How to Learn Mathematics and Computer Science

A Personal Perspective

2024-12-27

Self-Guidance

This is an un-organized list of thoughts that I’ve had about how to learn more effectively.

• Anything I want to learn deeply, or think about carefully, should probably be worked out on paper or on a whiteboard first, before typing it on a computer.
  • If I’m just making a note of something then it’s probably fine to skip the above and just type it into a computer.
• I should probably periodically review things on my website, and resources I use to learn, e.g., textbooks, to promote deep learning.
• Maybe I should also try to explain things I want to internalize. This probably means recording videos of myself explaining things, but I’m not sure how to do this without spending a lot of money and time.
• I should try to make connections between topics and resources. For example, if I’m reading a computer science textbook and I see a connection to a topic in a mathematics textbook, then I should briefly describe that connection in my notes.
• I should revisit textbooks and notes. For example, if I’m reading SICP, then when I get to (say) Chapter 3, I may be able to link material in Chapter 1 to what’s happening in Chapter 3, so I should go back to Chapter 1, in the book and my notes, and see if I can make any connections.
• I should probably be reluctant to use the word “obvious” when I explain something. If it really is obvious, then it should fit in a footnote. If it takes more space than that, then it should still be explained, but perhaps in an appendix.
• A lot of my work will first be done on paper (or on a whiteboard), and then typed up for the website. So maybe I should try to “batch” this kind of work. Like let’s say I’m reading book A and B, and I’ve solved the exercises for a section A, but it’s all on paper. Then I could switch to working on book B and solving its exercises (on paper), while also typing up the accumulated work from A. Then continue alternating like that, roughly.
  • Also it’s worth noting that typing up exercise questions from a textbook is really easy work that I can do whenever I’m bored or looking to kill time.
• Here is a stock list of questions that I can ask myself when trying to understand a theorem/proposition/etc. in a book (proven by the author or even one I proved myself in an exercise):
  • What is the key idea in the proof?
  • Can I draw a picture(s)s that communicates the idea of the proof?
  • Which results are implicitly used in the proof? E.g. does Theorem 3.2.2 implicitly use Proposition 2.6.1? If so, make explicit note of that in my notes.
  • (Perhaps slightly less important) I could also look for counterexamples that clarify the need for the assumptions/conditions required in the statement of the theorem/result. If I do this, maybe it should be geared towards using the counterexamples to hone my intuition. In other words, it’s not just about “knowing” that we need a certain condition to be true, but the counterexample should help refine my intuitive idea (or picture in my mind) for whatever object we’re considering.
  • Where is this proof used? where is it applied? (As an example, I think the Weierstrass M-test is used by Le Gall when constructing Brownian motion.) What does it allow us to do or conclude?
  • Where in applied math is this theory used?
• Here is a stock list of questions that I can ask myself when trying to understand a definition.
  • What behavior or phenomenon is this definition trying to capture? What is the intuition for the thing this definition is trying to capture?
  • Can I draw a picture that communicates the concept of the definition?
  • Can I give an example where this definition is used? Ideally at least one example each in pure and applied math.
• Tentative motto: I am less interested in the answer than in finding the right way to think about the problem.

Notes from Mathematica

• Mathematics is the interplay between logic, intuition and creativity. (p. 6)
• We must try to train and improve our abilities in each of these three areas, and to move between them. (p. 6)
• Written mathematics is sheet music. We must try to see and feel what the writing represents, akin to hearing the music. (p. 7)
• Logic roughly corresponds to rigor and proof.
• Intuition is seeing, feeling and embodying the ideas hidden within the writing.
• Try to translate from logic to intuition, and from intuition to logic. Each improves the other.
• Creativity is involved in all areas, because creative ideas can help anywhere, e.g., in writing proofs. But it’s also involved in deciding what to explore and what to define.
• “See” and “feel” should be taken literally. We must use our minds, bodies, and imaginations to experience and understand the mathematics. For example, get up and walk around while learning math.
• To nurture your intuition and creativity, try to translate from the formal (e.g., the written mathematics) to the informal (e.g., see and feel the concepts).
  • Maybe there are specific exercises to do, to train e.g., spatial sense. Maybe also take drugs, e.g., LSD and/or psilocybin.
  • Examples, counterexamples, logic and rigor can help us determine whether our intuition is right or wrong, thereby improving our intuition. Use logic to improve your intuition. Refine and internalize the intuition.
  • Logic and formalism are used to record and communicate our intuition, but also to explore, test and refine our intuition.
• Try to be curious. Try to want to understand. Study a mixture of what is interesting and what is useful.
  • Similarly, try to find good resources and study them closely (e.g., specific textbooks), but also feel free to read different resources (e.g., other textbooks, notes on the Web, StackExchange posts, etc.).
  • Similarly, don’t necessarily feel pressure to read a book entirely in order, or to learn subjects entirely in order. Mathematical knowledge is in some ways linear and cumulative, but in other ways it’s not linear.
• Notice a general principle from the foregoing: you shouldn’t look at things from just one perspective. You need multiple, complementary, sometimes even conflicting perspectives, and you need to know how to move between them.
• So maybe as a corollary to the discussion about reading books in order, or not, you should have a small number of core books you study closely, and then use anything else as a supplement in any way that seems interesting or useful.
  • “interesting or useful” meaning simply that you are interested, and/or you think you might have a need for the knowledge in some book, e.g., you open a book/section on Fourier series because you might want to try to apply that knowledge in some other context.
• You shouldn’t fear looking stupid, or trying to understanding elementary topics.
• Try to cultivate your inner child and take pleasure in exploring and playing. Reread Chapter 7.
• Maybe when learning anything with a creative component, in addition to learning from an external source (e.g., a cookbook), you should also try to listen to what’s inside yourself and try to explore and express that as well. For example, playing with your own ideas of flavors and combining foods, or making your own explorations in math.
• In math, a “trick” or a “magic” step suggests that you haven’t found the right way of thinking about or visualizing a problem or concept. (Ch. 12)
• Practice intense, embodied, immersive, religious visualization and imagination. (Ch. 12)
• You can probably practice the techniques described here in different areas and at different levels concurrently. (I don’t have a good example right now)
• Could you design a neural network activation function that was 0 or 1, depending on whether the majority of its input neurons were 0 or 1? Does that make any sense? (p. 257)
• Could you appoly a “mushroom trip” to a neural network by scrambling, i.e., applying a small amount of noise to, its later neurons? Could you “randomly” make a bunch of (weak) new connections in the network to simulate the effect of a mushroom trip?