Why does mathematics describe the world?

In a lecture at New York University in 1959, published the next year, Wigner rested the puzzle on two observations. Mathematical concepts are chosen for their amenability to clever manipulation, and nothing in experience suggests the advanced ones. And theories built on them predict quantities to accuracies far beyond the data they were constructed from.

His cleanest example was Newton on gravitation. Newton could check the law to about four per cent, and by 1960 it had held to less than a ten thousandth of a per cent. He gave two more. Quantum mechanics got the lowest energy level of helium right to one part in ten million, the accuracy of the measurements, although the rules it was abstracted from cannot be applied to that atom at all. And the theory of the Lamb shift, into which experiment fed little more than the fact that there was an effect to measure, agreed with measurement to better than one part in a thousand.

A weak version of the puzzle dissolves easily: mathematics is a large enough body of structures that some of them are bound to fit, and physicists select the ones that do.

The strong version does not dissolve that way, because the fits are predictive. A theory chosen to match one set of measurements goes on to predict phenomena nobody had looked for, which selection cannot explain.

In 1980 Richard Hamming took up Wigner's puzzle, under four headings of his own. We see what we look for, and more physical law than the textbooks allow turns out to follow from how we think rather than from what was measured. We select the kind of mathematics to use, and invent a new kind when the old one fails. Science in fact answers comparatively few of the questions people ask. And the evolution of man provided the model, selecting for minds that find this sort of regularity.

He judged all four inadequate to the size of the puzzle, and said so in his opening paragraphs as well as at the end.

The value of the paper is its honesty: partial answers, set out by someone who wanted a whole one and did not find it.

That mathematics is discovered rather than invented, and the fit reflects the world being mathematical in its structure. That mathematics is invented but shaped by minds evolved in this world, so its abstractions inherit its regularities. That the puzzle is a selection effect operating on which parts of mathematics we notice and on which parts of the world we can describe at all.

Each has serious defenders. Max Tegmark holds the first in its strongest form, that the physical world is a mathematical structure, and says it makes testable predictions. Others question how unreasonable the fit really is: Ivor Grattan-Guinness argued that, with a wide and growing repertoire of mathematics, theory-building “can be seen as reasonable to a large extent”. No test has settled any of it, which is why this sits where it does rather than in physics.

Mathematics describes physics extraordinarily well and most other things poorly. Wigner said so himself. The laws he was admiring are of almost fantastic accuracy and strictly limited scope, and the regularity behind them applies to a very small part of what we know even of the inanimate world. He noted that the laws of heredity and of physics had, at present, nothing in common, and believed that something close to a conflict between them already existed.

That is evidence about which parts of the world are simple rather than about mathematics. Hamming made the same point in the third of his four headings.

The question of whether mathematical objects are real, in any sense other than being consistently describable, is genuinely unresolved and probably not empirical. One line of argument, from Quine and Putnam, disputes even that: if our best physical theories cannot be stated without quantifying over numbers, then believing those theories commits you to numbers.

It matters here because the boldest explanation of the fit, that the world is mathematical, requires mathematics to be the kind of thing a world could be made of.