Review of "Is God a Mathematician" by Mario Livio

 

Is God a Mathematician?

The short answer is “no.” But like all answers that imply a binary set of definite possibilities, the short answer is more misleading than leading. This is an excellent book. No one capable of understanding it would fail to profit from reading it. So I’m not going to focus on its myriad virtues, which you can easily find in other reviews, but on it’s few but essential problems.

The principle question the book ask is this: Was math invented or discovered? This translates to “Was the universe created (or better ‘is it governed’) by mathematical principles that are ‘ideal’ in the Platonic sense?” Is there a math that is outside the universe (in some sense not perfectly understood) whose principles are the principles that makes the universe work as it does? Along the way he also asks “Is math a language?”

The book’s slightly longer and somewhat more helpful answer is “both.” Math itself is created, but what math reveals is discovered.

I say somewhat more helpful answer because this answer maintains the uninvestigated distinction between “invention” and “discovery” that, had it been investigated, would have yielded a richer answer.

Mario Livio does not adequately define his terms. He assumes we know what it means to be invented—like the microphone—or discovered—like a new planet. The new planet was always there, but someone found it for the first time. The microphone never existed before. Someone made it. This may seem as obvious to you as it does to Livio, but even within his book, unnoticed problems arise.

First problem: prime numbers, he says, were invented. Any number of cultures had numbering systems, but most of the cultures did not have a concept of prime numbers, and they got along fine with out them. Western Mathematicians uniquely decided to invent this concept. That 9/3 = 3 was however discovered.

Second problem: No one, Livio says, would say that Shakespeare “invented” Hamlet.

But that there are numbers that are only divisible by themselves and one is true in any counting system. They exist even if unnoticed. So do they really differ in a completely different way from the answer to a problem of division? Does it really make sense, in this dualistic thinking, to say that the concept of prime number has to be invented rather than that it has to be recognized? It may be that either option can be supported. And this brings into question the very distinction between invented and discovered.

And as for Shakespeare, one person who would have said that Shakespeare invented Hamlet was Shakespeare. “I’ll give you a verse to this note that I made yesterday in despite of my invention,” says Jaques in As You Like It to show how clever he is before having to make something up. In fact any educated person would have said so. That is what the word meant in Shakespeare’s day. What Shakespeare would never have said was that he created Hamlet.

Of course we can say that words change their meaning. So this doesn’t count against Livio.

It’s true that words change their meaning. They do so when the concepts that they supported in their previous meanings are no longer supportable. The very “invent/discover” distinction which we tend to believe is too obvious to need strict definition is one that Shakespeare would not immediately wrap his mind around. The corollary for us, who profit so much from reading Livio’s book, is that we too may need to rethink our very distinction between invention and discovery.

To come at this from another angle, Is God a Mathematician is a book build on two conundrums: if math is invented how can it predict facts about the universe that were not even suspected at the time the math was invented? How can the mathematical theory of knots, useless the purpose for which it was invented, explain the structure of molecules? The conundrum will go along way toward taking care of itself when we understand that discovery and invention do not describe two sets without common elements—a fact that this book needs to maintain that within the sent of “invented” things is a member called math, in which there are things that were discovered. If we have to have it both ways, or one way at one moment and another way at another, then the problem is certainly in the question or the model that gives rise to the question and not in the thing the question is posed to explain.

The second conundrum, which the book brings up several times but is not deeply interested in is this: Is Math a language? Livio will tell us it sort of is and sort of isn’t. He doesn’t believe much depends on a rigorous answer to this question, and he does not give one. This makes sense given the structure of thought in which the book operates. When it gets to the point where something can be or not be a language at the same time, it closes the door and goes better lighted hallways.

It does seem like a troublesome question not admitting of easy answer. If math is a language, how come small children, who are so good at acquiring languages have such trouble learning math? On the other hand, it is a symbolic structure made of signs representing concepts. It works by rules of syntax and grammar.

The problem however is only an apparent one, like a knot that is just a tangle that disappears with a tug. Math is not a language. Math is something we do in language. When I do math, I do it in English. When a French person does math, they do it in French. Math appears at first glance to be a language only because we use the same representations, the same words with the same spelling to represent the same concepts “2” is two in English and deux in French and er in Mandarin, but we all spell that concept as 2 when we do math. (A side note, Livio’s short but illuminating excursions into the history of math leave out the essential observation of the indebtedness of math to Arabic numerals.) Why do children have trouble with math? For the same reason they have trouble with logic (which no one calls a language) and with diplomacy and with any of the more complicated functions we do in language. What children acquire easily is vocabulary and syntax. Whatever it is they are capable of expressing they easily learn to express from one natural language to another.

Finally then my point is that Livio’s question about the discover v. invention of math is of the same type as his question of whether math is or is not a language. If a better vocabulary for thinking about math is developed (and I’m sure it has already been developed, though I can’t point to it at the moment), then the problem itself goes away. At least I suspect that is so.

All that said, this is a terrific book. For someone who gave up on math after three semesters of calculus it makes me re-think my choice. What I do not know because I did not get into higher mathematics is a field of wonder that I would love to explore. But life only allows us so many loves. And this peek at what I cannot explore further was infinitely worth the time I spent in the doorway.