91av

Christopher Zeeman is a British mathematician who inspired generations of young people, built a world-class maths department from scratch and still manages to find time to correct theorems developed by Euclid in ancient Greece. He spoke to Justin Mullins about what makes maths beautiful and why he will never understand it all.

What’s wrong with Euclid?

I’ve always been fond of The Elements, ‘s book on mathematics, but one definition in it is pretty corny. It says: “a ratio is a sort of relation in respect of size between two magnitudes of the same kind”. To say “sort of relation” seems terribly feeble compared with the other definitions Euclid uses. Also, having defined a ratio, it’s surprising that he didn’t introduce the idea of a ratio of two ratios.

Why is that a problem?

It’s the mathematical tool you need to develop projective geometry. You also need it to define the product of two ratios, which is key to group theory. And velocity is the ratio of distance over time, so acceleration, which you need to define dynamics, is a ratio of two ratios. That’s three huge branches of mathematics the Greeks couldn’t develop.

You built a world-class mathematics department from scratch at the University of Warwick when it was founded in the 1960s. How?

I wrote to the six best topologists in the world asking them to join me. They all said no. So I wrote again saying the other five had agreed, and all replied to say yes. Having enough people to attract other researchers was crucial.

You were also the first mathematician to give the for children at the Royal Institution, in 1978. How did you approach it?

Faraday invented the lectures in 1825 because kids in his day weren’t taught science at school. They’ve been given every year since, but maths was always thought too difficult.

In my lectures I proved 20 theorems rigorously. For example, I proved one of Euclid’s, that there is an infinite number of prime numbers. You start by assuming there is a finite number of primes. Multiply them all together and add one. Call this new number n. Now n is not divisible by any of the primes you started with so it’s another prime. The original assumption of a finite number of primes must be wrong.

What was the response to the lectures?

The BBC tried to stop me giving proofs, but I insisted. Then the head of the Science and Engineering Research Council, which I worked for, wrote to me saying how dare you do all this TV stuff when you should be doing research. I wrote back saying how dare you say I shouldn’t take mathematics to the public. After my lectures, the institution received more letters from young people than after any previous lecture series. In maths you can explain the whole lot from beginning to end, so children totally understand it.

You say that you find maths beautiful. Can you give an example from your work?

One of my best theorems was about untying knots in spheres in five dimensions. I had been trying, in vain, for seven years to prove such knots could be tied. One Saturday morning I thought I’d try to prove the opposite. A few hours later I had done it – I was amazed. Then at 2 am on Sunday night it hit me that I could generalise the theorem to n dimensions, as long as the difference between the number of dimensions in the knot and the sphere was 3 or more. Once I saw that, I reduced the proof for spheres in five dimensions to just 10 lines.

So beauty stems from economy of thought?

Yes. If a proof is very short compared to the statement, you admire its beauty, but other proofs are very long and you admire them for getting to the result after such a tortuous path.

Seven years is a long time to study one thing. Do time constraints limit the maths you can cover?

I once told the biologist Lewis Wolpert that 90 per cent of mathematics was a closed book to me. He was very surprised and said all of biology was open to him. The trouble is there is so much mathematics. It might take me three years to understand one theorem. So understanding the lot is impossible. I just choose the bits I like.