Accessible mathematics achievements

The twentieth century is now over a decade away. Time enough for some degree of perspective to be obtained. There are so many lists: top films, top personalities, top sporting achievements, that another one might seem out of place, but I want to add one.

Open a book in the mathematics section of a bookshop, or try searching Wikipedia for a mathematical topic and you will quickly be confronted with words and symbols that seem designed to put up an impenetrable barrier between the casual reader and the subject matter. So I wanted to try to put together a short list of mathematical achievements from the twentieth century that are accessible.

Let’s look at the first item. Suppose we take a map of Europe; and we paint France in blue, Germany in red, Belgium in green and so on, making sure that each time we colour in a new country it doesn’t touch a different country painted in the same colour. What is the fewest number of colours you need to paint Europe complete? Any map maker will tell you the answer - four. Try it out for yourself if you don't believe it. What about a different map, say one of Africa or Asia (no, I’m not going to fall into the trap of the asking about Australia). What about the counties of England? The answer, in each case, is four, but is it still four for any map you can imagine?

Before we go any further, we need to talk about proof. Proof, in the mathematical world, means that there is sequence of reasoned steps deducing a proposition from a set of assumptions and already proven propositions. The logical reasoning leaves no room for doubt. AT ALL. Why is this so important? Because, once a proof has been found, it is true for ever.

Now let’s go back to our map colouring problem. Finally, towards the end of the twentieth century, and with the help of computers, a proof was found showing that any map needs just four colours. The search for the proof inspired numerous forays into “tough” mathematics, the sort that would make most of us hastily put down that book referred to earlier. But the end result, buy four colours only if you want to make maps, is accessible to all of us.

That links us to the second mathematical achievement on my list, the stored program computer, perhaps more an achievement by mathematicians, specifically John von Neumann and Alan Turing, than a mathematical achievement. Nevertheless, it is included because whatever machine you’re running your browser on, it operates according to a theoretical model developed by the two aforementioned gentlemen. What vision they had!

Step up to the podium number three please. You have a lot of oranges and one large crate. What is the best way to pack the oranges so as to maximize the number packed into the crate? Are you crazy, I hear you ask, every green grocer and fruiterer knows how to do that. Here’s how it’s done: make a line of touching oranges on one edge at the bottom, make the second line interleaved so that the centres of three touching oranges form an equilateral triangle. When the bottom layer is finished, make the second layer so that each orange in that layer rests on three oranges below it. Continue layer by layer until the crate is full. It’s actually harder to describe than to do. But it’s much harder to prove than to describe. Indeed the proof wasn’t found until the last decade of the twentieth century. Once again it was the search for the proof that was the inspiration.

Painting maps, stacking oranges and making calculating machines, is that it then? Not at all, the theme running through each of these was that the search for a certainty, a proof, an absolute to a simple problem, revealed much more than could have been imagined at the start of the quest. Here is a quote from David Hilbert’s famous address to the Paris International Congress of mathematicians:-
“It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: ‘A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.’ This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.
“Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution”.
He said that to launch the twentieth century hunt for problems – and solutions.