Scott Aaronson in Aeon:
In one sense, there’s less mystery in mathematics than there is in any other human endeavour. In math, we can really understand things, in a deeper way than we ever understand anything else. (When I was younger, I used to reassure myself during suspense movies by silently reciting the proof of some theorem: here, at least, was a certainty that the movie couldn’t touch.) So how is it that many people, notably including mathematicians, feel that there’s something ‘mysterious’ about this least mysterious of subjects? What do they mean?
There are certainly mysteries that exist within math. For starters, there are the thousands of unsolved problems, assertions that no one can prove or disprove, sometimes despite decades or centuries of effort. Although many of these problems are deep and important, a small example will do for now: no one has proved that, as you go further out in the decimal digits of π=3.141592653589…, the digits 0 through 9 occur with equal frequency.
Yet, for reasons that apply to many other unsolved mathematical problems, it’s debatable whether to call this a ‘mystery’. What would really be mysterious, one wants to say, would be if the digits didn’t occur with equal frequency! The whole challenge is to give airtight proof that what does happen is what anyone with common sense, after thinking the matter over for a bit, would conclude almost certainly must happen. As Jordan Ellenberg, a mathematician at the University of Wisconsin, wrote, a dirty secret in mathematics is that many unsolved problems have a similar flavour: they’re less about mysterious coincidences than about the lack of them.
Take, for example, the Twin Primes Conjecture, which holds that there are infinitely many pairs of prime numbers separated by 2 (such as 3 and 5, or 11 and 13). Ellenberg explains that, for this conjecture to be true, there doesn’t need to be any mysterious ‘force’ pulling primes together; there just needs not to be a mysterious force pushing them apart. Or take the Riemann Hypothesis, which states that the infinitely many non-trivial roots of a certain complex function all lie on a single line. When it’s stated that way, the hypothesis does sound like a mystery. Why should infinitely many numbers all happen to line up like that?