[2 February 2010]

Lately I’ve been spending some time of an evening reading E. T. Bell’s classic collection of biographies of mathematicians: *Men of Mathematics* (1937; rpt. New York: Simon and Schuster, 1985). (As the title suggests, Bell is thoroughly pre-feminist; a collection called *Women of Mathematics* has been published more recently, which may be intended as a kind of pendant to Bell, though sadly it’s nowhere like as much fun to read; its authors are for the most part better historians and much less opinionated.) And something rather puzzling caught my eye the other evening.

Bell writes (p. 357 of the Simon and Schuster paperback reprint) in his sketch of the Irish mathematician William Rowan Hamilton:

So long as there is a shred of mystery attached to any concept that concept is not mathematical.

Granted, Bell was probably tired of having to tell generations of incredulous undergraduates that there is nothing especially imaginary about imaginary numbers; one might grow peevish on the subject over the years. But still: did Bell seriously believe that the natural numbers (say) are not mysterious?

Can anyone not dead to all sense of wonder contemplate the commutative property of integer addition without feeling themselves to be in the presence of mystery?

more modern usage of mystery means just something you don’t understand, and mathematics has plenty of that;-) but older usages seem to be specifically relating to secret rites and religious faith, and Bell would be quite reasonable in asserting that mathematics was distinct from such things.

eg

http://www.thefreedictionary.com/mystery

definition 1 is

One that is not fully understood or that baffles or eludes the understanding;

which could be applied to mathematics but most of the other definitions are more related to definition 6 which is

A religious truth that is incomprehensible to reason and knowable only through divine revelation.

Still there is a genuine mystery in art, and a real place for wonder. In

Sartor ResartusCarlyle distinguishes extrinsic symbols, like the cross or the national flag, which are without value in themselves but are signs or indicators of something existential, from intrinsic symbols, which include works of art. On this basis we may distinguish two kinds of mystery. (A third kind, the mystery which is a puzzle, a problem to be solved and annihilated, belongs to discursive thought, and has little to do with the arts, except in matters of technique.)The mystery of the unknown or unknowable essence is an extrinsic mystery, which involves art only when art is also made illustrative of something else, as religious art is to the person concerned primarily with worship. But the intrinsic mystery is that which remains a mystery in itself no matter how fully known it is, and hence is not a mystery separated from what is known. The mystery in the greatness of King Lear or

Macbethcomes not from concealment but from revelation, not from something unknown or unknowable in the work, but from something unlimited in it.–Northrop Frye,

Anatomy of CriticismClearly mathematics is full of mysteries of the third kind (how ’bout that Riemann zeta?), and I understand you to be saying that it’s full of the second kind as well (I agree; mathematics is an art). I understand Bell to be denying the presence of the first kind.

David and John are right, I think, in their interpretation of what Bell wanted to say.

I am not persuaded, however, that Bell can be acquitted of tone-deafness, or at least grave carelessness, in phrasing his remark as he did. Meaning something plausible and saying something plausible are different things. If you say not what you mean but only something kind of close to what you mean, you expose yourself to all sorts of dangers, including impertinent comments from would-be copy editors with blue pencils and itchy fingers. Fortunately, Bell will survive my observations and blue pencil. And he deserves some thanks from me for reminding me to think, from time to time, about the mathematical mysteries of the second kind.

I really enjoy reading Bell’s books