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What Is Mathematics, Really? 1st Edition

4.0 out of 5 stars 14 customer reviews
ISBN-13: 978-0195130874
ISBN-10: 0195130871
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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; 1 edition (July 8, 1999)
  • Language: English
  • ISBN-10: 0195130871
  • ISBN-13: 978-0195130874
  • Product Dimensions: 9 x 1.1 x 6.1 inches
  • Shipping Weight: 1.4 pounds (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (14 customer reviews)
  • Amazon Best Sellers Rank: #598,009 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

Format: 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.
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Format: Paperback
Brint Montgomery is spot-on calling this "a choppy rough draft," but I'd say s/he gives it one star too many.

Hersh wanders through the landscape of ideas about math without much of a plan: he visits some regions so often the reader loses patience, while wholly ignoring others. He harps to the point of total tedium on his not-very-exciting notion of mathematics as a social undertaking, without really examining what that implies. Are we to think the idea "two" for * * is some sort of social construct? I'll bet anything Australopithecus was hard-wired to know the difference between one lion and two. Anyway, does counting up to five really belong in the same category with group theory? Hersh just assumes so, leaving a whole slew of fascinating questions out of his purview.

But Hersh writes clearly, if sometimes a bit too cutely, covers a lot of ground, and has some fascinating proofs and explanations in the back of the book. I'd be happy to have bought it just for the explanation and short proof of Gödel's Incompleteness Theorem. Bottom line: glad I bought it, hope something a lot better comes along soon.
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Format: Kindle Edition Verified Purchase
I feel Reuben Hersh has done for math what Kuhn did for science: show the "back room" of the enterprise. Here is an non idealized, messed up if you will, version of math that takes into account what mathematicians really do, and not just what philosophers have to say and adjudicate about it. For a non mathematician like myself, Hersh's book has been a revelation, for now I have a much better grasp of what the field is about. I highly recommend this book to anyone interested in philosophy and/or math.
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By S.Z. on August 13, 2013
Format: Paperback
I'm a math major who became interested in math via philosophy. It is not easy to find a philosophical approach to what math means from the outside. How true and real are mathematical objects? Just what does a proof prove? I've generally been dismayed at how little this is discussed within mathematics. Hersh calls his view "humanism." His view is related to that of Doron Zeilberger, though not as radical or indulgent. Hersh strikes me as an anti-foundationalist in regards to math.I've come around to this view myself, partly thanks to Hersh. I'm also personally more interested in applied than pure mathematics, especially because I enjoy math as a language with which to describe reality. Before shifting to an anti-foundationalist perspective, I was looking in constructive analysis as presented by Errett Bishop. In the end, the foundationalist has to build math out of sets, integers, or logic; yet, as Popper mentioned, the edifice of science is still seen to rise out of the swamp. What is a set? What is an integer? At some point one has undefined terms, rules justified by instinct. For this reason, a more holistic, social, and empirical understanding of math seems appropriate. I've just paraphrased my understanding of Hersh's general position. The book is generous in its tour of many philosophers' treatment of mathematics, though, not just Hersh's. The typeset is good and the writing style is clean and direct.
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This book has four parts: In the first the author discusses his ideas about his philosophy of mathematics. The second and longest part is historical, divided into mainstream philosophies of mathematics and "humanists and mavericks". There follows a short summary and some interesting and more technical notes.

There are basically three philosophies of mathematics: Platonism, Formalism and Constructivism. Reuben Hersh proposes an alternative: Humanism. The three basic philosophies deal mainly with the problem of foundations and view mathematics as a source of indubitable truth. The problems with foundations (the paradoxes), the failure of Hilbert's program (Gödel's theorem) and recent controversial proofs, such as the Four Colour Theorem, breathe air to this new kind of philosophy, perhaps not so new, since we can find its origins already in Aristotle. The humanist philosophy looks at what mathematicians do. It is no so different from what other scientists do. Mathematics is fallible and corrigible and mathematical rigour varies with the ages(remember A. Wiles first proof of Fermat's conjecture, classical calculus infinitesimals or Pasch missing gap in Euclid's axioms). Mathematics is not so different from music. Music exists by some biological or physical manifestation, but it makes sense only as a mental and cultural entity. RH defines mathematics as "the study of the lawful, predictable, parts of the socio-conceptual world". Mathematics is part of our culture and history and mathematical ideas match our world for the same reason that our lungs match earth's atmosphere.

Solving problems and making up new ones is the essence of mathematics. It is the questions that drive mathematics.
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