I am grateful for the example of Seamus Heaney, both man and poet, fully engaged in his time and his art with its visions of timelessness. He was never distracted by trends and trivialities. If the world seemed, implausibly, a better place when he lived, at least we now have his living work, its humane erudition, vitality and growth. Heaney said that Robert Frost inhabited the world at body heat. He could have been talking of himself. The biographer R. F. Foster refers to Heaney’s “overpowering but benign authority,” and anyone who knew him remembers an affable presence. His was an earthly civility. A farmer’s son, he led the life of a privileged intellectual and writer, “always politic / And shy of condescension” (as he put it in a great poem, “Casualty”). He worked hard, and the result of the labor was not just his productivity, but also his ability to change.
‘As long as there has been such a subject as philosophy, there have been people who hated and despised it,’ reads the opening line of Bernard Williams’s article ‘On Hating and Despising Philosophy’ (1996). Almost 30 years later, philosophy is not hated so much as it is viewed with a mixture of uncertainty and indifference. As Kieran Setiya recently put it in the London Review of Books, academic philosophy in particular is ‘in a state of some confusion’. There are many reasons for philosophy’s stagnation, though the dual influences of specialisation and commercialisation, in particular, have turned philosophy into something that scarcely resembles the discipline as it was practised by the likes of Aristotle, Spinoza or Nietzsche.
Number theory is a branch of mathematics that deals with whole numbers (as opposed to, say, shapes or continuous quantities). The prime numbers — those divisible only by 1 and themselves — are at its core, much as DNA is core to biology. Quadratic reciprocity has changed mathematicians’ conception of how much it’s possible to prove about them. If you think of prime numbers as a mountain range, reciprocity is like a narrow path that lets mathematicians climb to previously unreachable peaks and, from those peaks, see truths that had been hidden.
Although it’s an old theorem, it continues to have new applications. This summer, Rickards and his colleague Katherine Stange, together with two students, disproved a widely accepted conjecture about how small circles can be packed inside a bigger one. The result shocked mathematicians. Peter Sarnak, a number theorist at the Institute for Advanced Study and Princeton University, spoke with Stange at a conference soon after her team posted their paper. “She told me she has a counterexample,” Sarnak recalled. “I immediately asked her, ‘Are you using reciprocity somewhere?’ And that was indeed what she was using.’”
Tokelau, a necklace of three isolated atolls strung out across the Pacific, is so remote that it was the last place on Earth to be connected to the telephone—only in 1997.
Just three years later, the islands received a fax with an unlikely business proposal that would change everything.
It dawned on me that running had begun to change not only my body and mind, but the way I travelled. As well as choosing hotels close to green spaces, had become mindful of things like air quality and stray dogs. In Tbilisi, during an intense heatwave, I used a hotel gym treadmill, up on the 18th-floor, which had air-conditioning and wrap-around floor-to-ceiling windows for city views. With my fitness so hard won, I found myself scared to lose it, even if it meant paying to jog on the spot.