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
... [Show More] 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 centra [Show Less]