Brouwer’s fixed point theorem states that if h is a continuous function mapping a closed unit ball (or disc) into itself, then it must have at least one fixed point. That means: at least one point x where h(x)=x.
In other words, if the women above are sloshing their coffee about gently as they walk, then between any two moments of time at least one molecule in each drink must end up exactly back where it started.
“Continuous” means that the function has no jumps. Or: you could draw it without taking your pencil off the paper. Or: if x1 and x2 are close, then h(x1) and h(x2) are also close.
The “unit” ball (or disc) means a ball (or disc) of radius 1. “Closed” means that we include the surface (or circumference).
Here is a proof for the unit disc. It is a proof by contradiction, that is, it works by assuming that the theorem is wrong, and showing that that is impossible.
If the theorem is wrong, then h(x) is different from x for all x. So we can draw a ray from h(x) through x to the circumference, as shown in the diagram.
Call the point where the ray hits the circumference r(x). Then if h(x) is continuous, so is r(x). And since r(x)=x for x on the circumference, r(x) must take values all round the circumference.
There is a smooth, continuous process (called a “retraction”) which shrinks the whole unit disc down to one point: just make its radius smaller and smaller, smoothly, until it is down to zero.
When x is going through that process, r(x) (i.e., the whole circumference) must also shrink smoothly within itself down to a point.
But imagine a wire circle covered by an infinitely stretchy, infinitely squeezable sheath. Even though the covering is infinitely stretchy and infinitely squeezable, there is no way of stretching and squeezing it down to a point. You would have to break it or cut it to do that.
The conclusion that r(x) must shrink smoothly down to a point must be wrong; and so the initial assumption, that h(x) is different from x for all x, must be wrong. ▇
This proof is interesting for several reasons.
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”.
If you want to solve a tricky mathematical problem, almost always you have to find some unexpected way of formulating the problem, some way of breaking it down into smaller steps, some zig-zag route.
For most A-level questions, going at them head-on is ok. But not for STEP, for example.
Here are three other neat examples of solving mathematical problems by an unexpected zig-zag route.
Historically, Brouwer’s fixed point theorem was one of the first results in an area of maths called topology, which is concerned with the properties of shapes which stay the same when they are stretched or squeezed, but not when they are broken or cut.
Issues which are now reckoned to be part of topology had been studied by Leonhard Euler back in the 18th century, such as the Königsberg bridges problem and Euler characteristics, but topology as an established area of maths (alongside arithmetic, algebra, geometry, calculus, etc.) is usually reckoned to date only from an article by Henri Poincaré in 1895.
This fixed point theorem was proved by the Dutch mathematician L E J Brouwer in 1910.
Although the proof above is generally reckoned by mathematicians to be sound, Brouwer himself came to think that it, and some other proofs of results in topology which had made him famous as a mathematician, were unsound.
He developed a philosophy which said that mathematical objects have no existence “out there”, and exist only as and when they are constructed by mathematicians. Since the proof does not tell us how to find the “fixed point” – it does not construct it – therefore it does not prove its existence.
In later years, according to another Dutch mathematician, Brouwer “never gave courses on topology, but always on – and only on – the foundations of [mathematics]. It seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of [his philosophy], and he judged everything he had done before, his greatest output, false according to his philosophy. He was a very strange person, crazy in love with his philosophy”.
Brouwer’s philosophy is still discussed among mathematicians, though it is a minority view. I taught for a while at Marsden State High School, near Brisbane in Australia, when the head of the maths department was an advocate of Brouwer’s philosophy.
The American mathematician Errett Bishop, in the 1960s, reconstructed proofs of a lot of the standard results of mathematics to be satisfactory according to Brouwer’s philosophy.