When the idea was recently floated that all children should study mathematics until the age of eighteen, it felt for a moment that Britain had become a classroom of baying teenagers, asking a question I am already well used to as a maths teacher: “Why do we need to learn this?” The cynicism was quite reasonable. Not only was the proposal senseless on practical grounds, but it also showed no appreciation for the real value of mathematics.
I am only an amateur mathematician, but the subject has been my main working concern for the best part of six years, from A-Levels to university and then to these early stages of my career. This is, I think, the first time that I have written about it. The occasion is that I have discovered why I like maths; or rather, I have discovered how to express it. My answer lay in a chapter of G. K. Chesterton’s characteristic book on Christian apologetics, Orthodoxy.
Chesterton describes any purely rational philosophy as being like a circle, and a small one at that. It is perfect in its completeness, yet it is narrow: ‘it is quite as infinite as a large circle; but, though it is quite as infinite, it is not so large.’ For instance, a materialistic account of the universe may plausibly be coherent (though I do not believe it is) but a sane man can never believe it to be large enough: large enough to explain love or beauty or G. K. Chesterton. This is clearer still for another entirely rational explanation: that everything is in my imagination or in a dream. It cannot be disproved, but it is not true. It is a full circle, but it is too small.
Mathematics is the greatest of these circles. To begin with, it is the nearest to actually being a full circle. It is not complete, but its incompleteness is in the very far distance. The foundations are sound. We may not know exactly where they come from, and we may vary how we describe them, but they are, and they are mathematical. The natural sciences, meanwhile, find themselves in the greatest muddle over the very basics, at the level of electrons and amoebae and beginnings. Each of these parts finds itself straying wildly away from the scientific circle, and into the supernatural.
Yet it is in the natural sciences that the circle is most trusted to explain everything. Even if it were a complete circle, it would be too small for life in its fullness. But it is widely believed that science rules all things. Thus we may not see the people we love or eat cake or worship our Creator because of Science. For Chesterton, this is an example of insanity. Mathematics does not suffer nearly as much from this ailment. Its students know that it has connections with the world around us, but they are amusing diversions from the great, solid object which is mathematics. I know that the equations of fluid dynamics describe water very well indeed, but it does not really matter if water exists. If it didn’t the equations would still hold, and I could imagine the liquid they describe. The scientist is obsessed with water itself – where it comes from and what it does. He will tell you how much you should drink. The mathematician does not care; though he may comment on how funny it is that here is a substance which obeys his equations.
(Incidentally, this is why I enjoy the company of other mathematicians, though of course I am not an impartial judge. The mathematician, while the most prone to witter on about his subject, does not expect you to care; he just enjoys talking about it. Hence, you may yawn without moral qualm, which I have always found quite liberating. Similarly, they are the most likely to have some entirely unrelated hobby, because they are aware that mathematics is not all of life, though it may be their overwhelming concern. They realise the necessity and the possibility of doing something quite separate, such as watching trains.)
Mathematics, then, is a game. It is played on a board, and on the board it is everything, and off the board it is nothing – except for the minor detail that some small parts of it describe all physical reality. It is not useful, in that it is not full of use; most mathematics has no direct importance for our lives. Its value lies elsewhere.
Chesterton’s contrast to the circle of rational explanations is the ‘infinite sea’ of poetry: while ‘the logician … seeks to get the heavens into his head, … the poet only asks to get his head into the heavens.’ Here, in poetry and religion and theology, is the largeness that we are looking for. The circles of mathematics and the natural sciences and economics and politics are contained within it. Their real value is in that context. Mathematics displays the order and beauty of that great sea which is the heavens and the earth; studying mathematics allows us to create order and seek beauty in that sea.
It is said that football is the most important of all the unimportant things in life. The events on the pitch do not matter. But on and around the pitch we see in astonishing colour the victories and the sadness and the brotherhood and the freedom of life. Perhaps we could say something similar of mathematics, with its harmonies and its infinities, which God has so elegantly fashioned to structure his creation: of all the un-useful things in life, mathematics is the most useful. [1]
[1] It should be noted that when my students ask why they must study mathematics, this is rarely the answer I give.
The Second Discourseman
The Four Discoursemen 