What is mathematics, really? /
Reuben Hersh.
- New York : Oxford University Press, 1997.
- xxiv, 343 pages : illustrations ; 25 cm
Includes bibliographical references (pages 317-334) and index.
Survey and proposals -- Criteria for a philosophy of mathematics -- Myths/mistakes/misunderstandings -- Intuition/proof/certainty -- Five classical puzzles -- Mainstream before the crisis -- Mainstream philosophy at its peak -- Mainstream since the crisis -- Foundationism dies/mainstream lives -- Humanists and mavericks of old -- Modern humanists and mavericks -- Contemporary humanists and mavericks Mathematics is a form of life. pt. 1 -- ch. 1. ch. 2. ch. 3. ch. 4. ch. 5. pt. 2 -- ch. 6. ch. 7. ch. 8. ch. 9. ch. 10. ch. 11. ch. 12. Summary and recapitulation -- ch. 13.
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.