“What does it mean to know mathematics?”
“Living in tropical coral reefs are species of sea slugs known as nudibranchs, adorned with flanges embodying hyperbolic geometry, an alternative to the Euclidean geometry that we learn about in school, and a form that, over hundreds of years, many great mathematical minds tried to prove impossible.”
“Sea slugs have at least the rudiments of brains; they generally possess a few thousand neurons, whose large size has made these animals a model organism for scientists studying basic neuronal functioning. This tiny number isn’t nearly enough to enable the slug to formulate any representation of abstract signs, let alone an ability to mentally manipulate them, and yet, somehow, a nudibranch materialises in the fibres of its very being a form that genius-level human mathematicians didn’t discover until the 19th century; and when they did, it nearly drove them mad. In this instance, complex brains were an impediment to understanding.”
“Nature’s love affair with hyperbolic geometry dates to at least the Silurian age, more than 400 million years ago, when sea floors of the early Earth were covered in vast coral reefs.”
“A head of coral is actually a colonial organism made up of thousands of individual polyps growing together; collectively, they grow a vascular system, a respiratory system and a crude gastrointestinal system through which all the individuals of the colony eat and breathe and share nutrients. Nothing like a brain exists, and yet the colony can organise itself into a mathematical surface disallowed by Euclid’s axiom about parallel lines. Strike two against ‘higher intelligence’.”
“Many species of corals, then and now, also have hyperbolic structures, which we immediately recognise by the frills and crenellations of their forms. Although corals are animals, they have only very simple nervous systems and can’t be said to have a brain.”
“Ask any fifth-grader what the angles of a triangle add up to, and she’ll say: ‘180 degrees’. That isn’t true on a hyperbolic surface. Ask our fifth-grader what’s the circumference of a circle and she’ll say: ‘2π times the radius’. That’s also not true on a hyperbolic surface.”
“Most of the geometric rules we’re taught in school don’t apply to hyperbolic surfaces, which is why mathematicians such as Carl Friedrich Gauss were so disturbed when finally forced to confront the logical validity of these forms, and hence their mathematical existence.”
“But can we say that sea slugs and corals know hyperbolic geometry? I want to argue here that in some sense they do.”
“Absent the apparatus of rationalisation and without the capacity to form mental representations, I’d like to postulate that these humble organisms are skilled geometers whose example has powerful resonances for what it means for us humans to know maths – and also profound implications for teaching this legendarily abstruse field.”
“I’m not the first person to have considered the mathematical capacities of non-sentient things. Towards the end of Richard Feynman’s life, the Nobel Prize-winning physicist is said to have become fascinated by the question of whether atoms are ‘thinking’.”
“Schrödinger equations (after the pioneering quantum theorist Erwin Schrödinger and his hypothetical cat), are so complicated that, when Feynman was alive, the best supercomputers could barely simulate even the simplest orbits. So how could a brainless electron be effortlessly doing what it was doing? Feynman wondered if an electron was calculating its Schrödinger equation. And what might it mean to say that a subatomic particle is calculating?”
“The world is full of mundane, meek, unconscious things materially embodying fiendishly complex pieces of mathematics. How can we make sense of this?”
“I’d like to propose that sea slugs and electrons, and many other modest natural systems, are engaged in what we might call the performance of mathematics.”
“Since at least the time of Pythagoras and Plato, there’s been a great deal of discussion in Western philosophy about how we can understand the fact that many physical systems have mathematical representations: the segmented arrangements in sunflowers, pine cones and pineapples (Fibonacci numbers); the curve of nautilus shells, elephant tusks and rams horns (logarithmic spiral); music (harmonic ratios and Fourier transforms); atoms, stars and galaxies, which all now have powerful mathematical descriptors; even the cosmos as a whole, now represented by the equations of general relativity.”
“The physicist Eugene Wigner has termed this startling fact ‘the unreasonable effectiveness of mathematics’. Why does the real world actualise maths at all? And so much of it?”
“Even arcane parts of mathematics, such as abstract algebras and obscure bits of topology often turn out to be manifest somewhere in nature.”
“Most physicists still explain this by some form of philosophical Platonism, which in its oldest form says that the universe is moulded by mathematical relationships which precede the material world. To Platonists, matter is literally in-formed, and guided by, a pre-existing set of mathematical ideals.”
“In the Platonic way of seeing, matter (the stuff of everything) is rendered inert, stripped of power and subordinated to ethereal mathematical laws. These laws are given ontological primacy with matter being effectively a sideline to the ‘true reality’ of the equations. Over the past half-century, this vision has been updated somewhat because now matter, or subatomic particles, have themselves been enfolded into the equations. Matter has been replaced by fields – as in electric and magnetic fields – and now it’s the fields that follow the laws. Still, it’s the laws that retain primacy and power; hence the obsession with finding an ultimate law, a so-called ‘theory of everything’.”
“Rather than being a remote abstraction, mathematics can be conceived of as something more like music or dancing; an activity that takes place not so much in the writing down as in the playing out.”
“Platonism has always bothered me as a philosophy in part because it’s a veiled form of theology – mathematics replaces God as the transcendent, a priori power – so if we want to articulate an alternative, we need new ways of interpreting mathematics itself that don’t also slip into deistic modes. Thinking about maths as performative points a way forward, while also offering a powerful pedagogic model.”
“It took a woman, the mathematician Daina Taimina at Cornell University, to discover hyperbolic crochet and to give mathematicians a tangible model of this form. I have conducted workshops about this with women all over the world delighting in how much geometry can be conveyed through acts of making.”
“There’s also a link here with general relativity, because the discovery of the hyperbolic plane opened up a whole new era in geometric thinking, leading ultimately to generalised Riemannian geometry, which can describe any complexly curved surface, and is the mathematics underlying Albert Einstein’s equations for the cosmos.”
“Via handicrafts, we can introduce people to concepts about curved spacetime and multidimensional manifolds, leading with our fingers, and out to questions about measuring the structure of the cosmic whole.”
“We can see this as a form of ‘digital intelligence’, and it’s worth noting that iterated handicrafts (knitting, crochet, weaving) were the original digital technologies: their algorithmic ‘patterns’ are literally written in code.”
“It’s no coincidence that computer punch cards were derived from the cards used in automated looms. Here, knowing emerges from hands performing mathematics: it is a kind of embodied figuring.”
“People talk about playing music but mathematics can also be a form of play.”
“One way of thinking about maths is as a language of pattern and form, so when you play with patterns you are doing maths.”
“All around us, nature is playing mathematical games and we too can join in the fun.”
“Mathematics need not be taught as an abstraction, it can be approached as an embodied practice, like learning a musical instrument.”
“By thinking about mathematics as performance, we liberate it from the straightjacket of abstraction into which it has been too narrowly confined.”
“If you ask professional mathematicians what they love about their work, a likely answer is its beauty. ‘Euclid alone looked on beauty bare’ wrote the poet Edna St Vincent Millay in 1923, while the mathematician André Weil (brother of Simone) claimed that solving a hard mathematical problem topped sexual pleasure.”
“The professionals know that mathematics swings; they delight in its playfulness, the plasticity of its forms, and (after some initial shock) the absurdities it throws up.”
“Hyperbolic surfaces, aperiodic tilings, Möbius strips, negative numbers and zero all generated alarm at first, yet were ultimately embraced as gateways to new continents of mathematical wonder.”
“Just as humans are endowed with an ability to dance and play music (even if education too often crushes this out of us), so we have innate form-making and pattern-playing proclivities.”
“Sea slugs, sound waves and falcons do mathematics; Islamic mosaicists and African architects do it too. So can you.”