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.
True, just like learning a foreign language, or to play the piano, or programming.
"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.
Don’t do the course until you have tried doing the thing without the course first and hit the pain points.
Then, when attending the course your brain is in a different mode. You are filling the gaps rather than laying the foundations. And you get much better value out of it.
(This is for in person courses where you can ask questions - so the value you get out of it might be how good your questions are)
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%.
Chattier authors are nice for providing context and intuition and sometimes details about the historical context, but I personally find them to be very distracting and a bit overwhelming. I don't like using them for much other than a reference or more casual reading. On the other hand, I loved reading a few sentences from Rudin that I didn't quite follow, then pulling out some pen & paper and doing a quick validation, or even going on a drive and munching on them until I understood
That's me though. I'm glad that there seems to be no shortage of introductory analysis texts written in all kinds of styles so that folks can find the ones that work best for them.
As an aside, I think it's a bummer that analysis classes often feel like hazing courses in the math curriculum, leading many mathematicians to despise it. I've been very lucky to have great analysis teachers, or at least ones that care very much about pedagogy over ruthless elitism, and conveying the beauty and fun that lies amid the ugly bit :)
Also, re Rudin: his autobiography is certainly worth reading, if for no other reason than for his account of surviving the Anschluss (the Nazi annexation of Austria during WWII) as a young Jew. One of my favorite bits:
"On the first day of school after the Anschluss several of our teachers and even some students strutted around in their shit-colored storm trooper uniforms. (The Nazi party had been illegal, but had obviously existed.) One of those was the gym teacher whom I had always disliked. He even had a pistol strapped to his belt. A few days later I heard that he had shot himself in the foot. This was one of the very few bits of cheerful news at the time."
It's a fairly harrowing read, and perhaps some of his best writing overall
>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?
A more realistic reader would have been more enjoyable. This "reader" is kind of a caricature.
The main ones are adjoint, norm, and fixed points. I wrote something about this https://github.com/adamnemecek/adjoint/
How would you prove the tychonoff theorem purely with adjunctions? How about the representation theorem of finitely presented abelian groups? And on and on and on...
But I've read the basic idea is there are "objects" and (directed) relations between them. The basic idea is you can use it with anything defining your own "nouns" which relate to each other via the "arrows".
It is like Humpty Dumpty words only mean what you define them to mean. BUT to define them you must define how they relate to other words. What matters is the structure of relations/arrows between the concepts. Instead of leaving definitions in rather vague natural language everything becomes "concrete" when you can draw it as a graph of vertices and arcs.
The best experience I had learning math was a linear algebra course during which I programmed most of the formulas that were taught into my programmable pocket calculator. I learned the things well and got straight A's on every test even though that usually didn't happen with me with other courses.
So my suggestion for aspiring math students would be: Create programs that do the calculations.