A Programmer's Introduction to Mathematics

Anecdote aside, there's a difference between reading something and understanding something. My friend who is really in tune with the nature of the subject, is also spending an incredible amount of time surrounding it.

"To understand any subject, above all, a mathematician SHOULD NOT pick up a book and read it.

It is the worst error!

No, a mathematician needs to look in a book, and to read it backwards. Then, he sees the statement of a theorem. And, well, he goes for a walk. And, above all, he does not look at the book.

He says, "How the hell could I prove this?"

He goes for his walk, he takes two hours ... He comes back and he has thought about how he would have proved it. He looks at the book. The proof is 10 pages long. 99% of the proof, pff, doesn't matter.

Tak!, here's the idea!

But this idea, on paper, it looks the same as everything else that is written. But there is a place, where this little thing is written, that will immediately translate in his brain through a complete change of mental image that will make the proof.

So, this is how we operate. Well, at least some of us. Math is not learned in a book, it cannot be read from a book. There is something active about it, tremendously active.

[...]

It's a personal, individual work."

[0] https://www.youtube.com/watch?v=9qlqVEUgdgo

I started with 3blue1brown's Youtube course[0] on Linear Algebra and loved it. I had already done a college course on LA, but this made me truly understand what I was doing.

MIT OCW Scholar(independent study) course on Linear Algebra by Prof. Strang[1] is really good and is designed for self-study. If you have the time, you could look up Coding the matrix[2] too. I read probability from Mathematics for Computer Science-MIT[3] and also referred Khan Academy[4] and PennState STAT 414/415 [5] for statistics and probability. StatQuest channel[6] on Youtube has handwavy but easy to understand videos on statistics for ML too. The Deep learning book[7] by Ian Goodfellow et al. has a couple of chapters at the beginning that gives you a fairly good idea of the mathematics required to get into Deep learning. Communities like r/AskStatistics and r/statistics on Reddit were really helpful when I got stuck.

I also chanced upon Mathematics for Machine Learning[8] book recently and it seems to be good. It has a chapter on optimization that is left out in most books but it skips statistics.

[0] - https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x....

[1] - https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algeb....

[2] - http://codingthematrix.com/

[3] - https://courses.csail.mit.edu/6.042/spring18/mcs.pdf

[4] - https://www.khanacademy.org/math/statistics-probability

[5] - https://onlinecourses.science.psu.edu/stat414/

[6] - https://www.youtube.com/user/joshstarmer/videos

[7] - https://www.deeplearningbook.org

[8] - https://mml-book.com

Copied from my comment here: https://news.ycombinator.com/item?id=18582022

Indeed it doesn’t.

As the author of the book in this post points out on his blog [1] in math you focus on learning the syntax first, just like in programming.

Off the top of my head I can recommend pre-reading a chapter before reading it: go through theorems, bold/italic text, graphs, tables, conclusions, exercises. Also, try proving theorems yourself before reading proofs in a textbook.

My general advice for you is to check out a book “How to Read a Book” by Mortimer J. Adler and Charles Van Doren. It’s a great manual for reading, which contains both general advice (in the form of 15 rules) applied to any reading material, and specific advices for math, philosophy, history, social science, etc.

> I try to go chapter after chapter and make notes

Notes are good, but only if they “active” notes. If you are able to formulate something in your own words it means you’ve done the work towards understanding, if you’re only able to express a thing in author’s words — you have not, and if you leave it at that the chances are you’ve wasted your time.

In math you also have to do problems. Your ability to solve problems w/o the help of the textbook is the ultimate test of whether you’ve learned what you’ve read.

Interleave: in your practice session work simultaneously on problems from different (usually sequential) chapters. IOW: don’t practice one topic during several study sessions until you’ve exhausted all exercises for that topic in a book — go read the next chapter (or chapters), and include its exerices in your next practice.

Interleaving is harder, but more efficient. The reasons being that (1) our brain is bi-modal processing system, it needs time to process things both consciously and unconsciously, it does the latter while your attention is busy with other things; (2) it also needs time to consolidate memories, grow and strengthen new synaptic connections (which it primarily does in sleep) — so spreading a topic/exercises over time is generally a good idea.

> but I often get bored because I don't see the usage in my real life coding > Also after few chapters it often turns into a "job" of finishing the book. I don't have the pleasure of learning new stuff anymore.

This is about psychology of study.

Don’t turn it into a job, it’s not a competition, there are no bosses, no deadlines. Don’t push too hard, allow yourself time to integrate new knowledge by keeping yourself busy with other things and studying several subjects/books simultaneously — when you are tired of one thing, switch to another. It’s more efficient way of learning (as explained above), despite increase in absolute time for mastering a single subject.

And work on your motivation. You can start by read what Paul Graham has to say on why study math [2]. Remember, math is hard; our brains are not wired for it.

[1] http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-...

[2] http://www.paulgraham.com/college.html