In general, no.
> It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant.
Perhaps to you, but not to most mathematicians, certainly not those working in academical settings. Have you ever looked at group theory or topology?
I don't think you've ever done the kind of math that mathematicians do. What you learn in school is not math, it's calculating. What mathematicians do is to invent constructs that have no basis in reality and prove statements about them, then come up with more constructs based on those statements, ad infinitum.
Sometimes those constructs may be designed to model real world problems, and getting funding is probably easier in those areas, but just as often the applicability is only discovered afterwards - or not at all.
The best example (because it's something we've actually all learned about) is complex numbers. They were first invented in teh 16th century and considered pointless and irrelevant at first. People soon discovered that they could be useful in proofs about non-complex numbers as well, but it took several centuries before they were found to be directly applicable in electrical engineering (many more applications have been discovered since).
Regardless of how abstract you get, how far you go, math is always tied to our physical reality. The basic operations are reflections on properties of our universe. The "kind of stuff" mathematicians do" allows us to model and reason about our world in ways that wouldn't be possible any other way (that we know of).
I'm sorry but even if this is objectively valid, this not a claim that you can support. There are entire sub-fields of maths in which there are no known physical attachments.
Now, that's not to say that some time in the future we won't discover the relationship between every mathematical concept and some physical system. But at this moment, the claim you are making has a numerable set of counter examples, with a very large order. For one example, take the Banach–Tarski paradox:
>Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball.[1]
This is, as far as we know, physically nonsense.
That being said, there is quite a lively debate among mathematicians as to whether or not math is tied to reality.
[1] http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
No, this is most definitely not true for all branches of mathematics.
To many mathematicians reality is largely irrelevant. For example G. H. Hardy, an extremely prominent British Mathematician, believed that true Mathematics is an art form and is not useful. He dismissed applications of mathematics as dull and boring.
And I can relate to him. It's incredible how complex, beautiful structures arise from a couple of simple axioms. It doesn't matter if what you study will be relevant or not, what matters is that it's fun and stimulating to explore.
A lot of my friends feel the same way, with some of them specifically avoiding having "real world" applications of their work, as if that makes it an even better sand castle.
As to why I have an applied degree instead of doing pure math, numerical analysis makes a weird intuitive sense to me, and I figured building decent sand castles on the beach was better than making terrible sand castles in the sky that could barely hold themselves up. It also gets the grant money.
Just to clarify: I am an expert too.
When it all works out as beautifully as it does, in say Euler's Identity, it's hard to remember the possibility that the axioms could turn out false, or logic as we know it flawed.
But assuming (heh) that the axioms are true and that our understanding of logic is valid, "pure" math is as much apart of reality as "applied" math. And I'll choose proving the Fundamental theorem of Galois theory over number crunching in Matlab as my exercise in experiencing reality every time.
Not to speak of Calculus...