18 Unconventional Essays on the Nature of Mathematics

Reuben Hersh is a kind of agent provocateur, probing and poking at mathematicians, provoking them to think more broadly about what they do when they do mathematics. 18 Unconventional Essays on the Nature of Mathematics is just what it says. In Hersh’s own introductory words:

This book comes from the Internet. Browsing the Web, I stumbled on philosophers, cognitive scientists, sociologists, computer scientists, even mathematicians! — saying original, provocative things about mathematics. And many of these people had probably never heard of each other! So I have collected them here. That way they can read each other’s work. I also bring back a few provocative oldies that deserve publicity.

This is not a book about foundations or formal logic. It is — at least in part — about the philosophy of mathematics, but perhaps more about the human practice of mathematics. One of my favorite essays from the collection is Bill Thurston’s “On Proof and Progress in Mathematics.” Here he writes of mathematics as a human activity that relies fundamentally on social communication of mathematical ideas and ways of thinking instead of simply the flat pronouncements of definition, theorem and proof that appear in our journals.

Some of the essays have a kind of shock value. Anthropologist Leslie White, in “The Locus of Mathematical Reality: An Anthropological Footnote” addresses the question of whether mathematical ideas are created or discovered. He’s unequivocal: “...mathematics is nothing more than a particular kind of primate behavior.” Further, “...mathematics in its entirety, its ‘truths’ and its ‘realities’ is a part of human culture, nothing more.” Brian Rotman, a mathematician-turned cognitive scientist, argues in “Toward a Semiotics of Mathematics” that a mathematician — the person sitting at a desk writing a paper — has at least two co-authors: a disembodied pure thinker, the impersonal voice who calls himself “we”, as well as an imaginary automaton who “in principle” carries out any calculations or algorithms that “we” mention.

Raphael Núñez writes in “Do Real Numbers Really Move?” of how we produce hand gestures — with millisecond-precise synchronization — as we talk mathematics. Maybe this is how we do our best and most effective teaching: literally by hand-waving!

Two essays discuss unfortunate effects mathematics has on other disciplines. Jack Schwartz in “The Pernicious Influence of Mathematics on Science” argues that mathematics, by concentrating our attention, makes us blind to its own omissions, an attribute he calls the “simple-mindedness of mathematics.” Furthermore, this simple-mindedness tends to inappropriately impose strictures of rigor on developing but immature scientific theories. Mathematics, he says, knows better what to do than why to do it. Similarly, in “The Pernicious Influence of Mathematics on Philosophy,” Gian-Carlo Rota suggests that philosophers who have put all their eggs in the basket of precision and definitiveness would do better to reconsider traditional philosophy with its two thousand year history of dealing realistically with ambiguity, uncertainty and inconsistency.

This is a terrific collection of essays. Everyone is certain to find something to hate. Almost all of the essays give us new insight into that curious thing we do when we do mathematics.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.