In 1931, Austrian-born mathematician Kurt Gödel shook the academic world when he announced that some statements are ‘undecidable’, meaning that it is impossible to prove them either true or false. Three researchers have now found that the same principle makes it impossible to calculate an important property of a material — the gaps between the lowest energy levels of its electrons — from an idealized model of its atoms....In fairness, there's also a huge gap in getting other more-or-less accurate predictive physical models to predict exactly once all the complications are worked in. As James and Eric H were remarking the other day regarding the Russian airplane, calculating for a vacuum is going to yield very different results than when you input calculations for air resistance on an crumpled airframe. The differences in what must be accounted for across the operation may be so immense as to make the calculations practically impossible.
Cubitt and his colleagues showed that for an infinite lattice, it is impossible to know whether the computation ends, so that the question of whether the gap exists remains undecidable.
For a finite chunk of 2D lattice, however, the computation always ends in a finite time, leading to a definite answer. At first sight, therefore, the result would seem to have little relation to the real world. Real materials are always finite, and their properties can be measured experimentally or simulated by computer.
But the undecidability ‘at infinity’ means that even if the spectral gap is known for a certain finite-size lattice, it could change abruptly — from gapless to gapped or vice versa — when the size increases, even by just a single extra atom. And because it is “provably impossible” to predict when — or if — it will do so, Cubitt says, it will be difficult to draw general conclusions from experiments or simulations.
Yet a practically impossible calculation is still different from one that turns out to be impossible in principle. The one we might hope to overcome with better tools. If it's impossible in principle, a better tool alone won't fix the problem. The principles have to change -- and changing the principles of mathematics while preserving its predictive capacity is not easy.