The Book of Numbers

Is God a Mathematician?

by Mario Livio

Simon & Schuster, 320 pages, $26

When it comes to mathematics, two things seem evident. First, a great many people have nothing but negative feelings about it. There is probably no other subject about which people so proudly proclaim their ignorance and even their basic incompetence (“I can’t even balance my checkbook!”). Second, those who like mathematics do so ­passionately. Some seem to love mathematics itself, finding it aesthetically pleasing and intellectually ­compelling.

In 1960, physicist Eugene Wigner wrote a paper that expressed a more complex reaction. He called it “The Unreasonable Effectiveness of Mathematics in the Physical Sciences,” thereby giving us a useful name for the underlying philosophical question. It is a mystery, he argued, that mathematics could actually be so useful in the physical sciences. Mathematicians, says Wigner, typically develop their theories out of pure curiosity and aesthetic appreciation. “It is sometimes suggested,” wrote British mathematician G.H. Hardy, “that pure mathematicians glory in the uselessness of their work, and make it a boast that it has no practical applications.”

So why is mathematics as useful as it is? We are not surprised that the calculus can be used to describe motion, or that statistics is useful for analyzing society; they were, at least to some extent, constructed for those purposes. But it seems amazing that group theory, invented in the 1830s to explain why certain kinds of equations were harder to solve than ­others, has turned out to have a fundamental role in describing the symmetries of quantum physics, or that number theory, once considered the quintessence of pure mathematics, has turned out to be crucial to cryptography. It is, as Wigner put it, “a wonderful gift which we neither understand nor deserve.”

The question of the unreasonable effectiveness of mathematics is the starting point for a new book by Mario Livio called Is God a Mathematician? The title reflects one possible answer to Wigner’s question: Mathematics is useful for understanding the universe because God, who designed the universe, did so mathematically. We, being made in God’s image, also think mathematically. Livio, who does not seem to believe there is a God, does not like that answer, and the goal of his book is to propose a different solution to Wigner’s conundrum.

Livio complicates the issue by introducing a far deeper (and older) problem in the philosophy of mathematics: the question of what mathematics is about. Is it more akin to ­science, in which we use our minds to explore an external reality, or more akin to poetry, in which we construct new realities? When mathematicians speak of regular polygons, five-dimensional spaces, symmetry groups, or, for that matter, the number seven , what are they actually talking about? When we prove a theorem, are we discovering something or creating something? If the latter, why do we feel that mathematical results carry so much certainty?

This is a big question. The mathematician Barry Mazur calls it “The Question” and argues that it is unavoidable: “The bizarre aspect of the mathematical experience”and this is what gives such fierce energy to The Question”is that one feels (I feel) that mathematical ideas can be hunted down, and in a way that is essentially different from, say, the way I am currently hunting the next word to write to finish this sentence. One can be a hunter and gatherer of mathematical concepts, but one has no ready words for the location of the hunting grounds. Of course we humans are beset with illusions, and the feeling just described could be yet another. There may be no ­location.”

Philosophers (and philosophically inclined mathematicians) have proposed many answers to The Question. The way that mathematicians speak about their everyday work seems to suggest a kind of naive platonism: They cannot be precise about where exactly these mathematical entities are, but they feel as if they are dealing with some sort of reality. Of course, other answers have been proposed. Logicism posits that the vast edifice of mathematics is nothing but a working out of logic, of the rules of reasoning. This was Bertrand Russell’s view, famously worked out with Alfred North Whitehead in Principia Mathematica . Most mathematicians and most philosophers of mathematics found the book unreadable and the argument unpersuasive.

Another alternative, formalism, claims that mathematics is a game: Choose your axioms and follow the rules to prove your theorems. (This view is often attributed to David Hilbert, though Hilbert’s view was, in fact, somewhat different.) Mathematical formalism intends to put the metaphysical question out of bounds by saying that mathematics makes no claims at all about reality.

Recently, in the philosophy of mathematics as in other fields, several kinds of social constructivism have been gaining popularity. These often start from the obvious fact that mathematics is something human beings do, and then they try to explain away the feeling of reality and solidity that seems to attach to mathematical objects and theorems. Some of these would argue that we should not expect an alien civilization to share our mathematics.

Several things drive all this discussion. Mathematics does seem to evoke a feeling of timelessness and certainty. We may not formulate geometry exactly as Euclid did, but none of Euclid’s theorems is now considered false. The proofs given by Apollonius and Archimedes still work as proofs for us, and the theorems they prove are, we say, true , not just agreed upon or universally accepted. Livio quotes Ian Stewart: “There is a word in mathematics for previous results that are later changed”they are simply called mistakes .” Indeed, mathematicians seem never to have to deal with the kind of debate about fundamentals with which other disciplines contend. Once a theorem has been proved and the proof has been ­examined and certified by those competent to judge, the result is just accepted as true, and everyone goes on to the next step.

The question of “unreasonable effectiveness” is by far the easier one to address. Convenient as it is for mathematicians to quote Wigner’s dicta, which provide the perfect justification for pursuing whatever interests us while assuring society that applications for it are bound to show up eventually, I think most mathematicians, in their more serious moments, don’t quite buy it. There are plenty of things that can be made to seem miraculous given the right point of view, from the ­physiology of trees to the fact that ­children learn to speak. There is a philosophical question there, discussed in detail, for example, in Mark Steiner’s 1998 book The Applicability of Mathematics as a Philosophical Problem , but in general philosophers of mathematics have not found it too difficult to deal with. The question of the nature of mathematics seems deeper.

In order to use these issues as the driving force for a book for the general public, Livio faces the crucial problem that most of the interesting examples are far too difficult for
that public. In order to explain, for example, why it is surprising that the theory of Lie groups is useful in quantum mechanics, it is necessary to be at least a little familiar with both Lie groups and quantum mechanics.

Livio’s solution is to proceed by way of history. He begins with Pythagoras and Plato, early proponents of the idea of mathematics as being about real things. (In fact, the Pythagoreans seem to have felt that mathematical objects are more real than the objects of sense experience.) As examples of mathematics applied to the world, he discusses the work of Archimedes, Galileo, Descartes, and Newton”a motley crew whose ideas about how mathematics relates to the world couldn’t be more ­varied.

Later in the book he looks at the development of statistics, the birth of non-Euclidean geometry, and the debate about the foundations of mathematics in the late nineteenth and early twentieth century. Livio’s account of these episodes is, for the most part, uncritical: He recounts what one might describe as the folk history of mathematics, the stories mathematicians tell each other. His bibliography cites few recent histories of mathematics, which results in many statements that will annoy careful historians.

All of these topics are standard fare in books about mathematics aimed at the general public, and few of them seem to be on point. Statistics, having been developed precisely for the purpose it serves, is the weakest of his examples. In the early modern period, the development of mathematics was so thoroughly intertwined with questions of natural philosophy that it is not too surprising to see mathematics turning out to be useful. The work of the logicians in the early twentieth ­century was important in the philosophical debate about The Question, but mostly in a negative way: It convinced most mathematicians that both Logicism and Formalism were inadequate as descriptions of ­mathematics.

Only two of Livio’s examples are really illustrative of unreasonable effectiveness. The story of non-Euclidean geometry has the element of surprise that drove Wigner’s article: Developed for rigorously internal reasons, it turned out to be the key element in the formulation of Einstein’s general theory of relativity. In addition, the discovery that one could get consistent geometries from different sets of axioms raised the deeper question of what geometry was about. Up to that point, it had been easy to say that geometry was the study of the properties of physical space. Now that there were alternative geometries, all of which seemed to share the peculiar solidity of mathematical reality, this was clearly no longer tenable.

The other good example is knot theory. Mathematicians started studying knots in the late nineteenth century because Lord Kelvin had proposed a theory of atoms based on knots. Kelvin’s theory was quickly discovered to be wrong, but mathematicians had found a neat subject to think about and kept right on ­proving theorems. Then, late in the twentieth century, knot theory was suddenly applicable again, both in biochemistry and, more spectacularly, in quantum physics. These links were so surprising and so deep that a publisher started a series of books on “Knots and Everything.”

After working for most of the book to convince his reader that both Wigner’s question and “The Question” are worth thinking about, Livio goes on, in the last two chapters, to propose answers. As to the nature of mathematics, he ends up arguing for a kind of middle ground: Mathematical concepts are created, but the theorems we prove about them are discovered. He thinks of these concepts as existing only in human minds. Influenced by cognitive science, he explains the apparent solidity of these mathematical objects as reflecting the structure of the human brain as shaped by natural selection. The end result is a kind of neurologically accented social constructivism.

Livio is not prepared, however, to dissolve physics itself in this solvent. The appeal to natural selection is made in order to allow the argument that, while mathematics is shaped by the structure of the human brain, that structure, having been shaped by the pressures of the real world, is adequate to explain reality. Of course, it is hard to see how having brains capable of quantum mechanics would have survival value to early hominids.

But Livio tries to deal with this by arguing that “nature has been kind to us by being governed by universal laws,” that our brains have evolved in such a way as to be able to grasp such laws, and that in fact we have selected out those parts of nature that are amenable to mathematical treatment. He seems to realize, in the end, that this is unsatisfactory. For one thing, isn’t the fact that there are such things as natural laws”and that they are mathematical”almost exactly the mystery Wigner was talking about?

The book does not end with a bang: “Have we then solved the mystery of the effectiveness of mathematics once and for all? I have certainly given it my best shot, but doubt very much that everybody would be utterly convinced by the arguments that I have articulated in this book.” We are left, after that, with a line from Bertrand Russell to the effect that philosophical questions are worth thinking about even though we can never find final answers to them.

Fair enough. Unfortunately, Livio does little to advance that thinking.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and author, with William P. Berlinghoff, of Math Through the Ages: A Gentle History for Teachers and Others.