I have some time this summer to spend learning maths, and I was going to begin studying real analysis with Rudin's "Principles of Mathematical Analysis". I have heard it is the best book if you have the time to study thoroughly, which I do (three almost uninterrupted months, although I plan to learn lots of other maths too). As someone who is mathematically able, but has not done Real Analysis (and will not study it at university) what is your recommendation that I read?

Added: My info comes from the incredivly positive amazon reviews, and the less positive Best Textbooks LW thread.

Could you say more about why you think real analysis specifically is good for this kind of general skill? I have pretty serious doubts that analysis is the right way to go, and I'd (wildly) guess that there would be significant benefits from teaching/learning discrete mathematics in place of calculus. Combinatorics, probability, algorithms; even logic, topology, and algebra.

To my mind all of these things are better suited for learning the power of proof and the mathematical way of analyzing problems. I'm not totally sure why, but I think a big part of it is that analysis has a pretty complicated technical foundation that already implicitly uses topology and/or logic (to define limits and stuff), even though you can sort of squint and usually kind of get away with using your intuitive notion of the continuum. With, say, combinatorics or algorithms, everything is very close to intuitive concepts like finite collections of physical objects; I think this makes it all the more educational when a surprising result is proven, because there is less room for a beginner to wonder whether the result is an artifact of the funny formalish stuff.

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 sense that you do not know much modern mathematics

... from what do you get this impression, and in what way is it relevant? Yes, there are many parts of modern mathematics I am not familiar with. However, nothing that had come up up to this point was defined in the last 100 years, let alone the last 50.

I have a PhD in physics. I know what an affine space is. If you were thrown off by my uses of basis changes to effect translations, which would signal ignorance since vector addition is not equivalent to change of basis... I did clarify that I was in a function space defined over time, and in the case of function spaces defined over vector fields, translations of the argument of the function are indeed changes of basis.

In physics, we set the origin to be whatever. All the time. This is because we need to do actual arithmetic with actual numbers, and number systems with no 0 are needlessly cumbersome to use. Moving 0 around all the time in context-dependent and arbitrary ways completely defuses the 'harm' of A-theory, as far as I can tell.

I do know how to characterize the affine line as a topological space without reference to the real numbers.

This is what I was referring to. The axioms of ordered geometry, especially Dedekind's axiom, give you the topology of the affine line without a distinguished 0, without distinguishing a direction as "positive", and without the additive structure.

However, in all the ways I know of to construct a structure satisfying these axioms, you first have to construct the rationals as an ordered field, and the result of course is just the reals, so I don't know of a constructive way to get at the affine line without constructing the reals with all of their additional field structure.

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).

I think that this analogy is accurate and reveals that A-theorists are attributing additional structure to time, and therefore that they take a hit from Occam's razor.

However, to be fair, I think that an A-theorist would dispute your analogy. They would deny that time "is" splayed out on a number line, because there is no standpoint from which all of time is anything. Parts of time were one way, and other parts of time will be other ways, but the only part of time that is anything is the present moment.

(I'm again using A-theorist as code from presentist.)

By the way, off-topic, but:

The only defined operation is the taking of differences, and the notion of affine line relies on a previously defined notion of real line.

This is true if affine space is defined as a torsor for the reals as an additive group, but you can also axiomatize the affine line without reference to the reals. It's not clear to me whether this means that you can construct the affine line in some reasonable sense without reference to the reals. Do you know?

In history-of-the-universe space, time-translations are just changes of basis. The difference between A and B is what time you assign to be '0'.

When you're thinking about yourself, it's appropriate to privilege facts pertaining to yourself. Like, if I'm on a roller-coaster, I will do most of my thinking about accelerations in my personal reference frame. This is a stupid reference frame to use for anything else, even for thinking about the person sitting next to me.

I guess the issue is, it's easier to think of a block universe in B-theory-mode?

I agree I was too brief there. The original motivation for math was to help figure out the physical world. At some point (multiple points, really, starting with Euclid), perfecting the tools for their own sake became just as much of a motivation. This is not a judgement but an observation. Yes, sometimes "pure math" yields unexpected benefits, but this is more of a coincidence then the reason people do it (despite what the grant applications might say).

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 heuristics. Is that not what pure math is?

The main reason is pure curiosity without any care for eventual applicability to understanding the physical world. Pretending otherwise would be disingenuous.

Theoretical physics is not very different in that regard. To quote Feynman

Physics is like sex: sure, it may give some practical results, but that's not why we do it.

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?