SUPPOSE we had a theory that could explain everything. Not just atoms and quarks but aspects of our everyday lives too. Sound impossible? Perhaps not.

It's all part of the recent explosion of work in an area of physics known as random matrix theory. Originally developed more than 50 years ago to describe the energy levels of atomic nuclei, the theory is turning up in everything from inflation rates to the behaviour of solids. So much so that many researchers believe that it points to some kind of deep pattern in nature that we don't yet understand. "It really does feel like the ideas of random matrix theory are somehow buried deep in the heart of nature," says electrical engineer Raj Nadakuditi of the University of Michigan, Ann Arbor.

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The matrix was a neat way to express the many connections between the different rungs. It also allowed Wigner to exploit the powerful mathematics of matrices in order to make predictions about the energy levels.

Bizarrely, he found this simple approach enabled him to work out the likelihood that any one level would have others nearby, in the absence of any real knowledge. Wigner's results, worked out in a few lines of algebra, were far more useful than anyone could have expected, and experiments over the next few years showed a remarkably close fit to his predictions. Why they work, though, remains a mystery even today.

From all of this I am forced to conclude both that mathematics is unreasonably effective and that all of the explanations I have given when added together simply are not enough to explain what I set out to account for. I think that we-meaning you, mainly-must continue to try to explain why the logical side of science-meaning mathematics, mainly-is the proper tool for exploring the universe as we perceive it at present. I suspect that my explanations are hardly as good as those of the early Greeks, who said for the material side of the question that the nature of the universe is earth, fire, water, and air. The logical side of the nature of the universe requires further exploration.

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Take prime numbers, for example, which as far as I’m concerned, constitute a more stable reality than the physical reality that surrounds us. The working mathematician can be likened to an explorer who sets out to discover the world…We run up against a reality every bit as uncontestable as physical reality.

The question about the nature of mathematics is not new, and the number of possible answers is small. I think that everyone would agree that mathematics has something to do with the physical world, since it has been so successfully applied in science. Regarding its precise relationship to nature, three answers are generally given, and two of these are wrong.

The first wrong answer identifies nature and mathematics. Mathematical space is the same thing as physical space. Physical quantity is the same as mathematical quantity. We could call this the Pythagorean answer.

The second wrong answer begins by assuming no relationship between mathematics and nature, mathematics being simply a human construct. Mathematics is then pasted onto nature, and provides models for understanding nature. It is a language for describing nature, and so on. We could call this the positivist error. Positivists can never explain why mathematics actually works.

The third and correct answer is found in Aristotle. It is the hardest to express precisely because it is correct, and it contains aspects of each of the wrong answers. According to this view, mathematical objects are "beings of reason" that is to say human concepts (as in answer 2) but they have been derived by abstraction from physical nature, and so bear an intrinsic relation to nature (as in answer 1).

So now we know what mathematics is; at least I have applied some labels to the issue. But the errors themselves typically have a history in the development of the student. For instance, the Pythagorean error is frequently committed by beginners with genuine mathematical aptitude. They get carried away by the equations, and then see the mathematics instead of nature. The second error is a sort of undergraduate, pseudo-sophisticated response to the discovery that nature and mathematics are not identical, typically occurring when one first meets up with non euclidean geometry. In anyone older than that, it's an intellectual cop-out; regretably a cop-out that you often find in those introductory chapters in textbooks all about the so-called scientific method: math is a language pasted onto nature, and we paste a different equation every hundred years or so when we have a paradigm shift. The third answer comes with
experience. Whether experience can be hastened by good teaching is something I don't know. It can sometimes be retarded through bad teaching.

I am quite confident that the statement 2 + 3 = 5 is true; I am far less confident of what it means for a mathematical statement to be true.

In "The Simple Truth" I defined a pebble-and-bucket system for tracking sheep, and defined a condition for whether a bucket's pebble level is "true" in terms of the sheep. The bucket is the belief, the sheep are the reality. I believe 2 + 3 = 5. Not just that two sheep plus three sheep equal five sheep, but that 2 + 3 = 5. That is my belief, but where is the reality?