Ask HN: Math books like SICP?
I'd like to start learning undergrad math at a rigorous level.
As a CS grad I already covered certain topics like abstract algebra at a respectable level. But I'm lacking consistent and broad knowledge throughout all areas.
I'm looking for math books covering undergrad topics like SICP, which Peter Norvig described as "[...] a way of synthesizing what you already know, and building a rich framework onto which you can add new learning over a career" [1].
[1] http://www.amazon.com/review/R403HR4VL71K8
91 comments
[ 3.2 ms ] story [ 133 ms ] threadAs some recreational reading, less suiting the original request, I very much enjoyed David Foster Wallace's 'Everything and More: A Compact History of Infinity' (http://www.amazon.com/Everything-More-Compact-History-Infini...). DFW is not for everyone, but I enjoyed it a lot. Maybe just check it out of the library first to see if it's for you.
http://mitpress.mit.edu/sites/default/files/titles/content/s...
The main content is just 'measure theory', and that's just freshman calculus grown up. Why? Because near 1900 it became clear that freshman calculus was clumsy for some important progress, especially cases of convergence of functions.
Measure theory? Well, first cut, 'measure' is just a grown up version of ordinary area. Simple.
Why interested in measure theory? Because want to cook up a new way to do integration, that is, what freshman calculus dues with the Riemann integral. Recall, the Riemann integral partitions the X axis and approximates the area under the curve with tall, thin rectangles. Measure theory partitions the Y axis: At first glance this seems a little clumsy, but in the usual cases get the same number for area under a curve and in bizarre cases, that can get from converging functions, get a nice answer that Riemann integration can't do.
Royden likes Littlewood's three principles, and they are cute. So, spend an evening on them. Yes, it's possible to use Littlewood's to do the subject, but there is a better way, also heavily in Royden, roughly called 'monotone class arguments' -- which are gorgeous and turn much of the whole book into something quite simple. So, prove the theorem for indicator functions. Then extend to simple functions by linearity. Then extend to non-negative measurable functions by a monotone sequence. Then extend to integrable functions by linearity. Can knock off much of the book this way, e.g., Fubini's theorem which is just interchange of order of integration grown up. The foundations of this little four step process is Fatou's lemma, the monotone convergence theorem, and the dominated convergence theorem.
About 2/3rds of the way through Royden is a single chapter that essentially compresses much of the rest of the book -- I would have to step into my library to find the chapter.
For the early exercises on upper and lower semi-continuity, they are a bit much and you likely won't see that topic again. So that exercise can be skipped.
Royden is elegant beyond belief; if you still have trouble finding the main themes, then chat for an hour with a good math proof who understand Royden well.
Then, don't miss the Radon-Nikodym theorem: It can be seen as a grown up version of the fundamental theorem of freshman calculus but, really, is much, much better. The role of the Radon-Nikodym theorem in 'modern' (i.e., Kolmogorov) probability theory, stochastic processes, Markov processes, martingales, etc. is astounding.
It starts off with metric spaces and Euclidean n-space. The main goal is to cover open and closed sets, convergence, and compactness.
Why? Because those topics are crucial in continuity and uniform continuity.
Why want those? Because continuous on a compact set means uniformly continuous and, thus, that the Riemann integral exists.
Any questions?
As a followup, Paolo Aluffi's "Algebra: Chapter Zero" is the best synthesizing text for abstract algebra for a beginning graduate student. The thing that makes it so amazing is the writing style: it introduces and demystifies category theory, and then discusses groups, rings, modules, linear algebra, fields, r-modules, and advanced topics (toward the end) with the unifying theme of how they work and relate as categories. It is very much a book that focuses on the why over the what and how. And there are many many many juicy exercises.
I would recommend against Spivak's Calculus on manifolds. It's too dense and too focused on advanced topics (unless you're an ivy league undergrad, you don't learn cohomology). That being said I don't have an analysis reference that fits your request.
As a CS person I can continue with book recommendations related to computing-related topics in math (computational algebraic geometry comes to mind). Let me know if you're interested.
I was looking for a real analysis companion, perhaps baby Rudin. I was also wondering whether it'd make sense to proceed directly to real analysis, or to step down a bit and read something like Spivak's Calculus.
I'd also like to hear about recommendations at undergrad level in the fields of logic and set theory, geometry, combinatorics, and probability theory. Those would complete my basic math curriculum.
I'd actually like to hear about alternatives to Rudin. My undergrad analysis class used it, but it's lack of diagrams was particularly bothersome. I'd spend an hour digesting a rat's nest of a paragraph only to discover that the underlying concept was simple enough that even a rough sketch ought to be able to get the gist of it across in seconds.
I also like Strang for Linear Algebra for undergrads. Beyond that it's Hoffman and Kunze.
Likewise I essentially learned all the probability theory and combinatorics I know from people and scraps, so I can't recommend a synthesizing text.
Undergraduate geometry can be a mess, so you should know what you're looking for. There are three kinds of undergraduate geometry classes: 1. Euclid's Elements (ugh), 2. The hyperbolic version of Euclid's Elements (meh), and 3. The "Erlangen Programme" style, which involves studying geometry via group theory and linear algebra. As you can probably tell, my recommendation is to study the last, because you already know group theory and with the other two you'll spend a lot of time wondering whether you can apply some basic obvious fact to prove some other basic obvious fact. The Erlangen style also allows you to describe projective and hyperbolic geometry via linear algebra (as well as the Euclid way), which is far more useful. See, for example, my post on projective geometry for elliptic curves [1]. I went through all three styles, but the last unfortunately had no textbook.
I'm not a huge fan of logic/set theory, but again the best treatment I can see for basic logic is to view it as algebra. In that vein, Halmos's "Logic as Algebra" was all I needed, and the prose is superb. This book does not contain any real set theory (say, about higher cardinals), but it's nice and short.
[1]: http://jeremykun.com/2014/02/16/elliptic-curves-as-algebraic...
Wilf - Generatingfunctionology (CRC 3rd ed.; free 2nd ed. pdf[1])
Lovasz - Combinatorial Problems and Exercises (AMS Chelsea)
The classic probability book is Feller (2 vol), but it's absurdly priced. There's also Sidney Resnick's Probability Path and Adventures in Stochastic Processes. Grinstead & Snell - Introduction to Probability Theory is free[2], and there's also Chung's A Course in Probability Theory (Academic Press/Elsevier).
Dover publishes at least three good books on counterexamples and pathological cases: Counterexamples in {Analysis, Probability, Topology}
[1] http://www.math.upenn.edu/~wilf/DownldGF.html
[2] http://www.dartmouth.edu/~chance/teaching_aids/books_article...
My suggestion would be to narrow it down to a specific field you're interested in like abstract algebra and ask for suggestions about that field. Even narrowed down the field might be too big - a basic introduction to abstract algebra would probably contain the same amount of information as SICP.
CS people tend to find discrete math and linear algebra most useful in their work, but maybe you want to learn something just for the sake of it and not for work.
Apart from that I wanted to expand on basic set theory, geometry, combinatorics and probability theory to have a well-rounded basic education.
My ultimate goal is to be able to digest advanced probability and statistics books.
Re: statistics and such though, I have heard very good things about http://www.indiana.edu/~kruschke/DoingBayesianDataAnalysis/
If you are interested in a Math 55 approach (yikes! I took 25), you could consider using their syllabus. http://abel.math.harvard.edu/~elkies/M55a.02/index.html
Unfortunately I have to hold my own tongue this time, because books I found useful are mainly in russian and they surely aren't like SICP. And I still lack the whole understanding of the area anyway.
* Set theory * Linear algebra * Geometry * Real analysis * Combinatorics * Probability theory
Topology, number theory, abstract algebra (I mean, real one, not CS-course basics), statistics, tensor analysis? Isn't that "undergrad math"?
For things like Set theory/combinatorics/logic basics I'd recommend Rosen's "Discrete Math and Applications"[1]. CS oriented, simple, interesting, broad. Covers all the basic stuff.
Linear algebra — two books, "Linear algebra done Right" and "Linear algebra done Wrong". Second one more math-oriented, the first one — pretty simple, pretty clear, fun to read.
Real analysis ("calculus" you mean?) — I personally learned from different sources and probably the most concise book I read is Fichtengolz's "differential and integral calculus", but I don't know if it's available in english. I guess, almost any book on topic is fine.
Geometry & Probability theory — not sure what to recommend, because books on topic vary in depth dramatically, I would appreciate myself if somebody would outline the borders for what to cover first. Anyway, most of what I read and found useful is in russian, unfortunately. But still, what do you mean by geometry and prob. theory? Differential geometry, Riemannian geometry, erlangen program covered or only basic euclidean/analytic geometry stuff? Same goes for probability. If you care only for very basics — Khan's academy (or any random youtube videos) is fine. Any intro book on statistics covers it as well.
[1] - http://www.amazon.com/Discrete-Mathematics-Applications-Kenn...
Update: yeah this list was last updated in 2000.
https://mitpress.mit.edu/sites/default/files/titles/content/...
Don't be fooled by the title and authors of the book!
Learning Lagrangian mechanics when all your life you only ever heard about the Newtonian approach is a lot like stumbling upon Scheme after years of programming in C.
I'm looking forward to cracking the cover of SICM once more this summer!
https://math.stackexchange.com/questions/62190/mathematical-...
(comparing to Feynman's Lectures on Physics rather than SICP.)
However, I agree with zodiac: Math is a much broader field and I think very few living people have the kind knowledge you're talking about: consistent and broad knowledge in all areas.
I have the same background as you and I started doing exactly this about 8 years ago. I'm just finishing my PhD in math. I'll give you some advice. First, studying math for its own sake by yourself is extremely hard. That's one reason I ended up back in academia. If you can't go back to school, still find some kind of community. Second, rather than studying generally, try to identify a goal to work towards. What are you trying to understand or figure out? Third, try to model your plan on a rigorous undergrad program, e.g., MIT. Then, in each main undergrad area (algebra, analysis, topology, geometry, etc.) try to find the "SICP" and study that. For general book recommendations, I like Fowler's A Mathematics Autodidact's Aid:
http://www.ams.org/notices/200510/comm-fowler.pdf
That's great one! Thank you for pointing this out very much.
For first-year stuff, I would recommend No bullshit guide to math and physics[1] and the No bullshit guide to linear algebra[2] of which I am the author.
[1] http://minireference.com/ [2] http://gum.co/noBSLA
CM covers Category Theory as a general tool for probing and exploring algebraic concepts beginning as simply as endomorphism and extending all the way out to how it can be the foundations of a generalization of set theory via Toposes... all in a cheery and explorative fashion which really illuminates why the ideas work instead of merely stating them.
I'm not sure what the required background might be, but it's probably pretty minimal.
Conceptual Mathematics puts a lot more time up front motivating the material with examples and tries to build your intuition before getting into the details. If all you've done previously was linear algebra and some discrete math, this book is probably better.
All Engineers go through both years, except Computer Science who don't do the common second year and they directly go to Computer Science.
In these two years, everyone goes through this (maybe it'll give you some ideas on what you want to add):
I'll only list the "Maths" we take first and second year:
First year: - Algebra: (a long course, bottom up. From Boole's algebra, to groups, sigma-algebra, yadda yadda), linear algebra(vector spaces, etc)..
- Probabilities and Statistics.
- Analysis: (Taylor series (Lagrange, Laplace, Young, Cauchy, Maclaurin), integrals, differentiations, different series, convergence/divergence kung fu), Riemann overall, proofs, etc.. Functions, multivariable, real and complex, etc.
Second year:
- Analysis I - Numerical Analysis (Equation systems, Gauss-Seidel, different algorithms(also calculating their speeds), Newton-Raphson, extrapolation, interpolation, etc).
- Analysis II - Integrals(up to 3rd - curves, areas/surfaces(Green) and volumes (Ostrogradsky)), Differential equations (Wronskian, etc).
This is the minimum (to be able to function in other modules, and some other modules are needed before you can function in these, so there's sort of bootstrapping of sort).
And then it depends what you take as specialty (if it's something involving Signal Processing, for instance, or Control Systems, you also need to do stuff).
Hope that helps and you can find some things.
PS: None of these are done with computers, so computing stuff with Newton algorithm and operations on big matrices are all done by hand. It takes a lot of time.
PPS: We don't have multiple answer questions. There's a question, and you answer it (and some answers take multiple pages).
Also, most tests are designed in a way that even if you have the answer sheet right next to you, it still takes you more time to copy the answers than the time of the exam itself. i.e: Even if you don't think and only "write", the time-frame is too tight.
So the test is impossible then? If you don't even have time to copy the answers how can you possibly have time to work out the problems?
For instance, in Control Systems .. We get a problem to design a system with certain characteristics. You then have to do Z transform, etc. Then design correction to match what you're asked for (a certain overshoot max, phase, etc).
Which involves a lot of matrice multiplication (Pontryagin, optimal control). Big matrices multiplied(and elements aren't numbers, they're bits of transfer functions. But I replace them with letters and double indices, many students recopy the same expression all the way :) ).
Or your design an RST control (polynomial, much, much complex than PID).
All of this takes a lot of time to write and requires a lot of concentration (if you screw up just one element in a matrix, all ensuing is wrong and gets you nothing, so you have to be rigorous and not the day-dreaming type).
And you end up pretty much not doing some part of the thing, which is something most students don't find frustrating, since most of them wouldn't touch a part of it anyway (they haven't studied it, they don't understand it, or something else).
They're not designed like that by some sadistic tendency, it's just some stuff that needs to go down the exam..
Although I remember a teacher who told us as he gave us the exam sheet: "Don't bother looking at the verso". Meaning "You'd be happy if you only did the recto to get your 10/20". It was a deliberate move.
And in addition, it's mostly end of year exams that are like that (content from the whole year). And not all modules are like this. Some are calculation intensive and next step is dependant on current step, so making a mistake in the beginning is "fatal".
But then again, I find the exam thingy slightly ... well, let's not even go there.
In high school algebra, the variables stand for numbers. In abstract algebra, the variables might stand for the rules of algebra.
It's long on proofs and short on numbers.
https://github.com/ystael/chicago-ug-math-bib (updated Univ of Chicago bibliography
http://math.ucr.edu/home/baez/books.html
http://www.maths.cam.ac.uk/undergrad/course/schedules.pdf
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and 2 i got from HN and /r/machineLearning
http://www.reddit.com/r/MachineLearning/comments/1jeawf/mach...
https://github.com/vhf/free-programming-books/blob/master/fr...
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finally, the "Maths for PHysics" texts
http://www.scribd.com/doc/156523189/Boas-mathematical-Method...
http://www.goldbart.gatech.edu/PG_MS_MfP.htm
http://www.scribd.com/doc/91670553/Arfken-Math-Physics
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(if i had to recommend only one book, it would be the Boas, or maybe Princeton Companion: http://press.princeton.edu/titles/8350.html
http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...
The best thing I like about this book is that it often first gives a real-life example of a problem, then gives some history of how the problem was solved, and what the solution was. Even the first chapter was highly illuminating.
I credit that book with much if not all my mathematical insight.
Thank you.
(Just seeing that cover leaves me all tear-eyed, reminiscing over that wonderfully irresponsible time.)
replace "that" with "those" ( http://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics ).
the dover books are also a good series, and pretty much anything by Artin is good. i also looked at the Halmos book a couple of people have mentioned.
perhaps one thing to be aware of, though, is that you're not always going to learn things in the linear way they're layed out on the page. moreover, it might be helpful to have more than one book for any given subject. Lang's Algebra, for example, is a really good reference, but a tome if you read it like a text. so you might pick up something small and subject-oriented with a lot of exercises like Artin's Galois Theory or Atiyah's Commutative Algebra, and supplement it with a reference like Dummit and Foote or Lang's texts on Algebra as a whole.
oh, right: whatever book you choose, do the exercises.
My warning as a math student is that a lot of book recommendations are just a tad bit elitist. Don't stick with a single book too long if it isn't cutting it for self-study. Exercises are good.
[1] http://press.princeton.edu/titles/8350.html
https://en.wikipedia.org/wiki/Concrete_Mathematics
Also Knuth's TAOCP series are very formal and dense, in a good way.
... something covering "how to do math". This might be an introductory real analysis or linear algebra book, or (alas, I haven't read much in this area) you could do no wrong looking for books on problem solving, contests, and inequalities, say by Polya or Andreescu.
... a higher-level view of day-to-day mathematical practice. There probably isn't one book for this, but I'd recommend Loomis and Sternberg's Advanced Calculus as a summation of linear algebra and calculus on manifolds; you'd also need to read on complex and functional analysis, algebra (Lang and the unofficial companion volume?), topology ...
... a broad but shallow introduction to several fields and applications unified but a common underlying approach (abstraction and programming language design). I'd recommend Geroch's Mathematical physics, which, as the name implies, studies algebra, algebraic topology, and functional analysis through the common lens of category theory.
... a somewhat quirky book on foundations? You could look for a book on naive or axiomatic set theory, categories, or type theory.
He takes the approach of starting right at the beginning of human history when we first came to look at the stars and seasons and try to understand why they work, and then builds the math knowledge with each lesson. Excellent book!