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The Mathematics Autodidact’s Aid (2005) [pdf] (ams.org)
216 points by kercker on March 23, 2017 | hide | past | favorite | 47 comments

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?

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.

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.

Pugh, by any chance?

Krantz. Seems to be at a decent pace. No idea if it's any good though - problem of not having a mentor!

Thank you :) I'm currently reading Calculus by Spivak too. Fantastic book.

I find getting a taste and moving forward breadth-first keeps things fresh.

I fully suggest this, but also some other things.

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).

It is also important to look for primers from multiple decades because what sometimes can be confusing is the current state of the art, thesis papers (PhD students are struggling to learn it the first time too, but often they hold the complete supporting material with references), and the original paper for the field. A lot of those papers are struggling to explain the new concept so they draw more analogies and you can see where they are going compared to a worked example.

It is also important to look for primers from multiple decades because what sometimes can be confusing is the current state of the art, thesis papers (PhD students are struggling to learn it the first time too, but often they hold the complete supporting material with references), and the original paper for the field.

True, but the fly-in-the-ointment here is that sometimes notation changes over the years, and comparing papers / textbooks across textbooks can add even more cognitive overhead in that sense. :-(

I agree but if you get enough books in the same area you can start to figure out the mapping. Even books released in the same subfield at the same time can have wildly different notation and only by having enough references can you have a chance.

That should have read:

... comparing papers / textbooks across decades can add even ...

perhaps one approach to this is a wiki linking to different approaches for describing the same concepts?

Plus, there are enough old math books (out of copyright) to just include scans directly. I actually collect math books, have a few infitesimal calculus books even.

Others have already given some good answers, but I'll add that a part of it is that this sort of thing is to be expected, especially when dealing with proofs. All proofs have at least some ambiguity because whether or not to leave certain parts out is subjective. Don't expect to just get a nontrivial proof on the first read. You have to convince yourself that the proof is valid. It's easy to think of this as a waste of time, because the author could have just been more clear, but to a certain extent it's a good thing. It forces you to understand the context around this proof, and ultimately is more illuminating than just reading and memorizing the proof.

You're not stupid. At least, there is very little chance that the math you're trying to work on right now requires super-normal cognitive abilities of any sort: it's more about learning to be comfortable with the ambiguity or lack of comprehension that the process of learning entails. Here's a grab-bag of stuff that have helped me (and still do) lift myself out of "damn I'm dumb" situations such as the one you seem to find yourself in.

"Does math have to be ambiguous or cryptically written? That doesn't sound right."

Well, no. Sometimes, the solution is just "get a better book" (although, see [-1]). However, even extremely well-written books will skip over certain details (which ones they elide depends on the target audience), for the simple reason that it

a) massively shortens the book

b) maintains the "flow" well, instead of stopping every few sentences to distract the reader with ideas not directly relevant to the argument

c) acts as a natural method to help people choose materials: if you're hitting things you can't understand or even parse every two lines, maybe you'd be better served by coming back to it a while later?

In other words, the skipped details and "why would that be the case?" are going to be there even in books that are considered masterpieces of mathematical writing, and everyone encounters them and learns to deal with them. So what do you do when you hit the inevitable "yeah, right, that's trivial, suuuuuure" next time? To paraphrase someone I can't place right now, you either go under it, over it, around it, and -- if all those fail -- then ($deity willing or no) you push right through it. (Exercise for the cough careful cough reader: what do I mean?)

Less faux-metaphorically, try

* skipping forward a little -- new ideas do not have to be learned and absorbed linearly; case in point: the next paragraph goes into detail on this :)

* checking out another book on the same topic

* asking a question on "MSE" / math.stackexchange.com (the /r/math subreddit is good too)

* checking out the extremely friendly MSE chatroom, which I credit with fixing a ton of my "I am just stupid" moments

* looking for lecture notes (shameless plug: [0]) online

* (failing everything above) spending a day walking around your house with your hands squeezing your head from the sides, trying to mimic the romantic picture of the tortured-genius lonewolf-mathematician-in-training and somehow trying to become more intelligent by increasing the density of your brain matter until you've triumphed over whatever puny margin-note was keeping you occupied

Also note that mathematical understanding and intuition always, always grow organically. You often end up understanding something "properly" years after you stop struggling with it, and it only gets better as you learn things and see how disparate ideas fit together or even become special cases of a general concept, or variations upon a basic theme. This is touched upon in Terry Tao's excellent career advice essays[3], which are aimed at students learning to make sense of the (mathematical) world: in particular, I'd recommend

* https://terrytao.wordpress.com/career-advice/there%E2%80%99s...

* https://terrytao.wordpress.com/career-advice/does-one-have-t...

Grothendieck talks about his journey (see [4]) himself, and his writing is beautiful (if also prone to creating a ton of hope in young people encountering this anecdote for the first time). It's profound, but not an authoritative account of what it means to do mathematics, by any means (just like Hardy's A Mathematician's Apology).

> Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects. In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

Some of the stuff on this page[1] is very helpful, whatever your current level of knowledge. Ctrl-F "tendrils". You might find Thurston's essay "On Proof and Progress in Mathematics" (arXiv link: [2]) interesting too. Good luck!

[-1] There are some well-known books that students are almost always advised to come back to later (and there are probably more that they're advised to not bother with, ha).

[0]: http://refinator.herokuapp.com/

[1]: http://math.stanford.edu/~vakil/potentialstudents.html

[2]: https://arxiv.org/abs/math/9404236

[3]: https://terrytao.wordpress.com/career-advice/

[4]: https://quomodocumque.wordpress.com/2010/07/05/the-capacity-...

You can try asking at http://math.stackexchange.com/

I run in the same problem all the time and appreciate you asking the question.

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.

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

Reading Mathematics, by John Hamal Hubbard: http://www.math.cornell.edu/~hubbard/readingmath.pdf

I don't mean to be that person, but it's Timothy Gowers, not James.

My bad, thank you for the correction!

Great read, thanks for the links

Appreciate the good vibes, but no need for these kinds of comments. An upvote is enough.

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.

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.

I'm mostly interested in "modern" mathematical logic then. I'm very interested in learning category theory and its connections to programming language theory. I already know a little bit of category theory, but am open to any good beginner sources. I'm also interested in classic recursion theory and a bit of proof theory with its connections to CS. I don't know many people doing any of this and it doesn't seem that popular in math departments.

Category theory is not logic.

Perhaps not quite what you are looking for, but I've found the book reviews etc on this site very useful


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.

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.

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.

Oh, differential equations.

I'm doing ECE at university now and I have an exam in about 7 weeks. I've found that it's quite hard to identify what sort of differential equation a given problem is, and then how to solve it. However I've found that by looking around for other material I can understand it better.

For example web searching "type 4 differential equation" bought up this gem: https://www.math.ucdavis.edu/~thomases/W11_16C1_lec_1_7_11.p...

I think I'll try out Khan Academy because I've found that the lecturer's problem sheets probably aren't enough on their own.

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.

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.

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.

Sometimes the problem is more important than the solution.

Yes, but if you're cramming for an exam it is so much more productive to just read through a fully worked out solution instead of trying to figure out how to do it yourself.

For another list of recommendations by topic, check out this (very popular) list:


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

There's lot sof other lists of recommended texts after calc3, https://www.math.ucla.edu/ugrad/courses/math/ and


and http://mathoverflow.net/questions/761/undergraduate-level-ma...

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!)

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

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).

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.

Note that the Mathematical Atlas was supposed to have been moved to 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.m...

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.

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-eq...

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