Chaitin's nightmare
Chaitin's nightmare consists of him going into a library of the
future, and opening up a math book and seeing just words, words, words.
I -think- that what's going on is:
Ultimately, (formal) knowledge stops having structure:
it's just a list of facts or such.
My interpretation:
Also, eventually you get -uncompressable knowledge-
[Once you learn it, all you can do is simply -repeat- it to another!]
(yes, this is obviously -my spin-)
That is: normally you read a big book to get a simple idea:
(it's big because of exposition)
you eventually have a simple, small, hard idea in your head.
...but imagine if future math is like the classification of finite
simple groups, and incompressibly so:
you can't even abstract papers -- they are -just- huge.
Indeed, it's sorta already like that:
we still have short -statements-, but the proofs get huge:
(and the time/effort to -understand- the proofs is -also- huge)
hence we rely on experts.
For instance, in analysis, apparently people rarely write papers:
they write monographs hundreds of pages in length.
This is what it means for a field to be exhausted.
[Non-classification results have this form.]
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The end of math?
The hope (mine) is that we can reach "closure" before chaos takes over:
that all the naive/basic enough questions have short enough answers
and proofs.
[This isn't true: you can't classify 4-manifolds given a cell
structure, for instance, or a f.g. group given a presentation.
So replace "all" by "most".]
IE, we know that (relatively) short questions have (arbitrarily) long
answers ("answer length" grows non-computably in "question length"),
but the hope is that questions become uninteresting before they
become insoluable.
Clearly we're discovering new fields, with new and simple and
beautiful patterns, and there are interesting questions and answers
left to find,
but this gives us a view to how math can end,
with a whimper: slowly petering out.
(and as noted above, *we've already seen this in many subfields*)
That said, it can also end with a bang, with a bow on top:
a grand theory that explains, not all of math (as there are too many
details to be subsumed by any simple theory), but all core ones.
Some form of algebraic geometry seems the best bet
(in a quip: "non-commutative infinite dimensional algebraic geometry")
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Does abstraction help?
IE, we can have short statements by using good, meaningful terms,
and we can have short proofs by using lemma / existing results.
Ultimately, no -- you still need to learn the terms, learn the lemmata;
the results are *fundamentally* incompressable.
You see this in the fact that to even understand some statements
requires 10+ years of study.
You can take things for granted,
but then you'll only ever understand a theory of a theory:
if you hierarchicalize knowledge,
you'll only understand "local" results (oh cool!),
which are the technical, field-specific ones,
*not* the deep, far-reaching ones.