Today is there any purely abstract math that has no link to use on real world problems?
There used to be, but reality caught up.
This is from memory, probably from Asimov, probably from the sixties or seventies, so assume I'm messing up:
Okay, I've done a little research, and determined that I am definitely messing it up. Nonetheless, I'll tell the story my way, the wrong way, because that's the way I know it. Then I'll post the inconvenient facts below.
So, Euclid made a list of axioms, and all geometry grew from those simple axioms.
But one of the axioms was troubling, because it seemed more like a theorem, something more complicated than an axiom, something that should be proven rather than assumed.
So, many people over the centuries (or whatever) tried to prove the fifth axiom from the other four. They always failed. It had to be assumed, because nobody could prove it, so it remained an axiom.
The fifth axiom, as I wrongly remember it, is this: A single straight line connects any two points. Exactly one straight line; neither more nor less. Five is right out.
So one day this guy gets a brilliant idea. He will assume that the fifth axiom is
wrong, and work until that produces an error, and thus he will have proved that the fifth axiom is true, and thus it will become a theorem (or whatever) rather than an axiom.
So, he assumes that, between any two points, there are
an infinity of different straight lines. Using this and the other four axioms, he develops an entire system of geometry. And it works! There are no contradictions!
So, briefly, the answer to your question was yes. There was an abstract math (if geometry counts as math) that had no bearing on reality.
But then Einstein, and suddenly the universe was expanding rather than flat, and it turned out that this new non-Euclidean geometry was useful for describing that weird expanding universe (the one we live in).
So then another guy did it again. This time he assumed that between any two points there are
no straight lines. And his system worked too. It was a whole nother system of geometry that contained no contradictions but that didn't describe anything at all.
But then, what if the universe (which, these days, people say isn't going to happen, but which back then seemed plausible) quits expanding and starts to fall back in on itself? Won't this third system of geometry describe that universe? Yes. So, once again, maybe this system does, in some degree, possibly apply to reality.
But, for a time, these two new systems were thought to be gloriously totally useless.
---
Okay, now the evidence that my memory fails, but this may not be the only way in which I butchered the story:
Euclid's Postulates
1. A straight
line segment can be drawn joining any two points.
2. Any straight
line segment can be extended indefinitely in a straight
line.
3. Given any straight
line segment, a
circle can be drawn having the segment as
radius and one endpoint as center.
4. All
right angles are
congruent.
5. If two lines are drawn which
intersect a third in such a way that the sum of the inner angles on one side is less than two
right angles, then the two lines inevitably must
intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the
parallel postulate.
Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("
absolute geometry") for the first 28 propositions of the
Elements, but was forced to invoke the
parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "
non-Euclidean geometries" could be created in which the parallel postulate
did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)
--
http://mathworld.wolfram.com/EuclidsPostulates.html
It looks like a confused #1 and #5. And I can't tell the story about #5 because I don't understand that at all.
