How to read mathematics (2015)
"Mathematics is not a spectator sport". No quote was more useful to get me through Uni, and enjoying it in the process. I'm a software engineer, not a mathematician.
That being said, a decade later I still catch myself often "glancing" over equations in papers and textbooks, and have to force myself to really look at them and check that I indeed "got them". I don't know what it is, maybe I just need more training/habit around it. There's a tendency for me to half-consciously say to myself "yeahyeah I'll get it from reading the text" or "I'll get to it later", which usually does not work.
For more important equations (Taylor, sinc function, all the variations of Fourier Series, Fourier Transform, DFT, DTFT) I actually write them down as flash cards in Anki and learn them verbatim. Yes, I have to understand them otherwise it's useless, but being able to just "make the equations appear" in my head to look at and work with them is invaluable.
Even after understanding, I won't derive the Taylor Series myself (and even if I did, I would not always want to repeat that), so the old adage that understanding is better than rote memorization is useless here.
As an experienced mathematician once told me:
"The way I read papers is by first reading the abstract. Then I try to state the results and prove them myself. When I get stuck I go to the paper to see what I got wrong."
Mileage may vary.
Good article.
I have a different opinion on how to Read/Study/Teach Mathematics. Contrary to popular belief, Maths can be a "spectator" sport. There is a very big difference between Reading/Understanding (to a certain extent) and Doing Maths. The latter is not a necessity unless you plan to be a Mathematician while the former is a necessity for every educated individual in our Scientific Society. To paraphrase Richard Hamming's quote; "The purpose of Mathematics is Insight, not Numbers".
Mathematics is a Language. Thus it is important to become fluent in the Notation of the Language first. The Notation is used to express Concepts (eg. Sets), Relationships(eg. Functions) and then build Complex Edifices(eg. Linear Algebra) using them. The metalanguage of Maths is Logic. The purpose of Maths is to Describe and Explain Nature (notwithstanding "Pure Maths"). As V.I.Arnold said in his essay "On Teaching Mathematics" (https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html) - "Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap". Hence always look for Applications of Maths in various fields. Abstraction should only happen after studying a lot of Concrete Examples else it becomes incomprehensible. Thus Read lots of Maths books, get comfortable with the notation, follow the worked out examples closely making sure that you comprehend every step and finally if you want to, do some exercises.
PS: I love this dig at 'Murica in the article :-)
P: For example, the author is not looking for a solution like this: everyone lives in Independence Land and is born on the 4th of July, so the chance of two or more people with the same birthday is 100%.
I find that reading math at all is sometimes not the best approach. When working out of a textbook I often find it more constructive to attempt problems first and then use the text as a guide to help me solve the problem, particularly when the textbook is quite dense. For example, even after taking years of analysis I still find Rudin impossible to simply "read" because the mathematics is so condensed and difficult to follow.
> “It follows easily that” does not mean (...)
>this shouldn’t take more than two minutes,
>but a person who doesn’t know the lingo might interpret the phrase in the wrong way, and feel frustrated.
I would have liked to know this before I started my master, not now that I'm finishing it. That's a weird bit of lingo, or is that just me? I've seen this expressed more fully like "result follows easily but tediously" or something to that effect. The second statement does not follow from the first, so leaving it out doesn't imply it in my reading.
Perhaps it's usually left out to not discourage students/readers from working it out themselves?
> That’s tedious, and the final exact value won’t even fit on my calculator.
A more realistic reader would have been more enjoyable. This "reader" is kind of a caricature.
Math became easier to read when I realized that there are relatively few ideas in math and most things are just permutations thereof.
The main ones are adjoint, norm, and fixed points. I wrote something about this https://github.com/adamnemecek/adjoint/
I actually just skimmed this and told my friend, "Hey, check this out! The sum of consecutive integers starting at 1 is the product of the final number and the number that is two before it!" So I totally ignored the lesson of the essay.
Good advice. Also applies to reading code.