Mathematics: a beauty warm and near

In a week when Daniel Strömbom showed us how beautiful mathematical models can be, Teddy Herbert-Read emailed me the above link. It’s a great video, but has a quote by Bertrand Russell that I’m not so keen on. So I thought I’d write about it.

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music, without appeal to any part of our weaker nature..." —Bertrand Russell

I know what Russell was getting at with his analogy, there is a purity in equations that distinguishes them from the mess of the natural world.  Mathematics can seem superior, in a way that rises above our everyday problems. But I think this is the wrong analogy, and so too did Russell later in his life.

The quote is taken from an essay he wrote in 1902, when he was thirty, on the “The Study of Mathematics”. At the time he was in the midst of a twenty year long mission, together with Alfred Whitehead, to capture this mathematical purity. They tried to provide an infallible logical basis to mathematics, which would ensure that everything that could be proved in mathematics could be derived from a small set of axioms. This was a doomed exercise. Although Russell and Whitehead published their results in three volumes of Principia Mathematica, Russell was not fully convinced that they had succeeded.  Then in 1931, Gödel’s Incompleteness Theory showed that any such set of axioms could not be used to prove their own consistency. Every piece of austere beauty ends up, in some way, contradicting itself. Often in an embarrassingly obvious way.

Gödel’s proof was the nail in the coffin for Russell and Whiteheads project, but I think the problems of trying to ground mathematics in truth went even deeper. Russell wrote much later in his life that “I wanted certainty in the kind of way in which people want religious faith……but as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.” There is nothing we can do to make mathematical models completely stable. The bigger the elephant, the harder it will fall.

Mathematics is not cold and austere artist with some higher purpose. It is a warm and friendly workman who arrives at your door with a few ideas about how to fix your boiler. Mathematics helps you tidy up, it helps you put things in order and to fix those awkward little problems. It can help you understand all sorts of little things: light waves, backgammon strategies, fractal snowflakes, spinning toys, magnetic fields, tomorrow’s weather, tree growth, spiraling DNA, stock prices, aerodynamics and electrical circuits. It’s the ultimate Mr-Fix-It.

Mathematics certainly shows you how to think clearly and sometimes you wonder at its incredible efficiency. But mathematics knows nothing more about the nature of reality than anyone else, and it would have no right to set itself above the real world. But that’s OK. Because used properly mathematics has no such pretentions. Mathematics rightly viewed, possesses a supreme beauty — a beauty warm and near, that appeals to every part of our nature.