7 ms·

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

216 points by kercker 10y ago

rrherr 10y ago
I've also found these to be helpful: The Language and Grammar of Mathematics, from The Princeton Companion to Mathematics, by James Gowers: http://press.princeton.edu/chapters/gowers/gowers_I_2.pdf http://press.princeton.edu/chapters/gowers/gowers_I_2.pdf Reading Mathematics, by John Hamal Hubbard: http://www.math.cornell.edu/~hubbard/readingmath.pdf http://www.math.cornell.edu/~hubbard/readingmath.pdf
- jimsojim 10y ago
  Great read, thanks for the links
  - vecter 10y ago
    Appreciate the good vibes, but no need for these kinds of comments. An upvote is enough.
- lunchladydoris 10y ago
  I don't mean to be that person, but it's Timothy Gowers, not James.
  - rrherr 10y ago
    My bad, thank you for the correction!
deleted 10y ago
[deleted]
MichaelMoser123 10y ago
maybe i am just stupid, but I find that many texts in mathematics like to skip over details, or I get stuck because there is an ambiguity in the text and there is no one to ask about it.. What do we do in this kind of situation?
- wolfkill 10y ago
  Oh the dreaded "details left as an exercise." Personally, if I can't fill in details readily myself, I use Google or "just take there word for it" and hope it doesn't bite me later.
- medymed 10y ago
  One approach is to look at a few different but overlapping primers on the same matter and do the exercises. If one treatment doesn't click, another might. And for well-known subjects, topics are often covered in roughly the same order. Maybe not for graduate classes, not sure, but linear algebra, set theory, group theory, analysis it's generally the case. Unless you can afford a tutor to teach it several different ways until it clicks, which is functionally similar.
  - jwdunne 10y ago
    I can verify this approach works for me. The other thing I suggest is use "getting stuck" as a way to identify gaps - go look for books that fill those gaps. Recently I cracked open Principle of Analysis by Rudin. I got stuck pretty much straight away and looked it up - I saw a few suggestions for a bridge between high school maths and the rigor required by that book. I'm now reading another book that took inspiration from the previously mentioned book but gives a bit shallower a slope towards the rigor required.
    - mrkgnao 10y ago
      Pugh, by any chance?
      - jwdunne 10y ago
        Krantz. Seems to be at a decent pace. No idea if it's any good though - problem of not having a mentor!
        gtani 10y ago
        That book got good reviews. /r/math has lots of threads on this, recommend Abbott, Tao, Pugh, Spivak etc https://www.reddit.com/r/math/comments/50m4ib/what_are_the_main_differences_between_rudin_and/ https://www.reddit.com/r/math/comments/50m4ib/what_are_the_m... https://www.reddit.com/r/math/comments/4jopme/is_it_premature_for_me_to_start_reading_baby_rudin/ https://www.reddit.com/r/math/comments/4jopme/is_it_prematur... https://www.reddit.com/r/math/comments/512p2p/analysis_text_for_selfstudy/ https://www.reddit.com/r/math/comments/512p2p/analysis_text_...
wolfkill 10y ago
In the area of numerical analysis, I'd recommend works by LeVeque and Trefethen.
- thearn4 10y ago
  Trefethen & Bau's "Numerical Linear Algebra" is a pretty good primer, though the last few chapters I remember being a bit lacking (it's been about 6 years though).
  - wolfkill 10y ago
    This book helped me to visualize Krylov space methods better than any other resource I have seen. It got me through my numerical analysis qualifying exam. I do agree that the last chapters are lacking in depth. I also recommend his book "Pseudo-spectral Methods in Matlab" for a thorough and very visual look at things like Gibb's phenomenon and spectral convergence.
lunchladydoris 10y ago
I enjoy a mathematics textbook as much as the next person, but what annoys me is the lack of solutions to the problems in so many of the books I've skimmed outside of classes.
- cle 10y ago
  I prefer solutions to some, but not all problems. Being able to independently and confidently verify your own solutions is an important skill to develop.
  - rguzman 10y ago
    having the solutions does not stop you from developing that skill. however, the lack of solutions makes it difficult to make progress when you have time constraints, which is the common case.
  - gms7777 10y ago
    This is absolutely a crucial skill. One of the most valuable exercises I had to do in college was during my introductory physics courses. Before solving any problem, we had to write what we expected the solution to look like, then after solving the problem, we had to write whether our solution seemed plausible. Did it fit our initial expectation? I not, could we explain the disparity? If we applied this value or equation to something else, would we get reasonable results (say, if the problem were estimating the gravitational force of the sun, what would this value give us for the length of the earth's year). Note that we'd get credit for this part even if our solution was wrong, as long as we recognized that it was in fact wrong. It was a huuuge pain at the time and often took longer than the initial problem took to solve, but it did force me into the habit of critically evaluating my work, and its been one of the most valuable life skills I learned in college. It also helped develop my intuition, and significantly improved my teaching and presentation skills.
- narrator 10y ago
  That's why Khan Academy is awesome. I'm working through their calculus and differential equations curriculum right now to refresh my math and the amount and quality of autograded excercises is fantastic! The other nice thing is it periodically tests knowledge of previously studied material.
imakecomments 10y ago
I'd like to see a list like this that included the field of mathematical logic. For whatever reason mathematical logic no longer seems to be a "popular" area of research, despite its deep connection to theoretical computer science. But there are distinction in study, as computer scientist tend not to go deeply into computability theory like a traditional mathematician would.
- mrkgnao 10y ago
  My Hackernews! http://www.logicmatters.net/tyl/ http://www.logicmatters.net/tyl/
- ExternalCenter 10y ago
  I think I understand why. Mathematical logic research has largely become focused on problems that, while important, seem very arcane from an outsiders perspective (even by the standards of other fields of math). The fundamental results of classical recursion/computability theory can be developed and presented with a fairly small amount of logic (see something like Cutland or Cooper). I'm not really super informed on the current state of the field though, so I may be misjudging the situation.
- plinkplonk 10y ago
  Perhaps not quite what you are looking for, but I've found the book reviews etc on this site very useful http://www.logicmatters.net/tyl/ http://www.logicmatters.net/tyl/
- fmap 10y ago
  What exactly are you trying to learn? Mathematical logic is a huge field in its own right, with plenty of topics that are of historical interest and a lot of active research areas. If you want to learn modern mathematical logic you're in for a rough time, since you'll basically have to learn category theory in order to understand the few really excellent textbooks which exist (e.g. Sketches of An Elephant). If you are interested in type theory you should try reading the Homotopy Type Theory book, which is (mostly) self contained.
  - 10y ago
mrcactu5 10y ago
one day I woke up and realized the mathematics curriculum is almost completely arbitrary. there is no reason to teach algebra, geometry, trigonometry in that order by the middle of graduate school everyone is self-teaching and you may know more than a professor from time to time about a given topic. And certainly about the basics since professors forget to do basic integrals at the board. any good study group has to retain momentum and keep the discussion moving forward.
nicklaf 10y ago
How long would it take to study 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. 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-equations-text-for-undergraduates-who-want-to-become-pure-math http://mathoverflow.net/questions/28721/good-differential-eq...
tribe 10y ago
For another list of recommendations by topic, check out this (very popular) list: https://github.com/ystael/chicago-ug-math-bib https://github.com/ystael/chicago-ug-math-bib
- nicklaf 10y ago
  Much better. Looking through the intermediate and advanced sections, I see that this guide suffers from none of the omissions which I had complained about in my post (specifically, the omissions of classic works by Arnold and Spivak). (I still think it would be crazy not to buy the PCtM somewhere before reaching the advanced part of this guide, and it may as well be before the intermediate as well. Consider that without the PCtM, you may never realize that your initial (probably false) assumptions about the scope of mathematics may mean that you will never even get past the beginner section at all!)
- gtani 10y ago
  I think the guy who wrote that original U of C list has stated that it's mainly of historical interest, given the many omissions noted, and that the lists by Baez and Univ. Cambridge that i pushed there are better starting points, or this one: https://www.reddit.com/r/math/wiki/faq https://www.reddit.com/r/math/wiki/faq There's lot sof other lists of recommended texts after calc3, https://www.math.ucla.edu/ugrad/courses/math/ https://www.math.ucla.edu/ugrad/courses/math/ and https://math.berkeley.edu/courses/archives/announcements/fall-2011-textbooks https://math.berkeley.edu/courses/archives/announcements/fal... and http://mathoverflow.net/questions/761/undergraduate-level-math-books http://mathoverflow.net/questions/761/undergraduate-level-ma...
sedachv 10y ago
Note that the Mathematical Atlas was supposed to have been moved to http://www.math-atlas.org/ http://www.math-atlas.org/ in 2016, but that has not happened yet. Latest snapshot from the Wayback Machine is from April 2015: https://web-beta.archive.org/web/20150424120057/http://www.math.niu.edu:80/~rusin/known-math/ https://web-beta.archive.org/web/20150424120057/http://www.m...

216 points by kercker 10y ago

I've also found these to be helpful: The Language and Grammar of Mathematics, from The Princeton Companion to Mathematics, by James Gowers: http://press.princeton.edu/chapters/gowers/gowers_I_2.pdf http://press.princeton.edu/chapters/gowers/gowers_I_2.pdf Reading Mathematics, by John Hamal Hubbard: http://www.math.cornell.edu/~hubbard/readingmath.pdf http://www.math.cornell.edu/~hubbard/readingmath.pdf

jimsojim 10y ago
Great read, thanks for the links
- vecter 10y ago
  Appreciate the good vibes, but no need for these kinds of comments. An upvote is enough.
lunchladydoris 10y ago
I don't mean to be that person, but it's Timothy Gowers, not James.
- rrherr 10y ago
  My bad, thank you for the correction!

deleted 10y ago

[deleted]

MichaelMoser123 10y ago

maybe i am just stupid, but I find that many texts in mathematics like to skip over details, or I get stuck because there is an ambiguity in the text and there is no one to ask about it.. What do we do in this kind of situation?

wolfkill 10y ago
Oh the dreaded "details left as an exercise." Personally, if I can't fill in details readily myself, I use Google or "just take there word for it" and hope it doesn't bite me later.
medymed 10y ago
One approach is to look at a few different but overlapping primers on the same matter and do the exercises. If one treatment doesn't click, another might. And for well-known subjects, topics are often covered in roughly the same order. Maybe not for graduate classes, not sure, but linear algebra, set theory, group theory, analysis it's generally the case. Unless you can afford a tutor to teach it several different ways until it clicks, which is functionally similar.
- jwdunne 10y ago
  I can verify this approach works for me. The other thing I suggest is use "getting stuck" as a way to identify gaps - go look for books that fill those gaps. Recently I cracked open Principle of Analysis by Rudin. I got stuck pretty much straight away and looked it up - I saw a few suggestions for a bridge between high school maths and the rigor required by that book. I'm now reading another book that took inspiration from the previously mentioned book but gives a bit shallower a slope towards the rigor required.
  - mrkgnao 10y ago
    Pugh, by any chance?
    - jwdunne 10y ago
      Krantz. Seems to be at a decent pace. No idea if it's any good though - problem of not having a mentor!
      - gtani 10y ago
        That book got good reviews. /r/math has lots of threads on this, recommend Abbott, Tao, Pugh, Spivak etc https://www.reddit.com/r/math/comments/50m4ib/what_are_the_main_differences_between_rudin_and/ https://www.reddit.com/r/math/comments/50m4ib/what_are_the_m... https://www.reddit.com/r/math/comments/4jopme/is_it_premature_for_me_to_start_reading_baby_rudin/ https://www.reddit.com/r/math/comments/4jopme/is_it_prematur... https://www.reddit.com/r/math/comments/512p2p/analysis_text_for_selfstudy/ https://www.reddit.com/r/math/comments/512p2p/analysis_text_...

wolfkill 10y ago

In the area of numerical analysis, I'd recommend works by LeVeque and Trefethen.

thearn4 10y ago
Trefethen & Bau's "Numerical Linear Algebra" is a pretty good primer, though the last few chapters I remember being a bit lacking (it's been about 6 years though).
- wolfkill 10y ago
  This book helped me to visualize Krylov space methods better than any other resource I have seen. It got me through my numerical analysis qualifying exam. I do agree that the last chapters are lacking in depth. I also recommend his book "Pseudo-spectral Methods in Matlab" for a thorough and very visual look at things like Gibb's phenomenon and spectral convergence.

lunchladydoris 10y ago

I enjoy a mathematics textbook as much as the next person, but what annoys me is the lack of solutions to the problems in so many of the books I've skimmed outside of classes.

cle 10y ago
I prefer solutions to some, but not all problems. Being able to independently and confidently verify your own solutions is an important skill to develop.
- rguzman 10y ago
  having the solutions does not stop you from developing that skill. however, the lack of solutions makes it difficult to make progress when you have time constraints, which is the common case.
- gms7777 10y ago
  This is absolutely a crucial skill. One of the most valuable exercises I had to do in college was during my introductory physics courses. Before solving any problem, we had to write what we expected the solution to look like, then after solving the problem, we had to write whether our solution seemed plausible. Did it fit our initial expectation? I not, could we explain the disparity? If we applied this value or equation to something else, would we get reasonable results (say, if the problem were estimating the gravitational force of the sun, what would this value give us for the length of the earth's year). Note that we'd get credit for this part even if our solution was wrong, as long as we recognized that it was in fact wrong. It was a huuuge pain at the time and often took longer than the initial problem took to solve, but it did force me into the habit of critically evaluating my work, and its been one of the most valuable life skills I learned in college. It also helped develop my intuition, and significantly improved my teaching and presentation skills.
narrator 10y ago
That's why Khan Academy is awesome. I'm working through their calculus and differential equations curriculum right now to refresh my math and the amount and quality of autograded excercises is fantastic! The other nice thing is it periodically tests knowledge of previously studied material.

imakecomments 10y ago

I'd like to see a list like this that included the field of mathematical logic. For whatever reason mathematical logic no longer seems to be a "popular" area of research, despite its deep connection to theoretical computer science. But there are distinction in study, as computer scientist tend not to go deeply into computability theory like a traditional mathematician would.

mrkgnao 10y ago
My Hackernews! http://www.logicmatters.net/tyl/ http://www.logicmatters.net/tyl/
ExternalCenter 10y ago
I think I understand why. Mathematical logic research has largely become focused on problems that, while important, seem very arcane from an outsiders perspective (even by the standards of other fields of math). The fundamental results of classical recursion/computability theory can be developed and presented with a fairly small amount of logic (see something like Cutland or Cooper). I'm not really super informed on the current state of the field though, so I may be misjudging the situation.
plinkplonk 10y ago
Perhaps not quite what you are looking for, but I've found the book reviews etc on this site very useful http://www.logicmatters.net/tyl/ http://www.logicmatters.net/tyl/
fmap 10y ago
What exactly are you trying to learn? Mathematical logic is a huge field in its own right, with plenty of topics that are of historical interest and a lot of active research areas. If you want to learn modern mathematical logic you're in for a rough time, since you'll basically have to learn category theory in order to understand the few really excellent textbooks which exist (e.g. Sketches of An Elephant). If you are interested in type theory you should try reading the Homotopy Type Theory book, which is (mostly) self contained.
- 10y ago

mrcactu5 10y ago

one day I woke up and realized the mathematics curriculum is almost completely arbitrary. there is no reason to teach algebra, geometry, trigonometry in that order by the middle of graduate school everyone is self-teaching and you may know more than a professor from time to time about a given topic. And certainly about the basics since professors forget to do basic integrals at the board. any good study group has to retain momentum and keep the discussion moving forward.

nicklaf 10y ago

How long would it take to study 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. 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-equations-text-for-undergraduates-who-want-to-become-pure-math http://mathoverflow.net/questions/28721/good-differential-eq...

tribe 10y ago

For another list of recommendations by topic, check out this (very popular) list: https://github.com/ystael/chicago-ug-math-bib https://github.com/ystael/chicago-ug-math-bib

nicklaf 10y ago
Much better. Looking through the intermediate and advanced sections, I see that this guide suffers from none of the omissions which I had complained about in my post (specifically, the omissions of classic works by Arnold and Spivak). (I still think it would be crazy not to buy the PCtM somewhere before reaching the advanced part of this guide, and it may as well be before the intermediate as well. Consider that without the PCtM, you may never realize that your initial (probably false) assumptions about the scope of mathematics may mean that you will never even get past the beginner section at all!)
gtani 10y ago
I think the guy who wrote that original U of C list has stated that it's mainly of historical interest, given the many omissions noted, and that the lists by Baez and Univ. Cambridge that i pushed there are better starting points, or this one: https://www.reddit.com/r/math/wiki/faq https://www.reddit.com/r/math/wiki/faq There's lot sof other lists of recommended texts after calc3, https://www.math.ucla.edu/ugrad/courses/math/ https://www.math.ucla.edu/ugrad/courses/math/ and https://math.berkeley.edu/courses/archives/announcements/fall-2011-textbooks https://math.berkeley.edu/courses/archives/announcements/fal... and http://mathoverflow.net/questions/761/undergraduate-level-math-books http://mathoverflow.net/questions/761/undergraduate-level-ma...

sedachv 10y ago

Note that the Mathematical Atlas was supposed to have been moved to http://www.math-atlas.org/ http://www.math-atlas.org/ in 2016, but that has not happened yet. Latest snapshot from the Wayback Machine is from April 2015: https://web-beta.archive.org/web/20150424120057/http://www.math.niu.edu:80/~rusin/known-math/ https://web-beta.archive.org/web/20150424120057/http://www.m...