How long would it take to <i>study</i> the books recommended here? Answer: years. Does it ever make sense to condense a comprehensive summary of mathematics into five page document? Perhaps, perhaps not.<p>I do know one thing: this was published in 2005, three years before the release of the <i>Princeton Companion to Mathematics</i>. 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.)<p>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 <i>and</i> 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.<p>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.<p>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.<p>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.)<p>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 <i>Calculus</i>, 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.<p>[1] <a href="http://mathoverflow.net/questions/28721/good-differential-equations-text-for-undergraduates-who-want-to-become-pure-math" rel="nofollow">http://mathoverflow.net/questions/28721/good-differential-eq...</a>