The beauty of mathematics is not obvious…but we can make it so
Where Paul Erdös got it wrong
I seem to keep encountering the following quote:
It appears on my Twitter feed every few weeks, is quoted in several maths books and, just this week, I found the words emblazoned on the office wall of a mathematician colleague.
The quote is attributed to Paul Erdös, the twentieth century Hungarian mathematician who was so prolific that mathematicians hence have identified themselves in terms of an Erdös number, a value that reflects their “collaborative distance” to the man himself. Erdös held a romantic view of mathematics, captured by his conception of The Book — a fictional and divine collection of the most ‘elegant’ mathematical proofs.
To anyone that has dedicated themselves to mathematics, as a professional or for leisure, Erdös’s tribute to the subject is alluring. It validates the investment we have made to maths problems, and liberates us of the need to justify our chosen endeavour to others. In Erdös’s mind, if you don’t see the beauty of mathematics then the problem lies with you — it is so damn obvious, after all.
But Erdös needs revision. It turns out that you can beautify music to the unitiated, and that the same is true of mathematics. In fact, not only can maths be made beautiful; it must be made so, since most people see the subject as a torrid mess of symbols.
Our impressions of mathematics are formed in school, where we are forced to consume an unsavoury diet of facts and algorithms, and where the performance-driven ritual of exams supersedes honest-to-god problem solving. Even the most hardened mathematician would struggle to find beauty in the ugly brand of school maths. Every mathematician I know found solace outside of the formal curriculum, through enrichment activities and extracurricular reading.
To convince one of the beauty of mathematics quite often requires an aggressive reconditioning of their mathematical worldview. People need to be made aware that an entirely different brand of mathematics exists to the one they endured at school, one that possesses beauty as a chief virtue and, subsequently, has mathematicians hooked.
The lid is being lifted on this enthralling variant of mathematics, thanks to the efforts of a handful of maths educators who strive to represent their beloved subject in ways unrecognisable from the formal curriculum. The popular maths genre exists because of such people and it’s only a shame their ideas have to be considered enrichment (as if it’s acceptable that students have to settle for a learning experience that is anything less than enriching).
Popularisers of mathematics strive to bring outsiders into the mathematical fold by explicitly showcasing the beauty of the subject. They realise that concepts can be presented in a multitude of ways; some representations as beautiful as others are ugly. Some even have the creative firepower to dream up new representations that illuminate old concepts in ways unprecedented.
To borrow from maths educator Chris Brownell:
It is as if you can hear [the popularisers of mathematics] saying, “Look, I did not decide to put all my passion aside, and study a lifeless, soulless, non-creative set of machine rules! Mathematics sparked curiosity, pleasure, passion, and yes JOY and Creativity in me. Let’s show this to children, and not bury the good stuff behind mindless stuff.”
Contrast this desire to enlighten others with the arrogant declarations of Erdös. It is the humble maths educator who assumes responsibility for sharing the beauty of mathematics with others. Realising that mathematical beauty is to be experienced, they champion inclusive pedagogies that afford all students a taste of mathematics proper. Erdös has more in common with the obstinate pedagogue who insists knowledge acquisition is the sole ingredient of mathematical development.
You may contend that some people — perhaps including yourself — will never find beauty in mathematics, whatever the representation or experience. You may be right, but consider this: I have never encountered a person who can resist finding beauty in patterns, nor have I met anyone who is unable to point to their favourite puzzles. Mathematics is nothing less than the most creative enterprise ever invented for pattern seeking and puzzling. If you manage to grab a taste of this version of mathematics, and its beauty proves elusive still, then fair play. But at least you weren’t fooled into thinking school maths was the only brand on offer.