>As of 2011, the "largest known prime number,"
as reported by GIMPS (the Great Internet Mersenne Prime Search),15 is p := 243112609 − 1. But
on reflection, what do we mean by saying that p is "known"?
what a rubbish. The "p" isn't "known". The "p" is "known to be prime". Professional philosopher must be able to feel the difference.
Not at all. This is ultimately an epistemic issue - what do we mean by "known"? What property does the largest known prime have that the next largest prime lacks? How can we articulate the difference? These are the types of questions philosophy asks.
Most philosophers know the difference between "known" and "known to have property P".
If Clinton is a well-known ex-President, and he is a peanut farmer, he might be called a "well-known ex-President" in the "known" sense, but not necessarily in the "known to have property P".
"known N" in the second sense is thus non-intersectional and must be analyzed as "P(x) & known(P(x))" rather than "P(x) & known(x)". Non-intersectionality is not as weird as you may think, as superlatives such as "largest" are also non-intersectional in the sense that if someone has the smallest green t-shirt, it's the case that they have a green t-shirt, but it's not (necessarily) the case that they have the smallest t-shirt, (e.g. when the smallest t-shirt is actually red).
If you want to put it back in language, put it like this:
As of 2011, the "largest number known to be prime", as reported by GIMPS (the Great Internet Mersenne Prime Search),15 is p := 243112609 − 1.
So, the "largest known prime number" is a way to describe our knowledge about numbers, rather than a property that you can attribute to numbers out of context.
>This is ultimately an epistemic issue - what do we mean by "known"? What property does the largest known prime have that the next largest prime lacks?
"known" is by "whom". Being "known" isn't an intrinsic property that one prime has and another lacks. "Knowing" some specific number as a prime is a property of the one who "knows". If some person (or alien race) knows p as the largest known [to them] prime and another person knows q as the largest known [to him] prime it has absolutely no bearing on the intrinsic properties of either p or q.
>These are the types of questions philosophy asks.
if it really so nowadays, then it would explain why there is no philosophers anymore
p is "known" by the mathematical community. As of 2011, p is the largest natural number such that (a) we know p is prime and (b) we know a polynomial-time algorithm to enumerate the digits in the decimal expansion of p.
I am often under the impression that many philosophers take linguistic constructions such as "meaning" and "morality" to be entities that can exist on their own, outside functioning brains, while on the other hand they claim to be materialists. They strive to find a way to make logic (syntax) produce meaning, forgetting that semantic meaning is all about context, emotion and behavior shaped by millions of years of evolution. Philosophers do care about complexity (they care about everything, it's their job), but their metaphors are always about metaphysics or notions of some kind of "x-liness" that, like an "elan vital" will give meaning to the soulless constructions of mathematics and physics (chinese room, waterfalls, qualia).
The paper gives numerous examples where a philosopher could use complexity theory, but doesnt go so far as to make any conclusions, thus showing that comp. complexity is actually useful for these problems.
