http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

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.

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.

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.

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...

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...

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.

I'd recommend the following three books to similarly cover the three main areas of higher mathematics (analysis, algebra, and topology):

1. Principles of Mathematical Analysis by Rudin (http://www.amazon.com/Principles-Mathematical-Analysis-Inter...)

2. Algebra by Artin (http://www.amazon.com/Algebra-2nd-Featured-Titles-Abstract/d...)

3. Topology by Munkres (http://www.amazon.com/Topology-2nd-James-Munkres/dp/01318162...)

Working through all three (including the exercises!) will give you a solid understanding of the basis of modern mathematics. If you don't have experience with proofwriting, you might find them difficult at first - the activity is very different from performing calculations or solving equations. It's also best to have someone trained in mathematics talk to you about the proofs, until you develop a feel for the needed level of logical rigor.

Rudin, in particular, leaves a lot of work to the reader; going through that book is the most intellectually difficult work I've ever done. If you find it hard-going (which is completely natural), you might want to try Artin first, especially since you have some background in algebra.

(Incidentally, these are the three books used to teach analysis, algebra, and topology to MIT mathematics majors. You can look up the assignments and exams for the three courses - 18.100B, 18.701, and 18.901 - for a good list of exercises to work through.)

4. Topics in Algebra by I.N.Herstein.

It has a slightly better (and tougher) set of problems than what you would find in the book by Artin.

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

As 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.

... 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.

After that, anything by Serge Lang. He is a first-rate mathematician and a fine educator. A master of mathematical exposition, a rare talent.

I started with two books: "Principles of Mathematical Analysis" by Walter Rudin, and "Abstract Algebra" by Israel Herstein. Keyword is "started". Nearly every paragraph of either book sent me to a goose-chase of research and reading up on supporting materials. Safe to say that by the time I was nearly done with either, I have acquired about 50 other math books and had about zero social life ;-)

Good news is that good math books are cheap, compared to anything computing. You can find classics for $1-$5 in most yard sales and 2nd hand bookstores. The trick is to buy thin math books; the giant, multi-color text books they use in schools are highly confusing, at least to me. It's 200 pages of math stretched to 1200, with a similar jump in price.

Mathematics is both style and substance. Once you get the hang of the basic language, the succinct delivery style, the proofs and generalizations, how notation is introduced and then elided when it becomes too apparent .. all these will add up to help you navigate more "advanced" texts. You will come to recognize what is a well-posed mathematical problem, and what is not, even if you don't understand the domain itself.

Finally, if you have the option to study this formally at an institution, do it. The 2nd best option is to find a friend that shares this interest and discuss things. I didn't study it formally, but I was a barista at a coffee shop with plenty of mathphile customers. I had a lot of 1:1 instruction for highly qualified people on my 10 minute breaks :-)

Update: yeah this list was last updated in 2000.

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

http://mitpress.mit.edu/sites/default/files/titles/content/s...

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?

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.

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://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://en.wikipedia.org/wiki/Concrete_Mathematics

Also Knuth's TAOCP series are very formal and dense, in a good way.

1. Analysis - The Elias Stein, Rami Shakarchi Analysis series, Meassure Theory by Terrence Tao, Lebesgue Integration on Euclidean Spaces by Frank Jones, A radical approach to Lebesgue Integration by Bressoud

2. Algebra - Can't do much better than the classic texts by Herstein & Artin here. I know Artin doesn't get much love but after a point, you really begin to appreciate it.

3. Complex Analysis - is there a better math book than Visual Complex Analysis by Needham?

4. Topology - i'm partial to a Dover book on Topology by Theodore Gamelin. I learnt the basics of point-set topology & homotopy form this book. Great exercises.

5. Number Theory - check out the 2 problem books on number theory by Ram Murty. Melvyn Nathanson's book is quite good as well.

6. Combinatorics - Combinatorics of Finite Sets by Ian Anderson. Extremal Combinatorics by Jukna.

7. Graph Theory - Diestel or Bondy/Murty.

8. Galois Theory - Rotman is great.

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...

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

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!

http://groups.csail.mit.edu/mac/users/gjs/6946/calculus-inde... http://www.amazon.com/gp/product/0262019345/

Edit: included Amazon link.

  "This extraordinary work takes the reader on a long and
   fascinating journey--from the dual invention of numbers
   and language, through the major realms of arithmetic,
   algebra, geometry, trigonometry, and calculus, to the
   final destination of differential equations, with
   excursions into mathematical logic, set theory, topology,
   fractals, probability, and assorted other mathematical
   byways. The book is unique among popular books on
   mathematics in combining an engaging, easy-to-read
   history of the subject with a comprehensive mathematical
   survey text."

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

______________________

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...

________________________

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

___________________

(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.inference.phy.cam.ac.uk/mackay/itila/

Mathematics: from the Birth of Numbers.

It's mainly pure math but is a great foundation. I mean really better than anything else I've read.

That would be just a list of title to browse. Buy the ones you find useful.