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The Essential John Nash

Harold W. Kuhn and Sylvia Nasar, editors
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Darren Glass
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If there is any mathematician about whom it can be said that they "need no introduction," then it would probably be John Nash. Between the (wonderful, in this reviewer's humble opinion) biography A Beautiful Mind released a decade ago and the (not-so-wonderful, in this reviewer's humble opinion) film adaptation that came out in 2001, the story of Nash as a brilliant mathematician haunted by his schizophrenia who goes on to win the Nobel Prize is now quite familiar to the general public. As is expected, however, the actual mathematics he has done is far less well-known than his biography, and that is the gap which The Essential John Nash attempts to fill.

Recently released in paperback for the first time, The Essential John Nash is a collection of many of Nash's mathematical writings, edited and slightly annotated by economist Harold Kuhn and by Sylvia Nasar, who wrote the book A Beautiful Mind . The book includes some biographical and background information, in the form of an introduction by Nasar, an autobiography and an afterword by Nash himself, and a wonderful photo essay, but most of the book is dedicated purely to mathematics.

Not all of Nash's papers are included, but Kuhn and Nasar have selected seven particularly important ones and collected them in a nice volume. There is a 1995 article by John Milnor describing the game of Hex, a version of which was invented by Nash while he was a graduate student. There is a 1954 paper written by Nash for the RAND Corporation containing ideas about parallel computing. There is Nash's 1958 paper on "Continuity of Solutions of Parabolic and Elliptic Equations." Also in the volume are two papers on the relationships between real manifolds and algebraic varieties, including Nash's proof that every Riemannian manifold can be isometrically embedded in some Euclidean space, a result now known as the 'Nash Embedding Theorem'.

And, of course, there is Nash's groundbreaking work in game theory, which eventually led to him being awarded the Nobel Prize. Nash's work was mostly in the area of non-cooperative game theory, and greatly extended the work done by von Neumann and Morgenstern, who limited themselves to cooperative game theory. Nash's three papers in the area are included here, and the editors include both a facsimile of his original PhD thesis in the area as well as a version that is typeset in a way that is more appealing to modern eyes. Kuhn's background as an economist is particularly on display here, as these papers have more annotation and background than some of the more theoretical and abstract mathematical works in the volume.

The book is a collection of serious research papers, and as such it is not the kind of light bedtime reading that one might expect from a paperback book with a flashy cover that you find at an endcap at your local Borders, even for mathematicians. (One can only imagine what non-mathematicians who pick up the book because they enjoyed the movie will think when they get to the chapter on Riemannian manifolds). For that, there is A Beautiful Math  and similar books. But the papers make for fascinating reading, both from a mathematical and a historical perspective and I hope that the success of this book inspires further such collections for the general public.

Darren Glass ( is an Assistant Professor at Gettysburg College.

 PREFACE by Harold W. Kuhn vii
INTRODUCTION by Sylvia Nasar xi
Chapter 1: Press Release--The Royal Swedish Academy of Sciences 1
Chapter 2: Autobiography 5
Photo Essay 13
Editor's introduction to Chapter 3 29
Chapter 3: The Game of Hex by John Milnor 31
Editor's Introduction to Chapter 4 35
Chapter 4: The bargaining problem 37
Editor's Introduction to Chapters 5, 6, and 7 47
Chapter 5: Equilibrium Points in n-Person games 49
Chapter 6: Non-Cooperative Games Facsimile of Ph.D. Thesis 51
Chapter 7: Non-Cooperative Games 85
Chapter 8: Two-Person Coooperative Games 99
Editor's Introduction to Chapter 9 115
Chapter 9: Parallel Control 117
Chapter 10: real Algebraic Manifolds 127
Chapter 11: The Imbedding problem for Riemannian Manifolds 151
Chapter 12: Continuity of Solutions of Parabolic and Elliptic Equations 211