What Is Mathematics, Really?
By addebook • Jun 29th, 2008 • Category: Mathematics •
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What Is Mathematics, Really?
What Is Mathematics, Really?
By Reuben Hersh
Publisher: Oxford University Press, USA
Number Of Pages: 368
Publication Date: 1999-05-10
ISBN-10 / ASIN: 0195130871
ISBN-13 / EAN: 9780195130874
Binding: Paperback
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.
Book Description:
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.
Summary: An alternative philosophy of Mathematics
Rating: 4
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. It is a pity that math teachers forget about this when they teach and professional mathematicians often forget it when they write their papers.
Is mathematics invented or discovered? It has been a long standing controversy subject of discussions such as Alain Connes and a French neurologist. Hersh thinks both. After you invent a new theory (example group theory) you must discover its properties (find, for example, how many simple finite groups exist). And you may have to invent a trick to discover the solution of a problem.
To sum up: this book a “dimythization” of mathematics. Mathematics is just a human endeavour,but a highly beautiful, interesting, sophisticated and applicable human endeavour.
Summary: a very good book
Rating: 5
It is a very good book.The scope of this book all inclusive
and philosophical ideas are very well described and put in perspective especially on foundations of mathematics.Plus,a very
clear exposition.Highly recomended.Dr.A.Gelman
Summary: At times full of empty rhetoric
Rating: 2
This book is interesting, and has made me think hard about my own views. Mostly that is because I disagree with a lot of the beliefs touted by Hersh.
Hersh starts out with an approach to the hypercube, always a fascinating topic for me. Pages later he is trying, very unsuccessfully though he doesn’t realize, to decimate the “old fashioned” views about numbers, physical objects, social conventions, and basically everything Intelligible. His arguments are so terrible they basically ruin the book. He uses every form of slimy rhetoric to be convincing. Read these pages with an open eye and you’ll see how often he plays on the reader’s biases and preconceptions to make his point seem clear.
For the rest, the book is quite inclusive. A lot of this is interesting. But all of these ideas have existed without the help of Hersh, who would seem to have problem accepting this fact unless we reduced the timespan of their existence to include Hersh’s closest biological relatives, the rest of us humans.
Summary: A choppy rough draft in philosophy of mathematics
Rating: 3
This book comes across as some kind of extended constructivist/pragmatist complaint. Disjointed in its execution, it gives the appearance of a bunch of lectures too-quickly thrown together. Some weak arguments appear here and there, a few even coming across as downright silly. Perhaps its because Hersh has a simplistic, even at times sophomoric understanding of philosophy. He also has the lazy-man’s habit of quoting huge tracts of other peoples writings without giving any sort of application or interpretation. On the up side, the book does have an encyclopedic breadth, so it’s not a complete waste of time, even given its weaknesses. I took down several references. Did I like the book? Yes. Hersh should have retained an editor, or perhaps spent another year tidying it up. One more thing: Hersh is very anti-theistic. He downgrades Platonism on the basis that nobody believes in God anymore. He really should get out more, or at least read some sociology. The vast majority of the human race and even westerners believe in God. Hence, maybe Platonism in mathematics isn’t so crazy after all.
Summary: Really philosophy of mathematics
Rating: 5
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.
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