AChaoticEnd? 253
++Fig. 7The ‘classical’ model of the helium atom: two electrons orbiting a nucleus. The quantum model, which fails to distinguish
between the electrons and smears them out over a wide radius, is too difficult to draw.
all ‘modernists’. In the 1930s John von Neumann wrote a book (1996) one of whose aims was
to show that the theory could be adequately developed without the bizarre functions which Dirac
had introduced.^12 Twenty years later, Laurent Schwarz developed a respectable mathematical
theory (‘distributions’) in which Dirac’s functions made perfect sense. But physicists could not wait
20 years to be allowed to proceed.
The situation grew worse in the 1940s with the ‘success’ of quantum electrodynamics—the
study of fields which were, at least potentially, infinite systems of particles. This is certainly a sign
of the unreasonable effectiveness of something, but not of mathematics as most mathematicians
would accept it. To quote an online encyclopedia’s summary:
It was immediately noticed, however, that self-interactions of particles would give rise to infinities, much as in classical
electromagnetism. At first attempts were made to avoid this by modifying the basic theory...But by the mid-1940s
detailed calculations were being done in which infinite parts were just being dropped—and the results were being found
to agree rather precisely with experiments. In the late 1940s this procedure was then essentially justified by the idea of
renormalization: that since in all possible QED processes only three different infinities can ever appear, these can in effect
systematically be factored out from all predictions of the theory. (www.wolframscience.com/reference/notes/1056a)
Confused? A more sophisticated form of the theory, ‘dimensional renormalization’, involved
writing the equations in 4+εdimensions (where they were not infinite), calculating the infin-
ite term asε→0 and removing it. It looks like nonsense to many mathematicians, but it gives
accurate predictions.
And yet, theoretical physicists still behave, as they used to, like a kind of mathematician, writing
down equations and manipulating them according to agreed rules to see if they work; the fact that
most conventional mathematicians do not understand or believe the rules is immaterial. Moreover,
they constantly find it useful to raid developing parts of ‘pure’ mathematics—Riemann surfaces,
knots, complex three-manifolds,...—for some idea which may be useful in a new model. Worse,
there has been traffic the other way; important theorems in ‘pure’ mathematics have been proved by
(in particular) Ed Witten by methods which many find suspect since they derive from the most high-
flown of infinite physical procedures. Have we returned to the dark ages of Newton and Leibniz? An
interaction of physics and mathematics is being preserved, but the power relations have shifted; and
- Specifically,δ(x), infinite whenx=0 and zero otherwise; and, worse, the derivatives ofδ(x).