from Sacred Mathematics: Japanese Temple Geometry
by Fukagawa Hidetoshi, Tony Rothman, Princeton University Press
Here is that quotation about elephants from Poincaré’s Science and Method,
mentioned in an earlier post:
Would a naturalist imagine that he had an adequate knowledge of the elephant if he had
never studied the animal except through a microscope?
Poincaré is arguing that logic misses the je ne sais quoi that “constitutes the unity” of a
mathematical proof. Here, however, I want to focus my microscope on the elephant. I wrote
that elephants keep popping up in books about mathematics, but I didn’t realize how pervasive the phenomenon really is. Here is a short list of books about mathematics, ordered by decreasing number of elephants:
Philip Davis and Reuben Hersh, The Mathematical Experience: 3
Keith Devlin, Introduction to Mathematical Thinking: 3
Jordan Ellenberg, How Not to Be Wrong: 2
John Allen Paulos, Innumeracy: 2
Barry Mazur, Imagining Numbers: 2
Mathematics without Apologies: 2
Steven Strogatz, The Joy of x: 1
Peter Woit, Not Even Wrong: 1
Max Tegmark, Our Mathematical Universe: 1
Wittgenstein, Lectures on the Foundations of Mathematics: 1
Poincaré, Science and Method: 1
Princeton Companion to Mathematics: 1
Hersh’s What is Mathematics, Really? would have topped the list with more than 6, but
most of these occur in reference to the poem about the blind men and the elephant.
On the other hand, there are no elephants at all in Timothy Gowers’ Mathematics: A Very Short Introduction, nor in Edward Frenkel’s Love and Math; this places them in the company of Bertrand Russell’s Principles of Mathematics and thus implicitly in the camp opposed to that of Poincaré. On the other hand, neither Lakatos in his Proofs and Refutations nor Hacking in his Why Is There Philosophy of Mathematics at All? feels the need to bring elephants onto the stage, so the elephant’s relevance to one’s philosophical position may not be as weighty as it seems.
And there are no elephants in Hardy’s A Mathematician’s Apology.
In most of these books the elephant is a metonym for something big and complicated that
can’t be easily classified. This is clear in Hersh’s citation of the poem, and even clearer
in the Princeton Companion, where mirror symmetry is called “the elephant”:
To date, no unified understanding of the phenomenon [of mirror symmetry]
has been achieved. To some extent, we are still “feeling the elephant.”
Elephants also appear in literally supporting roles when authors refer to the myths of
the World Elephant and the World Tortoise holding up the earth. Here, if the Wikipedia article
is correct, these authors are continuing a longstanding tradition to which John Locke, David Hume, and William James, as well as Russell, have contributed.
By now elephant imagery is so deeply ingrained in writing about mathematics that a book lacking elephants calls for an explanation. This must be why I felt such a sense of relief when my first elephant finally showed up in Chapter 7. This is not the only secret rule I followed. Each of the expressions “my mother,” “my father,” and “my wife” appears exactly once, in Chapters 2, 4, and 9 respectively.
Update April 13, 2015: More elephants are on the way. Here’s a preview.