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How does anyone learn advanced math. so hard

21 points by paulpauper ↗ HN
I am trying to read papers and answer on stack overflow and despite many years of education and other learning, still find it immensely hard to follow most of the stuff. Beyond basic calculus and linear algebra, it's close to impossible to understand anything. The main problem is that mathematicians tend to omit a lot of steps, as those are taken for granted, which means days or even weeks spent trying to recreate the results, even with my background knowledge and the internet it's very hard to do, if not impossible in some cases. Sometimes the most important steps are omitted that are the entire crux of the problem yet trivial steps are included in detail.

All these viral posts about "how to learn math" and I don't think anyone quite grasps what this entails if you plan to go beyond calculus or linear algebra. Basically you're stuck, and maybe 2-3 people can help you who actually know this stuff or are patient enough to help.

10 comments

[ 2.9 ms ] story [ 17.6 ms ] thread
xanderlewis
I think to answer this it’d be very helpful to have some more detail. What sort of thing exactly have you been trying to learn?
vanboxel
I'm a very technical person, but I never feel dumber than when doing mathematics. Self-teaching it is especially challenging. In many fields, one can learn to recognize general patterns and quickly apply them elsewhere. In math, I find that general pattern matching can be helpful for getting ideas, but there's so shortcut to working through the logic required in a rigorous mathematical proof. It's almost a different kind of thinking, and it takes a ton of effort.

If you're interested in self-teaching, I recommend an excellent Abstract Algebra book designed explicitly for that: https://today.williams.edu/books/abstract-algebra-a-student-.... This was the book that got me into rigorous mathematics and started me on the journey of doing a math PhD (defending this semester!). It only requires high-school algebra, and honestly, even that isn't really needed.

eltonlin
From what I'm guessing (correct me if I'm wrong), the "2-3 people who can help you" are so busy so you feel bad asking / it's just an inefficient arrangement for them. Moreover, tutoring for truly advanced math basically doesn't exist, or would be insanely priced (proven Wyzant tutors charge $100-150 undergrad.level)

How would you feel about a hybrid between pure Youtube & pure tutoring? Where a proven mathematician creates detailed visual explanations (that don't omit steps) on fundamental advanced topics based on what you ask? And you pay, say $30/month, which is much cheaper because of economies-of-scale for the teacher?

My incentives is that I'm the founder of https://explanations.app. I'm a MIT grad, ex-Apple, raised $500K. I'm pivoting desperately and will spend all my energy and focus on what you want if you give me a chance to work with you.

BOOSTERHIDROGEN
I would like to understand attention is all you need paper.
atrettel
I totally agree that many mathematicians omit steps and make it really hard to follow what they are doing. As with most advanced subjects, what comes off as "obvious" to experts isn't always obvious to others (sometimes called the curse of expertise).

For more advanced subjects in mathematics, I like to find books with lots of problems that I can work on. That is difficult, especially at the graduate level, but when I find a good book with good problems, I can usually learn the material quite well. Many books only have the answers but some have full step-by-step explanations. I always first do a problem without looking at the answer or explanation, so that I learn by making mistakes (a good way to learn). This then gives me a much more hands-on feel of the material from start to finish.

It would be helpful if you could say what subjects you are interested in learning more about. Then somebody could recommend resources, etc.

akasakahakada
Yea the main issue of advanced math is that there are more gatekeepers than real people. They will prove themselves immediately by flagging/ downrate and then promote their left as exercise mentality.
b20000
imagine if we all published source code by just writing the first lines and last lines of our code and then with great smugness write a comment saying something like “writing the rest of the code is left as an excercise to the reader”… and then got away with it all and got praised as geniuses
a_tartaruga
What you are pointing out is real and I think unavoidable. Math is just denser than most other things. A few high level things I've come up with while self studying for the past few months

- Focus on what is achievable. Learning difficult things is like building a tall pile of sand at the beach. You need a very wide base of experience to be able to climb high. Don't expect to learn algebraic geometry without mastery of abstract algebra etc. Reading research papers before working through relevant textbooks is a lot harder.

- Use quality resources. Some books are really good for self study at a less advanced level (Abbot, Sipser, Pinter) while others are bad or need more maturity. Research papers from earlier on in fields and further back in time feel more approachable.

- Try harder to get help from people. I have found reddit's r/math biweekly Quick Questions threads to be helpful for specific, advanced questions. Reach out to colleagues and acquaintances. There are people out there who love this stuff.

- There's no substitute for going back to school and taking an actual class with office hours and someone who will look over your work every week and correct your errors, it probably speeds things up 10x

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