The only justification ever needed for mathematics
Mathematics or the arts?
That question is a mainstay of education debate. It surfaces every few years, with maths invariably winning out. Maths sits comfortably atop curricula agenda all around the world, rivalled only by English language learning.
The focus of the English Baccalaureate in the UK, the US Common Core State Standards and the PISA international comparisons reveals the status quo of education policy: Maths is king of the school curriculum.
The calculus of curriculum planning leaves many subjects — the arts in particular — out to dry. In the most extreme cases, even playtime and physical education give way to the pressures of ‘raising maths standards’.
Logic suggests that as a mathematician and educator, I should welcome the dedicated focus garnered by my favoured subject, and that I should staunchly defend mathematics against accusations of bias. In fact, I despair, because the mathematics of school is a butchery of the mathematics I fell in love with.
There is a sad irony to education policy. In an attempt to harness our human potential for reasoning and problem-solving — those skills vital to the future of our economy — policymakers neglect the most important of all mathematical truths:
Mathematics is art.
I am not alluding to Geometry. Sure, my heart stirs at the sight of fractals. I have even partaken in a walking tour of Oxford that celebrates the coming together of maths and design. But mathematics transcends these visual forms.
The Penrose tiles outside the Mathematical Institute of Oxford — a literal case of ‘maths is art’
No, I speak of the emotion that mathematics evokes. Have you ever felt a tingle while wrapped in the dreamy embrace of a maths problem? I have; the satisfaction of turning an unknown into a known is immense. It is also deeply innate.
But don’t take my fuzzy word for it — virtually every mathematician that has reflected on the nature of their subject has spoken of its artistic appeal.
In his note to students of mathematics, Michael Atiyah advises:
“The art in good mathematics, and mathematics is an art, is to identify and tackle problems that are both interesting and solvable.”
The term “interesting” is not free of subjectivity. A trained mathematician can certainly draw on their deep knowledge and experience to filter through problems, but determining whether a problem is worth your time also requires a personal judgement. If this is ‘good mathematics’, then the ‘bad mathematics’ of schooling disempowers students from posing their own questions, or to consider what makes a problem interesting or solvable.
In his famous Apology, G H Hardy professed plainly that
“I am interested in mathematics only as a creative art.”
This is an astonishing statement when you consider that Hardy was a pure mathematician. Hardy was fiercely attached to the formalities of rigour and proof. His manuscripts, replete with symbolic abstractions, do not immediately strike one as creative or artistic. Yet for Hardy, a mathematical argument, while bound to logic, had to be crafted with artistry.
This hardened thinker was not without feeling:
“Like creative art, maths promotes and sustains a lofty habit of mind, increases happiness of mathematicians and other people.”
Hardy found delight in mathematical reasoning. His towering intellect was matched by a simple understanding of a mathematician’s primary purpose:
“A mathematician, like a painter or a poet, is a maker of patterns.”
Hardy even laid down a criterion for what constitutes ‘serious’ mathematics:
“Beauty is the first test. There is no permanent place in the world for ugly mathematics.”
Most of school mathematics, with its rigid focus on tedious procedure, would fail Hardy’s test. Bertrand Russell, a contemporary of Hardy’s, defines mathematics in terms of its beauty:
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere…sublimely pure, and capable of a stern perfection such as only the greatest art can show.”
Where Hardy apologizes, mathematician turned teacher Paul Lockhart laments. His two-part essay begins with a scathing attack on school mathematics:
“No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression.”
Lockhart too cannot escape the characterization of mathematics as art itself:
“Mathematics is the purest of the arts, as well as the most misunderstood…The mathematician’s art [is] asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations.”
We can quibble over the precise definition of art. But whatever it is that makes music, poetry and painting art forms must also apply to mathematics. Art lies in process, not outcome. We experience art with emotion and intellect. We have mental and physical — even spiritual — reactions to it. Mathematics, in its true form, elicts the very same. While mathematical truths are absolute, the manner in which we discover and engage with those truths is anything but.
Euclid’s proof of the infinitude of primes — cited by Hardy (and many others) as his favourite piece of mathematics. The beauty lies in the simplicity and power of the argument.
These conclusions pose new challenges for educators. They suggest that the purpose of maths education is not rooted solely in skills and understanding, but also in beauty. For teachers to convey the beauty of mathematics to their students, they must first experience it for themselves, shaking off their own impressions of mathematics as a collection of rigid mechanical truths.
At the policy level, the rationale for mathematics needs a rethink. Mathematics is prized for its utility, yet the content and form of school mathematics, crammed thick with rote procedures, defeats this purpose.
Curriculum and assessment must evolve to take account of the holistic nature of mathematics.
The best way to cultivate a generation of critical thinkers is to get students addicted to solving problems. Mathematics is only able to achieve this once it is beautified - when it is embraced as an art. The desired economic outcomes will follow.
Just as the arts deserve more attention in the school curriculum, mathematics demands more attention to its artistic form. The justification for mathematics is no different to the arts. And that is because mathematics is art — nothing more, nothing less.
I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.
Come say hello on Twitter or LinkedIn.
If you liked this article you might want to check out my following pieces:
My Nephew Brought Home This Menacing Maths ProblemWhat happened next is a sad indictment of school maths.medium.com
The time I ‘nearly’ solved the Twin Prime Conjecture_My first authentic mathematical experience_hackernoon.com
I no longer understand my PhD dissertation (and what this means for Mathematics Education)_Earlier this week I read through my PhD dissertation. My research was in an area of Pure Mathematics called Functional…_medium.com