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Comment by kinai | Hacker News Reader

Does anybody know a good guide on where to begin? Resources are not the issue here, but usually the overwhelming amount of information regarding all those topics and areas of mathematics. I was always very interested but got discouraged rather quickly, even after a semester at university. So far my favorite access to math was through philosophy.

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Step 1: Read Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament....

Step 2: Download the Book of Proof: http://www.people.vcu.edu/~rhammack/BookOfProof/ You read through it and do all the odd numbered exercises (the solutions are at the end of the book).

Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, with a view not to rush through it, but to expand your mind through each problem.

Step 4: Pick any of these books that interest you the most and do the same:

- Calculus by Spivak

- Algebra: Chapter 0 by Paolo Aluffi

- Linear Algebra Done Right by Axler

By then you should have enough mathematical maturity to know what to do next.

I'd also recommend Hubbard & Hubbard for a beautiful and beginner-friendly mix of algebra and analysis.

My preferred starter kit is Rudin plus Halmos or Axler, but treating Rudin as a summary. So a helper would be needed, like Counterexamples in Analysis. This is what Math 55 used to do.

I started with Hubbard & Hubbard. It is truly wonderful (and based on Spivak's other calculus book), but I couldn't imagine learning it without either significant background or a formal class. Even with a 2 semester class devoted to it in college, I ended up not understanding the most advanced concepts (eg. differential forms) until years later when it finally came up again in graduate-level courses. Of course, it was really an excuse to learn to think mathematically, not to learn calculus on manifolds.

I don't really recommend Rudin for a true beginner at all (unless, by "as a summary," you mean not really digging into the proofs themselves, in which case any good analysis book will do). Rudin will always try to take the most elegant route to the theorem, regardless if that route goes anywhere near where the rest of the text has been. The result, for me, has been that many of his proofs seem to just meander about for a little while until, at the very end, you arrive at the theorem. It's a bit like driving to work on auto pilot, and just as disconcerting to me.

I should have said as an outline, instead of a summary.

I think the beauty of Rudin is how compact it is. But of course you need an alternative book to be able to digest it.

I don't find calculus and lin alg very interesting, school ruined those for me forever I guess. I am mostly interested Logic and Information Theory but I guess the latter doesn't go with a good calculus base.

Thanks for the links, I will read the book of proof and try the exercises, though mostly studied those topics already.

You may be interested in Paul Taylor’s “Practical foundations of mathematics”.

Caveat Emptor: Aluffi and Axler are texts meant for vastly different levels of maturity. The former for first-year graduate students, and the latter for first/second-year undergraduate students.

Where to begin depends on what your goal is, mine was to be able to complete Sussman's SICM book (and later, his Functional Differential Geometry book).

The book I used to do this was Advanced Calculus by Loomis & Sternberg because it covers classical mechanics, potential theory, differentiable (Banach) manifolds, differential equations, (multi)linear algebra, fundamental theorum of calculus and the Fourier transform. The exercises are not very difficult compared to a lot of other texts (Spivak's Calculus) so this is an accessible math book as I don't have any formal math training.

I also liked the Mathematical Preliminaries crash course in the Art of Programming Vol 1 because it led me to looking into probability which has turned out to be an infinite rabbit hole of discovery. (a grad students highly opinionated list) https://www.amazon.com/gp/richpub/listmania/fullview/1F85VWN...

The Second Edition of Courant & Robin's 'What is Mathematics?', revised by Ian Stewart is good. This is from the preface:

" In short, it wanted to put the meaning back into mathematics. But it was meaning of a very dif- ferent kind from physical reality, for the meaning of mathematical ob- jects states "only the relationships between mathematically 'nndefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. "

https://www.amazon.com/Mathematics-Elementary-Approach-Ideas...

I am totally biased, but I think my book is a good introduction to math fundamentals (high school) and calculus: https://minireference.com/

preview: https://minireference.com/static/excerpts/noBSguide_v5_previ...

As an added bonus, the book also covers mechanics (Physics 101), which is really important to understand to build up you general modelling skills. (Physics is all about coming up with math models to describe reality.)

I read a lot of math in ML papers, but I don't do any exercises, and maybe that's why I am not sure of myself. A very gradual collection of problems pertaining to linear algebra, calculus and statistics would be great. Something like Khan Academy, but the key point is to be gradual, so as not to terrify me before I get through it. Ideally such a training ground could guide students through problems based on exercise logs collected from other students, by creating an optimized curriculum. Anyone knows where to look?

I made a 3-page overview of calculus as an extra section of some notes for a class I'm teaching. Mostly to see if it could be done.

Here it is. https://www.dropbox.com/s/yp8ijkzir4h8rz9/Calculus%20in%20e%...

I'm REALLY sorry for the picture quality. I'll use a proper scanner later. My phone's camera and app are terrible.

In case you check back on this, I made a better resolution copy. Hopefully it's useful to someone :) https://drive.google.com/open?id=0B2FKX2JZ061dTzRNTjl6SFJVaV...

If I were you, I'd start here (free book on the basics of math without which it's very easy to drown in the so called proof-based math):

http://www.people.vcu.edu/~rhammack/BookOfProof/

Since philosophy was your favourite, you might enjoy the novels of Greg Egan. As far as I can tell, he puts good mathematics in his stories.

I recently read my first Greg Egan and enjoyed it (Quarantine). Got any recommendations for specifically the ones he put good math into?

Perhaps Incandescence, though it's less about mathematics and more about a society working out the physics of its environment. His website has more maths on it than I recall being in the novels.

Schild's Ladder is terrific.

http://us.metamath.org

a) get started on something - if you can't decide, roll the dice and pick something at random

b) ignore everything else until you are done

c) repeat

Don't worry that at your pace it will take ages - very soon you will develop an idea how to rank what you should look at next. If you don't, go back to a).

Obviously it's useless to think about it too much when your knowledge about a subject consists mostly of holes and gaps, so first gather data (a).

I can't quite see how concrete your question is, but try Khan Academy. Follow the suggested order if you have no preferences.

pretty much this.

The last math class I took was in high school and I dropped out of college because my CS degree required me to do a metric fuckton of math, which I hated at the time.

A few years ago I got interested and started working on Integer Factorization. It is not that likely that I will solve that problem but in the past few years I have learned a lot more math by attempting to solve that problem that I learned in college!

Moral of the story: Just have fun and try to solve some problem(s).

But some areas require knowledge base of others

Which you won't find out unless you start - somewhere, anywhere. How are you going to find out when you just sit there and think about something you know nothing about? Wait for sudden enlightenment? Math is one off the easiest subjects to get started, soooooo many starter books and by now even great online courses (again: Khan Academy, for the basics, probably many more). So much guidance. There is so much out there - free and easily accessible thanks to the Internet, the problem isn't "how do I get started" but "what resource do I use to get started".

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