What Is Mathematics, Really? 1st Edition by Reuben Hersh (PDF)

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book being reviewed here, “What is Mathematics, Really?”, is engagingly written. I found the literary style to be highly palatable. However, I do not concur with the author’s philosophy of mathematics. Admittedly, he was a professional mathematician, while I’m a mere amateur mathematician and amateur philosopher. Nevertheless, the stance I take on the philosophy of mathematics is not idiosyncratic. I am essentially Platonist in my worldview – with respect to the ultimate nature of mathematics. Platonism in mathematics is the most widely held view of modern mathematicians – or, so I believe. This puts me in favorable company among professional mathematicians.Rueben Hersh’s philosophy of mathematics is “humanist-social-historical”. He stated that his favorite philosopher in college was David Hume. There’s little wonder that Hersh is a left-leaning humanist with an (apparently) atheistic worldview. On pages 248-249, Hersh states the following: “…Mathematics is another particular, special social-historical phenomenon. Its most salient special feature is the uniquely high consensus it attains.”My reply to this assertion is this: Is that consensus not because mathematics asymptotically approaches the objectively existing perfect and infallible mathematics? Note that the study of physics leads physicists deeper and deeper into a better, more precise, and valid philosophy of physics. Newton’s theories of space and time were “corrected” by Einstein’s deeper insights into the true nature of space and time. In like manner, mathematicians learn more and more about the true nature of mathematical realities, even as physicists learn more about the true nature of mass-energy and space-time.Overall, I give Hersh’s book moderately high marks, notwithstanding his unpalatable atheist-humanist-social philosophy that’s espoused in his engaging book.

⭐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.

⭐the original conception of “the truth in mathematics” ….

⭐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.

⭐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.

⭐This book is a masterpiece on the philosophy of Mathematics… Reuben delivers a masterclass about humanistic mathematics, a must read for everyone interested in thinking about the nature of Mathematical reality and its objects. Fantastic book!

⭐More a philosophy book than a mathematics book… though the author traces histories of several lines of mathematical thought, he assumes more knowledge than I came to the book with. Perhaps better for those already engaged in philosophy of math than for curious newcomers.

⭐This is an incredibly readable book for a layperson, despite a bit of ‘math jargon’. The writing style is journalistic. Reuben Hersh leads his reader at a comfortable pace through a well-structured argument. As a layperson I couldn’t put the book down until I had read from cover to cover.

⭐Hersh sets out to define the parameters of a philosophy of maths, his best answer being a socio-historiic-cultural context, resolving the Platonist / formalist split, the ethereal real zone of Platonism being replaced by the collective human mind/brain.The delight of the book though is the later sections on the history of philosophy of maths where he namechecks many’s a classic; and the section on basic principles of maths – I say basic, but the piece on Godel’ s incompleteness theorem …?!

⭐This is really a recreational mathematics book for mathematicians. It has a light-hearted approach but it deals with some very deep topics.. It is extremely well written and was a joy to read. I think it is aimed at the general lay-person and indeed the non-mathematician would get something from it but in the main you have to be a mathematician to get the most from it as it deals with some very fundamental features of the subject.

⭐Excellent book by a renowned Mathematician. You do not need high powered mathematical background to follow the book. I recommend it.

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