maths for fun and profit

Someone who had begin to read geometry with Euclid, when he had learned the first proposition, asked him "What shall I get by learning these things?"

Euclid called his slave and said "Give this person a penny, since he must make a profit out of what he learns"

-- Stobaeus, as quoted by George Simmons in his "Calculus Gems: Brief Lives and Memorable Mathematics" [0]

Euclid has come and gone, but the question remains - a plaintive "but when will I ever use this?" from generation upon endless generation of bored students, a skeptical "but what use is all this, then?" from their hard-headedly pragmatic elders, and the general feeling that the question is but rhetorical, that "all this" is patently of no use to anyone but the eggheads.

One easy answer, of course, is that the universe runs on mathematics, a truism whose startling nature is beautifully expressed by the physicist Eugene Wigner in his "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". But this misses the point - everyone knows that mathematics is of use to someone, to scientists and engineers and 'that sort of person' - indeed, Wigner's argument is precisely that everyone does know that; that we internalise that assumption without ever stopping to wonder why.

Another common answer is that it may not be obvious now, but "sooner or later" you'll find out what it's good for. Children are told "you'll realise how useful all this is when you grow up", often by parents who have no real idea either, but vaguely feel that it is their duty to see that their offspring learn some maths. The skeptic is diverted with "applications turn up in the most unexpected places", a favourite supporting fact being that number theory, that most impractical of fields, has suddenly proven of immeasurable worth to cryptography, and hence to online commerce in general. But this, while true, is a somewhat defensive tack to take - it implicitly assumes that mathematics is only worthwhile because it is useful, and that in questioning its utility the sole error lies in not realising why it is useful.

While the two arguments above are, indeed, fundamental to the field of mathematics, they do little to address the real question: what shall I get from learning these things? And this is a question that must be addressed - Euclid's dismissal of his student may make for an entertaining anecdote, but it does very little to further the cause of either mathematics or humanity-in-general.

In the conclusion of their paper "How to Differentiate a Number", Ufnarovski and Åhlander provide one answer: "This article is our expression of the pleasure [of] being a mathematician. We have written it because we found the subject to be very attractive and wanted to share our joy with others." [1] Now this "pleasure of being a mathematician" may sound like it is confined to the very few, but the truth is, it is more accurately "the pleasure of mathematics", and is accessible at every level of the subject. [2] Whether it be a mathematician assisting the birth of a new theorem or a student suddenly understanding Euclid's marvellous proof of the infinity of primes, mathematics is a human pleasure on the same level as poetry, art and literature - an appreciation of beauty and the joys of thinking. And, of course, there is the entire field of recreational mathematics - puzzles, games, magic tricks, and all the other diversions of the mathematician at play - again, capable of appealing to people across the gamut of mathematical sophistication.

The other good reason to study mathematics is that, quite simply, it changes the way you think. Or, more specifically, it changes the way you can think about things - it adds an entire range of tools to your mental toolbox. The "unreasonable effectiveness of mathematics in the natural sciences" is only half the picture - a far more compelling half is the unreasonable effectiveness of mathematical thought in day-to-day life. Topics like arithmetic, probability, geometry, basic calculus[3] and set theory are invaluable not just for the explicit solving of problems, but for the powerful way in which they shape and aid your intuition in situations that are not, on the face of them, mathematical in nature.

A final objection is "yes, that's all well and good, but I've actually tried to learn mathematics, and hated it". In my experience, this is inevitably the result of bad teaching, an entrenched school of thought that stresses a pedantic teaching of facts with no appeal to the power and beauty of the underlying structure (because that, of course, is advanced mathematics).

So what is my solution? Well, I certainly don't propose to reform the teaching of mathematics in a single post! There have been several fascinating proposals in this area, some of which I shall touch upon in future posts - my intent in this one is more of an evangelical plea to go see for yourselves. Good starting points are recreational mathematics websites and books, and articles dealing with the "mathematics of everyday things" (again, more about these topics later). And above all, remember, mathematics not only can be, but is fun.

[0] Simmons goes on to add "The reliability of these smug little stories can be judged from the fact that their authors (Proclus and Stobaeus) lived in the fifth century AD, more than 700 years after the time of Euclid

[1] This struck me all the more forcibly because I had been thinking "okay, this is pretty, but why is it worth a paper?"

[2] At this point, I'd like to recommend Yakov Perelman's tragically-out-of-print "Mathematics Can Be Fun" - should you ever run across a used copy, grab it!

[3] The word 'calculus' has scary, 'advanced maths' connotations, but at its heart it deals with some very basic concepts like the way things change over time, or the way quantities add and accumulate