractness (participants and processes)

• THEME (unmarked choice is Subject of the clause with marked case indicates steps in simplification) • Multiple Theme (textual element spatially placed) • Ellipsis (spatial positioning) • Dependent clauses (thematic or otherwise spatially separated) • Conventional spatial organization • Rule of Order and use of brackets for unfolding of Operative processes

EXPRESSIONS [[ EXPERIENTIAL

LOGICAL

//

]]

INTERPERSONAL

TEXTUAL

• Rhetorical 'temporal' • Intricacy (degree and • Rule of Order and • Operative processes (condensation occurs conjunctive relations explicitness of use of brackets for unfolding of realized through embedding) via high level of conventionalized rankshift within and Operative processes • Degree of between participants) Rule of Order of abstractness (nature of participants and • Degree of rankshift operations and use of brackets processes) indicated by [ [ ] ] • Circumstantial elements (through processes and fused participant structures • Ellipsis of Operative processes • Rule of Order of operations

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Table 4.3(2) - cont COMPONENTS EXPERIENTIAL

LOGICAL

• Conventionalized • Restricted range of combinatory units in the nominal practices group (the absence of DEIXIS, attitudinal and experiential epithets) • Qualifiers (form part of the nominal group without the need for embedding as phrases) • Classifiers • Conventionalized use of specific symbols (numerals, Roman, Greek, Hebrew alphabet)

INTERPERSONAL

TEXTUAL

• Degree of abstractness • Degree of modification

• Function of constituents (spatial, serial position and brackets)

DISPLAY PLANE EXPERIENTIAL

LOGICAL

• Variations in the form • Spatial organization of symbolic text of case, font, scripts and size for special symbols, abbreviations, icons, punctuation, brackets, and combinations of symbols • Use of spatial and positional notation

INTERPERSONAL

TEXTUAL

• Style of production • Spatial arrangement (hand written, of text at each rank computer generated) • Font style and format • Contrasts in font, • Ellipsis of process script and size

The SF framework in Table 4.3(2) provides insights into the ways in which the grammar of mathematical symbolism is organized to fulfil the functions of mathematics, and the ways in which the systems and lexicogrammatical strategies in the symbolism depart from those found in language. Further research is needed in the analysis of mathematical texts, however, in order to fully document the systems, which remain at a preliminary stage of theorization. The framework presented in Table 4.3(2) is best viewed as a first step towards a comprehensive SFG for mathematical symbolism. The metafunctional systems in the SF framework for mathematical symbolism are discussed with reference to the mathematical symbolic text displayed in Plate 4.3(1). This mathematics problem is concerned with

EXAMPLE 4

If f ( x ) = J x - 1 , f i n d t h e d e r i v a t i v e o f / . S t a t e t h e d o m a i n o f / ' .

SOLUTION

Here we r a t i o n a l i / e the numerator

We see that f'(x) exists if x > 1, so the domain of/' is (1, o°). This is smaller than the domain of/, which is [1, °°). Plate 4.3(1) Mathematical Symbolic Text (Stewart, 1999: 139)


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finding the derivative/'(x) of a function/(x) by finding the limit off(x) as x is approached; that is, as h tends to zero. The derivative is the rate of change of the function, which may be interpreted geometrically as the slope of the curve at the point (x,f(x)). The visualization of the derivative in the form of a graph for the geometrical interpretation of the derivative is examined in Chapter 5. 4.4 Contraction and Expansion of Experiential Meaning

Experiential meaning in mathematical symbolism is largely concerned with a semantic field in the form of the description and manipulation of relations. The semantic field of mathematics therefore includes a limited experiential domain compared to language. With the narrowing of the semantic domain, an expansion of meaning took place in mathematics with respect to the description of relations and patterns of variation. The ways in which mathematical symbolism achieves this simultaneous contraction and expansion of the experiential meaning are discussed below. One major innovation in mathematical symbolism is the evolution of a new process type, the Operative process, which takes the form of arithmetic operations and other processes found in the different fields of mathematics. Operative processes initially arose in early numerical systems, which were among the earliest forms of mathematical symbolism. Numerical notation appeared in different cultures arising from practical needs such as recording quantities and marking time intervals for social and economic activities. The nature of the early numerical systems in cultures which include the European, Egyptian, Mesopotamian, Indian, Arabian and Chinese, and independent traditions such as the Mayan in South America, depended upon the functions which were required to be fulfilled and the availability of material resources. Once established, numerical systems circumscribed mathematical activities and new developments in much the same way that grammatical systems in language function to structure reality through the nature of the linguistic choices which are available. As we have seen, the adoption of the Hindu-Arabic numerical system, for example, had a major impact on the development of mathematics in Europe. Symbolic processes in early numerical systems developed from Material processes (Lemke, 1998b; O'Halloran, 1996) which were concerned with counting, adding, multiplying, subtracting, dividing and measuring. However, new mathematical Operative participants and processes began to appear with the development of numerical systems. For example, new participants in the form of very small and very large numbers, which could not materialize in concrete form, arose in the symbolism. Moreover, Operative processes replaced the semantics of Material processes. That is, Operative processes of adding, multiplying, subtracting and dividing symbolic numbers initially paralleled existing Material processes of combining, increasing, decreasing and sharing physical objects. However, the complexity of Material processes undertaken by human participants in a physical

104


world had practical limitations and intuitive expectations which did not necessarily extend to the semiotic Operative processes performed using symbolic notation. It became possible to perform complex combinations of Operative processes which were not otherwise feasible or even conceivable, and to obtain results unlike those previously expected. An example of this type of semantic extension occurs in the case of multiplication of fractions where the product is less than the numbers which are multiplied. This contravenes the common-sense understanding where 'to multiply' means 'to increase'. A similar situation arises with the division of fractions where the result is larger than the number which is being divided. The limits of Operative processes within early numerical systems were dependent upon parameters such as the base of the system, the existence of place value, the inclusion of a symbol for zero, a means of separating fractional components and the intricacy and number of symbols. When calculations became complex, material computational devices based on the number systems, such as counting boards, table reckoners and the abacus, were employed. With the development of symbolic algebra, attention turned to generalized descriptions of relations using algebraic methods. The success of these descriptions meant that mathematical symbolism developed as a semiotic resource with grammatical systems which were unique to that resource. These systems developed in accordance with the aim of mathematics: the descriptions of patterns and the means to solve problems relating to those descriptions. This largely involved capturing and rearranging generalized descriptions of relations between variables through Operative processes. With the evolution of mathematical symbolism as a semiotic resource, arithmetic Operative processes were supplemented with processes concerned with powers, roots, complex numbers, limits and other processes found in different branches of mathematics, as seen in the calculus example in Plate 4.3(1) where the limit as h —> 0 is derived for Operative processes are typically performed by human agents on symbolic semiotic participants in the form of numbers and later variable participants, as seen in Plate 4.3(1) where the reader is instructed 'If f(x) = J x — l , find the derivative of/'. In the development of symbolic algebra, the human agent was not included in the mathematical symbolic statements which were more concerned with describing the nature of relations based on established mathematical results, rather than encoding the rhetorical commands which accompanied the solution to the problem. As a result, the human agent tends to be located within the linguistic part of the text which is concerned with the commands (for example, ' [you] find the derivative of/' in Plate 4.3(1)) and statements such as 'We recognize this limit as being the derivative of /at xl} that is,/' (xj' (Stewart, 1999:132). This statement takes the form of the metaphorical projecting clause 'we recognize' for 'this limit is the derivative o f / a t xlt that is, /'(*i)' (see Section 3.2). As we have seen, the human agent also disappeared in math-


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ematical visual images as the concern with lines and curves grew during the seventeenth and eighteenth centuries. At the same time, the notion of agency, where one participant impacts on another, appears to have developed in the symbolism in rather a different fashion from that found in language. For example, in a mathematical function the value of the independent variable x 'impacts' on the value of the dependent variable y in so much as the value of y depends on the value of x. However, the grammatical strategies for encoding such relations take the form of interactions between multiple participants rather than direct impact of one participant on another participant. This idea is developed below through an examination of the way in which Operative processes and participants are configured in mathematical symbolic statements. Operative processes appear to be grammatically different from the linguistic processes documented in Halliday's systems of TRANSITIVITY and the related system of ERGATTVTTY which is concerned with agency. The process types in language are Material, Mental, Behavioural, Verbal, Relational and Existential processes. Halliday (1994: 163) explains that in language there is a key participant, the Medium, which is associated with each process. In the clause 'Jack opened the door', the verb 'opened' is a Material process with 'Jack' as the Actor/Agent who acts on 'the door' which is the Goal/Medium. In this case the Medium is 'the door'. Without 'the door', the action of opening could not have been performed by Jack, the Agent. Halliday (1994: 163) calls the key participant the Medium in the ergative interpretation of the clause: Every process has associated with it one participant that is the key figure in that process: this is the one through which the process is actualized, and without which there would be no process at all. Let us call this element the MEDIUM, since it is the entity through the medium of which the process comes into existence.

Every process in language has an associated Medium, and only in some cases is there an Agent. For example, 'Jack talked' is a Verbal process with the Sayer/Medium 'Jack'. In this example there is no Agent. In the case of 'Jack and Jill walked up the hill', the Medium is 'Jack and Jill', realized as a complex nominal word group and the Range is 'up the hill'. There is no Agent associated with the process of walking up the hill. In a clause such as 'the best idea [ [that Jack and Jill had all day] ] was [ [to walk up the hill'] ], the Token/Identified (Medium) in the Relational process is 'The best idea [ [that Jack and Jill had all day] ]', as evidenced by the probe 'what is the best idea [[that Jack and Jill had all day]]?' The Value/Identifier (Range) is ' [ [to walk up the hill] ]'. In this case, the Medium is the rankshifted clause 'The best idea [[that Jack and Jill had all day]]'. Within this rankshifted clause Jack and Jill' function as the Medium for the process 'had'. The typical nuclear configuration of functional elements for experiential meaning in language has the form: Participant (Medium) + Process ± Participant (Agent) ± Range ± Circumstance/s

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However, Operative processes in mathematical symbolic clauses do not appear to replicate the nature of the experiential meaning in language. While the notion of a Medium and an Agent exists at the rank of clause in mathematical relational processes (for example, realized through '=') in the form of the 'Token (Medium or Agent) = Value (Range or Medium)', the corresponding mathematical participants do not take the correlate form of a word, word group/phrase or a rankshifted clause (with one embedded Medium) as discussed in the case of 'Jack', 'Jack and Jill', and 'the best idea [[that Jack and Jill had all day]]'. Rather, the Medium and other participants in relational symbolic statements are most typically configurations of Operative processes with multiple participants which appear to play equally key roles. For example, the notion of 'a single key participant' in the configuration of Operative processes which constitute the value of the derivative

does not seem to apply. Rather,

there appear to be several key participants, x, h and 1, in the algebraic expression for the derivative/' (x). In a similar fashion, there appear to be multiple key participants which are central to the Operative process configurations in the examples given below. Arithmetic Operations: Exponents: (xyz) n—x"y" zn Factoring Special Polynomials: x* — y2 = (x + y)(x— y) Geometric Formulae: Cosine Law: a? = tf +

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Visual Explanations: Images and Quantities, Evidence and Narrative

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The Language, Discourse, Society Reader

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Working Images: Visual Research and Representation in Ethnography

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The Language of Images (A Phoenix Book)

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