FOM: review of book (fwd)

Reuben Hersh rhersh at
Thu Jan 29 12:52:18 EST 1998

---------- Forwarded message ----------
Date: Thu, 29 Jan 1998 13:29:10 GMT
From: Paul Ernest <P.Ernest at>
To: Reuben Hersh <rhersh at>
Subject: review of book

Here is text of review I have just sent to Times Higher Ed. Suppl. (Our
Chronicle of HE)

Thanks for your generous mentions of my work!

best wishes

>Jonathon Cape, 343pp, £18.99
>ISBN 0-224-04417-6 (cloth
>Published 4 September  1997.
>For many, mathematics is the paradigm of perfect and infallible knowledge.
For over two thousand years Euclid's Elements was taken as the paradigm of
absolute truth. In the twentieth century philosophers became aware of its
flaws and the leading schools in the philosophy of mathematics tried to
justify the certainty of mathematics in different ways. This was through the
logical foundations of mathematics in Frege and Russell's logicism, as a
meaningless formal system according to Hilbert's formalism, and as based on
infallible intuition in Brouwer and Heyting's intuitionism. Each of these
schools tried to promote a big idea about what mathematics really is.
However it is now accepted that these attempts failed. Although recent
philosophers of mathematics offer virtuoso technical arguments, few have
dared to address the big issue of what mathematics really is. The present
book takes up this challenge again.
>In the past quarter century a new movement has emerged in the philosophy of
mathematics, called the 'maverick' tradition by Philip Kitcher. This
movement is naturalistic, concerned to describe the nature of mathematics
and the practices of mathematicians, both in the present and in the past. It
has a fallibilist epistemology, rejecting the claims for the absolute
timeless certainty of mathematical knowledge. It also rejects the
Platonistic view that the objects of mathematics exist in some perfect,
timeless and superhuman realm. Members of this movement include Philip J.
Davis, Reuben Hersh, Imre Lakatos, Philip Kitcher, Tom Tymoczko, and others,
and Hersh was one of the first to call for the reform of the philosophy of
mathematics. He has been working at the leading edge of this movement ever
since, and the present book is his magnum opus. In it he takes a calculated
risk. Instead of writing an academic book aimed only at persuading scholars
he also makes his arguments and erudition in mathematics and its philosophy
accessible to the well informed public. Any taxing mathematical examples are
collected together in an appendix at the end of the book. The risk is that
the philosophical community will take the book less seriously. But it is a
serious book which succeeds in putting a strong case for a socio-historical
view of mathematics, which Hersh calls 'humanistic'. The book is easy to
read but it has a subtle and complex structure. Ideas, issues and questions
are first touched upon lightly, illustrated by examples, and then returned
to repeatedly, from different angles, until by the end thorough analyses
have been provided, seemingly without effort.
>After a some illuminating preliminaries the book proper starts with a quick
introduction to traditional philosophies of mathematics, as well to Hersh's
own humanistic philosophy. This is followed by an inquiry into the nature of
the philosophy of mathematics itself, and essential and desirable
characteristics of the field are discussed. The central part of the book is
taken up with two accounts. The first is a discussion of key concepts of
mathematics, such as its overall image, intuition, proof, infinity, change,
and existence. The second is an historical overview of the philosophy of
mathematics organised into themes and encompassing almost fifty philosophers
from Pythagoras, Plato and Aristotle to Lakatos, Kitcher and other present
day thinkers. This account humanises the history of the philosophy of
mathematics right up to the present, and provides the central axis of the
account. Finally Hersh's own humanistic philosophy of mathematics is
evaluated in terms of the criteria he defined earlier.
>One of the most important arguments in the book, hinted at by the title,
concerns the reality of mathematical objects. Hersh argues that these
objects are part of social-historic reality, but they have no sense or
existence beyond their cultural meanings. He argues that there are two
technical senses of the existence of objects in mathematics, constructive
and indirect. Constructive existence means that a mathematical object is
made from known objects by a determinate and normally finite sequence of
operations. Thus the 'recipe' for constructing the object is given in
mathematical language. Indirect existence means that within a mathematical
system it is possible to derive a contradiction by deductive reasoning from
the assumption that the object does not exist. The difference between these
two senses is what led to the conflict between Brouwer and Heyting's
intuitionism and the rest of classical mathematics, for classical
mathematicians accept both senses but intuitionists only the former. Hersh
argues that both senses are important and meaningful, but what they really
show is the fictional nature of mathematical objects. Thus he proposes a
form of nominalism. Mathematical objects only exist in the system of
mathematics. However, because of the power of the structures and system of
mathematics as a tool for describing the world about us, which mathematical
humanism shows to be its historical origin, mathematics has real
consequences. These consequences, like those of music, economics, and the
gap in a doctors appointment schedule into which we are fitted when we have
an urgent complaint, are really experienced by us, even if the objects
themselves are not real in the down-to-earth sense that stones are.
>Overall, this is a well written, well argued and fascinating book. It
contains important arguments that push back the boundaries of debates in the
philosophy of mathematics. It proposes an important new position in the
philosophy of mathematics: Hersh's humanism. It offers both a challenge to
philosophers of mathematics and an accessible account for the
non-specialist. It deserves to be read by all interested in mathematics and
its philosophy, from philosophers and mathematicians to school teachers and
interested members of the public at large. (922 words)
>Dr Paul Ernest
>Reader in Mathematics Education
>University of Exeter
>Paul Ernest specialises in mathematics education and the philosophy of
mathematics, and his most recent book is Social Constructivism as a
Philosophy of Mathematics, SUNY Press, New York, 1997.   
>He edits The Philosophy of Mathematics Education Journal which is published
electronically at   

  |                                                         |
  |   Paul Ernest            Tel. +44-1392-264857           |
  |   University of Exeter   Fax +44-1392-264736            |
  |   School of Education    PErnest at               |
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  |   New book: Social Constructivism as a Philosophy of    |
  |   Mathematics, Albany, New York: SUNY Press, 1997.      |
  |   (ISBN 0791435873 h/b; 0791435881 p/b)                 |

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