VISUALIZATION, EXPLANATION AND REASONING …download.e-bookshelf.de/download/0000/0032/87/L-G-0000003287... · TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES,Stanford

VISUALIZATION, EXPLANATION AND REASONING STYLES

IN MATHEMATICS

SYNTHESE LIBRARY

STUDIES IN EPISTEMOLOGY,

LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Editor-in-Chief:

VINCENT F. HENDRICKS, Roskilde University, Roskilde, DenmarkJOHN SYMONS, University of Texas at El Paso, U.S.A.

Honorary Editor:

JAAKKO HINTIKKA, Boston University, U.S.A.

Editors:

DIRK VAN DALEN, University of Utrecht, The NetherlandsTHEO A.F. KUIPERS, University of Groningen, The Netherlands

TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A.PATRICK SUPPES, Stanford University, California, U.S.A.JAN WOLENSKI, Jagiellonian University, Kraków, Poland

VOLUME 327

VISUALIZATION,EXPLANATION AND

REASONING STYLES INMATHEMATICS

Edited by

PAOLO MANCOSUUniversity of California,

Berkeley, U.S.A.

KLAUS FROVIN JØRGENSENRoskilde University,

Denmark

and

STIG ANDUR PEDERSENRoskilde University,

Denmark

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CONTENTS

Contributing Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

P. MANCOSU, K.F. JØRGENSEN AND S.A. PEDERSEN / Introduction . 1

PART I. MATHEMATICAL REASONING AND VISUALIZATION

P. MANCOSU / Visualization in Logic and Mathematics . . . . . . . . . . . . . . 13

1. Diagrams and Images in the Late Nineteenth Century . . . . . . . . . . . 132. The Return of the Visual as a Change in Mathematical Style . . . . 173. New Directions of Research and Foundations of Mathematics . . . 21Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

M. GIAQUINTO / From Symmetry Perception to Basic Geometry . . . . . 31

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311. Perceiving a Figure as a Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312. A Geometrical Concept for Squares . . . . . . . . . . . . . . . . . . . . . . . . . . 393. Getting the Belief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444. Is It Knowledge? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

J.R. BROWN / Naturalism, Pictures, and Platonic Intuitions . . . . . . . . . . . 57

1. Naturalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572. Platonism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593. Godel’s Platonism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60¨4. The Concept of Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625. Proofs and Intuitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646. Maddy’s Naturalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667. Refuting the Continuum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . 67Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Appendix: Freiling’s “Philosophical” Refutation of CH . . . . . . . . . . . 71References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

v

vi

M. GIAQUINTO / Mathematical Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

1. Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762. Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773. Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814. Refining and Extending the List of Activities . . . . . . . . . . . . . . . . . . 845. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

PART II. MATHEMATICAL EXPLANATION AND PROOF STYLES

J. HØYRUP / Tertium Non Datur: On Reasoning Styles in Early Mathema-tics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

1. Two Convenient Scapegoats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912. Old Babylonian Geometric Proto-algebra . . . . . . . . . . . . . . . . . . . . . 923. Euclidean Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034. Stations on the Road . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055. Other Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076. Proportionality – Reasoning and its Elimination . . . . . . . . . . . . . . 109Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

K. CHEMLA / The Interplay Between Proof and Algorithm in 3rd CenturyChina: The Operation as Prescription of Computation and the Operation asArgument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

1. Elements of Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252. Sketch of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263. First Remarks on the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314. The Operation as Relation of Transformation . . . . . . . . . . . . . . . . . 1325. The Essential Link Between Proof and Algorithm . . . . . . . . . . . . . 1346. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

vii

J. TAPPENDEN / Proof Style and Understanding in Mathematics I: Visual-ization, Unification and Axiom Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

1. Introduction – a “New Riddle” of Deduction . . . . . . . . . . . . . . . . . 1472. Understanding and Explanation in Mathematical Methodology:

The Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493. Understanding, Unification and Explanation – Friedman . . . . . . . 1584. Kitcher: Patterns of Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685. Artin and Axiom Choice: “Visual Reasoning” Without Vision . . 1806. Summary – the “new Riddle of Deduction” . . . . . . . . . . . . . . . . . . . 187Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

J. HAFNER AND P. MANCOSU / The Varieties of Mathematical Explana-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

1. Back to the Facts Themselves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2152. Mathematical Explanation or Explanation in Mathematics? . . . . 2163. The Search for Explanation within Mathematics . . . . . . . . . . . . . . 2184. Some Methodological Comments on the General Project . . . . . . 2215. Mark Steiner on Mathematical Explanation . . . . . . . . . . . . . . . . . . . 2226. Kummer’s Convergence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247. A Test Case for Steiner’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

R. NETZ / The Aesthetics of Mathematics: A Study . . . . . . . . . . . . . . . . . 251

1. The Problem Motivated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2512. Sources of Beauty in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2543. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

CONTRIBUTING AUTHORS

In order of appearance:

Paolo Mancosu is Associate Professor of Philosophy at U.C. Berkeley. Hismain interests are in mathematical logic and the history and philosophy ofmathematics. He is the author of Philosophy of Mathematics and Mathema-tical Practice in the Seventeenth Century (OUP 1996) and From Brouwer toHilbert (OUP 1998).

Marcus Giaquinto teaches philosophy and logic at University College Lon-don, and is a member of UCL’s Institute of Cognitive Neuroscience, partici-pating in its Numerical Cognition research group. His book, The Search forCertainty, on foundations of mathematics, was published by Oxford Univer-sity Press in 2002. The areas of his current research are the epistemology ofvisual thinking in mathematics, and numerical knowledge.

James Robert Brown is a Professor of Philosophy at the University of To-ronto. His books include: The Rational and the Social (Routledge 1989),The Laboratory of the Mind: Thought Experiments in the Natural Science(Routledge 1991), Smoke and Mirrors: How Science Reflects Reality (Rout-ledge 1994), Philosophy of Mathematics: An introduction to the World ofProofs and Pictures (Routledge 1999), and Who Rules in Science: A Guideto the Wars (Harvard 2001).

Jens Høyrup is docent (∼British “Reader”) at Roskilde University, Sectionfor Philosophy and Science Studies. His main research field is the conceptualand cultural history of pre- and early Modern mathematics. He has recentlypublished Human Sciences: Reappraising the Humanities through Historyand Philosophy (SUNY Press, 2000) and Lengths – Widths – Surfaces: aPortrait of Old Babylonian Algebra and Its Kin (Springer Verlag, 2002).

Karine Chemla is Directeur de recherche at the CNRS (French nationalcenter for scientific research), where she is the director of the research groupREHSEIS (Univ. Paris 7 & CNRS). Her main research area is the historyof mathematics in China, from the perspective of both an international his-tory of mathematics and the relationships between mathematics and culture.Together with Prof. Guo Shuchun (Academia sinica), she is completing thecritical edition and the translation of The nine chapters on mathematical pro-cedures and the earliest commentaries composed on this Canon. Moreover,she is also completing, as the main editor, the section on the history of sci-ence in China in the Encyclopedia of history of science, in preparation at

ix

x CONTRIBUTING AUTHORS

the Istituto dell’Enciclopedia Italiana. Since 1992, together with FrancoisMartin, she has edited the journal Extreme-Orient, Extreme-Occident.

Jamie Tappenden is Associate Professor of Philosophy at the University ofMichigan (Ann Arbor). His main interests are in the history and philosophyof mathematics and the philosophy of language. His most recent publicationshave explored the nineteenth century mathematical context of Frege’s logicalfoundations.

Johannes Hafner is a graduate student in the Group in Logic and the Method-ology of Science at U.C. Berkeley, where he is currently finishing his PhDthesis. His main interests are in philosophy of mathematics and logic and inthe history of analytic philosophy.

Reviel Netz is Professor of Classics and Philosophy at Stanford. His mainresearch field is Greek Mathematics. Among his published books are TheShaping of Deduction in Greek Mathematics: a Study in Cognitive His-tory (CUP 1999) and Adayin Bahuc (A volume of Hebrew poetry, Shufra1999). He publishes a new translation with commentary of the Works ofArchimedes (the first volume of which has appeared from CUP in 2004),and is also a member of the team editing the Archimedes Palimpsest.

P. MANCOSU, K.F. JØRGENSEN AND S.A. PEDERSEN

INTRODUCTION

In the 20th century philosophy of mathematics has to a great extent beendominated by views developed during the so-called foundational crisis in thebeginning of that century. These views have primarily focused on questionspertaining to the logical structure of mathematics and questions regardingthe justification and consistency of mathematics. Paradigmatic in this re-spect is Hilbert’s program which inherits from Frege and Russell the projectto formalize all areas of ordinary mathematics and then adds the require-ment of a proof, by epistemically privileged means (finitistic reasoning), ofthe consistency of such formalized theories. While interest in modified ver-sions of the original foundational programs is still thriving, in the secondpart of the twentieth century several philosophers and historians of mathe-matics have questioned whether such foundational programs could exhaustthe realm of important philosophical problems to be raised about the natureof mathematics. Some have done so in open confrontation (and hostility)to the logically based analysis of mathematics which characterized the clas-sical foundational programs, while others (and many of the contributors tothis book belong to this tradition) have only called for an extension of therange of questions and problems that should be raised in connection with anunderstanding of mathematics. The focus has turned thus to a considerationof what mathematicians are actually doing when they produce mathematics.Questions concerning concept-formation, understanding, heuristics, changesin style of reasoning, the role of analogies and diagrams etc. have becomethe subject of intense interest. These historians and philosophers agree thatthere is more to understanding mathematics than a study of its logical struc-ture and put much emphasis on mathematical activity as a human activity.How are mathematical objects and concepts generated? How does the pro-cess tie up with justification? What role do visual images and diagrams playin mathematical activity? In addition to these cognitive issues one mightalso investigate how mathematics interacts with the natural sciences, andhow mathematical thinking might depend on the culture it is embedded in.

This book is based on the meeting “Mathematics as Rational Activity”held at Roskilde University, Denmark, from November 1 to November 3,2001. The meeting focused on recent work in the study of mathematicalactivity understood according to the outline given above. The lectures, bysome of the most outstanding scholars in this area, addressed a variety ofissues related to mathematical reasoning. Despite the variety of the con-tributions there were strong unifying themes which recur in these lecturesthereby providing a strong sense of unity and purpose to the present book.

1P. Mancosu et al. (eds.), Visualization, Explanation and Reasoning Styles in Mathematics, 1-9.

© 2005 Springer. Printed in the Netherlands.

2 P. MANCOSU, K.F. JØRGENSEN AND S.A. PEDERSEN

The title of the book “Visualization, Explanation, and Reasoning Styles inMathematics” is indeed an accurate description of these recurring themes.

The volume is divided into two parts. The first part is called Mathemati-cal Reasoning and Visualization.

One question which arises pertaining to mathematical reasoning is towhat extent, if any, diagrams and visual imagery can provide us with mathe-matical knowledge. Most of the contributions in the book touch upon thisquestion but the first part of the book is fully devoted to it. In “Visualizationin Logic and Mathematics”, Paolo Mancosu provides a broad introductorydiscussion of visualization and diagrammatic reasoning and their relevancefor recent discussions in the philosophy of mathematics. Mancosu beginsby outlining how visual intuition and diagrammatic reasoning were discred-ited in late nineteenth century and twentieth century analysis and geometry.While diagrams and visual imagery were considered heuristically fruitfultheir role for justificatory purposes was considered to be unreliable and thusto be avoided. However, recent developments in mathematics and logic havebrought back to the forefront the importance of visual imagery and diagram-matic reasoning. Mancosu describes how many mathematicians are callingfor more visual approaches to mathematics and the recent developments inlogic related to diagrammatic reasoning. In the final part of the paper hediscusses how these recent developments affect the traditional foundationaldebates and describes some recent philosophical attempts to grant to visual-ization (Giaquinto) and diagrammatic reasoning (Barwise and Etchemendy)an epistemic status which goes beyond the mere heuristic role attributed tothem in the past.

With this background the reader can then move on to Marcus Giaquinto’s“From Symmetry Perception to Basic Geometry”. In Frege’s approach to thefoundations of mathematics, Frege explicitly excluded that psychological in-vestigations might be relevant to the foundational goal. This was basicallymotivated by the idea that experience, whether physical or psychological,could not warrant the generalizations drawn from it and thus in this way onecould not account for the objectivity of mathematics. The Fregean approachhad no account of how our psychological processes relate to our graspingof mathematical truths. Frege’s position rests on the assumption that theonly role a perception or an experience of visual imaging can play is that ofevidence for a further generalization. By contrast, Giaquinto proposes thatexperiences of seeing or visual imaging might play the role of “triggers” forbelief-forming dispositions which in turn give us geometrical knowledge. Hegives an account, meant to be empirically testable, of how we could come tothe knowledge of a simple geometrical truth, such as that in a perfect square

INTRODUCTION 3

the two parts either side of the diagonal are congruent. The account pro-ceeds by stages. First, a description of how we perceive squareness (and therole that visual detection of symmetries plays in the process) and how thisresults in a category representation (in the form of a description set) for theperceptual concept of square. Second, how a modification of the perceptualconcept of square gives rise to the concept of a perfect square. Third, it isargued that the possession of these concepts is tantamount to having certainbelief forming dispositions which can be triggered by experiences of visualseeing or imaging. However, the role of the visual or imaging experienceis not that of evidence but rather of “trigger” for the belief-forming disposi-tions. Finally, if the mode of acquisition of such triggered beliefs is reliableand there is no violation of epistemic rationality, Giaquinto claims that thebeliefs thus obtained constitute knowledge. What type of knowledge? Be-ing non-logical and non-empirical (since the role of experience is not that ofproviding evidence) the beliefs thus obtained are synthetic a priori. Whetherany individual or we as a community of mathematical learners come to havethe beliefs in question through the process described ought, according toGiaquinto, to be subject of empirical investigation. If he is right the gap be-tween experience and mathematical knowledge would finally be filled by anaccount that does justice both to the role of psychological processes and tothe objectivity of mathematics.

In a more traditional philosophical context, namely the perennial dis-pute between platonism and naturalism, James Robert Brown also addressesthe role of “seeing” and “intuition” in mathematics and the relevance of dia-grams in this context. While providing a defense of Platonism, Brown agreesthat with respect to epistemological questions traditional Platonism has al-ways been problematic: How are we to have access to the mathematical enti-ties which exist in an abstract non-causal world? Modern Platonists typicallyclaim that we can “see” or “intuit” the mathematical entities with a specialnon-sensible intuition. K. Godel and G. H. Hardy are two of the most well¨known mathematicians holding this view. Thus, Hardy for instance, sees allmathematical evidence as some sort of perception. But in this respect, ac-cording to Brown, he goes too far. We only need, Brown says, to commitourselves to the perception of some basic mathematical objects and facts.These can then serve as grounds for more advanced mathematics which wecannot directly “see” and this is quite close to Godel’s position. By stress-ïng the fact that much of what is called “seeing” in natural science is quiteremote from visual perception, Brown goes on to describe how we can have“seeing” in mathematics through diagrams. Using a simple example of apicture-proof, Brown claims that a diagram can function as a “telescope”


that allows us to “see” into the Platonic realm. The diagram displays only aspecific case but through the diagram we are able to intuit a general truth andthis intuition cannot be confused with the sensory information given by thediagram. Brown’s position on intuition is then elaborated further through ananalysis of how Freiling’s well known informal disproof of the ContinuumHypothesis (by throwing darts at the real line) affects Maddy’s naturalism inphilosophy of mathematics. The resulting claim is that, according to Brown,there is “some sort of mathematical perception which cannot be reduced toeither physical perception or to disguised logical inference”.

In his second contribution, Marcus Giaquinto addresses the varietiesof mathematical activities which are encountered in mathematical practice.These include, to name only some paradigmatic examples, discovery, expla-nation, formulation, application, justification and representation. All of theseactivities provide rich material for a philosophical analysis of mathematics.Unfortunately, until recently philosophers of mathematics have mainly paidattention to only a few of these and, moreover, the of attention has oftenbeen too narrowly focused. The extension proposed by Giaquinto concernsnot only the proposal to take into account the above mentioned activities butalso the various aspects in which mathematics is done and communicated(making, presenting, taking in).

Three important ingredients of mathematical activity are discovery, ex-planation and justification. The discussion of discovery through visual imag-ing nicely ties up with the previous material on visualization and again Gi-aquinto points out that although we might reach knowledge by such meansthis need not be a proof. Explanation is also a theme touched upon by manycontributors in the book. Giaquinto points out that there are proofs whichare not explanations and explanations which are not proofs. Of course, thereare also examples of proofs which are explanations, and Giaquinto refersto Chemla’s paper for an important historical example. Moreover, explana-tions might play a role in motivating definitions, as illustrated by the movingparticle argument which gives a satisfactory account of the use of Euler’sformula

eiπ = cosπ+ isinπ.

as a definition in extending the exponential function to complex numbers.Motivating a definition through an explanation is thus an important type ofmathematical activity and it can be seen as a form as justification which isdistinct from proving a theorem. Another such activity is motivating or ‘jus-tifying’ the axioms. Giaquinto concludes that extending the philosophicalanalysis of mathematics to all these aspects, and the many more discussed in

INTRODUCTION 5

his paper, ‘would restore to philosophy of mathematics its ancient depth andsucculence’.

Part II of the book is entitled Mathematical Explanation and Proof Styles.Jens Høyrup in “On Reasoning Styles in Early Mathematics” discusses

aspects of reasoning in Babylonian and Greek mathematics. Høyrup’s essaytakes its start from a criticism of those historians of mathematics who for-mulate a distinction between Babylonian and Greek mathematics claimingthat the former is basically a collection of ad hoc rules whereas the latter isa reasoned discipline. In addition to show that this is an incorrect charac-terization of the situation, Høyrup also wants to characterize the reasoninginvolved in Old Babylonian mathematics in constant comparison with Greekmathematics. What he points out is that there are certainly mathematicaltablets in which solutions to problems are given where “no attempt is madeto discuss why or under which conditions the operations performed are le-gitimate and lead to correct results”. The situation is made worse by thefact that most of these clay tablets contain no diagrams. But obviously theremust have been more that accompanied the process of instruction and learn-ing. Høyrup argues that a few remaining texts from Susa allows us to seethe kind of explanations that would have been given orally in a learning con-text. Moreover, these explanations are ‘critical’, i.e. provide reasons forthe extent of the validity of the procedure under discussion and for why theprocedure works. Thus, Old Babylonian mathematics displays its own char-acteristic style of thought. The real difference with Greek mathematics isthat in the Old Babylonian school “the role of critique had been peripheraland accidental; in Greek theoretical mathematics it was, if not the very centrethen at least an essential gauge”. Høyrup concludes that we cannot count asmathematics any activity that is devoid of understanding and that when thehistorian works on a mathematical culture for which the sources do not re-veal an appeal to reasoning then either we are not understanding the sourcesor the sources are not an accurate mirror of the mathematical practice.

Another area in history of mathematics which has traditionally beenjudged against the yardstick of Greek mathematics is Chinese mathemat-ics. Karine Chemla’s “The Interplay between Proof and Algorithm in 3-rdCentury China” dovetails well with Høyrup’s contribution by showing thatChinese mathematics also presents reasoning styles which differ from Greekmathematics but should nonetheless be seen as part of the history of proof. Akey case study in this connection is Liu Hui’s commentary on The nine chap-ters on mathematical procedures. In particular, Chemla focuses on Liu Hui’scommentary on the measurement of the circle. The commentary, made up oftwo parts, reveals Liu Hui’s concerns for explaining why a certain algorithm


is correct and it functions as an explanation of the algorithm. The work bearswitness to a sophisticated practice of proving mathematical results in ancientChina which differs from proving practices in Greek mathematics. Liu Huiwas a commentator and this is significant as proofs seem to emerge in Chi-nese mathematics as the result of such activity. Chemla claims that whereasin Greek mathematics proofs seem mainly aimed at establishing the truth ofpropositions in the case of Liu Hui what is at stake is the establishment ofthe correctness of a certain algorithm or possibly showing why the algorithmis correct. Without entering into the details of the analysis let us point outthat we have here another interesting case study of mathematical practicewhich highlights two important facts. First, it shows that the role of proofmight be explanatory, in addition to that of certifying a result. Moreover,the reasoning style displayed in these texts represent a characterizing featureof Chinese mathematics and thus it reminds us about the importance of themathematical culture in which different proof practices are embedded.

Thus, Høyrup’s and Chemla’s case studies, in addition to their intrin-sic importance for the historiographical characterization of Old Babylonianmathematics and Chinese mathematics vs. Greek mathematics, raise impor-tant issues concerning mathematical understanding and mathematical expla-nation and show that these notions are also context-dependent.

Jamie Tappenden’s article “Proof style and Understanding in Mathemat-ics I” touches on several topics central to the book such as visualization,explanation, justification, and concept formation. The article focuses on thedifferent styles within complex analysis represented by Weierstrass and Rie-mann. Weierstrass’s methodology was computationally motivated: it aimedat finding explicit representations of functions and algorithms to computetheir values. Riemann, by contrast, was more abstract in his approach, more“conceptual”. With the introduction of the concept of a Riemann surface,Riemann not only reorganized the subject matter of complex analysis but in-troduced a whole new style in the area. This new approached yielded newdiscoveries, new proofs and it deepened our understanding of the subject inunexpected ways. Tappenden explores here the important role that the vi-sualization allowed by Riemann’s approach played in this reconfigurationof the subject. The unification yielded by Riemann’s approach is also ana-lyzed by Tappenden with reference to contemporary debates on the nature ofunification, understanding, and explanation (Friedman, Toulmin, Kitcher).The topic of unification is intimately tied to the discussion of ‘fruitful’ con-cepts. Fruitful concepts have unifying and explanatory roles but it is oftendifficult to say what makes a concept fruitful in mathematics. Tappendenmentions the unification of the theory of algebraic functions of one variable

INTRODUCTION 7

and the theory of algebraic numbers given by Dedekind and Weber. In thiscase, notions like “ideals” and “fields” turned out to be extremely fruitfulconcepts: they help us understand “what is going on” and as such dischargean explanatory role. In connecting the topics of visualization, explanation,and unification, Tappenden notes that often, as in the case of Riemann’s ap-proach to function theory and Artin’s Geometric Algebra, a contributor tothe “fruitfulness” or “naturalness” of the approach is that the arguments andcategories characterizing the approach can be visualized. These aspects ofmathematical practice (visualization, explanation, fruitfulness) are often rel-egated to ‘subjective’ matters of taste but Tappenden makes a strong case thatthey can be the topic of fruitful philosophical and methodological analysis.However, he concludes, one should not hope to provide an a priori accountof such notions; rather, only detailed case studies of mathematical practicewill be able to enlighten us on these complex issues.

The notion of explanation in mathematics, which has appeared in manyof the articles discussed above, is the focus of Johannes Hafner and PaoloMancosu’s “The Varieties of Mathematical Explanations”. While Hafnerand Mancosu emphasize that explanations in mathematics need not be proofs(for instance, theories as a whole might be explanatory), in this paper theyrestrict attention to proofs. They begin by providing evidence for the claimthat mathematicians seek explanations in their ordinary practice and cherishdifferent types of explanations (for instance, many mathematicians are oftencritical of proofs that only show that something is true but do not give anhint of why it is true). They go on to suggest that a fruitful approach to thetopic of mathematical explanation would consist in providing a taxonomy ofrecurrent types of mathematical explanation and then trying to see whetherthese patterns are heterogeneous or can be subsumed under a general ac-count. In the literature on explanation in mathematics there are basicallyonly two philosophical theories on offer. One proposed by Steiner (1978)and an account based on unification due to Kitcher (discussed in the previ-ous article by Tappenden). Mancosu and Hafner provide a case study of howto use mathematical explanations as found in mathematical practice to testtheories of mathematical explanation. The case study focuses on Steiner’stheory of mathematical explanation. This theory singles out two criteria fora proof to count as explanatory: dependence on a characterizing propertyand generalizability through varying of that property. The authors argue thatPringsheim’s explanatory proof of Kummer’s convergence criterion in thetheory of infinite series defies both criteria and thus cannot be accounted forby Steiner’s model of explanation. This can be seen, as it were, as a case


study of how to show that the variety of mathematical explanations cannotbe easily reduced to a single model.

The last article of the book is devoted to a very neglected part of mathe-matical activity: the role of aesthetical factors in mathematics. Obviously,mathematics displays aesthetic features. Mathematicians often talk aboutthe elegance of certain constructions or the beauty of various geometricalfigures. But is the ‘aesthetic dimension’ a rational feature of mathematicalactivity or a completely subjective and non-analyzable aspect of the mathe-matical experience? Reviel Netz’s “Towards an aesthetic of mathematics”develops several analytical tools required for a productive discussion of thisdifficult topic. Netz argues from the outset that every type of human expres-sion possesses an aesthetic dimension. Moreover, the aesthetic dimensionis an objective fact, although a difficult one to analyze. Whereas most driv-ing factors in mathematical activity are epistemic in the case of beauty wehave a clear case of a non-epistemic value which is intrinsic to mathematicalresearch. Netz thus expands the range of topics addressed in the other con-tributions of the book: “The thrust of the articles collected in this volume is,I believe, to widen our picture of the field of mathematical practice as a ra-tional activity: one that appeals to the visual and not merely to the symbolic,that aims at explanation and not merely at proof. It also appeals, I suggest,to the aesthetic. Among other things – and still as rational practitioners –mathematicians aim at beauty.” Netz’s paper proceeds by giving a typologyof sources of mathematical beauty. Mathematical beauty can be predicatedof states of minds, of the products of mathematical activity (say theoremsas embodied in texts), and of the objects studied in the previous categories.Netz’s analysis focuses on mathematical texts and he proposes to bring tobear for the task a body of theory already developed in poetics. In order tolimit the scope he discusses Greek mathematical texts and explores the sensein which techniques of “narrative” and “prosody” can be fruitfully exploitedfor an analysis of the aesthetic dimension of these texts. In this approach“narrative” will account for the content and “prosody” for the form of themathematical text. Netz claims that just as in literature one source of beautyin mathematics is the interaction (he calls it “correspondence”) between formand content.

Given the emphasis on the heterogeneity of mathematical practice dis-played in most of the articles in the present collection, the outcome of thework is not that of claiming that some unique model or theory will accountfor the great wealth of mathematical activities. Even if such a theory were tobe found in the future, it would be premature to suggest anything of the sort atthis stage. Rather, through their mathematical, historical, and philosophical

INTRODUCTION 9

richness, these contributions show that there is a wide virgin territory opento investigation. Our hope is that others will also embark in its exploration.

Department of PhilosophyU.C. BerkeleyUSA

Section for Philosophy and Science StudiesRoskilde UniversityDenmark

PART 1

MATHEMATICAL REASONING ANDVISUALIZATION

PAOLO MANCOSU

VISUALIZATION IN LOGIC AND MATHEMATICS

In the last two decades there has been renewed interest in visualization inlogic and mathematics. Visualization is usually understood in different waysbut for the purposes of this article I will take a rather broad conception ofvisualization to include both visualization by means of mental images aswell as visualizations by means of computer generated images or imagesdrawn on paper, e.g. diagrams etc. These different types of visualizationcan differ substantially but I am interested in offering a characterization ofvisualization that is as broad as possible. The article describes and explains(1) the way in which visual thinking fell into disrepute, (2) the renaissanceof visual thinking in mathematics over recent decades, (3) the ways in whichvisual thinking has been rehabilitated in epistemology of mathematics andlogic.

This renaissance of interest in visualization in logic and mathematicshas emerged as a consequence of developments in several different areas,including computer science, mathematics, mathematics education, cognitivepsychology, and philosophy. When speaking of renaissance in visualizationthere is an obvious implication that visualization had been relegated to a sec-ondary role in the past. One usually refers to the fact that the development ofmathematics in the nineteenth century had shown that mathematical claimsthat seemed obvious on account of an intuitive and immediate visualizationturned out to be incorrect on closer inspection. This went hand in hand witha downgrading of Anschauung and specifically visuo-spatial thinking fromthe exalted status it had in Kant’s epistemology of mathematics. The effectswere also felt in pedagogy with a shift of emphasis away from visualization(for instance, in Landau’s diagram-free text on calculus).

1. DIAGRAMS AND IMAGES IN THE LATE NINETEENTHCENTURY

Some of the standard cases mentioned in this connection are the belief thatevery continuous function must be everywhere differentiable except at iso-lated points, or the tacit assumption in elementary geometry that the twocircumferences drawn in the construction of the equilateral triangle over anygiven segment in Euclid’s Elements I.1 meet in one point (the vertex of theequilateral triangle).1

In both cases the claim seems to be obvious from the visual situation(diagrams or mental imagery) but turns out to be unwarranted. In the for-mer case this is the consequence of the discovery of continuous nowheredifferentiable functions. In the latter case, this was due to the realization

13P. Mancosu et al. (eds.), Visualization, Explanation and Reasoning Styles in Mathematics, 13-30.

© 2005 Springer. Printed in the Netherlands.

14 PAOLO MANCOSU

that only a continuity axiom can guarantee the existence of the intersectionpoint of the two circles. Such results and many other concomitant factors,led mathematicians to formulate more rigorous approaches to mathematicsthat excluded the recourse to such treacherous tools as images and diagramsin favor of a linguistic development of mathematics. Of course, the use ofimages and diagrams was still allowed at a heuristic level. The careful math-ematician was however supposed to resist the chant of the visual sirens whenit came to the context of justification:

For the appeal to a figure is, in general, not at all necessary. Itdoes facilitate essentially the grasp of the relations stated in thetheorem and the constructions applied in the proof. Moreover,it is a fruitful tool to discover such relationships and construc-tions. However, if one is not afraid of the sacrifice of time andeffort involved, then one can omit the figure in the proof ofany theorem; indeed, the theorem is only truly demonstratedif the proof is completely independent of the figure. (Pasch,1882/1926, 43).

In short, visualization seemed to lose its force in the context of justifica-tion while being allowed in the context of discovery and as something thatsimplifies cognition (but cannot ground it). Pasch is well known for beingone of the pioneers of a development of geometry characterized by the rejec-tion of diagrams as relevant to geometrical foundations. In the Foundationsof Geometry (1899) Hilbert is not explicit about the role of diagrams in ge-ometry. However, in a number of unpublished lectures he raises the issue. Inlectures on the foundations of geometry from 1894 we read:

A system of points, lines, planes is called a diagram or figure[Figur]. The proof [of the theorem he is discussing] can in-deed be given by calling on a suitable figure, but this appealis not at all necessary. [It merely] makes the interpretationeasier, and it [the appeal to diagrams] is a fruitful means ofdiscovering new propositions. Nevertheless, be careful, sinceit [the use of figures] can easily be misleading. A theorem isonly proved when the proof is completely independent of thediagram. The proof must call step by step on the preceding ax-ioms. The making of figures is [equivalent to] the experimen-tation of the physicist, and experimental geometry is alreadyover with the [laying down of the] axioms. (Hilbert, 1894, 11).

And in other lectures from 1898 and 1902 Hilbert provides examples ofhow one can be misled by diagrams by going through a proof of the claimthat “every triangle is equilateral”. In introducing the example he says:

VISUALIZATION IN LOGIC AND MATHEMATICS 15

One could also avoid using figures, but we will not do this.Rather, we will use figures often. However, we will never relyon them [niemals auf sie verlassen]. In the use of figures onemust be especially careful; we will always have care to makesure that the operations applied to a figure remain correct froma purely logical perspective. (Hilbert, 1902, 602).

These motivations, emerging from the foundational work in geometryand analysis, led to a conception of formal proof that has dominated logic inthe past century (and it is usually attached to the names of Frege, Hilbert, andRussell). This conception of formal proof relies on a linguistic characteriza-tion of proofs as a sequence of sentences. We find the essential elements ofsuch conception already in Pasch:

We will acknowledge only those proofs in which one can ap-peal step by step to preceding propositions and definitions. Iffor the grasp of a proof the corresponding figure is indispens-able then the proof does not satisfy the requirements that weimposed on it. These requirements are fulfillable; in any com-plete proof the figure is dispensable [. . . ]. (Pasch, 1882/1926,90).

These attitudes towards diagrammatic reasoning and visualization havethus a complex history, which still calls for a good historian. Certainly onewould have to take into account the importance of the development of pro-jective and non-Euclidean geometries in the nineteenth century and of thearithmetization of analysis.2

However, I am not convinced that we can tell a linear story where theheroes finally attained a level of rigor hitherto unprecedented, thus leavingthe opposition in disarray. For instance, I think there is much to learn fromtaking a look at the opposition between Klein and the Weierstrass school orthe debate that opposed the “rigorist” Segre to the “intuitionists” Severi andEnriques in algebraic geometry.3

I will limit myself to a remark on one of the main paradigmatic exam-ples that were used to discredit the role of geometric intuition in analysis,e.g. Weierstrass’ discovery of a continuous nowhere differentiable function.Weierstrass’ result was announced in 1872 (and published by du Bois Rey-mond (1875, 29)). The function in question was given by the equation

f (x) =∑bn cos(anx)π

with a odd, b ∈ [0,1) and ab > 1+ 3π/2.

16 PAOLO MANCOSU

FIGURE 1. von Koch’s snow-flake.

Weierstrass’ result was given in a strictly analytic way (du Bois Rey-mond (1875, 29-31)) and it left obscure what the geometrical nature of theexample might be. This was somehow characteristic of the school of Weier-strass, which – as Poincare poignantly puts it – “ne cherche pas a voir maisá comprendre” (Poincar´` e, 1898, 16). However, there were mathematicians´who did not accept this distinction between seeing and understanding. Acase in point is Helge von Koch. Von Koch is now well known for his snow-flake, one of the earliest examples of fractals and up to this day one of theparadigmatic examples of fractals.

What is not well known is the motivation that led von Koch to his dis-covery of the snowflake. In his 1906 von Koch begins by remarking that untilWeierstrass came up with his example of a continuous nowhere differentiablecurve it was a widespread opinion (“founded no doubt on the graphical rep-resentation of curves”) that every continuous curve had a definite tangent(with the exception of singular points). But then he adds:

Although Weierstrass’ example has once and for all correctedthis mistake, it is insufficient to satisfy our mind from the ge-ometrical point of view; for the function in question is definedby an analytic expression which hides the geometrical natureof the corresponding curve so that one does not see, from thispoint of view, why the curve has no tangent; one should sayrather that the appearance is here in contradiction with the re-ality of the fact established by Weierstrass in a purely analyticmanner. (von Koch, 1906, 145-6).

Restoring geometrical meaning to the analytic examples was at the sourceof the work:


This is why I have asked myself – and I believe that this ques-tion is of importance when teaching the fundamental princi-ples of analysis and geometry – whether one could find a curvewithout tangent for which the geometrical appearance is inagreement with the fact in question. The curve which I foundand which is the subject of this paper is defined by a geomet-rical construction, sufficiently simple, I believe, that anyoneshould be able to see [pressentir] through “naıve intuition” theïmpossibility of a determinate tangent. (von Koch, 1906, 146).

Von Koch’s project must be seen against the background of the philo-sophical discussion among mathematicians on the demarcation between “vi-sualizable” (or “intuitable” ) and “non-visualizable” curves. This discussion(see Volkert (1986)), to which Klein, du Bois Reymond, Kopke, Chr. Wieneränd others contributed, should draw our attention to the fact that a detailedhistory of attitudes towards visualization in the twentieth century might re-veal a more complex pattern than a simple and absolute predominance of alinguistic, non visual, notion of proof.

2. THE RETURN OF THE VISUAL AS A CHANGE INMATHEMATICAL STYLE

But granting the predominance of a linguistic, non visual, notion of formalproof in mathematics – and examples such as Bourbaki make clear that thisnot a myth – let us now try to characterize the salient features of this ‘returnof the visual’.

One of the most important aspects is certainly the development of vi-sualization techniques in computer science and its impact on mathematics.Here there has clearly been a two ways influence as mathematical techniqueshave helped shape techniques in computer science (including those leadingto great progress in visualization techniques). Conversely, developments invisualization techniques developed by computer scientists have had impor-tant effects on mathematics. Computer graphics has allowed researchers todisplay information (say, analytic or numerical information) in ways that canbe represented in the form of a graph, or a chart or in other forms but in anycase in a form that allows for a quick visual grasp. Two areas are usuallysingled out as paradigmatic of the powerful role displayed by visualizationin the mathematical arena. The first one is the area of chaos theory, and inparticular fractal theory (see Evans (1991)). In “Visual theorems”, PhilipDavis emphasizes that “aspects of the figures can be read off (visual the-orems) that cannot be concluded through non-computational mathematical

18 PAOLO MANCOSU

FIGURE 2. Mandelbrot and Julia sets.

devices” (1993, 339). For delightful examples of visual proofs see Roger B.Nelsen (1993, 2000).

Here one can point at the dramatic case of the relationship between theMandelbrot set and all the Julia sets sitting inside it. It would have been im-possible to recognize analytically, without the visual support offered by thecomputer, that the Julia sets are present inside the Mandelbrot sets. More-over, the connectedness of the Mandelbrot set became apparent to Mandel-brot on the basis of its graphical appearance.

Another area where the benefits of computer graphics have been greatlyexploited is differential geometry. The visual study of three-dimensionalsurfaces was pioneered in the late seventies by T. Banchoff and C. Strauss.Through the use of computer graphic animation they were able to constructsurfaces and gain a better grasp of them by the application of transforma-tions. However, the two most eventful results obtained in this way werethe problem of everting the 2-sphere and the discovery of new minimalsurfaces.4 Palais aptly summarizes the situation:

Two problems in mathematics have helped push the state ofthe art in mathematical visualization – namely, the problemof everting the 2-sphere and of constructing new, embedded,complete minimal surfaces, especially higher-genus examples.In the case of eversion, the goal was to illuminate a process socomplex that very few people, even experts, could picture thefull details mentally. In the case of minimal surfaces, the vi-sualizations actually helped point the way to rigorous mathe-matical proofs. (Palais, 1999, 654).

In his account of the latter discovery David Hoffman says:


FIGURE 3. Costa’s surface.

In 1984, Bill Meeks and I established the existence of an in-finite family of complete embedded minimal surfaces in R3.For each k > 0, there exists an example which is homeomor-phic to a surface of genus k from which three points have beenremoved. Figure [3] is a picture of genus-one example. Theequations for this remarkable surface were established by Cel-soe Costa in his thesis, but they were so complex that the un-derlying geometry was obscured. We used the computer to nu-merically approximate the surface and then construct an imageof it. This gave us the clues to its essential properties, whichwe then established mathematically. (Hoffman, 1987, 8).

Hoffman emphasizes the importance of computer generated images as“part of the process of doing mathematics”. However, in his paper he alsoemphasizes the importance of proving ‘mathematically’ the properties of thesurface which can be ‘seen’ directly in the visualization:

Also it [the surface] was highly symmetric. This turned outto be the key to getting a proof of embeddedness. Withina week, the way to prove embeddedness via symmetry wasworked out. During that time we used computer graphics asa guide to “verify” certain conjectures about the geometry ofthe surface. We were able to go back and forth between theequations and the images. The pictures were extremely usefulas a guide to the analysis. (Hoffman 1987, p.17)

We thus see that the ‘return of the visual’ has led to new mathematicaldiscoveries, which, it might be argued, could not have been obtained withoutthe application of computer generated images. Nonetheless, these ‘images’are not taken at face value. The properties they display must then be verified‘mathematically’.5

20 PAOLO MANCOSU

The reaction against a purely symbolical conception of mathematics hasalso found its way in new presentations of certain mathematical subjects thatemphasize the visual aspects of the discipline. Paradigmatic examples areFomenko’s ‘Visual geometry and topology’ (1994) and Needham’s ‘Visualcomplex analysis’ (1997).

Both of them recognize the importance of the influence of computer sci-ence in the recent shift towards more visual methods but their call for a returnto intuition and visualization runs deeper and it is rooted in an apprecia-tion of the importance of visual intuition in areas such as geometry, topol-ogy, and complex analysis. Fomenko quotes Hilbert approvingly to the ef-fect that notwithstanding the importance of analytical and abstract reasoning“visual perception [Anschauung] still plays the leading role in geometry”.Fomenko however does not consider a visual presentation to be logicallyself-sufficient:

Many modern fields of mathematics admit visual presenta-tions which do not, of course, claim to be logically rigorousbut, on the other hand, offer a prompt introduction into thesubject matter. (Fomenko, 1994, preface p. vi).

And later:

It happens rather frequently that the proof of one or anothermathematical fact can at first be ‘seen’, and only after that (andfollowing the visual idea) can we present a logically consistentformulation, which is sometimes a very difficult task requiringserious intellectual efforts. (Fomenko, 1994, preface p. vii)

Thus, Fomenko’s emphasis is on the pedagogical and heuristic value ofvisual thinking and he does not seem to ascribe to results obtained by visualthinking a justificatory status comparable to that obtained by a ‘logicallyconsistent formulation’.

Needham is also very strongly critical of the tendency of modern math-ematics to downplay the importance of visual arguments. In a ‘parable’ hecompares the situation in contemporary mathematics to that of a society inwhich music can only be written and read but never be ‘listened to or per-formed’. He says:

In this parable, it was patently unfair and irrational to havea law forbidding would-be music students from experiencingand understanding the subject directly through ‘sonic intu-ition’. But in our society of mathematicians we have sucha law. It is not a written law, and those who flout it may yetprosper, but it says, Mathematics must not be visualized!6


Just like Fomenko, Needham concedes that “many of the arguments [inthe book] are not rigorous, at least as they stand” but that “an initial lack ofrigor is a small price to pay if it allows the reader to see into this world moredirectly and pleasurably than would otherwise be possible” (p. xi).

In concluding this section then I would like to point out that many con-temporary mathematicians are calling for a return to more visual approachesto mathematics. However, this return of the visual does not seem to upset thetraditional criteria of rigor. In all the cases mentioned above all the authorsremark on the cognitive importance of visual images in doing mathematicsbut also seem to recognize that images do not satisfy the criteria of rigor nec-essary to establish the results being investigated. In this sense this new trendtowards visualization, while marking an important shift in style of researchand mathematical education (on mathematics education see Zimmerman andCunningham (1991)), does not seem to me to bring about a radically new po-sition on the issue of the epistemic warrant which can be attributed to argu-ments which rely on visual steps. In any case, this problem is not addresseddirectly by any of the authors mentioned above.

At this point several problems could be raised. First, it would be inter-esting to know more about the cognitive visual roots of our mathematicalreasoning and the exact role that mental imagery plays in our mathematicalthinking. Second, a number of classical foundational and epistemologicalquestions can still be raised about the warrant afforded by diagrammatic orvisual reasoning. In the next section I will try to mention what seem to bethe most promising directions of research in these areas at the moment.

3. NEW DIRECTIONS OF RESEARCH AND FOUNDATIONS OFMATHEMATICS

Despite the great revival of interest for visual imagery in cognitive psychol-ogy (Kosslyn, 1980, 1983; Shepard and Cooper, 1982; Denis, 1989) researchin the specific field of mathematical visualization has still a long way to go.Some interesting results are emerging in the study of diagrammatic reason-ing (Larkin and Simon, 1987; Glasgow et al., 1995) and problem solving(Kaufmann (1979); for a survey and extensive references see Antonietti et al.(1995)). However, the two most up to date treatments of how the brain doesarithmetic (Butterworth, 1999; Deheane, 1997) contain very little on visual-ization in arithmetic. These investigations in cognitive psychology have in-fluenced a number of researchers active in foundations of mathematics, whoare interested in addressing the complex web of issues related to perception,imagery, diagrammatic reasoning and mathematical cognition.

22 PAOLO MANCOSU

Let me begin by mentioning the project “Geom´ etrie et cognition” led by´G. Longo, J. Petitot, and B. Teissier at the ENS in Paris. Their approachemphasizes the need to provide cognitive foundations for mathematics, inopposition to logical foundations a la Hilbert. Their research is strongly in-`fluenced by the dramatic developments in cognitive psychology, especiallyin the area of perception theory. And although mental imagery is not stressedin their ‘manifesto’ (where the emphasis is on perception)7, it is obvious thattheir program calls for an account of the cognitive role of mental imageryin mathematics. This ‘cognitive’ approach to the foundations is less con-cerned with the traditional goals of logical foundation, as it had been pur-sued in the tradition of proof theory. Rather, it goes back to the traditionrepresented by Riemann, Poincare, Helmholtz and Weyl. Moreover, they ap-´peal to Husserl’s phenomenological analyses and in particular the work ongenesis of concepts. For this tradition the foundational task is essentially togive an epistemological analysis of the constitutive role of the mind in theconstruction of mathematics and geometry in particular.

Of great interest is also the epistemological work on visualization car-ried out by Marcus Giaquinto.8 One can read Giaquinto’s project as tryingto account for the epistemological status of certain experiences of visual-ization which, he argues, are substantially different both from observationand from conceptual reasoning. There is obviously a “Kantian” flavor to theproject. The thesis, as presented by Giaquinto, is that the epistemic func-tion of visualization in mathematics can go beyond the merely heuristic oneand be in fact a means of discovery. We are used to associate discoverywith the heuristic context. But discovery is taken here in a technical senseaccording to which “one discovers a truth by coming to believe it indepen-dently in an epistemically acceptable way”. The independence criterion ismeant to exclude cases in which one comes to believe a proposition just bybeing told. One of the conditions on the requirement of epistemic acceptabil-ity is that the way in which one comes to believe a proposition is reliable.Giaquinto then proposes a case of visualization (a simple geometrical factabout squares) for which he claims that through that process of visualizationone could have arrived at a discovery (in the sense above, which does notentail priority) of the result. However, the justification provided by the vi-sualization need not be a demonstrable justification, i.e. a justification thatcan be checked by intersubjective standards of proof. And nonetheless thisis, he concludes, a legitimate way to come to know a mathematical proposi-tion. The upshot of his investigations is the claim that whereas in elementaryarithmetic and geometry it is possible to discover truths by means of visu-alization this is not the case in elementary real analysis, except perhaps in


extremely restricted cases. It is important to point out here some importantfeatures of Giaquinto’s approach. In traditional philosophy of mathematicsthe emphasis is mostly on major theories, such as arithmetic, analysis, or settheory. The main question that has been pursued is whether these theoriesare true and, if so, how do we know them to be true. Since deduction canpreserve knowledge, usually the question becomes that of the epistemologyof the axioms of such theories. Giaquinto asks the analogous question for thecase of the individual and his or her mathematical beliefs. How do peopleknow their initial (uninferred) mathematical beliefs? And more generally,how do they acquire their beliefs, whether initial or derived? His strategyis then to investigate how people actually acquire their beliefs, and this iswhere cognitive psychology comes in. Once we have isolated the cognitivemechanisms of belief acquisition then we can subject them to epistemolog-ical analysis and ask whether they are in fact knowledge-yielding. While itis beyond the scope of this paper to address the argumentative line defendedby Giaquinto, let me just grant him the thesis and see how it fits within thespectrum of positions in foundations of mathematics. Obviously, his projectdovetails quite well with the issues raised by those philosophers of math-ematics who insist that foundations of mathematics should address issuesconcerning the epistemology of mathematics. Giaquinto’s last writings onthis issue in fact (see this volume) put forth an account of the interactionbetween perception, visual imaging, concepts and belief formation in therealm of elementary geometry. But, as mentioned before, it is not central toGiaquinto’s claims that the types of visualizations he discusses would countas proofs in the traditional sense. He thus moves away from the traditionalconcerns in philosophy of mathematics in two ways. First of all, he shiftsthe focus from the community to the individual. Second, since justificationwill ultimately depend on some unjustified premises that we must hold to betrue, the question becomes how do we know these ultimate premises to betrue. And that, Giaquinto argues, cannot be done by giving another justifica-tion (which would involve a regress) but rather by an epistemic evaluation ofthe way we come to believe those premises. But this is a question that eventhose who focus on major mathematical theories will have to address, as thestarting point of those theories would have to be accounted for epistemolog-ically. And how could that be done, without going back to the mechanismsof belief acquisition of the individual? In this way, Giaquinto’s work showsits relevance also for traditional programs in the philosophy of mathematics.

By contrast, the work carried out by Barwise and Etchemendy on visualarguments in logic and mathematics is motivated in great part by the proof-theoretic foundational tradition.9 While Giaquinto was mainly concerned

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VISUALIZATION, EXPLANATION AND REASONING …download.e-bookshelf.de/download/0000/0032/87/L-G-0000003287... · TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES,Stanford