What Is Mathematics, Really? [Paperback]

Reuben Hersh (Author)

Book Description

ISBN-10: 0195130871 | ISBN-13: 978-0195130874 | Publication Date: July 8, 1999

Most philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Reuben Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of his book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Descartes, and Spinoza, to Bertrand Russell, David Hilbert, and Rudolph Carnap--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, and Lakatos.
What is Mathematics, Really? reflects an insider's view of mathematical life, and will be hotly debated by anyone with an interest in mathematics or the philosophy of science.

Editorial Reviews

Amazon.com Review

In What Is Mathematics, Really?, author Reuben Hersh proposes a philosophy of mathematics that he calls "humanism" and uses this philosophy to analyze age-old questions of proof, certainty, and invention versus discovery. He also surveys the history of the philosophy of math. Readers of all levels of mathematical experience will be stimulated by the fascinating and perspicacious discussions Hersh has to offer. --This text refers to an out of print or unavailable edition of this title.

From Library Journal

Hersh, mathematician and coauthor of The Mathematical Experience (1983), attempts to answer here the philosophical question, "What is mathematics?" Many practitioners think of themselves as "platonists," discovering truths about ideal, eternally existing, abstract objects. The principal alternative to this concept is the "formalist" notion that mathematics is a game in which theorems are developed logically, starting from a set of axioms chosen almost arbitrarily. Hersh's humanistic position is, in essence, that mathematics is what mathematicians do. This is hard to disagree with but does not really explain how the subject evolves. Many feel that all mathematics begins with real-world applications, from which we try to extract the common properties and thence create a universe of abstract objects that can reveal unexpected beauty. In this somewhat disjointed book, Hersh reviews the history of the philosophy of mathematics, discusses the major players, and convincingly sets forth his thesis while undermining those of the competitors. For academic collections.?Harold D. Shane, Baruch Coll., CUNY
Copyright 1997 Reed Business Information, Inc. --This text refers to an out of print or unavailable edition of this title.

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Product Details

• Paperback: 368 pages
• Publisher: Oxford University Press, USA (July 8, 1999)
• Language: English
• ISBN-10: 0195130871
• ISBN-13: 978-0195130874
• Product Dimensions: 9.2 x 6.1 x 1 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 3.9 out of 5 stars  See all reviews (10 customer reviews)
• Amazon Best Sellers Rank: #358,389 in Books (See Top 100 in Books)

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29 of 32 people found the following review helpful:

5.0 out of 5 stars Really philosophy of mathematics, January 4, 2001

By 

Colin McLarty (Chardon, OH USA) - See all my reviews

This review is from: What Is Mathematics, Really? (Paperback)

The book offers the best kind of live, seriously thought out, philosophy of mathematics--in real contact with mathematical practice and teaching. Hersh writes from a deep love of mathematics and a deep concern to make it accessible to others, and for him both of those motivate philosophic reflection on the nature of mathematics.

Hersh notes that mathematics is a social enterprise. People may pursue it alone in their rooms, and even do the greatest thinking that way (as Andrew Wiles did some great thinking in near secrecy on the way to proving the Fermat theorem). But what they think about is not their sole creation (witness the many enthusiastic citations Wiles gives to what he owes others). What we call "proofs" in actual practice are not complete deductions in formal logic, nor simply "whatever persuades you". They are reasonings that live up to a socially recognized standard.

Hersh believes, and argues, that students who understand the social nature of mathematics will approach it with more interest and less fear than those who think it is inhuman perfection. Actually, I think he is wrong about that. Students today generally believe literature is a social product, but they still too often think that "getting it" is an arcane and uninteresting skill of English teachers. But Hersh's view deserves careful consideration and you can learn from him whether you agree in the end or not.

I will also say that Hersh's descriptions of earlier philosophies of mathematics are not always historically very accurate. And though he has genuine concern to give sympathetic accounts of them (before giving his own refutation) he does not always succeed. But neither are his versions notably worse than the versions in other similar books. For accurate accounts of Plato or the 20th century giants Poincare, Hilbert, Brouwer, and so on, you have just got to read the originals.

Anyone interested in philosophic thought about math, and not just solutions to one or another specific technical problem in the philosophy of math, should read this book. But don't only read this one.

9 of 11 people found the following review helpful:

5.0 out of 5 stars Tremendously Thought-Provoking, December 27, 2007

By 

John P. (Kennett Square, PA USA) - See all my reviews

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This review is from: What Is Mathematics, Really? (Paperback)

First, I need to disclose that I'm not a mathematician or a philosopher. I'm a lawyer with an interest in jurisprudence (philosophy of law) and the nature of legal reasoning.

"What Is Mathematics, Really?" is one of the most though-provoking books I've ever read. It has helped me to make progress on jurisprudential problems that I had formerly been attacking in largely fruitless ways.

The book thus filled a particular need for me. But I think anyone interested in intellectual history or the placing of math in context with other fields will find this book fascinating.

5 of 6 people found the following review helpful:

5.0 out of 5 stars An excellent text for an upper-level course in the philosophy of mathematics and is engaging reading for all practicing mathema, September 29, 2008

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: What is Mathematics, Really? (Hardcover)

Since the time of Plato, a major area of discussion within mathematics has been over whether mathematicians create or discover new mathematics. Those who favor the discovery side believe that mathematical objects and concepts already exist in some abstract book of knowledge and the discoverer simply turned to the right page in the book. On the other side are people who believe that the mathematical objects and concepts have no independent pre-existence and appear for the first time in any form when a mathematician expresses them. Of course, with any two widely disparate positions, there are many people who take a middle ground. Some have invoked the "mind of God" as the location of the pre-existence, yet invocation of a deity is not necessary to argue the pre-existence position.
Hersh puts forward complete descriptions of both these positions and then lists the mathematical principals who have put forward arguments on one side or the other. It is a list of most of the significant figures in the history of mathematics, which is an indication of how dynamic the field has been. Some of the major discoveries in mathematics have been counter-intuitive and led to alterations in the very foundations of mathematics.
This is not a book that can easily be digested by the non-mathematician. To understand how significant a new discovery was, it is necessary to have a solid grasp of the mathematics. For example, the consequences and significance of Georg Cantor's discovery that there are different levels of infinity cannot be understood without knowing how mathematicians struggled with the concept for centuries.
This book would make an excellent text for an upper-level course in the philosophy of mathematics and is engaging reading for all practicing mathematicians. It does all people working in any field good to take time out on occasion to study exactly what that field is and how it relates to the world.