As a current Harvard math grad student I think you should read many different easy books to learn a subject whenever possible, especially if you can find them for free. When you say you are mathematically able it is unclear what level you are at. All of my favorite books for learning involve huge number of exercises, and I recommend you do all of them instead of reading ahead.
For basic real analysis, my favorite book is
Rosenlicht's Introduction to Analysis
but baby Rudin is pretty good too, and I recommend you flip back and forth between them both.
For learning math in general, I think real analysis is a poor place to start, but that may be personal preference because I have a more algebraic slant. I highly recommend books like
Herstein's Abstract Algebra,
Mathematical Circles: A Russian Experience,
I.M. Gelfand's Trigonometry,
and Robert Ash's Abstract Algebra: The Basic Graduate Year,
mostly for the wealth of exercises. Some of these are books for small children and I think those are the best sort of books to first learn from.
Personally I think real analysis is an awkward way to learn mathematical proofs, and I agree discrete mathematics or elementary number theory is much better. I recommend picking up an Olympiad book for younger kids, like "Mathematical Circles, A Russian Experience."
I'm sure not only "elite" mathematicians intuit the interest of problems like the unsolvability of the quintic. That one can prove a construction impossible, the very concept of an invariant, is startling to the uninitiated. So many classic problems of this nature are held up as paradigms of beauty--the Konigsberg bridge problem, ruler and compass constructions of cube roots, the irrationality of sqrt(2),..
I'm doing my math PhD at Harvard in the same area as Qiaochu. I was also heavily involved in artofproblemsolving and went to MathPath in 2003. I hoped since 2003 that I could stake a manifest destiny in mathematics research.
Qiaochu and I performed similarly in Olympiad competitions, had similar performances in the same undergraduate program, and were both attracted to this website. However, I get the sense that he is driven quite a bit by geometry, or is at least not actively adverse to it. Despite being a homotopy theorist, I find geometry awkward and unmotivated. I cannot form the "vivid" or "bright" images in my mind described in some other article on this website. Qiaochu is also far more social and active in online communities, such as this one and mathoverflow. I wonder about the impact of these differences on our grad school experiences.
Lately I've been feeling particularly incompetent mathematically, to the point that I question how much of a future I have in the subject. Therefore I quite often wonder what mathematical ability is all about, and I look forward to hearing if your perspective gels with my own.
I think it's very important in understanding your first Grothendieck quote to remember that Grothendieck was thrown into Cartan's seminar without requisite training. He was discouraged enough to leave for another institution.
I apologize for the snipy remark, which was a product of my general frustrations with life at the moment.
I was trying to strongly stress the difference between
(1) an abstract R-torsor (B-theory), and
(2) R viewed as an R-torsor (your patch on A-theory).
Any R-torsor is isomorphic to R viewed as an R-torsor, but that isomorphism is not unique. My understanding is that physicists view such distinctions as useless pendantry, but mathematicians are for better or worse trained to respect them. I do not view an abstract R-torsor as having a basis that can be changed.
You might be able to do it with some abstract nonsense. I think general machinery will prove that in categories such as that defined in the top answer of
there are terminal objects. I don't have time to really think it through though.
I have always heard the affine line defined as an R-torsor, and never seen an alternative characterization. I don't know the alternative axiomatization you are referring to. I would be interested to hear it and see if it does not secretly rely on a very similar and simpler axiomatization of (R,+) itself.
I do know how to characterize the affine line as a topological space without reference to the real numbers.
Torsors seem interesting from the point of view of Occam's razor because they have less structure but take more words to define.
I think that the distinction may be clarified by the mathematical notion of an affine line. I sense that you do not know much modern mathematics, but let me try to clarify the difference between affine and linear space.
The A-theorists are thinking in terms of a linear space, that is an oriented vector space. To them time is splayed out on a real number line, which has an origin (the present) and an orientation (a preferred future direction).
The B-theorists are thinking in terms of an affine line. An affine line is somewhat like the A-theoriests real line, but it doesn't have an origin. Instead, given two points a & b on the affine line, one can take their difference a-b and obtain a point on the real line. The only defined operation is the taking of differences, and the notion of affine line relies on a previously defined notion of real line.
I guess I always took the phrase "unreasonable effectiveness" to refer to the "coincidence" you mention in your reply. I'm not really sure you've gone far toward explaining this coincidence in your article. Just what is it that you think mathematicians have "pure curiousity" about? What does it mean to "perfect a tool for its own sake" and why do those perfections sometimes wind up having practical further use. As a pure mathematician, I never think about applying a tool to the real world, but I do think I'm working towards a very compressed understanding of tool making.
So what does "gone wild" mean? Your paragraph about this is not very charitable to the pure mathematician.
Say that mathematics is about generating compressed models of the world. How do we generate these models? Surely we will want to study (compress) our most powerful compression heurestics. Is that not what pure math is?