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.
