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Comment by pm90 | Hacker News Reader

I’m fascinated by these things but afraid of the formalism involved in the papers that define concepts and prove things. Am I doomed to only enjoy math through the generosity of people like the author? Or is there a way for me to dive into math without the help of an actual mathematician using only books?

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For a bunch of topics in math broadly, this is a nice book, with an emphasis on intuition rather than thorough proofs(hence a compact and readable book) : All the Mathematics You Missed by Thomas A. Garrity https://www.goodreads.com/book/show/967329.All_the_Mathemati...

There's of course Roger Penrose's "The road to reality" which has a completely different emphasis (more focused on fundamental physics and therefore introduces a lot of the math used there) but is also very interesting to read. This however, is a real time -- will take some effort to go through, but probably worth it :-)

You can absolutely learn math on your own. Many of the most well-known mathematicians, engineers, and scientists were largely self-taught. Examples include da Vinci, Watt, Edison, the Wright Brothers, Heaviside, Ampere, Boole, Galileo, Pascal, Leibniz, and Ramanujan. The learning materials available to those guys were vastly inferior to what we have today. In fact, you could take yourself very far using nothing more than Wikipedia coupled with a divide-and-conquer approach. If you want to understand a topic, do a Wikipedia search on it. There are usually some good references and links at the bottom of the page that can help with further background and intuition. Never be intimidated by the complexity. Every complex topic can be broken down into simple, easy-to-understand pieces. For every subtopic in the description that you don't understand, click on its link, read, and repeat. Eventually, your subtopic dive should reach a low enough level that you can grasp the concept and use it to understand the previous higher-level topic. For each concept, you should develop an intuitive understanding that goes along with the formalism. Try to visualize every concept in some way. Use mental animations, graphs, sounds, colors, etc. One of the tricks is that you should only focus on learning a small number of new concepts every day (maybe 5 to 7 at most). A night's sleep will help you consolidate those concepts and build on them. For even the most accomplished mathematicians, there are papers that they don't understand at first glance. They must go through their own divide-and-conquer process to grasp unfamiliar concepts.

I have been semi-diligently (4-5 hours/week as adult life allows) following this approach coupled with working my way through math texts. Although I zero problem grasping concepts and parsing propositions and examples, I find it very difficult to solve end-of-chapter problems. No more than 4 or 5 problems a week, and hence my advancement is agonizingly slow.

I started with Spivak & Apostol but reverted to Riley and Hobson's Foundations text because I was struggling too much with the A&S's problems. Given that I did very well on symbolic-logic proofs in an undergrad course, I figured it would be easy enough to get into math if I had diligence and sincere interest. After having been a 'D' student, my recognition of arithmetic and algebraic expressions and manipulations is hopelessly sub-par. Even many of the R&H problems are beyond my grasp.

My approach has been informed by a sincere desire to engage with mathematics (one I unfortunately did not have through my formal education) as well as several "How do I self-teach maths" threads on HN, Reddit, and /sci/. Previously I had chalked up my poor grades to an undisciplined, unmotivated youth. Now I'm beginning to suspect I may simply not have the requisite intelligence, or am at least outside the age where I had enough time and neuroplasticity to pick math up in earnest.

Given that 1) you have zero problem grasping the concepts and understanding the examples, and 2) your difficulty is in solving the end-of-chapter problems; I think it is almost certain that your issue is one of making small mistakes that throw off your solutions (for example, didn't change a plus to a minus sign, left off a term, made an incorrect assumption, etc.). You just need to learn how to double-check your work and figure out where you went wrong. This is a skill that everyone doing nontrivial math has to learn. In a formal learning environment, you constantly get feedback on your mistakes through the grading process, which you obviously don't get when self-learning. My advice is to work the example problems before looking at the solution. After you have worked it yourself, you can then easily see where you went wrong. You should also get books of solved problems that let you practice this more extensively. There are problem books for pretty much every undergrad math topic as well as some grad-level ones. A lot of textbooks even have solution manuals that will show you how to solve the end-of-chapter problems in the textbook. An Amazon search with the word "solutions" and whatever math topic you want should turn up many examples. With practice, you learn how to detect errors and rectify them even when there is no solution manual.

It depends what you mean by "dive into math". To get to the level where you understand the proof of the theorem mentioned in this blog post (Theorema Egregium)? This, I think, is roughly where non-elementary math begins. My opinion may prove unpopular, but I don't think this is actually doable by self-study. Well, except that I did something like this myself, I spent 3 months one Summer going page by page through the wonderful "Riamannian Geometry" by Manfredo Perdigao do Carmo [1].

So, it's certainly possible, but also kind of not. It depends where you are in your life, and why you want to do that. If you have a job, maybe a family, and think you'd like to invest one-two hours of day for this hobby, I'm not sure you can get there. If you are willing to invest substantially more time and effort, you need to ask yourself why? Do you want to enhance your career? I think there are better areas of math/CS/Machine Learning that you can attack, with a much better return on investment. Do you already have enough money, that career advancement is not a concern, and you simply want to pursue truth and beauty, and you find advanced math to fit your taste? Then you can go solo, and I think you can succeed, but I think it's much more efficient to actually get feedback from other people (via tutoring or attending courses, or even MOOC).

Anyway, not sure if you know about 3brown1blue [2]. Check it out, I hope you'll enjoy it.

[1] https://www.amazon.com/Riemannian-Geometry-Manfredo-Perdigao...

[2]https://www.3blue1brown.com/

You can only go so far without doing/understanding the mathematics.

It is possible to learn the math through self study.

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