How to read mathematics (2015) | Hacker News Reader

How to read mathematics (2015)

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137 pointspenguin_booze31 comments

"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.

> "Mathematics is not a spectator sport".

True, just like learning a foreign language, or to play the piano, or programming.

And not everyone can be a Mozart but that's OK there's room between zero and genius. It took me too many years to realize that.

With the obvious difference that you can enjoy listening to a piano ... or even understand a foreign language without being able to speak it.

Makes sense in the context of software engineering: when reading math, you are the interpreter. You can run code to see if it works. When reading math, your brain is running it - if you don't understand it you'll get runtime errors lol

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.

Do you remember who that was?

This is like some advice I figured out from experience about software courses.

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)

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.

When you say "Rudin" which one do you mean? Big Rudin? Baby Rudin?

I know he gets a lot of hate, but I personally love Rudin's writing style. There are more chatty analysis textbooks like Abbott and Carothers, but the conciseness of Rudin plays nicest with the way I think, and the exercises (in Baby Rudin) are really wonderful. They're hard but I usually felt a genuine sense of accomplishment when I finished one

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

The first time I attempted Rudin I gave up after the second chapter. I had been through Abbot, and still didn’t truly understand Analysis (or the style of analysts). After letting it sit for a few months, I picked Rudin back up and it instantly became my favorite text. I have no clue why this change happened so suddenly, but as a convert I will point out that he is a master at including exactly the right amount of detail: he does not leave details out, but neither does he commit the equally (if not more) harmful sin of over explaining a concept. This leaves every concept explained as concisely and simply as possible, but no more, and is extremely helpful as a person with ADHD. The book is worth it’s weight in gold: one can read a chapter of Rudin in a day and have a better understanding of the concept than reading twice or three times as much material in a competing analysis text.

I asked Walter about this when I was a grad student at University of Hawaii. He and his wife Mary Ellen would visit regularly and I was attempting to make small talk at a pau hana party. He said he made an effort to find the shortest proofs of all the basic theorems so students could get to the more interesting results quicker.

> “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/

That's just one subfield of mathematics. During my PhD, I barely touched adjoints and fixed points. Though, many fields of math so indeed have this sort of repeating pattern. It's just not the same for all of them.

What subfield do you think I'm talking about? Which subfield doesn't involve adjoints and fixed points?

Most "hard results" in any subfield of math cannot be derived from pure categorical nonsense like adjunctions. At some point, you have to roll up your sleeve and do the work.

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...

Interesting post. In "Linear Algebra" section what do you mean by `a'`? Do you mean `inv(a)`? It seems like this was never explained in your own formalism.

It’s some Julia pseudo code. inv is inverse. Tick is adjoint.

Category theory?

Constructivism. Linear logic, geometry of interaction, transcendental syntax.

I love Category theory, even though I know very little about it.

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.

I would argue that something like induction is probably more universally relevant.

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.

I think it's the product of the final number and the middle number. (given an odd length sequence)

And it's the final number +1 times the lower of the middle two if it's an even length sequence! So with modulo mapping between even and odd, we can easily construct a \sum that works for any n.

Yeah, that was the irony. I read a paper about reading math carefully, I ignored the intent of the paper entirely and skipped straight to the example, I saw 1+2+3+4+5=3x5, and I told my buddy, "Look, it's the product of the final number and the one that is two before it! So for 435, it's 435 times 433! Isn't that cool?"

Good advice. Also applies to reading code.

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