
The Mathematics Autodidact’s Aid (2005) [pdf] - kercker
http://www.ams.org/notices/200510/comm-fowler.pdf
======
MichaelMoser123
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?

~~~
medymed
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
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
Pugh, by any chance?

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

~~~
gtani
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_m...](https://www.reddit.com/r/math/comments/50m4ib/what_are_the_main_differences_between_rudin_and/)

[https://www.reddit.com/r/math/comments/4jopme/is_it_prematur...](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/512p2p/analysis_text_...](https://www.reddit.com/r/math/comments/512p2p/analysis_text_for_selfstudy/)

~~~
jwdunne
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.

------
rrherr
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)

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

~~~
rrherr
My bad, thank you for the correction!

------
imakecomments
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.

~~~
fmap
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.

~~~
imakecomments
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.

~~~
wolfgke
Category theory is not logic.

------
lunchladydoris
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
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.

~~~
gms7777
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.

------
tribe
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)

~~~
gtani
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/fal...](https://math.berkeley.edu/courses/archives/announcements/fall-2011-textbooks)

and [http://mathoverflow.net/questions/761/undergraduate-level-
ma...](http://mathoverflow.net/questions/761/undergraduate-level-math-books)

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

~~~
thearn4
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
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.

------
sedachv
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.m...](https://web-
beta.archive.org/web/20150424120057/http://www.math.niu.edu:80/~rusin/known-
math/)

------
mrcactu5
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
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...](http://mathoverflow.net/questions/28721/good-differential-equations-
text-for-undergraduates-who-want-to-become-pure-math)

