Social Constructivism as a Philosophy of Mathematics
Paul Ernest
STATE UNIVERSITY OF NEW YORK
... [Show More] PRESS
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Published by
State University of New York Press, Albany
© 1998 State University of New York
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Library of Congress Cataloging-in-Publication Data
Ernest, Paul.
Social constructivism as a philosophy of mathematics / Paul
Ernest.
p. cm. (SUNY series in science, technology, and society)
(SUNY series, reform in mathematics education)
Includes bibliographical references and index.
ISBN 0-7914-3587-3 (hardcover : alk. paper). ISBN 0-7914-3588-1
(pbk. : alk. paper)
1. MathematicsPhilosophy. 2. Constructivism (Philosophy)
I. Title. II. Series. III. Series: SUNY series, reform in
mathematics education.
QA8.4.E76 1997
510´.1DC21 97-3515
CIP
10 9 8 7 6 5 4 3 2 1
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Contents
List of Tables and Figures vii
Acknowledgments ix
Introduction xi
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1. A Critique of Absolutism in the Philosophy of Mathematics 1
2. Reconceptualizing the Philosophy of Mathematics 39
3. Wittgenstein's Philosophy of Mathematics 64
4. Lakatos's Philosophy of Mathematics 97
5. The Social Construction of Objective Knowledge 131
6. Conversation and Rhetoric 162
7. The Social Construction of Subjective Knowledge 206
8. Social Constructivism: Evaluation and Values 247
Bibliography 279
Index 305
Page vii
Tables and Figures
Table 4.1. A Comparison of Popper's LSD and Lakatos's LMD 100
Table 4.2. A Comparison of Lakatos's LMD with His MSRP 109
Table 4.3. Cyclic Form of Lakatos's Logic of Mathematical
Discovery
117
Table 5.1. Dialectical Form of the Generalized Logic of
Mathematical Discovery
151
Table 7.1. Harré's Model of 'Vygotskian Space' 209
Table 7.2. Dowling's Model of Contexts for Mathematical
Practices
236
Figure 7.1. The Creative/Reproductive Cycle of Mathematics 243
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Acknowledgments
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The following publishers have kindly given permission for the use of quotations for
which they retain copyright. In each case I indicate within the text the relevant source and
page references.
Basil Blackwell of Oxford has given permission to quote from Philosophical
Investigations by Ludwig Wittgenstein, 1953, translated by G. E. M. Anscombe.
Harvard University Press of Cambridge, Massachusetts has given permission to quote
from Mind in Society by Lev Vygotsky, 1978.
Massachusetts Institute of Technology Press, of Cambridge, Massachusetts and Basil
Blackwell of Oxford have given permission to quote from the 1978 revised edition of
Remarks on the Foundations of Mathematics by Ludwig Wittgenstein, translated by G. E.
M. Anscombe.
Routledge of London has given permission to quote from The Archaeology of Knowledge
by Michel Foucault, 1972.
I wish to express my gratitude to the Leverhulme Trust for supporting my initial work on
this book through the award of a senior research fellowship for the period 199193.
I am very grateful to a number of people for kindly reading part or all of a draft of this
book and offering helpful comments, criticisms or suggestions. These include David
Bloor, Stephen I. Brown, Randall Collins, Bettina Dahl, Philip J. Davis, Ray Godfrey,
Reuben Hersh, Vibeke Hølledig, Stefano Luzzato, Jakob L. Møller, Jacob Munter, Lene
Nielsen, Alan Schoenfeld, Susanne Simoni, Ole Skovsmose, Robert S. D. Thomas, Lars
H. Thomassen, Thomas Tymoczko and others.
Sal Restivo has been very encouraging throughout the development of this book and has
influenced the final outcome considerably. I am also very
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grateful to the editorial staff at SUNY Press for the care they have taken in the preparation
of the text for publication.
Finally, I wish to record my appreciation and thanks to Jill and our daughters Jane and
Nuala for their love and support which sustained me, as always, while I worked on the
book.
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Introduction
Mathematics is one of the great cultural achievements of humankind. Every schooled
person understands the rudiments of number and measures and sees the world through
this quantifying conceptual framework. By these means mathematics provides the
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language of the socially all-important practices of work, commerce, and economics. In
addition, digital computers and the full range of information technology applications are
all regulated by and speak to each other exclusively in the language of mathematics, and
they would not be possible without it. Thus mathematics is essential to the modern
technological way of life and the social outlook that accompanies it.
In contrast, some of the deepest and most abstract speculations of the human mind
concern the nature and relations of objects found only in the virtual reality of
mathematics. Infinities, paradoxes, logical deduction, perfect harmonies, structures and
symmetries, and many other concepts are all analyzed and explored definitively in
mathematics. Thus mathematics provides the language of daring abstract thought. Related
to this, mathematics is the language of certainty. For over two thousand years thinkers
have regarded mathematics as the only self-subsistent area of thought that provides
certainty, necessity, and absolute universal truth. So mathematics might be said to have,
in addition to a mundane utilitarian role, an epistemological role, an ideological role, and
even a mystical role in human culture
Despite being partly familiar to all, because of these contradictory aspects, mathematics
remains an enigma and a mystery at the heart of human culture. It is both the language of
the everyday world of commercial life and that of an unseen and perfect virtual reality. It
includes both free-ranging ethereal speculation and rock-hard certainty. How can this
mystery be explained? How can it be unraveled? The philosophy of mathematics is meant
to cast
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some light on this mystery: to explain the nature and character of mathematics. However
this philosophy can be purely technical, a product of the academic love of technique
expressed in the foundations of mathematics or in philosophical virtuosity. Too often the
outcome of philosophical inquiry is to provide detailed answers to the how questions of
mathematical certainty and existence, taking for granted the received ideology of
mathematics, but with too little attention to the deeper why questions. Thus, for example,
there are still real controversies in the philosophy of mathematics over whether the history
of mathematics has any bearing on its philosophy, and whether the experiences and
practices of working mathematicians can shed any light on questions of mathematical
knowledge. In the philosophy of science such questions have long been settled
affirmatively. But this is not yet the case in the philosophy of mathematics. One of my
goals in writing this book is to try to lift the veil and to demystify mathematics; to show
that for all its wonder it remains a set of human practices, grounded, like everything else,
in the material world we inhabit.
In the philosophy of mathematics a number of voices have been heard calling for a more
naturalistic account of mathematics. In differing ways Davis and Hersh (1980), Kitcher
(1984), Lakatos (1976), Tymoczko (1986a), Tiles (1991), Wittgenstein (1956), and others
have argued for a critical re-examination of traditional presuppositions about the certainty
of mathematical knowledge. Kitcher and Aspray (1988) suggest that these voices make
up a new "maverick" tradition in the philosophy of mathematics which is concerned to
accommodate current and past mathematical practices in a philosophical account of
mathematics.
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Outside of the philosophy of mathematics there has been more progress. First of all, a
number of different traditions of thought in sociology, psychology, history and
philosophy have been drawing on the central idea of the social construction of
knowledge as a way of accounting for science and mathematics naturalistically. Second, a
growing number of researchers have been drawing on other disciplines to account for the
nature of mathematics, including Bloor (1976), Livingston (1986) and Restivo (1992),
from sociology; Ascher (1991), D'Ambrosio (1985), Wilder (1981) and Zaslavsky (1973)
from cultural studies and ethnomathematics; Rotman (1987, 1993) from semiotics,
Aspray and Kitcher (1988), Joseph (1991) and Gillies (1992) from the history of
mathematics, and Bishop (1988), Ernest (1991) and Skovsmose (1994) from education.
This book can be located at the intersection of these traditions. It draws its central
explanatory scheme from the interdisciplinary social constructionist approaches currently
burgeoning in the human sciences. It gains confidence from the parallels in
multidisciplinary and multidimensional accounts of mathematics. But it draws its central
concepts and inspiration from the
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emerging maverick tradition in the philosophy of mathematics.
The book begins with a strong critique of absolutist views of mathematical knowledge in
the philosophy of mathematics (chap. 1) and traditional approaches to the philosophy of
mathematics in general (chap. 2). It argues that the philosophy of mathematics needs to
be reconceptualized and broadened to accommodate the social and historical factors
mentioned above.
In the next part, the philosophies of Wittgenstein (chap. 3) and Lakatos (chap. 4) are
critically reviewed and then used as a basis for an account of the social construction of
mathematical knowledge (chap. 5). This involves redefining the concept of mathematical
knowledge to include tacit and shared components, as well as developing an account of
the ''conversational" mechanism for the social genesis and justification of mathematical
knowledge. This is the generalized logic of mathematical discovery, extending Lakatos's
heuristic.
Chapter 6 develops the central idea of conversation which underpins social
constructivism. This requires breaking new ground in exploring the textual basis of
mathematical knowledge and the rhetorical functions of mathematical language and
proof. The role of conversation in the formation of mind and in social construction of
subjective knowledge of mathematics is also developed (chap. 7), together with the role
of semiotic tools and rhetoric in the learning of mathematics. A surprising analogy is
revealed between the social genesis and justification of "objective" mathematical
knowledge, on the one hand, and that of subjective mathematical knowledge, on the
other. It is argued that the philosophy of mathematics must consider the social
construction of the individual mathematician and her/his creativity, if it is to account for
mathematical knowledge naturalistically.
The book concludes by evaluating its proposals in the light of its critique of the
philosophy of mathematics and argues that, contrary to traditional perceptions, a socially
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constructed mathematics has a vital social responsibility to bear (chap. 8).
Followers of my work will know that I have been working on social constructivism for
more than a decade, and it will come as no surprise that this account builds on an earlier
version (Ernest 1991). The greatest similarities between the two versions occur in
chapters 1 and 2 of this book, where I felt it was necessary to go over and improve the
arguments against absolutism in the philosophy of mathematics and for the
reconceptualization of the field. In addition to the goal of making the argument
self-contained, there is enough novelty in these chapters to justify including them in their
own right, even for seasoned readers of the earlier work. For example, there is a new
argument that a reconceptualized philosophy of mathematics should offer an account of
the learning of mathematics and its role in the onward transmission of mathematical
knowledge.
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The present work is not merely an extension and elaboration of the earlier version of
social constructivism in Ernest (1991). In addition to being almost three times the length
there are a number of significant conceptual differences between this and the earlier
version, including the following improvements:
1.
Deeper analyses of Wittgenstein's and Lakatos's thought
2.Less reliance on language as an explicit foundation of subjective knowledge
of mathematics, with more emphasis on tacit knowledge, and on language and
rhetoric in accounting for "objective" mathematical knowledge
3.Recognition of the semiotic basis of mathematics and mathematical
knowledge
4.A shift from a Piagetian/constructivist view of mind to a social view based on
Mead, Vygotsky, and others (see also Ernest 1994b)
5.Greater recognition of the culture-boundedness of all knowledge, and the
necessity of identifying its material basis
6.A diminished concern to maintain the boundaries between history, sociology,
psychology and the philosophy of mathematics
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Chapter 1
A Critique of Absolutism in the Philosophy of Mathematics
Historically, mathematics has long been viewed as the paradigm of infallibly secure
knowledge. Euclid and his colleagues first constructed a magnificent logical structure
around 2,300 years ago in the Elements, which at least until the end of the nineteenth
century was taken as the paradigm for establishing incorrigible truth. Descartes ([1637]
1955) modeled his epistemology directly on the method and style of geometry. Hobbes
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claimed that "geometry is the only science bestow[ed] on [hu]mankind" (Hobbes [1651]
1962, 77). Newton in his Principia and Spinoza in his Ethics used the form of the
Elements to strengthen their claims of systematically expounding the truth. 1 This logical
form reached its ultimate expression in Principia Mathematica, in which Whitehead and
Russell (191013) reapplied it to mathematics, while paying homage to Newton with their
title. As part of the logicist program, Principia Mathematica was intended to provide a
rigorous and certain foundation for all of mathematical knowledge. Thus mathematics has
long been taken as the source of the most infallible knowledge known to humankind, and
much of this is due to the logical structure of its presentation and justification.
With this background, a philosophical inquiry into mathematics raises questions
including: What is the basis for mathematical knowledge? What is the nature of
mathematical truth? What characterizes the truths of mathematics? What is the
justification for their assertion? Why are the truths of mathematics necessary truths? How
absolute is this necessity?
The Nature of Knowledge
The question, What is knowledge? lies at the heart of philosophy, and mathematical
knowledge plays a special part. The standard philosophical
Page 2
answer, which goes back to Plato, is that knowledge is justified true belief. To put it
differently, propositional knowledge consists of propositions which are accepted (i.e.,
believed), provided there are adequate grounds fully available to the believer for asserting
them (Sheffler 1965; Chisholm 1966; Woozley 1949). This way of putting it avoids
presupposing the truth of what is known, although traditional accounts require it, by
referring instead to adequate grounds, which also include the justificatory element. The
phrase "fully available" circumvents the difficulty caused when the adequate grounds
exist but are not in the cognizance of the believer. 2 [Show Less]