Andrew Wiles: what does it feel like to do maths?

What do you do when you get stuck?

The process of research mathematics seems to me [to be] that you absorb everything about the problem, you think about it a great deal, all the techniques that you use for these things. Usually [the problem still] needs something else – so yes, you get stuck.

Then you have to stop, let your mind relax a bit and then come back to it. Somehow your subconscious is making connections and you start again, maybe the next afternoon, the next day, the next week even and sometimes it just comes back. Sometimes I put something down for a few months, I come back and it’s obvious. I can’t explain why. But you have to have the faith that that will come back.

The way some people handle this is they work on several things at once and then they switch from one to another as they get stuck. I can’t do that. I get manic about it. Once I’m stuck on a problem I just can’t think about anything else. It’s more difficult. So I just take a little time off and then come back to it.

I really think it’s bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it’s a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that’s what actually enables you to get round [the problem].

So if you remembered all the failed attempts before, you wouldn’t try them again. But because I have a slightly bad memory I’ll probably try essentially the same thing again and then I realise I was just missing this one little thing I needed to do.


After Prime Proof, an Unlikely Star Rises SHARE


What is your approach to math beyond what you’ve said in other interviews — being patient and focused?

Do not easily say, “Oh, I really understand everything, so I have no problem.” You try to discover problems, to ask yourself the problems. Then you can find a correct direction to solve the problem.

How game theory can help you do a better job of parenting

In 1944, the economist, physicist, mathematician and computer scientist John von Neumann published a book that became a sensation, at least among mathematicians – Theory of Games and Economic Behavior. Written with a colleague, Oskar Morgenstern, this volume of nearly biblical proportions is so dense and littered with mathematics that only a game-theory specialist could understand it – and some of them struggled.

Even so, game theory has spread far beyond the boundaries of mathematics to become a valuable tool for explaining human behaviour. It’s used by diplomats, biologists, psychologists, economists and many others in business, research and global politics.

Game theory can also be a useful tool for parents. Children can be tough negotiators, as parents know. And the stakes are high: the outcome of negotiations between parents and their children can affect a family’s happiness and the children’s futures.

Despite its sometimes complicated mathematics, game theory is simple to explain: it’s the science of strategic thinking. Game theory does not cover all games, but only those in which an opponent’s or negotiator’s strategy affects your next move. It has nothing to do with solitaire, in which your ‘opponent’ – a deck of cards – has no strategy. Chess, on the other hand, is a beautiful example of a game-theory game, where two crafty strategists are continually trying to anticipate and block the other’s likely moves.

Encouraging cooperation between children is a wonderful game-theory example. Some years ago, Robert Axelrod, a game theorist at the University of Michigan, asked the following question: when should a person cooperate, and when should a person be selfish, in an ongoing interaction with another? He set up a computer competition among game theorists, and he was astonished at what he found. The most sophisticated solutions failed to beat something called tit-for-tat, in which each player responds by doing what the other did. If the first cooperates, so does the second. And so on and so forth. If the first does not cooperate, neither does the second.

Auctions are yet another subject of game-theory research, and useful for parents. Suppose your children all want to control the TV remote. Set up what’s known as a sealed-bid, second-price auction. Each secretly writes down what he or she is willing to pay. When the papers are opened, the highest bidder wins the right to buy the remote at $1 plus the second-highest bid. It’s far superior to a coin flip because the person who most wanted the remote got it.

Game-theory deals depend upon fairness and, often, so do dealings between parents and children. Children are consumed with the idea of fairness. If a candy bar meant to be shared by two isn’t broken exactly in half, the one who gets the smaller piece will howl. Game theory offers parents a way around this.

Suppose you break the candy bar into two pieces that are almost the same size, but not quite. And your children can see that they are different. You could do something that seems eminently fair: toss a coin. Your children can recognise the fairness in tossing a coin; nobody controls the outcome. What could be simpler? You toss the coin into the air. The winner gets the slightly bigger piece of candy; the loser gets the other. But something changes when the coin hits the floor. The winner now believes the decision was completely fair; the loser demands that he get a do-over. To him, it doesn’t seem fair at all.

The problem here turns on the meaning of fairness. The coin toss is fair, as we usually understand that. So what is the problem? In game-theory terms, the outcome was not envy-free. The loser desperately envies the winner. It’s not a very satisfactory solution for you or for one of your two children.

Here’s a way to get a much better outcome. Suppose you have the remains of a birthday cake you want to divide between your two children. You have the same problem as you did with the candy: it’s difficult to cut two equal pieces. So you turn to the technique we call ‘I Cut, You Pick’: your daughter cuts the cake, and your son picks the half he wants.

Your daughter will be as careful as possible to cut the cake into identical halves, because if she doesn’t, she will get the smaller one. Because it might not be possible to cut the cake into two exact halves, you designate your son to make the cut the next time you have cake. And you continue to take turns. This is fair, and game theory shows that people will recognise it to be fair. It’s far superior to the brutal coin toss.

Now let’s make it a bit more difficult. The cake is half chocolate and half vanilla. Your son loves chocolate; your daughter prefers vanilla. If your daughter cuts the cake in a way that gives each of them half of the chocolate and half of the vanilla, the cut is fair. Each piece is the same size. But neither child is entirely happy, because each got some cake they didn’t want. Turn the cut the other way – and divide it into a chocolate half for your son and a vanilla half for your daughter, and both are far happier. Both cuts were fair, but the cut into chocolate and vanilla halves demonstrated what’s called Pareto optimality. Each was not only fairly treated, but also got the best possible outcome.

Game theory can also be used to help the family to decide where to go on vacation, what to have for dinner, how siblings can learn to cooperate without mum or dad’s intervention – and many other problems that routinely turn up in the family.

Game theory is a gift of evolution, which sculpted us to behave according to precise and illuminating mathematical rules. We use it all the time. Game-theory parenting is a way to help parents explicitly understand the rules and reflect on what the rules say about raising healthy and successful children. An understanding of game theory helps us become the parents we were meant to be. Parents and children might not be able to make their way through von Neumann’s often opaque book, but they don’t have to. They are all game theorists already. All they need are a few good rules. And that’s what game theory provides.


Interview with a mathematician: Fan Chung Graham

AB: What advice would you give students studying mathematics, especially young women studying in mathematics?

FCG: My answer is simple: don’t be intimidated. In mathematics, you can build up one step at a time. Once you do it, it’s yours. It is a big area, and no one knows everything.

The Intrepid Mathematician

This week, I am presenting the second of my ongoing series of interviews with influential mathematicians.

Fan Chung Graham (born in Taiwan in 1949) is one of the world’s leading graph theorists and combinatorialists, with major contributions to spectral graph theory, random and quasi-random graphs, Ramsey theory, extremal graph theory, and complex networks.

Fan with Russell, a watercolor by Maria Klawe.

Fan is a professor at University of California, San Diego, where she holds the Paul Erdős Chair in Combinatorics. She spent a sizeable portion of her career working in industry at Bell Laboratories and Bellcore.

Doc1 Fan’s books: Spectral Graph Theory, Erdős on Graphs: His Legacy of Unsolved Problems, and Complex Graphs and Networks

She published hundreds of papers and three books in mathematics, and remains very active in research. Fan won numerous awards for her mathematical work, and was inducted as a fellow of the American Mathematical Society in 2012.

I first met Fan in Vancouver…

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Summer Books

Life Through A Mathematician's Eyes

During the summer holiday I never enjoy reading hard math text books and most of the time I enjoy reading more easy ones. Normally I chose them carefully, to find out more about a subject I am not really confident with. This is a great way to find inspiration and gain curiosity for a though math-subject. Also, these books are great for math-lovers that don’t want to read textbooks. Moreover, I have recommended these books for everyone that wants to start doing mathematics but doesn’t know where to start exactly. Let’s get started:

1. Fermat’s Last Theorem by Simon Singh: Fermat’s last theorem is one of those extremely known theorems out there. Also, it is a theorem easy to understand for everyone, but it showed up that being easy to understand doesn’t make it easy to prove it. In this book, Simon Singh has crafted a remarkable tale of intellectual endeavour…

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What a Mathematician should visit in Bucharest

Life Through A Mathematician's Eyes

The series ‘What a Mathematician should visit’ continues. For this summer I have decided to write about a city from as many European countries as possible. They will be mostly countries I have visited in the past ^_^ For today I have chosen Bucharest, Romania. Romania is my home country and I am really excited to write something about its capital. Lets see what a math-lover can visit in Bucharest:

1. The 1st on the list is Museum of the National Bank of Romania. Reading about the currency a country uses is always interesting and if you are interested in economy and the history of this subject this is the museum for you. The fundamental theme of the museum is the history of the circulation of money on the territory of Romania, as well as a brief review of the Bank’s history. You can see coins minted in the Greek colonies of Sinister…

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What a Mathematician should visit in Stuttgart

Life Through A Mathematician's Eyes

June is just a couple of days away and the holiday season is not an excuse to forget about math. I know that summer is the time when we forget about school, but for me it is also the time to visit a little more and see how cities around the world embrace the beauty of math. So, with this idea in mind, lets see what a math-lover can visit in Stuttgart, Germany:

1. First of all Stuttgart the city and region has a lot of technology and industry related museums, which is great. Thus, you can choose the ones you think are more related to you. If you ask me a technology or industry related museum should never be omitted from your list because mathematics is at the base of evolution. Kärcher Museum in Winnenden takes you in the journey through the development of cleaning technology and the over…

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