## What is Mathematics, Really?Virtually all philosophers treat mathematics as isolated, timeless, ahistorical, inhuman. In What Is Mathematics, Really? renowned mathematician Reuben Hersh argues the contrary. In a subversive attack on traditional philosophies of mathematics, most notably Platonism and formalism, he shows 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. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. 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. The humanist standpoint helps him to resolve ancient controversies about proof, certainty, and invention versus discovery. The second half of the book provides a fascinating history of the "mainstream" of philosophy - ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, Hilbert, Carnap, and Quine. Then come the mavericks who saw mathematics as a human artifact - Aristotle, Locke, Hume, Mill, Peirce, Dewey, Wittgenstein. In his epilogue, Hersh reveals that this is no mere armchair debate, of little consequence to the outside world. Platonism and elitism fit together naturally. Humanism, on the other hand, links mathematics with people, with society, and with history. It fits with liberal anti-elitism and its historical striving for universal literacy, universal higher education, and universal access to knowledge and culture. Thus Hersh's argument has educational and political consequences. |

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#### What is mathematics, Really?

User Review - Not Available - Book VerdictHersh, mathematician and coauthor of The Mathematical Experience (1983), attempts to answer here the philosophical question, "What is mathematics?" Many practitioners think of themselves as ... Read full review

#### Review: What Is Mathematics, Really?

User Review - Glenn Kriger - GoodreadsA great book for a philosophical nerd. Read full review

### Contents

Survey and Proposals | 3 |

Criteria for a Philosophy of Mathematics | 24 |

MythsMistakesMisunderstandings | 35 |

IutuitionProofCertniuty | 48 |

Five Classical Puzzles | 72 |

Mainstream Before the Crisis | 91 |

Mainstream Philosophy at Its Peak119 | 119 |

Mainstream Since the Crisis | 137 |

Foundationism DicsMainstream Lives | 165 |

### Common terms and phrases

abstract algebra analytic angles answer arithmetic axiom of choice axiomatic set theory axioms believe Bertrand Russell Brouwer calculation called certainty Chapter complex numbers concepts construction contradiction David Hilbert definition derivative Descartes entities equation Euclid Euclidean Euclidean geometry example exist experience finite formal formalist formula foundationist foundations Frege function gazillion Godel graph Hilbert human humanist idea ideal indubitable integral intuition intuitionism Kant Kitcher Kurt Godel Lakatos Leibniz logic logicist mathe mathematical knowledge mathematical objects mathematical proof mathematicians matics meaning mental method mind myth natural numbers nature of mathematics non-Euclidean geometry notion number system philosophy of mathematics physical Platonism Platonist polynomials problem properties proved pure Pythagorean question rational numbers real numbers reality reasoning rigorous rules Russell Russell's paradox sense set theory set-theoretic space Spinoza square statement teaching theorem There's things thought tion triangle true truth understand universe What's Wittgenstein zero