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18 Unconventional Essays on the Nature of Mathematics

Reuben Hersh, editor
Publisher: 
Springer Verlag
Publication Date: 
2006
Number of Pages: 
326
Format: 
Paperback
Price: 
49.95
ISBN: 
0-387-25717-9
Category: 
General
[Reviewed by
William J. Satzer
, on
01/3/2006
]

Reuben Hersh is a kind of agent provocateur, probing and poking at mathematicians, provoking them to think more broadly about what they do when they do mathematics. 18 Unconventional Essays on the Nature of Mathematics is just what it says. In Hersh’s own introductory words:

This book comes from the Internet. Browsing the Web, I stumbled on philosophers, cognitive scientists, sociologists, computer scientists, even mathematicians! — saying original, provocative things about mathematics. And many of these people had probably never heard of each other! So I have collected them here. That way they can read each other’s work. I also bring back a few provocative oldies that deserve publicity.

This is not a book about foundations or formal logic. It is — at least in part — about the philosophy of mathematics, but perhaps more about the human practice of mathematics. One of my favorite essays from the collection is Bill Thurston’s “On Proof and Progress in Mathematics.” Here he writes of mathematics as a human activity that relies fundamentally on social communication of mathematical ideas and ways of thinking instead of simply the flat pronouncements of definition, theorem and proof that appear in our journals.

Some of the essays have a kind of shock value. Anthropologist Leslie White, in “The Locus of Mathematical Reality: An Anthropological Footnote” addresses the question of whether mathematical ideas are created or discovered. He’s unequivocal: “...mathematics is nothing more than a particular kind of primate behavior.” Further, “...mathematics in its entirety, its ‘truths’ and its ‘realities’ is a part of human culture, nothing more.” Brian Rotman, a mathematician-turned cognitive scientist, argues in “Toward a Semiotics of Mathematics” that a mathematician — the person sitting at a desk writing a paper — has at least two co-authors: a disembodied pure thinker, the impersonal voice who calls himself “we”, as well as an imaginary automaton who “in principle” carries out any calculations or algorithms that “we” mention.

Raphael Núñez writes in “Do Real Numbers Really Move?” of how we produce hand gestures — with millisecond-precise synchronization — as we talk mathematics. Maybe this is how we do our best and most effective teaching: literally by hand-waving!

Two essays discuss unfortunate effects mathematics has on other disciplines. Jack Schwartz in “The Pernicious Influence of Mathematics on Science” argues that mathematics, by concentrating our attention, makes us blind to its own omissions, an attribute he calls the “simple-mindedness of mathematics.” Furthermore, this simple-mindedness tends to inappropriately impose strictures of rigor on developing but immature scientific theories. Mathematics, he says, knows better what to do than why to do it. Similarly, in “The Pernicious Influence of Mathematics on Philosophy,” Gian-Carlo Rota suggests that philosophers who have put all their eggs in the basket of precision and definitiveness would do better to reconsider traditional philosophy with its two thousand year history of dealing realistically with ambiguity, uncertainty and inconsistency.

This is a terrific collection of essays. Everyone is certain to find something to hate. Almost all of the essays give us new insight into that curious thing we do when we do mathematics.


Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Introduction by Reuben Hersh ........................................................................... vii

About the Authors............................................................................................. xvii

Chapter 1

A Socratic Dialogue on Mathematics ................................................................ 1

Alfréd Rényi

Chapter 2

“Introduction” to Filosofia e matematica........................................................... 17

Carlo Cellucci

Chapter 3

On Proof and Progress in Mathematics ............................................................. 37

William P. Thurston

Chapter 4

The Informal Logic of Mathematical Proof ...................................................... 56

Andrew Aberdein

Chapter 5

Philosophical Problems of Mathematics in the Light

of Evolutionary Epistemology........................................................................... 71

Yehuda Rav

Chapter 6

Towards a Semiotics of Mathematics ................................................................ 97

Brian Rotman

Chapter 7

Computers and the Sociology of Mathematical Proof....................................... 128

Donald MacKenzie

Prelims.qxd 9/9/05 6:24 PM Page v

Chapter 8

From G.H.H. and Littlewood to XML and Maple:

Changing Needs and Expectations in Mathematical Knowledge Management....... 147

Terry Stanway

Chapter 9

Do Real Numbers Really Move? Language, Thought, and Gesture:

The Embodied Cognitive Foundations of Mathematics .................................... 160

Rafael Núñez

Chapter 10

Does Mathematics Need a Philosophy? ............................................................. 182

William Timothy Gowers

Chapter 11

How and Why Mathematics Is Unique as a Social Practice .............................. 201

Jody Azzouni

Chapter 12

The Pernicious Influence of Mathematics upon Philosophy.............................. 220

Gian-Carlo Rota

Chapter 13

The Pernicious Influence of Mathematics on Science........................................ 231

Jack Schwartz

Chapter 14

What Is Philosophy of Mathematics Looking for? ............................................ 236

Alfonso C. Ávila del Palacio

Chapter 15

Concepts and the Mangle of Practice Constructing Quaternions...................... 250

Andrew Pickering

Chapter 16

Mathematics as Objective Knowledge and as Human Practice.......................... 289

Eduard Glas

Chapter 17

The Locus of Mathematical Reality:

An Anthropological Footnote ........................................................................... 304

Leslie A. White

Chapter 18

Inner Vision, Outer Truth.................................................................................. 320

Reuben Hersh

Dummy View - NOT TO BE DELETED