Ask HN: Learn Math the Hard Way

It's not comprehensive by any means, and you probably need to know at least calculus to be ready for most of this, but it covers some pretty cool stuff, including RSA.

Generally in advanced calculus I avoided the discussion of, and much connection with, geometry, Stokes theorem, and exterior algebra. So, I avoided Buck, Fleming, Spivak, and of course also, now in English, Henri Cartan, 'Differential Forms'. I even avoided the classic applied advanced calculus text, long used at MIT, Francis B. Hildebrand, 'Advanced Calculus for Applications'.

For Loomis and Sternberg, I agree with you, and have both the hard copy and the PDF.

Since I've mentioned such advanced calculus, I will try to save many students: Students, there's a secret. The secret is that vector analysis, Stokes theorem, etc. are important in physics and engineering; they will also be important in computing when computing concentrates on such physics and engineering. But still mostly what you will find in physics and engineering is vector analysis much as it was done in the 19th century which the late 20th century math departments liked about as much as a skunk at a garden party.

If you read the modern treatments, complete with differential forms, then you will be at the head of the class in an advanced class in general relativity (e.g., Misner, Thorne, and Wheeler) but will still be lost in much of old physics and engineering!

So, what to do? Sure, go to Tom M. Apostol, 'Mathematical Analysis: A Modern Approach to Advanced Calculus', Addison-Wesley, Reading, Massachusetts, 1957. The good thing about this book is the lie in the title -- it's mostly a 19th century treatment and not "modern"! So do whatever you have to do to get a copy. And get the 1957 edition and NOT a more recent edition where he omitted the 'good stuff'!

Then, in about 20 pages of the sweetest dessert you ever tasted, with line integrals, conservative force fields, and potentials, volume and surface integrals, nice stuff like that, you will find a charmingly clear presentation of what you need. Right: The treatment is not up to the precision of Rudin and actually needs pictures. Still it's what you need for much of physics and engineering. It's, uh, 'intuitive' math; trying to make that material as precise as Rudin could take you, well, a long time.

And it's EASY -- can take it with a couple of beers and have a really fun evening. Then don't tell anyone where you learned it! Besides, at its core, it's just nice uses of the fundamental theorem of calculus you saw in freshman calculus! Did I mention, it's easy?

The key point about Rudin's 'Principles' is the care with which he covers the real numbers, compactness, continuity and uniform continuity, sequences and series, and the Riemann integral (yes, patched up with the Stieltjes extension which isn't much different). So, he concentrates hard on the foundations. For someone like the OP, getting those foundations solid is likely more important than rushing into many of the more famous topics in 'advanced calculus' -- Fourier series, the heat equation, Lagrange multipliers, vibrating strings (boundary value problems), the Navier-Stokes equations, series solutions to ordinary differential equations, etc.

While I like Rudin, 'cut many of my math teeth' on Rudin, and really like some of his treatments of some topics, I omitted some notes on how to read Rudin; some such notes could be helpful. In particular, Rudin has some places where it's easy to get stuck, and students should be advised not to get stuck (don't assume that just because you can't see how to solve some one exercise must be missing something important) and if necessary just to look for other sources, ask for help, skip over and come back, or just f'get about it. Rudin was one of the best writers of his material, but he was not perfect, varied, got easier to read as he wrote more, but still is relative...

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I would also suggest more emphasis on logic, sets, and type theory. Category Theory is also something fairly common in CS. So I would recommend (Free!): www.cs.unibo.it/~asperti/PAPERS/book.pdf

Abstract algebra: http://abstract.ups.edu/

And while I am skeptical as to the need of linear algebra for CS, it is such a key requirement for mathematical maturity I will suggest: http://www.math.miami.edu/~ec/book/. If Linear algebra is a building then abstract algebra is the frame. The two should really be taught at the same time.

Right. The only advanced calculus text I listed was 'Baby Rudin', and the main contribution there is just to get some of the more important properties of the real numbers, Euclidean n-space, infinite sequences and series, and Riemann integration solid.

If a CS student is to stop before these topics, okay, but if they are going to go on then these topics will be part of what is generally assumed.

For CS, as we know, it is doing what EE did -- moving beyond its core tools and into what to do with those tools. E.g., EE got into nonlinear filtering and stochastic integration. E.g., CS is now getting into both optimization and statistics. Then being handy with that Baby Rudin material, at least the early chapters, will get to be important.

With Baby Rudin already done, there can be a good course in differential equations, and such a course can be a good way to see some of the value of what did in linear algebra and Baby Rudin and to exercise that material. At some point in the future, a CS guy might well get into some work involving differential equations -- viral growth models, flight of airplanes and space vehicles, and much of mechanical engineering.

One of the main themes in the future of computer applications is handling 'randomness', and my view is that serious work in that direction should have the measure theory foundations. I did an A/B on that! Early in my career I tried the easy way. After measure theory, Neveu, Breiman, Loeve, Dynkin, Lipster, Shiryaev, etc., I concluded that the measure theory approach to probability, stochastic processes, and statistics was essential.

In particular, without measure theory, people too easily get totally stuck in the mud on what 'random' means, while with measure theory Kolmogorov has a really nice answer. My view is that people may not like Kolmogorov's answer but that, with some really simple assumptions, we get forced into that answer anyway!

For the rest, the Hilbert and Banach spaces won't go away! E.g., a huge dessert buffet of really finger lick'n good applications of the Hahn-Banach theorem is David G. Luenberger, 'Optimization by Vector Space Methods', and that book will be a nice source of methods for a lot that computing people might encounter.

> You left out graph theory and combinatorics, both of which are extremely important to CS.

For combinatorics, I assumed that one would get enough, in some ways deeper than in Knuth's TACP, from abstract algebra and elementary probability. E.g., combinatorics is a lot about counting, and so is group theory in abstract algebra.

For graph theory, I assumed that one would get enough from optimization on networks. E.g., a 'basis' in the network simplex method on a network is a minimum spanning tree! And the max flow/min cut theorem can follow just from linear programming. Dynamic programming, which I mentioned, can be viewed as graph theory.

I did mention linear programming (and so does CLRS), and one reason is that in the CS study of algorithms the algorithms for linear programming are important, and surprising, benchmarks. Also integer linear programming is one of the more important motivations for the question of P versus NP.

For some of the surprise, the simplex algorithm has low degree polynomial expected performance (K. Borgward) but exponential worst case performance (Klee and Minty), but the polynomial algorithms (Khachiyan) when they are faster than simplex have both of them too slow for practice! So, at one of the first places we looked at computational complexity for problems more challenging than, say, heap sort and AVL trees, an exponential algorithm turned out to be superior in nearly every sense of interest to a polynomial algorithm! There never was a guarantee that the study of computational complexity would be easy!

But there's a lot of overlap: I didn't mention courses in 'finite mathematics', combinatorics, or grap...

But there are some risks.

First, broadly a big risk is getting off track. There are many poor books, and dig into one of those and might not ever come out. And there are side streets; might go down one of those and never get back to the main road. Don't do those things.

So, the lessons: Pick subjects carefully. In any subject, use only very highly recommended books, and use more than one in parallel although likely one book as your 'primary' source and the others as 'supplementary'.

For book recommendations, get those from a better source than Hacker News! I suggest getting recommendations from some of the best courses and professors in the departments of at some of the world's best, and I mean at top 20 or top 10, research universities. So, for a subject you want to learn, find what books such courses, profs, departments, and universities are using and recommending.

Second, another big risk is getting stuck. One way is to encounter an exercise you can't solve, believing that you are missing something important, and then just grinding to a stop, for days, weeks, or forever, on that exercise. Don't do that.

(A) There are a lot of 'misplaced' exercises. Just because an exercise is in a book doesn't mean that it is reasonable to be able to solve it yet. Yes, in a good book you should be able to solve 90% of the exercises, but one book may have 1-20 exercises that are just nasty chuckholes in the road. I know one beautiful book where the exercises are so difficult, and apparently usually beyond the book, that one should get a prize for each doable exercise they find! (B) There are some tricky exercises that are just tricky and not really important. (C) There's no way you are going to miss much if you skip a few exercises. (D) You can continue on, maybe ask for help, look at your supplementary sources, etc. and then either get a solution or conclude that you haven't missed much. Yes, do work hard, but definitely don't get stuck. After enough hours at one spot, just move on.

Going on will sometimes give you the tools you need to resolve the place you were having trouble. Generally it's no sin, and sometimes helpful, to rush ahead quickly, get an overview, and then return to a more complete pass, and the overview can help you judge if the place you were having trouble was important or not. Also there's no way just one pass over some material can be as effective as three or so. E.g., one goal is a 'synthesis' of the material, and that needs several passes and several sources.

Third, there are occasional actual errors in the sources. So, if something doesn't make much sense, then it's wise to suspect an actual error someplace. You might be able to resolve the issue by consulting supplementary sources, doing some derivations on your own, say, to find a counterexample, or asking for help. By now the Internet may actually be a good place to ask for help. And, there's no law against writing the author of a book!

Fourth, some books are just much better than others. And even a good book might make a mess out of a few topics. So, again, consider several sources.

Fifth, there is a broad point about 'standards of quality'. Math is by a wide margin the most precise of all academic subjects. Still, in the total body of math that is done currently or already on the shelves of the libraries, the standards of quality vary widely.

Some of the most highly polished math ever put on paper is by (the collection of largely anonymous mathematicians) Bourbaki; very little of the rest of math is so highly polished; some of the best math and math writing is not as highly polished as Bourbaki.

So, there is a danger: It's possible to work to standards that are, in some or nearly all respects, too high. Beware that in the best work, the best of that work is not some high polish but in the real 'content' ...