If in other sciences, we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics. – Roger Bacon
10,000. They say that’s how many hours it takes to get really good at something. You and I may argue over the suspiciously round number, or the meaning of “really good”, or who exactly the “they” are that say this. But the main point is simple and incontrovertible: if you want to get really good at something, you’ve got to practice a lot.
With sporting pursuits, you need to start training very young, so that you’ve already gotten enough time on the clock when your body hits its peak. We know this, which is why pee-wee leagues are such a big deal: almost all world-class athletes either began in their sports as children, or spent their younger years practicing a directly transferable skill; already being a talented sprinter, for instance, helps a lot if you’re trying to become an impossibly cool, wise-cracking Olympic bobsledder.
In academic pursuits, on the other hand, how late you get started doesn’t seem to matter: Descartes didn’t begin writing philosophy until he was in his forties. But for mathematics (and its sisters, logic and physics), the situation is a little different.
Albert Einstein was only 26 years old when he had his Annus Mirabilis: the “miracle year” in which he published four papers that changed the world forever. Einstein got all his really great work done early, and spent the rest of his adult life coasting on the fame that that earned him: holding a cushy position at Princeton, working on pet projects that went nowhere, and posing for photos with his tongue out. And while Einstein’s hilarious gag shots may well set him apart from the otherwise reserved ranks of great physicists and mathematicians, the tender age at which he made it big certainly does not.
Tarski published his first important paper on set theory when he was just 19. Dirichlet did his work on Fermat’s Last Theorem at 20. Gödel completed both his incompleteness theorems by 25. Dedekind cut the numbers at 27, and Reimann invented his eponymous geometries at 28.
Why is it so? As with the case of sports, a physiological explanation seems most attractive: our brains, like our bodies, physically degrade with time. Throughout the life of a human brain, neurons die off and are not replaced, so their numbers steadily dwindle. Meanwhile, as we live and learn, the number of synaptic connections between neurons steadily increases. Some cognitive tasks, especially those concerning language skills, tend to come more easily to older minds with more neural connections. Maths seems to be in the other camp: the sheer calculating power demanded by the field seems to require one’s neurons to be fresh and plentiful.
Whatever the cause, maths is a young man’s game. Despite occupying opposite ends of the jock-nerd spectrum, athletics and mathematics have this in common: if you’re not going to come to the party early, you may as well not bother showing up at all.
But we don’t have little leagues for maths – at least, not in the same way we have little leagues for sports. Harvard and MIT aren’t scouting for brilliant five-year-olds. World class coaches aren’t being brought in to help them train exactly the right amount, day after day, year after year, so that they’ll hit their primes at 19.
It’s not merely an absence of infrastructure for the precocious elite that’s holding us back: standards of education are declining all round. It’s as if the entire system of formal education simply can’t be bothered teaching as zealously and effectively as once it did. But while the shortcomings in kids’ understanding of grammar and history can reasonably be patched up later in life, a lack of proper mathematical foundations laid young can easily deprive an otherwise promising child of his chances at ever getting a Nobel Prize.
Perhaps the most worrying implication in all of this is not for mathematics itself, but for what mathematics supports upon its shoulders. Physics, for starters, has always depended heavily upon good maths, and for the past century or so – since Minkowski & Co. rewrote the fundamental way we understand space and time – the nature of the dependency of physics upon maths has moved from coffee to heroin. And all the sciences – soft and hard – rely on these intimate bedfellows: chemistry, biology, engineering, medicine, statistics, economics, electronics – the whole set. When maths suffers, everything suffers. Today, it’s our ability to properly manage a global economy; tomorrow, it might be our ability to construct a ship that can get us to Alpha Centauri before the meteor hits.
So what is to be done? It is often said that thinkers ought to be lauded and esteemed in the way that athletes are. I say that thinkers ought to be trained in the way that athletes are. There exist now sports laboratories which put millions of dollars into trying to reduce a cyclist’s drag coefficient by a couple of percentage points – imagine what could be accomplished if we put that sort of effort, and that sort of erudition, into our systems of education. For with that greater education, we could create a greater civilisation – and that, after all, is what really counts.