**Julian Baggini meets the Argentine novelist with maths on his mind**

It doesn’t sound very promising: A professor of mathematics writes novels in which logic and numbers play a central part. Those who remember *Satan in the Suburbs*, Bertrand Russell’s foray into fiction, might well be tempted to turn away at this point. But that would be a mistake. For Guillermo Martínez turns out to be even better with words than he is with numbers.

In the Spanish speaking world, in particular his native Argentina, he is an established, multi-award winning author. However, when his debut, *Regarding Roderer* (*Acerca de Roderere*), was published in the USA in 1994, it made little impact. His follow-up *La mujer del maestro* (1998), was not even translated. But in 2005 his breakthrough in the English-speaking world came with his third novel, *Crímenes imperceptibles*, published as *The Oxford Murders*. It’s an ingenious and original detective novel which became a film directed by Alex de la Iglesia, and starring Elijah Wood.

Given his moody author photos and evidently formidable intellect, I was pleasantly surprised to find Martínez an affable and highly approachable interviewee when I caught up with him in Buenos Aires, where he now lives. It soon became evident talking to him that the marriage of fiction and mathematics is much more natural than you would have thought, because it allows him to explore themes common to both: the limits of logic and the role of imagination and aesthetics in reason.

Martínez’s appetite for abstract ideas was whetted in a home in which ideas and learning surrounded him. “My father was an amateur writer,” he told me. “He wrote over three hundred short stories, some plays, four or five novels. He was very interested in some fields of philosophy. He would try to explain all those topics to me and my brother.” In addition, “When I was a child, my mother was finishing her degree in literature. So my house was full of books of different sorts.”

When it came to going to the Universidad Nacional del Sur at Bahia Blanca, in his home town, however, Martínez opted for something more practical. His father had been an agricultural engineer, and in electrical engineering Martínez saw something that might similarly provide him with a sustainable livelihood. But “after two years I saw that I was not going to finish that course. During those first two years I discovered higher mathematics. I learned about different kinds of infinite sets and logical paradoxes. I was interested in mathematical logic, so I switched to that.” He then went to Buenos Aires to pursue a doctorate “I studied some of the many kinds of valid logics first developed by Lukasiewicz, the Polish mathematician. He developed those logics as counterparts corresponding to the concepts of necessity and contingency.”

Although he was also interested in the existential philosophy of Jean-Paul Sartre, clearly the world of Frege and Gödel was a different kind of philosophy altogether. “They were very different fields, of course. You look at Jean-Paul Sartre’s books, they are about literature and philosophy, existentialism, how to deal with ethics, the notion of freedom. When I had to study Gödel and Frege, that was a piece of maths. I was very grateful to mathematics because I think there is depth in mathematics in the possibility and the ability of building and defining very subtle differences. Mathematicians are very keen in establishing very thin and slight differences and having a concept for each one of them. It is not a surprise that many of the most important philosophers have been mathematicians. Spinoza, Kant in some sense, Leibniz, Descartes – all of them have this kind of mathematical mind.

“I think there is a deep connection between the mathematical thinking and philosophy. I have translated a book that was written by a colleague when I was in Oxford, Vladimir Tasic, called *Mathematics and the Roots of Postmodern Thought*. He tried to trace some of the simple discussions back to discussions in logic in the thirties. He tried to prove that the core concepts and discussions are very similar. I was very impressed by that book, it was very smart, and I translated it into Spanish.”

In Bertrand Russell’s autobiography and also in *Regarding Roderer*, you get a strong sense of people being attracted to mathematics because they see in it the possibility of discovering the true nature of reality. Mathematics promises a glimpse into the real underlying order of things. Martínez feels something of this urge, but is keenly aware that maths too has its limits.

“You get the notion of truth in mathematics but it is like a game in which you fix rules. In mathematics the rules are clearly fixed, so that anyone can agree what is true. Of course, then things blur a little because once you have the notion of truth you have also have the notion of what is provable, and that is a very different notion, Not everything that is true can be proved in a mechanical way. So there is this gap between the notion of truth and what can be really proved by the axiomatic machinery.

“There are some axioms that mathematicians use in their professional lives which cannot be proved, and a whole different mathematics arises if they are not allowed to use them. So in some sense you can see that a proportion of what mathematicians are constructing relies upon some faith.”

However, appreciating the limits of proof does not make Martínez sceptical about the power of reason and argument. “At each fork in the road, what happens is that you see why this limit is reached,” he says, alluding to Nietzsche. “For example, I think it is the same kind of problem the Greeks had about the square root of the number two. It is a problem because quotients and integers were the only tools they had and using those tools there weren’t able to find the square root of the number two. But with the notion of limit, which was, I don’t know, 300 years ago or something like that, there was now a way of understanding why there is a limit and which way that limit can be overcome. Now we have a way of thinking and understanding that number. I think this is the way that reasoning expands itself. It is a kind of act of act of imagination. That’s why I always say in my essays that human reason is not something that is given once and forever, it is a historical development, something that is elastic.”

What’s particularly interesting for Martínez about such acts of imagination is that, to work, they must be rationally explicable, but the means by which they arise is often mysterious.

“The way that you reach the truth in mathematics, for example, or the way a new novel comes to your thoughts – is it just coming out from a leap or is it something that comes step by step? Do you see a kind of inspiration which is the end of some hidden reasoning?” He makes great use of this idea in *The Oxford Murders*, where intuition and logic provide two routes to the same conclusion. “You can make a comparison with the way a chess player thinks. In some sense you know which paths cannot be taken and you are thinking of some different possibilities and all of sudden you see the best way. But you see it before you can actually say why that is the right line.”

But, of course, until you’ve shown why it is the right line, you don’t actually know if the flash of insight is genuinely insight at all. This is another theme which is dealt with imaginatively in *Regarding Roderer*, subtitled in the American edition “A novel about genius”, but actually much less clear-cut than this. “There is a theme of ambiguity running through the book. Maybe he was just some guy, he thought he had this kind of inspiration but it didn’t have a solid background, and so probably he was wrong.”

This way of thinking helps close the gap between art and mathematics. Martínez had reminded me of an interview I had recently read with the guitarist Robert Fripp, who reiterated an idea commonly voiced by musicians that technique is something you learn and then throw away. You practice your scales and so on until you reach a stage when you don’t have to think about it any more, it becomes intuitive. Is Martínez saying maths is like this?

“There are people like Oliver Sacks who think that mathematical thinking is connected with artistic ability. I’m not talking about the computer part, the analysis. That is clearly the left part of the brain. But the right part, which is connected with the more primitive part, he thinks that thinking about numbers and counting and all that – not the ability to perform operations, but the number patterns – is connected with musical ability. I think there is some kind of musical intuition connected with number. There is a gift, like you have the gift of perfect pitch, that allows you to think in a mathematical way. I have met many mathematicians and the way they think is very interesting. They don’t have the absolute proofs but they know if things can be proven. They think in a more Zen way, like the archer.”

Perhaps Martínez’s most intriguing notion, voiced by a character in *The Oxford Murders*, is that the judgement that something is right, in mathematics and philosophy, is at least in part an aesthetic one.

“There is something that happens to nature with chess players, mathematicians, writers. Nature doesn’t try every possibility. There are some patterns that are clear to you or nice to you in some way. Marx said that cats don’t study mice objectively, they study them to eat them. There is no such thing as an objective study, there is always an interest in the things you study. A machine will prove every theorem, but a mathematician doesn’t want to prove every theorem. He chooses which theorems which are interesting.”

Martínez then launches into a complicated example concerning whether or not there is chance in the universe. The way he sees it, since we’re always studying finite series of events and never the totality, we can never be sure whether the fully rational account reflects a genuine absence of chance, or whether it is a mere rationalisation. “To each finite piece of knowledge, you have the machinery of explanation, but you still don’t know what happens with the whole series. Both ways of seeing are right in some sense. So people who say everything is chance could be right, but who knows if the whole series is the flipping of a coin; and those who say, no, everything is rational, they have part of the truth because everything we know can be explained.”

But isn’t it true that, although any finite series can be given a rationale, it is still the case that sometimes there are good reasons for saying that one is the better one?

“Again, it is an aesthetic one. Many times people say the second one is not as elegant, not as accurate perhaps, but it suits better. But there’s no way of really explaining that aesthetic judgement.”

The concern is that we like elegance, but reality may not be elegant at all.

“I think it is an acute and difficult problem with the way that proofs appear in mathematics now. Before a proof was something that a normal person in a normal life could check from beginning to end. Now a proof can be something run by a program, so the complexity, the kind of calculation, is totally different. Now a person in his whole life is not able to reach the end of the proof. What is elegant for a computer is no longer elegant for a person.”

In his fiction, Martínez explores the way in which we are able to impose structure on reality, precisely because the true structure is never fully given to us. This can happen retrospectively as well: something happens that we can later describe in ways we couldn’t at the time. Though in some sense this is inevitable, it is clearly also open to abuse.

“I was in the States when the attacks on the twin towers happened. When I went back one year after that I was very interested in all the fuss about Saddam Hussein and the way the media were super-imposing that reality on the citizens – all that discussion of weapons of mass destruction. All those things which were false and I knew that. I had the intuition that everything was just an excuse to go into Iraq. I was impressed to see how from one year to another they managed to change reality, and to impose something which was a total lie on 250 million people.”

But isn’t it the case that Martínez shows the possibility of doing something much more subtle than merely spreading lies? In *The Oxford Murders*, there is a true sequence of events about what happens. But the significance of what happens changes according to the interpretation, and the significance is the important thing. So it’s not just the ability to rewrite history falsely, which governments do, but it’s more subtle – the ability to rewrite history accurately, if you like.

“It’s to lie with truth,” he agrees. “Not just to lie – everyone can do that – but to lie like a magician, with all the cards on the table. That is the trick in the book.”

Martínez is justly pleased with the way his books have made maths interesting. “I am very proud that many people who really hate maths, after reading my book for a couple of hours even had the feeling that they understood some of that. Many times in maths that is what happens – you don’t really understand line by line but you have the feeling that you see something, you almost understand, but you get the impression that you almost reach some truth. And I think many people for the first time in their lives gave maths this kind of second opportunity after high school.”

But his books are not mere mathematical versions of *Sophie’s World*. Fiction is not just a vehicle for maths, the two fit much more closely together, illuminating something of the nature of both, and more.

“In some sense I believe like Henry James that fiction competes with reality. For example, this novel changed my life, in a very material way. And many books I read changed my way of thinking.”

**Julian Baggini‘s latest book is Should You Judge This Book by Its Cover?**

My comments to the quoted passages follow the *** sign.

“Mathematics promises a glimpse into the real underlying order of things. Martínez feels something of this urge, but is keenly aware that maths too has its limits.”

*** I agree with Martinez. Mathematics is one of those things that should make one wonder about how incredibly the universe is put together, along with the atom and DNA. From the lowly counting integers has come the magnificence of math(s). But in spite of its use to explain all sorts of phenomenon, mathematics has its limits, or at least our ability to use it has its limits. Except for the most trivial of examples it is used to approximate.

“There are some axioms that mathematicians use in their professional lives which cannot be proved, and a whole different mathematics arises if they are not allowed to use them. So in some sense you can see that a proportion of what mathematicians are constructing relies upon some faith.”

*** Axioms are not statements to be proved; they are tools similar to the rules of grammar. Whether the axioms ultimately have anything to do with real life situations is something else. I don’t see where faith enters into this.

“Martínez had reminded me of an interview I had recently read with the guitarist Robert Fripp, who reiterated an idea commonly voiced by musicians that technique is something you learn and then throw away. You practice your scales and so on until you reach a stage when you don’t have to think about it any more, it becomes intuitive.”

*** This analogy between the way mathematicians do math and musicians, music, doesn’t work for me, even assuming what you say about musicians is true. Mathematicians don’t have to exercise and limber up by running over the multiplication table or taking some derivatives of hyperbolic functions to start work. Their math is constantly at their disposal.

I’d be willing to bet quite a lot that before Perlman tackles a Pagannini caprice he practices it like mad and not all for the sake of committing the piece to memory. While we’re at it, consider the etude. Many of these are written specifically for very advanced players for learning some aspect of technique. Often, among other things, the brilliance needed for execution makes the etude a worthy concert piece.

“There is no such thing as an objective study, there is always an interest in the things you study.”

*** This is one of those type of definitive statements that turns me off. How do you prove this? What does he mean by “interest?” Can’t one study something in the hope of finding something interesting to study?

“Martínez then launches into a complicated example concerning whether or not there is chance in the universe. The way he sees it, since we’re always studying finite series of events and never the totality, we can never be sure whether the fully rational account reflects a genuine absence of chance, or whether it is a mere rationalisation. “

*** Isn’t this just another way of questioning inductive reasoning?

“Again, it is an aesthetic one. Many times people say the second one is not as elegant, not as accurate perhaps, but it suits better.”

*** Mathematicians are big on the elegance of a proof but never to the detriment of accuracy, nor can I see it should be in any other case.

“Before a proof was something that a normal person in a normal life could check from beginning to end.”

*** I assume he doesn’t really mean a normal person in a normal life. The time has long passed that one could explain to even a very intelligent normal person what mathematicians are doing now. Like everything else, mathematicians are so specialized their work is often unintelligible to even most other mathematicians

“Now a proof can be something run by a program, so the complexity, the kind of calculation, is totally different. Now a person in his whole life is not able to reach the end of the proof. “

*** There’s an awful lot to argue about here. For example, knowing the power of the computer I might ask it to prove (or show or find) something it would be impossible for a human to handle, e.g prime numbers of a particular type. The problem is built specifically for the computer and the result is the important part not the process.