Friday, June 19, 2009

mere calculation

But do we even know the meaning of a single comet's mathematical prowl?
- Robert Penn Warren, After Restless Night

Mathematics is often criticized for missing the point. It describes of all kinds of phenomena, but somehow it cannot provide insight into the meaning that is assumed to lurk behind the formal descriptions that it offers up. Walt Whitman's When I Heard the Learn'd Astronomer shows us that this is a particular affront to the romantic sensibility:
WHEN I heard the learn’d astronomer;
When the proofs, the figures, were ranged in columns before me;
When I was shown the charts and the diagrams, to add, divide, and measure them;
When I, sitting, heard the astronomer, where he lectured with much applause in the lecture-room,
How soon, unaccountable, I became tired and sick;
Till rising and gliding out, I wander’d off by myself,
In the mystical moist night-air, and from time to time,
Look’d up in perfect silence at the stars.
Long before Whitman, Seneca faulted mathematics for missing the point, and for failing to provide guidance where it really matters:
Oh the marvels of geometry! You geometers can calculate the areas of circles, can reduce any given shape to a square, can state the distances separating stars. Nothing is outside your scope when it comes to measurement. Well, if you are such an expert, measure a man's soul; tell me how large or how small that is. you can define a straight line; what use is that to you if you have no idea what straightness means in life?
- Seneca, from Letter LXXXVIII
To the romantic (or moral) sensibility, mathematics is worse than wrong; its formal descriptions kill the true meaning of things and renders them impersonal. Not only does it not evoke the personal, it requires no 'personality' - it is mere calculation: work fit for machines, not thinking humans. As Schopenhauer disdainfully noted,
That arithmetic is the basest of all mental activities is proved by the fact that it is the only one that can be accomplished by means of a machine. Take, for instance, the reckoning machines that are so commonly used in England at the present time, and solely for the sake of convenience. But all analysis finitorum et infinitorum is fundamentally based on calculation.
These two ideas, that mathematics misses the 'deeper meaning' of things and requires no 'personality' are closely related. The idea that there must be an underlying or hidden meaning beneath the surface of phenomena structures much of our thinking about the world and is connected closely to the concept of the self. Just as we have an inner personality and identity, so to do the heavens (as we see in the poems cited above). Consequently, to deny meaning in the world beyond what can be described formally seems equivalent to a denial of personal identity.

Simone Weil, while accepting the formal and impersonal nature of mathematics, sees things differently. To her, the idea of the personal is a distraction which keeps us from true understanding. Rather than missing the point, mathematics provides a means for attaining something beyond the merely personal:
Gregorian chant, Romanesque architecture, the Iliad, the invention of geometry were not, for the people through whom they were brought into being and made available to us, occasions for the manifestation of personality.
. . .
It is pure chance whether the names of those who reach this level are preserved or lost; even when they are remembered they have become anonymous. Their personality has vanished. Truth and beauty dwell on this level of the impersonal and the anonymous...

If a child is doing a sum and does it wrong, the mistake bears the stamp of his personality. If he does the sum exactly right, his personality does not enter into it at all. Perfection is impersonal...
In many ways the quotations above represent an old-fashioned view of mathematics, now generally understood as highly creative rather than mechanical. And yet even in its creative aspects, mathematics seems to confound our usual understanding of what creativity means, as mathematical creativity somehow joins the personal with the impersonal. There is something special and mysterious about mere calculation that makes us wonder about what we are doing when we calculate, makes us wonder who is doing the calculating, and makes us wonder how calculation connects to the world.