It's hard to tell exactly who has "natural" math ability. John Mighton has stories of tutoring severely math phobic people who later went on to get a PhD in math. To some extent, that seems to have been his own story as well. There are also stories of people failing calculus and then bouncing back to become math professors later on. I think most mathematicians would admit that they weren't great at math when they started (maybe it comes easily to some, but there were probably still gaping holes in their grasp of how to learn math at first).
In my own case, although I did show some signs of having some natural talent for math, my performance before becoming serious about it was pretty uneven, partly due to not caring about it or trying very hard, but partly because I didn't have a very good idea of how to learn it effectively.
So, there is a lot of evidence for the existence of people who are good at math, but their abilities are blocked. This doesn't mean that everyone is like that. Perhaps, some people are just not going to be good at it. But I don't think we can really tell that. So, I always remain agnostic about people's abilities. They could eventually learn to be great at math, or maybe they wouldn't. You just never know. As a math tutor, it helps me to imagine that maybe my students are secretly great at math. Even in cases where this might seem unrealistic, it is a helpful way to think, and there's always at least some truth to it. John Mighton has had success bringing entire classes of ordinary students up to the level of the top students, at least as far as elementary math is concerned.
When you get to very high level math in theoretical physics, does there come a point where you NEED to have a natural math ability to succeed in the field, or can anyone learn how to do math like this?
Well, the first point to make is that there are wildly differing levels of math that are used by different physicists. Some physicists are basically high-level mathematicians, and others are not very mathematical at all. Of course, there is a certain minimum level you have to have. You have to be able to handle calculus, differential equations, and linear algebra.
Here's a famous quote I made up: "Anything makes sense eventually, if you think about it long enough."
So, it's my conjecture that most people can understand anything in principle, given enough time to think about it. But maybe one person can learn it in a year, and for someone else, it could take 20 years. That's my theory, anyway.
Some say artistic ability is something you're born with and is something you can't necessarily learn. Is this true for math?
That's complete hogwash, and I'm surprised the people who say that still haven't heard of books like Drawing on the Right Side of the Brain that completely refuted that idea a very long time ago. Granted, that book is an extremely basic one. But I think the reason that people think it's hard to learn to draw is that the way people teach it is terrible. If more people were willing to work hard on REAL drawing books like The Natural Way to Draw, I don't think drawing would be considered such a magical thing that only really talented people are capable of doing quite so much. But how many people actually know about The Natural Way to Draw? Not that many. I suspect part of the reason for its lack of popularity compared to how good it is, is that the book asks you to work very hard to learn to draw and very few people are really willing to do that. They just want to take the magic pill and be able to draw. That's sort of what Drawing on the Right Side of the Brain does, which can take you from stick figures to realistic portraits pretty quickly, but it's really quite superficial and will not make you a real artist--it just teaches you how to copy what you actually see with some accuracy and removes some of the barriers that normally prevent that from happening for a lot of people.
Similar things could be said for math teaching/learning.
I'm an abstract reasoner and visual spatial, but math was never my top skill...
I think you might find this interesting (preface to visual complex analysis):
http://usf.usfca.edu/vca//vca-preface.html
Although, I agree with him, and I'm an extreme visual thinker, maybe he's over-stating it a bit, so take it with a grain of salt. It's possible to go a little too far with visualization if you try to visualize absolutely everything. Plus, you have to be careful with pictures, just as you have to be careful with any other kind of reasoning.
