Forgive the perhaps poor phrasing. This example is lifted from Scott Aaronson's Why Philosophers Should Care About Complexity Theory (pg. 9-10) and it poses an interesting question. Consider the two statements:
- 2^(43112609)-1 is prime.
We say this value is a "known" prime because we can prove it selects a unique positive integer and it's prime - but this is also true of:
- p' is the first prime larger than 2^(43112609)-1.
Most people's intuition suggests that 2^(43112609)-1 is a "known" prime whereas p' is "not known" - despite the fact we know p' exists and is prime (and we could, in principle, compute it).
(Aaaronson gives computational complexity reasons for the distinction, but that bypasses the issue since I suspect most of us would still have the known/unknown distinction even if p' could be computed in polynomial steps).
My question is, is the distinction just social, or is there a meaningful difference between the two expressions? Or is there an obvious distinction I'm missing? If it's not obvious, is there any writings on related problems or relevant writings on this question available?