If you want to build some piece of electronics, it helps if you know the value of pi to a certain precision. But does it really matter for someone making a guitar amp what is the 100th digit of pi?
How many times have you done a Google search and went beyond the first page to check if you find what you were looking for?
If Michael Phelps were to live forever, how many Olympic Games and Gold medals would it take for him to say "you know what? I am tired of spending 30 hours per week on a swimming pool. Gimme a hit of that bong."?
If historians had access to the recorded history of 1% of the population, we certainly would have a better understanding of the past. 10%, even more. But would this understanding change in any significant way if we had access to 50%, 80%, 100%?
Does it? At the very least, the relationships between prime numbers "seems likely" to be infinite, and that's only taking into account patterns in a countably infinite set, which likely have measure 0 as a subset of patterns in any uncountably infinite set (and there is an infinite progression of such sets uncountable sets such that each is of measure 0 compared to another).
This attitude seems very "all science has been discovered and it's only a matter of precision" to me.
It would certainly be a feat to formalize any of this, and we'd still be debating definitions and assumptions if we could formally prove anything.
> Learning the next thousand thousand prime numbers is not very interesting if you already know the first few million, and of very limited practical use.
Maybe your intuition here is right but my hunch is that if you were able to prove this conjecture, it would absolutely blow the socks off the mathematics community.
I look at it as a Pascal's Wager w/ regards to knowledge: either there's more to know that we don't know we don't know, or there isn't. If there is, we'll likely only find it out if we believe there is more knowledge to uncover and spend our time trying to uncover it.
My overall hunch here is that a lot of our current knowledge is constrained by our mental semantic structure, which is bounded by the working set of objects we can mentally manipulate at an instant in time. That set is probably O(10) for most of the population, and likely O(100) for everybody. We can increase the number of semantic objects we can operate on by working on them temporally using logical rules, but those rules themselves are bound by this O(100) limit, and so on for any meta-rules.
I would find it very surprising if all interesting things that could be stated were relationships between O(100) semantic objects, however abstract.