Reuben Hersh's What is Mathematics, Really? — A review

Reuben Hersh’s What is Mathematics, Really? attempts to explain the nature of mathematics through an exhaustive survey of different schools of thought through the ages. The bulk of the book is a historical survey of philosophy of mathematics with an encyclopedic breadth not unlike Bertrand Russell’s A History of Western Philosophy.

Since the time of Plato, a major bone of contention within mathematics has been whether new mathematics is created or discovered. Those on the side of discovery believe that all mathematical objects and concepts exist independently of human thought and the mathematician simply discovered what has been there all along. Meanwhile, those who believe that new mathematics is created claim that all mathematical objects and concepts have no independent pre-existence and appear for the first time when a mathematician expresses them. Hersh proposes a different philosophy: humanism. According to Hersh, mathematics is a social enterprise and must be contextualized (e.g. history); mathematical objects and concepts are socio-cultural constructs. In humanism, mathematics is a human endeavor that is fallible and corrigible.

While the book touts itself as “easily comprehensible to anyone”, it simply isn’t easily digested by the average person. One needs an above-average background in mathematics, philosophy, and history to fully grasp and appreciate the book. The book itself feels like rough first draft, desperately in need of a good editor. Hersh seems to haphazardly jump from one philosophy-chapter to another, seriously discussing some while ignoring others; for long stretches the book is tiresome to read except to the most dedicated philomath. The title itself is a question, one that is left unanswered. On the whole, however, Hersh succeeds in elucidating his novel views as well as long-standing views on the nature of mathematics. Engrossing in some parts, tedious in others. Exceptionally thought-provoking at its best, confused and drearily convoluted at its worst. Despite its shortcomings, it is one of the most interesting books I’ve read in years. Its discussion of philosophy and its history alone is enough reason to read the book. It just needs a bit of polishing.