Comparing mathematics with poetry is an infinitely rich game. For every opinion you express, there is an equally valid counter-opinion. Contrasted to Hilbert’s dismissal of a student who had left mathematics for poetry, “I always thought he didn’t have enough imagination for mathematics”, someone said to me recently that the early death of Schubert was a greater tragedy than that of Galois, since what Galois could have achieved would sooner or later be done by someone else, whereas Schubert’s potential was lost forever.
So it isn’t so surprising that a book by Ron Aharoni, newly translated into English, doesn’t come to a definite conclusion one way or the other. The best we can do in a book entitled Mathematics, Poetry and Beauty is to give many examples of beautiful mathematics and beautiful poetry and discuss what the similarities and differences are.
Ron Aharoni is a mathematician whose field is combinatorics. He has collaboration distance 2 from me (we are both co-authors of Paul Erdős). I hadn’t heard from him for a while. In the book he explains that he made a deliberate move from university to elementary school.
Many, though not all, of his examples of poetry are taken from Israeli poets. I don’t know whether Hebrew is a particularly good language for poetry, but some of these poets pack many layers of meaning into a few words. But other poets appear, including Johann Wolfgang von Goethe, John Donne, Emily Dickinson, Federico Garcia Lorca, Constantine Cafavy, William Carlos Williams, and Matsuo Basho.
Here is an example of the argument. Displacement is a mechanism which, according to Freud, occurs in almost every area of human thought. By focussing on a subsidiary idea, the main message slips through almost unnoticed, although it may be too painful to face directly. Aharoni suggests that this is a technique used by poets for diving inside themselves, and for mathematicians stuck on a problem who look at a seemingly irrelevant detail in the hope of a breakthrough. He gives several examples, both poetic and mathematical.
One of his telling comparisons, expanded over four chapters, is that both a poem and a mathematical proof constitute a game of ping-pong between the abstract and the concrete. A poem can have several such switches in a few lines, as in this example “Written in pencil in the sealed railway-car” by Dan Pagis (translated by Stephen Mitchell):
Here in this carload
I am Eve
with my son Abel
if you see my older boy
Cain son of Adam
tell him that I …
In mathematics, both finding a proof and (if you are kind to your readers) presenting it involve frequent shifts of focus between logical argument and examples. But Aharoni’s conclusion is
… the heart of the poem is given to the concrete, and it is in this direction that the poem goes. This is the diametric opposite of the ping-pong of mathematics, in which the last shot is always towards the abstract.
It is also true that, as he remarks, in many published proofs (most notoriously, those of Gauss, the “fox who effaces his tracks in the sand with his tail”, according to Abel), all traces of the concrete are covered and only the shots to the abstract remain.
This points to another important difference. A finished poem can convey its beauty to any open-minded reader; knowledge of the poet’s biography often gets in the way. But the beauty in mathematics lies in the experience of the discovery of the proof; this can be reproduced to some extent in a reader who follows the argument carefully, but does not reside in the published proof, and still less in the statement of the theorem (in most cases).
And on the same theme, Aharoni invites us to watch the mathematician and the poet at work. The striking secret he reveals is that the mathematician spends most of his time staring into space.
Both poets (as many thinkers have observed) and mathematicians are concerned, not with ever more florid invention, but with the truth. It seems like a different kind of truth. Poets remind us of things we already know. But almost every mathematician, no matter which side they take on the “discovered or invented” question, are in their ordinary work Platonists, and act as if that their mental constructions are “out there”. However, the means they have for diving inside themselves, described in detail by Hadamard in his book, and divided into four stages (preparation, incubation, illumination, and verification) go well beyond the subjective ways that poets operate.
There is much more thought-provoking material in the book, many more dimensions on which mathematics and poetry can be compared. The chapter titles give some indication, including “The miracle of order”, “The power of the oblique”, “Reality or imagination”, “Unexpected combinations”, “Symmetry”, “Content and husk”, and “Change”.