
How to Read Mathematics - ColinWright
http://web.stonehill.edu/compsci/History_Math/math-read.htm
======
gtani
There's a few books on how to approach books and problems, e.g. don't strain
on first seeing new material. Lara Alcock wrote 2 books on studying math which
seem to have the same ToC, a lot of undergrad programs have materials on
proofs and how to transition from high school problem solving:
[http://web.archive.org/web/20150319020039/http://www.maths.c...](http://web.archive.org/web/20150319020039/http://www.maths.cam.ac.uk/undergrad/studyskills/text.pdf)

Mathoverflow and math.stackexchange soft-questions tag has lots of mini-
essays, e.g. [https://mathoverflow.net/questions/143309/how-do-you-not-
for...](https://mathoverflow.net/questions/143309/how-do-you-not-forget-old-
math)

and [https://math.stackexchange.com/questions/617625/on-
familiari...](https://math.stackexchange.com/questions/617625/on-familiarity-
or-how-to-avoid-going-down-the-math-rabbit-hole)

____________________

For proofs, books by Polya, Velleman, Hammack etc

Prof Baez' reading lists for math and physics:
[http://math.ucr.edu/home/baez/books.html](http://math.ucr.edu/home/baez/books.html)

~~~
forgotpwagain
It would be remiss to not mention the well-curated list by Gerard 't Hooft
[1999 Nobel Laureate in Physics]. A lot of books on the list, but also a lot
of links to freely available lecture notes or online books as well.

[http://www.staff.science.uu.nl/~gadda001/goodtheorist/](http://www.staff.science.uu.nl/~gadda001/goodtheorist/)
(sidebar for specific areas).

------
rrherr
I've also found these to be helpful:

The Language and Grammar of Mathematics, from The Princeton Companion to
Mathematics, by Timothy 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)

------
maroonblazer
As someone who struggled with math and hated it in high school and now loves
it but still struggles with it I got a lot out of this article. It brought to
mind this EdX course: "Effective Thinking Through Mathematics"[0] which I also
found valuable.

Has anyone read the author's book[1] and can recommend it (or not)? It only
has 2 reviews, a 5-star and a 3-star and the latter review is not helpful.

[0][https://www.edx.org/course/effective-thinking-through-
mathem...](https://www.edx.org/course/effective-thinking-through-mathematics-
utaustinx-ut-9-01x-0) [1][https://www.amazon.com/Rediscovering-Mathematics-
Classroom-R...](https://www.amazon.com/Rediscovering-Mathematics-Classroom-
Resource-Materials/dp/0883857804)

------
cstrahan
If you haven't heard of Shai Simonson I suggest checking out his ArsDigita
University[0] lectures[1] (from which I studied Theory of Computation[2] back
in late 2011).

He's a superb lecturer, and I owe him a debt of gratitude for his recorded
classes.

[0]: [http://www.aduni.org/](http://www.aduni.org/)

[1]: [https://adunivids.neocities.org/](https://adunivids.neocities.org/)

[2]:
[https://www.youtube.com/watch?v=Pt6GBVIifZA](https://www.youtube.com/watch?v=Pt6GBVIifZA)

~~~
dpdp
Can vouch on his lecture skills as well, he was a visiting professor at UIC
and taught my intro to combinatorics class. He was also very funny, his
frequent stories would always make the class laugh.

The most amusing part about him was that he wore those tight biking shorts and
shoes while he taught.

Here's an audio clip of Shai introducing Richard Stallman at a talk where
Stallman gets pretty annoyed[0]

[0]: [http://audio-video.gnu.org/audio/rms-speech-
arsdigita2001.og...](http://audio-video.gnu.org/audio/rms-speech-
arsdigita2001.ogg)

------
danidiaz
I like this quote I found in a mathematical analysis book:

"It is an unfortunate fact that proofs can be very misleading. Proofs exist to
establish once and for all, according to very high standards, that certain
mathematical statements are irrefutable facts. What is unfortunate about this
is that a proof, in spite of the fact that it is perfectly correct, does not
in any way have to be enlightening. Thus, mathematicians, and mathematics
students, are faced with two problems: the generation of proofs, and the
generation of internal enlightenment. To understand a theorem requires
enlightenment. If one has enlightenment, one knows in one's soul why a
particular theorem must be true."

[https://www.goodreads.com/book/show/5760666-foundations-
of-a...](https://www.goodreads.com/book/show/5760666-foundations-of-analysis)

------
dshanko
I also found this to be helpful - Write Mathematics Right, by L. Radhakrishna
[https://www.amazon.com/Write-Mathematics-Right-
Professional-...](https://www.amazon.com/Write-Mathematics-Right-Professional-
Presentation/dp/1842657399)

~~~
ivan_ah
Could you say a few more words about this book? It looks interesting, but I
can't find a table of contents or a review of it. Is it intended for advanced
mathematics, or can it be useful for basic math too?

------
onuralp
I think it would be quiet interesting to crowdsource a math-to-english
interpreter (a la 'explain shell' [0]) for math equations.

[0] [https://explainshell.com/](https://explainshell.com/)

~~~
jordigh
This is not possible, at least not without a very sophisticated AI.

One thing that novices seem to not get about mathematics, especially those
coming from programming languages, is that the syntax of mathematical formulae
is not divorced from the natural language that surrounds it. Different authors
use different symbols for the same concept or the same symbol with different
meanings. That is, mathematical symbols have synonyms and homonyms, so to
speak. You can read someone's "accent" when reading mathematics. The French
consider 0 to be a positive number; the Americans don't. The Russians call it
the Bunyakovsky inequality, most of Western Europe and America calls it the
Schwarz or Cauchy-Schwarz inequality. The way they arrange their equations,
the particular details of the notation, their preferred term for a particular
concept, even tiny things like preferring upright or slant font for dx in an
integral (and by the way, often fonts carry semantic meaning, but not in this
case); all of these vary from author to author.

Mathematics is a very human activity. It is written by humans who write
ambiguous things because it's meant to be read by humans who deal well with
ambiguity. Reading human-written mathematics cannot be mechanised, at least,
not anymore than reading a natural language can be mechanised.

~~~
logicchains
It would certainly be possible if mathematics standardised on writing proofs
in a proof assistant like Coq. Then anyone with a basic functional programming
background could understand the most complex of proofs, given enough time to
step through and understand all steps of the proof in the Coq IDE.

~~~
jordigh
You would be trying to reverse a couple millenia of mathematical tradition if
you wanted to convince all mathematicians to only write machine-readable
mathematics. As appealing as the idea may sound to computer people,
mathematicians would find it quite repugnant if they were no longer "allowed"
to write mathematics in a casual style (as if it were even possible to forbid
them from doing so).

------
bluetwo
I've always loved math, but as things get more complex I stop thinking in math
and start thinking in code.

~~~
cosinetau
I think your conflating math as all-descriptive of all properties. It's
unfortunately not. Mathematics describes patterns, not procedures. You started
thinking in code, because you started thinking about problems as procedures,
not patterns.

~~~
guscost
Maybe this person is thinking in Haskell ;)

~~~
imcoconut
I wish i could give 2 upvotes

------
badpizza
I will add to the lists this another resource:

Handbook for spoken Mathematics (Larry's speakeasy) by Lawrence Chang:
[http://web.efzg.hr/dok/MAT/vkojic/Larrys_speakeasy.pdf](http://web.efzg.hr/dok/MAT/vkojic/Larrys_speakeasy.pdf)
from The Lawrence Livermore Laboratory

------
amelius
I guess how to read mathematics depends strongly on the context or the
subfield you are looking into.

A simple example: a superscript can mean exponentiation or an index.

Also, sometimes it is preferred to leave out symbols that occur often, to
avoid clutter. One example is dropping the sigmas out of equations, and infer
they exist by looking at indices.

------
bluenose69
I thought this was going to be about how to read Mathematics aloud, e.g. to
describe it over the telephone. It seems that even something as common as $f :
a \rightarrow b$ might be spoken aloud in different ways.

~~~
jordigh
This is in fact somewhat of a real problem. Mathematics is kind of meant to be
learned while interacting other humans, not in pure cold isolation. A lot of
what you absorb with other humans is how to pronounce things. A simple example
for example is learning to pronounce binomial coefficients as "5 choose 2" or
knowing that ŷ is pronounced "y hat".

Something kind of similar happens in hackerdom, where you may learn to
pronounce "/etc" as "slash etsy" or "!" as "bang".

~~~
urathai
I totally agree with you! Learning the language of mathematics, specifically
spoken in this case, is an important part in learning mathematics. I think it
is really important to discuss proofs and ideas with your friends because of
this.

Another problem is that most of the literature and the lectures are in English
but you often discuss in another language. So you have to both do the
translation from written to spoken mathematics and the translation from
English to, in my case, Swedish. This easily leads to half translated English
words but I try to avoid that as much as possible.

