*Mathematicians discussing in the new maths building at Cambridge University*

Lola asked me during a recent lesson: what do mathematicians do? If by “mathematicians” we mean people whose job it is to produce new mathematics, the answer is that they work at defining and solving unsolved mathematical problems.

Usually they are employed by a university and also work at teaching mathematics. Some have positions at universities which mean they can spend almost all their time on mathematical research and do little or no teaching. A few work for commercial research institutes big enough to have maths sections (Bell Labs, IBM in the past, Microsoft).

There are a few exceptions. The well-known American mathematician Ron Graham used to work as a circus performer, and is still a world-standard juggler. Arthur Cayley, Britain’s most-important-ever mathematician, did most of his maths in the evenings after a day job as a lawyer. The great Carl Gauss had as his formal job title “professor of astronomy”.

Most people who study maths at university do not become research mathematicians. They use their maths degree as the basis for training as engineers or scientists of many different sorts; finance geeks; economists; statisticians; IT people; and more.

No other university degree stream provides such a good starting basis for so many paths of specialist training. On the basis provided by a maths degree, you can become skilled in many fields by training and experience; but, since maths is the most social of sciences, the basis of mathematical knowledge is very difficult to get without a university education in maths.

People with mathematical training who become engineers or finance geeks or whatever generally don’t expect to produce new mathematics, but only to use their mathematical knowledge in a particular field.

The way they use maths is different from the way we do maths at school, because at school you tend to do maths by way of practice problems each of which comes with a clear hint for the mathematical method to use for it. Real-life problems come without a label saying “use this method”. Engineers, scientists, or finance geeks have to use imagination and cunning to decide what mathematical method to use.

Look at the website http://i-want-to-study-engineering.org/ for examples of what I mean.

As for the mathematicians whose job it is to produce new maths, they do maths in a way even more different from how we do maths at school.

In the first place, very little of what they do is *calculations*, and most of school maths is calculations. Much more of what they do is *proofs*. That may include finding new and better (shorter, more vivid) proofs of results already proved. Carl Gauss discovered and wrote a proof of his favourite theorem, the Law of Quadratic Reciprocity, at the age of 19. Over the next 59 years of his life, he produced seven other proofs which he thought were better.

Other things mathematicians do include *defining and formulating new problems*, producing *conjectures* (ideas which look as if they might be true and are worth trying to prove), and simply developing new mathematical concepts.

The mathematicians Evariste Galois and Frank Ramsey are famous because they opened up whole new fields of maths which just didn’t exist before, and now are called “Galois theory” and “Ramsey theory”. John von Neumann opened up a whole new field called the theory of games. Benoit Mandelbrot is famous for opening up the theory of fractals (though he himself thought it wasn’t really a new theory).

When they work at a research problem in maths, mathematicians do it not because someone has told them the method to solve the problem, but because so far no-one can find a mathematical method to solve the problem! They expect to start off having no idea how to solve the problem.

And sometimes it stays that way. A cleaner at Princeton University commented on the mathematician John von Neumann: “He’s very nice, but he’s a little strange…. All day he just sits at his desk with a pencil and a big stack of paper and all day he scribbles, scribbles, scribbles. And then at the end of the day, he takes all these sheets of paper that he’s covered with his scribblings and he throws them all away.”

Another mathematician comments: “This is in fact the way all mathematicians work. Mathematical thinking is not a neat orderly process. One tries all sorts of things, most of which don’t work, and some of which, one eventually realises, are totally stupid. One spends days chasing down a blind alley”.

You may fail to solve the problem completely, but solve part of it. Or solve a different, but connected problem. Or discover something quite aslant from the problem.

For example, we talked about how the “infinity” of real numbers (or of points on a line) is “bigger” than the “infinity” of whole numbers, 1, 2, 3, 4….

To discover that fact, Georg Cantor had to work out a sensible way of defining how two infinities can be the same size, or one can be bigger than the other. He developed a whole new method of proof (called “Cantor’s diagonal”, and used on many other problems since) to prove the result.

Then mathematicians asked: is the “infinity” of real numbers the next-biggest infinity after the “infinity” of whole numbers? They thought probably yes. For 73 years mathematicians worked on trying to prove that.

Then in 1964 Paul Cohen made a discovery which cut across all that. He proved that from what we know about whole numbers and real numbers, it is *impossible to tell* whether the “infinity” of real numbers is the next biggest infinity after the “infinity” of whole numbers. And, again, to get his result he had to develop a new method of proof (called “forcing”).

At any given time, there is a set of unsolved problems which are reckoned to be particularly important. In 1900 the mathematician David Hilbert gave a famous list of 23 unsolved problems: ten have been solved, and others have been put in a new light by discoveries which cut across them, like Cohen’s. In 2000 a new list of seven was published as “the Millennium Problems”.

The mathematician Andrew Wiles became famous in 1995 because, after seven years’ solid labour, he (and Richard Taylor) solved a problem which mathematicians had worked on without conclusion for 358 years, writing a proof of “Fermat’s last theorem”. The great Carl Gauss, at the age of 19, wrote a proof about constructing a 17-sided polygon which other mathematicians had tried and failed to get for over two thousand years.

Wiles was unusual because he worked for several years on his own without discussing with other mathematicians. Most mathematicians spend a lot of time discussing with other mathematicians and learning what other mathematicians are doing. For example, the British mathematician Tim Gowers was recently working on one of the seven “Millennium Problems”, the “P=NP” problem. He didn’t solve it, but he wrote an account of his efforts and attempts, so that it might help others or attract helpful comments from others.

Most mathematicians don’t work on the “big” problems, but more modestly try to find a few problems specialised enough and small enough to give them a better chance of progress (“a few” problems, rather than just one, so that they increase their chances of making progress on at least one of them).

The mathematician’s life is not just one day after another of scribbling and then throwing away failed attempts at solving problems. The mathematician will spend a lot of time learning about what other mathematicians have been doing, reading what they write, and talking with them face to face in their university or at conferences.

And teaching mathematicians at university is not quite like teaching maths in school. In school, maths teachers usually more or less have to follow a textbook which in turn is designed to fit an exam. You probably find it unusual that I often say that the textbook is wrong or does things a clumsy way.

At university level, however, the mathematician will be expected to think about what she or he is teaching, and to present it in a fresh way, using textbooks only as and when they may help. The mathematician David Hilbert used to design his teaching so that he never lectured on the same subject twice, and so he could use his lectures to rethink the widest possible range of maths.

Maths has expanded enormously in the last hundred years or so. Every mathematician now has to be specialised: she or he knows in detail only about a particular field of maths.

Our fortnightly maths prize for 13 October 2015 is about a problem in Ramsey theory, which is part of an area of maths called combinatorics. It’s a very active area now, important for computer science. When I studied maths at university, though, combinatorics was an obscure corner; the university offered no course in it. I’ve asked maths teachers at CoLA who’ve studied maths at university more recently, and none of them did a course in combinatorics either. You could do a university maths degree without even starting on this important area.

Here’s another story about how specialised. In Australia a few years ago, I was on a temporary contract at Brisbane State High School, which reckons itself the top academic school in the city. I discovered that an assignment on differential equations used for many years by the school had a chunk in it based on a mathematical error. I was sure it was an error; in fact I could prove it.

But I was worried that the school maths department would not want even to start reading an objection from someone there only on a temporary contract. I was wrong to worry: the other teachers read my proof. In the meantime, though, I phoned the specialist on differential equations at the local university to seek authoritative support.

Yes, she replied, what I’d written looked right. But it wasn’t quite her part of the theory of differential equations. I should ask someone else more specialised in that field, who was away on holiday… This, remember, was just for a high school assignment.

The learning about calculations which you do in school is important for maths, just as learning how to write legibly, how to touch-type, how to spell, how to punctuate, how to expand your vocabulary, is important for becoming a writer, poet, novelist, journalist, whatever.

But the maths which working mathematicians do is as much more an effort of imagination, beyond methods of calculation, as good poetry is beyond good handwriting and spelling. Or a brilliant performance at sport beyond practice routines.

David Hilbert remarked, when told that one of his students had given up maths to become a poet: “Good. He didn’t have enough imagination to be a mathematician”.

Carl Gauss, asked how he got his results, replied simply: “By systematically trying things out”. The modern French mathematician Alain Connes says that in mathematical research: “The main error to be avoided is trying to attack the problem head-on”.

What makes a good mathematician is imagination, to imagine different concepts and different ways of approaching problems. Wide knowledge, to be able to review in your mind a lot of different angles on a problem. Deft and accurate calculation, to be able to test quickly and accurately whether a method will work. And stubborn persistence, to be able to keep on trying different ideas, day after day.