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are called symmetries and they reflect immunity to changes in location, orientation, or the time you start your clock. If not for these (and other) symmetries, any hope of ever deciphering nature’s grand design would have been lost, since experiments would have had to be continuously repeated in every point in space (if life could emerge at all in such a universe). Another feature of the cosmos that lurks in the background of mathematical theories has become known as locality. This reflects our ability to construct the “big picture” like a jigsaw puzzle, starting with a description of the most basic interactions among elementary particles.

We now come to the last element in Wigner’s puzzle: What is it that guarantees that a mathematical theory should exist at all? In other words, why is there, for instance, a theory of general relativity? Could it not be that there is no mathematical theory of gravity?

The answer is actually simpler than you might think. There are indeed no guarantees! There exists a multitude of phenomena for which no precise predictions are possible, even in principle. This category includes, for example, a variety of dynamic systems that develop chaos, where the tiniest change in the initial conditions may produce entirely different end results. Phenomena that may exhibit such behavior include the stock market, the weather pattern above the Rocky Mountains, a ball bouncing in a roulette wheel, the smoke rising from a cigarette, and indeed the orbits of the planets in the solar system. This is not to say that mathematicians have not developed ingenious formalisms that can address some important aspects of these problems, but no deterministic predictive theory exists. The entire fields of probability and statistics have been created precisely to tackle those areas in which one does not have a theory that yields much more than what has been put in. Similarly, a concept dubbed computational complexity delineates limits to our ability to solve problems by practical algorithms, and Gödel’s incompleteness theorems mark certain limitations of mathematics even within itself. So mathematics is indeed extraordinarily effective for some descriptions, especially those dealing with fundamental science, but it cannot describe our universe in all its dimensions. To some extent, scientists have selected what problems to work on based on those problems being amenable to a mathematical treatment.

Have we then solved the mystery of the effectiveness of mathematics once and for all? I have certainly given it my best shot, but I doubt very much that everybody would be utterly convinced by the arguments that I have articulated in this book. I can, however, cite Bertrand Russell in The Problems of Philosophy:

Thus, to sum up our discussion of the value of philosophy; Philosophy is to be studied, not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe which philosophy contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good.

NOTES

Chapter 1. A Mystery

As the British physicist James Jeans: Jeans 1930.

Einstein once wondered: Einstein 1934.

he singled out geometry as the paradigm: Hobbes 1651.

Penrose identifies three different: Penrose beautifully discusses these “three worlds” in Emperor’s New Mind and Road to Reality.

Physics Nobel laureate Eugene Wigner: Wigner 1960. We shall return to this article many times in this book.

that he emphatically declared: Hardy 1940.

One of his works was reincarnated: For a discussion of the Hardy-Weinberg law in context see for example Hedrick 2004.

the British mathematician Clifford Cocks: Cocks invented in 1973 what has become known as the