The Mathematics Autodidact’s Aid (2005) [pdf]

I do know one thing: this was published in 2005, three years before the release of the Princeton Companion to Mathematics. I simply can't overstate how helpful owning a copy of the PCtM will be to any budding mathematican. Princeton University Press made a beautiful book which is worth its price several times over. (Don't think you can get by with Wikipedia! At over 1000 pages, the PCtM is stunning in its clarity and breadth of coverage.)

As for the paper, there are still a lot of gems here, which are worth considering, assuming you really are serious about teaching yourself mathematics and have purchased the PCtM. I studied mathematics in university but have always been an autodidact and something of a bibliophile, and I can say that most of the recommendations made here are the one's I would make as well. If nothing else, this list should save you a lot of time on Amazon and in the library chasing recommendations and references.

However, I do think it would be a strange thing indeed to hand this list over to somebody expecting to go out and buy a subset of it, and expect to be on his or her way to becoming a mathematician.

One of the reasons I recommended the PCtM instead is that there are clearly some missing perspectives that inevitably resulted from compressing all of mathematics into a short five page summary. The librarian who compiled this list did a fine job overall, but I think this list really bites off more than it can chew, in the sense that it would be impossible to convey all the different and conflicting perspectives that would need to inform a comprehensive summary of mathematics.

I would take each section of her paper as a starting point that ought to be supplemented by additional sources (or better yet, supplemented by reading the relevant section in the PCtM). As it stands, the list is overly academic and not sufficiently pedagogical, and too quickly jumps into advanced territory to be too useful to undergraduates. In particular, logic, geometry, representation theory, and physics are all incredibly important topics that can invigorate the subject, but the paper does not give them the attention they deserve. The author does mention Arnold's ODE book as an alternative that emphasizes "geometric ideas", which quite frankly is short shrift. (Look at the Mathoverflow [1] thread which discusses the topic of choosing an undergraduate text on differential equations, and you will see Arnold's ODE book mentioned several times over.)

I also feel the need to point out a complete absence of anything written by Michael Spivak, which is a crying shame. I would have expected to see at least his beautiful Calculus, to say nothing of his encyclopedic and highly pedagogical works on differential geometry. Also notably missing are books written by John Hubbard and Charles Pugh.

[1] http://mathoverflow.net/questions/28721/good-differential-eq...

Try different media. I've found, at least for me, sometimes just reading it won't click. I pull up some ocw or a random video on youtube and presto. It is important to remember that when it clicks that what you've been reading makes sense and to reread. Sometimes I'll go through like 5/6 videos, it is just finding the right one.

Try problems. Struggling is the key to success in mathematics. There is no doubt in my mind that this is true.

If you REALLY have problems look up a professor at a local university or community college. Find out when their office hours are and send a polite email asking that if they have time you'd appreciate the help. This will be hit or miss, but many are glad to help (no one does academics for the money). Many will even let you sit in on their classes (less will grade your work).