The joy of sets

Why won’t our artistic-literary establishment recognise that there’s mystery, beauty and humanity in maths?

January 13, 2023
Alan Turing. Photo: GL Archive / Alamy Stock Photo
Alan Turing. Photo: GL Archive / Alamy Stock Photo

In the autumn of 1900, Bertrand Russell felt that he stood at “the highest point of my life”. The young Cambridge logician and political reformer believed that he had codified the laws of arithmetic, put the discipline of mathematics on a firm foundation, and conquered “regions formerly abandoned to the vagueness of philosophy” for “the precision of exact formulae”. After such hubris, nemesis struck fast. The next year Russell began to realise that maths could not fully explain itself. The inadequacy of its axioms stemmed in part from contradictions in set theory that could be expressed through something as trivial—childish, even—as the ancient paradox of Epimenides: the one about the Cretan who says that all Cretans are liars. 

Soon after, Russell underwent a “mystical illumination” that overturned all his thinking. He decided that he no longer loved his first wife, and broke German philosopher Gottlob Frege’s heart by undermining the whole basis of his work on the foundations of arithmetic. Around Europe, other mathematical thinkers faced a crisis of belief as (in the words of the great Austrian engineer-turned-novelist Robert Musil) “they actually looked all the way to the bottom and found that the whole building was standing in midair”.

Who dares call maths dry? The year of Russell’s vision of disintegration, 1901, saw Picasso’s first major exhibition, Freud’s Psychopathology of Everyday Life, Thomas Mann’s Buddenbrooks, HG Wells’s The First Men in the Moon, Booker T Washington’s Up from Slavery… at this hinge point, creative pioneers in many fields began to rebuild numbers, words, images and identities from the smashed remnants of their former bedrock. In maths, this “foundational crisis” led via David Hilbert’s “decision problem”, and the Incompleteness Theorem that Kurt Gödel proposed in 1931, to the musings of a callow outsider in Cambridge. This young maverick began to think about an all-purpose mechanism to test the validity of any mathematical proposition. Naively, the dreamer even imagined his “universal machine” not as a notional device but a physical contraption. In 1937, Alan Turing published his paper “On Computable Numbers”. And the world we inhabit began to take shape…

Mathematics belongs firmly within, not outside, the Modernist revolution in art and thought that reconfigured minds and lives. So why would any writer who cares about the origins of the ideas that frame our existence not wish to understand—and see others understand—the epic stories that flowed from a catastrophic meltdown in the intellectual core of arithmetic to the technological transformation of our world? Yet the response of many traditional literati to any suggestion that mathematics should form part of a shared culture remains denial, derision and outrage. 

Rishi Sunak may well have meant his new-year proposal for the compulsory extension of maths education to age 18 as a diversionary stunt, or at best a utilitarian scheme to equip school leavers with the competence required of corporate drones. His fuzzy plan may indeed be cynical, incoherent or impractical. Recruitment of maths teachers runs many thousands below official targets. A 2017 review of the prospects for post-16 provision by Adrian Smith pointed to shortfalls in capacity and, crucially, the need to partner maths expansion with “wider, high profile action to improve local outcomes and increase social mobility”.

That said, the braying chorus of “no more boring sums” from many authors has offered a dismaying glimpse into the wilful ignorance that passes for cultivation in British literary life. More than 63 years after CP Snow deplored the “two cultures” that split scientists from humanists, too little has changed. Many “arts” folk still fondly believe creativity belongs exclusively to their tribe. They sneer at maths teaching as numbing formal exercises, or rigid training for careers of mindless servitude. Why can so few see that maths provision for every 16-to-18-year-old might actually open up possibilities to make Sunak squirm? 

Forget advanced calculus for all: that will never happen. But, on a practical level, a serious classroom bid to tackle statistical innumeracy could help protect students against commercial and political liars armed with fake figures—such as the weekly £350m supposedly earmarked for transfer from the EU to the NHS by the Leave campaign that the prime minister endorsed. Piquantly, mathematical awareness as a boost to active citizenship then edges close to the aims of that bugbear of every Tory blowhard: media studies.

In a wider frame, modern mathematics may deliver narratives of crisis, doubt, quest and discovery equal to any story the humanities can tell. Expert popularisers such as Marcus du Sautoy and Simon Singh have let thousands hear the music of the primes or touch the riddles of infinity. In the heartland of literature itself, novelists have traced the mathematician’s journey towards understanding in non-trivial ways that far surpass the tortured-genius trope beloved of Hollywood (in movies such as Good Will Hunting and A Beautiful Mind). From Janna Levin’s A Madman Dreams of Turing Machines (inspired by Turing and Gödel) to David Leavitt’s The Indian Clerk (based on the prodigiously original Srinivasa Ramanujan) and the smart mathematical mysteries of “Catherine Shaw” (penname of the Paris-based American mathematician Leila Schneps), the dance of forms, patterns and numbers has assumed elegant—and humanly convincing—narrative guises. 

The Greek polymath Apostolos Doxiadis has not only enriched the maths-fiction genre with his novel Uncle Petros and Goldbach’s Conjecture. Along with computer scientist Christos Papadimitriou and artists Alecos Papadatos and Annie di Donna, he spun that turn-of-the-century crisis into a mind-stretching graphic novel: Logicomix. Now there’s a precious resource to beguile mutinous teenagers during the compulsory classroom sessions in the future.  

“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful,” wrote Ramanujan’s Cambridge patron GH Hardy in the famous defence of his art, A Mathematician’s Apology. For Hardy, his discipline was (and is) an art, in which “Beauty is the first test: there is no permanent place in this world for ugly mathematics”. In the late 1930s, Hardy believed that he had pursued a useless kind of perfection (“Judged by all practical standards, the value of my mathematical life is nil”). Within a few years, however, Turing and others would prove that the most seemingly “pure” lines of enquiry might generate applications to win global wars and revolutionise economies. 

Politicians will always exploit education to score points and strike poses. But Sunak’s initiative has also exposed yet another culture-war division—this time the old rift between arts and sciences, so much more entrenched on the humanistic side. Our knee-jerk literary maths-haters still can’t see over their self-erected fences to relish the beauty, and wonder, on the other side. That’s their loss, and their shame. Creative joy has never been a zero-sum game.