Mathematical Coincidences II

String theory is rather successful when it doesn’t try to provide us with laws of physics. Here is an example from p.913 of ‘The Road to Reality‘ by Roger Penrose.

The Norwegian mathematicians Geir Ellingstrud and Stein Arilde Strømme were working on counting the number of rational curves in quintic threefolds. [It’s all right for these terms to be double-Dutch: their meaning isn’t important for the story]. Counting curves of successive orders, the numbers of curves were

2875, 609250, 2682549425,…

One of the beliefs of string theorists, based [at that time] on nothing more than a hunch and a desire for it to be true, was – well, let Penrose say it:

These complex 3-manifolds [the quintic threefolds] turn out to be Calabi-Yau spaces that ought [according to the physicists’ hunch] to be related to certain other Calabi-Yau spaces. The mirror symmetry, in a certain sense, interchanges complex structure with symplectic structure; accordingly, the problem of counting rational curves (which is technically a very different problem) is converted to a much simpler and quite different-looking problem in the ‘mirror’ Calabi-Yau space.

So Philip Candelas and his collaborators looked in the mirror and counted, and they came up with the numbers

2875, 609250, 317206375,…

But 2682549425 doesn’t equal 317206375, you will say! Maths 1, stringy maths 0.

Except that it emerged that there was a bug in the Norwegians’ computer program, and Candelas et al. really had got their total right. Maths 0, stringy maths 100.

Which is to say that the slapdash intuitive approach of string theorists, so different from mathematicians who won’t pick up their right foot unless they can first prove both that their left foot is on the ground and that the ground exists, sometimes makes imaginative leaps which land in the right place. They are more like Euler, back in the 18th century, who did maths because it was fun. Try anything that looks likely even if it can’t officially make sense. See if it works, and if it does, leave it to a future generation of theorists to prove why. (Or like a ground-breaking saint, who finds new ways of loving and leaves it to the theologians to explain it all later).