1We’ve encountered a curious, aggressive response to this fear of math on the part of a few primary and secondary school teachers. “I know all about math,” one of them told us, “and don’t need that fancy stuff they do in college. This is the real thing, which those people in their ivory towers haven’t a clue about.” That is, she knew what 7 × 8 was, without having to have it dressed up in the language of sets or of axiom systems. If we take what she said seriously, and not just as living in denial, it might well follow from her recognizing that math is as much our native language as is English—for which you don’t need grammar books in order to speak correctly. But of course her view stops short at the surface of math’s usage—the “number facts”—and never enters the depths where the reasons for them swim.

2Another Inca theme, blended with this one, holds that the world will one day be remade as a single body, its mind in Europe, its will in North America, and its heart in South America. Perhaps a similar fusion of will, thought, and feeling account for insight.

3Some recent experiments may suggest that linguistic and mathematical skills are distinct. These seem, however, to equate “mathematical skill” with calculation and familiarity with “number facts.” In any case, we’re speaking here of language in the broader sense of making and manipulating symbols for structures.

4When we posed this problem to an audience of 500 at the Perimeter Institute for Theoretical Physics, in Waterloo, Ontario, one of those in the audience (the distinguished knot theorist Louis Kauffman) came up in the course of the evening with a proof that such an infinite tiling works only if the proportions of the original rectangle are 1 × ɸ, where ɸ is the Golden Mean. See his paper “Fibonacci Rectangles” (eprint: arXiv:math/0405048).

5b can’t be –1, for if it were, 3a2 –2b2 would also have to be –1 (since –1 times –1 equals 1). But this would force a to be images or images. Were a to be either of these, however, with b = –1, then x = a3 – 6ab2 would not be a natural number, as we require it to be. So b must be 1.

6An odd number is one greater than an even number: 2n + 1. Square that and you get 4n2 + 4n + 1, which is an odd number.

7It is worth remarking that many mathematicians are as much put off by literary language as nonmathematicians are by math symbols. What uses are these turns and tropes, these allusions, indirections, and ambiguities, except to obscure the world and dim our sight? Each sort of language unfits us for the other, although the aim of the arts that employ them is the same.

8Variants on this phrase include “It can be shown,” “It is beyond the scope of this paper to show,” and “This margin is too narrow to contain the proof.”

9The most recent explanations (as of nonlinear phenomena now) always seem arcane, and what they describe, fundamentally complex. If the past is any guide, these too will simplify in time, and as the enriched context of our thought accommodates them, will in turn reinforce our sense that understanding’s ultimate surprise is its simplicity—not in the sense of a reductivist’s empty formalism, but because complex phenomena in simple realms become simple in complex realms.

10“Fringe” is due to William James. The notion of an antechamber to consciousness is, prior to him, in John Locke, and later, Francis Galton, and the Freudian “foreconscious,” follows him.

11The history of mathematics, and one’s own mathematical biography, are woven around these exceptions. Ancient lists would have had the five regular polyhedra on them, for example; the current list includes the sporadic simple groups and the exceptional Lie groups. John Stillwell says of them: “In the mind of every mathematician, there is tension between the general rule and exceptional cases. Our conscience tells us we should strive for general theorems, yet we are fascinated and seduced by beautiful exceptions.” Any problem is an exception to the current state of our knowledge. Think of the problem about the number of regions into which n chords divide a circle. We understood the sequence 0, 1, 2, 4, 8, 16 perfectly well, until the next number, 31, opened the door to a wholly different sort of counting.

12Noah Rosenblum, in Matt Paley’s 2001 The Math Circle.

13 Our information comes from an article by Larry Cahill, “His Brain, Her Brain,” in the May 2005 Scientific American.

14A visitor to a class early on in the semester might come away thinking: “They’ll never get anywhere at this rate!” Were that visitor a teacher, she might add: “All well and good for an after-school activity, but with a set syllabus and external exams, this would never work in schools.” It could; it has. The acceleration in a course, after such thoughtful beginnings, is astonishing. People come away knowing not only more than they would ordinarily, but with a confidence based on firm knowledge that school gymnastics never give. But of course it would be ideal were today’s hurdles removed and understanding math made the goal, rather than passing tests on the names of things and the application of rules.

15“Two Letters from N. N. Luzin to M. Ya. Vygodskii,” in Mathematical Evolutions (ed. Shenitzer and Stillwell, Mathematical Association of America, 2002, p. 36).

16 “Tribute to Vladimir Arnold” (Boris Khesin and Serge Tabachnikov, Coordinating Editors), AMS Notices, March 2012, 379.

17If there are to be competitions, might they not be in a more fruitful style? Barry Mazur asks why all—or the most telling—questions couldn’t be of the form: “What do you think about this?” He calls the present form of competitions “closed”: here is the problem we have made for you, solve it. He suggests instead an “open form”: tell us something of your own; surprise us. This would put competitions more in line with those of ancient Greece, where playwrights submitted their works for the prize of being performed on a great holiday.

18Kaprekar’s constant: take any four-digit number (not all digits the same); arrange it twice, once from highest to lowest, once from lowest to highest. Subtract. Did you get 6174? If not, repeat the process with what you did get, and go on: you will inevitably arrive at 6174.