There’s a famous paradox which is classically called the “uninteresting number paradox”. To make it more topical in today’s world of hipst, I’ll call it the “obscure number paradox” instead. It goes like this:
- Theorem: There is no obscure counting number.
- Proof Assume there is an obscure counting number. Then there is a smallest one. But the fact that it’s the smallest obscure number, is pretty non-obscure! Contradiction.
One way to try to find obscure numbers is to take big databases of numbers and find the smallest missing number. For example, Nathaniel Johnston found that 11630 was the smallest number missing from the Online Encyclopedia of Integer Sequences. However, 11630 later showed up (it happens to be a “5 times centered pentagonal number”) and Nathaniel’s Number was revised to 12067. That, too, soon made an appearance (something to do with generating sequences), and Johnston’s Constant swelled up to 12407. And now 12407 has appeared in the encyclopedia (it is the numerator of Hermite(3,19/26), whatever that means). Johnston apparently gave up, and hasn’t updated his blog post since.
I thought to myself: what’s an even bigger database of numbers than the OEIS? The answer screams itself out: Google. Unfortunately, unlike the OEIS, you can’t just download an exhaustive list of all the strings Google has indexed. The only way to find the smallest obscure number according to Google would be to enter all the numbers in until you finally got a results page with no results. Unfortunately, such a brute force attack would take forever (and you’d probably be banned from Google before you reached your conclusion).
Still, you can use Google to check whether a given number is obscure or not. For example, I randomly chose the number 553634475998 by mashing my keyboard, and sure enough, Google found no matches, confirming the number’s obscurity– at least, until I publish this blog post, at which point its obscurity will vanish!
Still, the Paradox of the Noninteresting Number relies somehow on an omniscient overseer deciding what’s interesting, what’s obscure: in the paradoxical argument, the fact that N is the smallest obscure number, is supposed to somehow make it non-obscure, and indeed, it WOULD, if we could actually detect what this number N is.
There really is (say) a smallest number N such that no human will ever write down N– assuming the human race eventually dies out in a finite amount of time. No human can predict what this number is, though, without breaking it. It is a designator which doesn’t concretely designate anything, at least not to us: it could designate something concrete to a more powerful being who outlives us all.
FURTHER READING
Philosoraptor Adventures
The Morning Star Paradox
Newcomb’s Paradox