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Ask HN: Good Reading/Immersion in Mathematics
48 points by jawee on Nov 14, 2010 | hide | past | favorite | 31 comments
I'm interested in getting a good solid basis for study of mathematics. Right now, I am interested in a broad and not necessarily too deep to start with; instead, I want to be acquainted with all of the different fields beyond what I've done in my mathematics classrooms.

What books and web pages do you recommend I read, as well as what blogs and podcasts are good to follow to learn more on a constant basis.

Thanks!




Pick up a discreet mathematics book. Learn logic, set theory, and writing mathematical proofs (induction, etc.)

I really wish they'd put a class like that right after basic symbolic algebra in normal school curriculum - it's far more useful in the modern world than trigonometry.


I agree whole-heartedly. I work at a large company and I find I can typically formulate business rules in set and function theory in mathematic notation, which typically blows away my peers and associates because they either 1) know it and are impressed to see it used in "business rules" or 2) think it looks like some space-alien language (which it may as well be to them).

Otherwise these rules come out as a set of vague half-instructions that always lead to rounds of revisions in UAT. Oh, and "we only scheduled a week of UAT".

A broader knowledge of discreet theory would be much more helpful than understanding a sine or cosine at a... trigonometric level.


I can't believe no one has mentioned Concrete Mathematics by Knuth & friends yet.

It's incredibly well-written. Very challenging, yet totally approachable and accessible to someone with even just a rather basic (even foggy) understanding of "high-school" math.


What is the prerequisite for reading this book. Can I read this book effectively with basic knowledge of high school algebra and some calculus?


Yes. I did. It will be very challenging.

I, quite a few times, spent more than four hours with a pencil and paper making my way through 2-4 pages of material, as since my background was lacking, there were parts where I had to figure out how they got from A to B. IIRC, I spent multiple days on a few paragraphs at one point.

That said, it is written how I think math books should be written, is considered part of the start of Discrete Mathematics, and is conversational, illuminating, well-layed-out, and approachable despite it's level of difficulty.

I can't recommend it enough.

Don't view it as something where if you're not making progress you're failing. Take the time to ensure you understand every bit of what you read before moving on, and do all the exercises you can muster.


thanks a lot.


Can't agree more about the value of Discrete Math. Any programming worth his salt can wax about combinatorics and counting problems at a moment's notice. I'm thoroughly convinced I'd be a few years further along as a programmer had I learned discrete math at an early age.

But anyway, the Teaching Company offers a phenomenal course on Discrete Math, taught by none other than the "Math Magician" Arthur Benjamin (seriously, this guy is magic. He can do 5 digit squares in his head!). It's easily torrent-able / available through more traditional means.


Potentially related threads :

Math for hackers : http://news.ycombinator.com/item?id=672067

Good books on mathematics for somebody who's only taken high school math? : http://news.ycombinator.com/item?id=299687

Recommendation for (re)learning Math Skills : http://news.ycombinator.com/item?id=1449799


"want to be acquainted with all of the different fields" may actually be quite hard - you can't do complex analysis, differential equations in multiple dimensions etc. without having a very firm grip on standard analysis (including all the proofs and definitions that they normally skip in high school).

If you want to build up your math muscles (as a good preparation for actually studying maths), you should have a look at some discrete mathematics books (the one I had was "Discrete Mathematics" by Norman Biggs) as they teach you to think in terms of proofs.

If you want to get a thorough foundation for non-discrete maths, you should start with a good (university math) analysis textbook (No idea what's a good one in English).

Another approach you could take is to take a math book that is targeted at physicists and EE people - those usually skimp on the proofs and don't contain enough detail to understand the fundamentals behind it all, but bring you to the interesting (to physicists and EE people) stuff much quicker than a real math course would.

Oh, and if you hang out on Youtube, be sure to watch the catsters - this is category theory, presented by actual working mathematicians, at an accessible level (and with a cute UK accent too).


Is there dependency graph of mathematical fields somewhere? That would be useful.


i'm not able to dig it up right away - but just a couple of days back there was a blog post by someone who gave a long, very comprehensive listing of topics that take you all the way from basic to postgraduate level math.

It was organized in a dependency graph kind of way.


You may find this useful: http://math-blog.com/mathematics-books/ (DISCLAIMER: It's my site.)


I like this list; thank you for taking the time to compile and annotate it.


This is hard to answer without knowing something about your current mathematical background and more about what you want to get out of learning some mathematics.

Assuming that you want to learn some 'university-level mathematics', then you'll really need to be prepared to study and work through problems rather than just read. Mathematics is an area where it's hard to get breadth of knowledge without also having at least some depth because things build very much on each other.

If you really do just want an overview of areas of different areas of mathematics to whet your appetite, there are books by people like Ian Stewart, Marcus du Sautoy and Keith Devlin, all worth reading. Just be aware that reading these is a bit like reading about different programming languages without ever having written a computer program.

If instead you just want to keep your brain engaged mathematically without learning more serious mathematics, there are also plenty of recreational mathematics books out there - Martin Gardner being the name that instantly springs to mind. On a similar vein, you may also enjoy the books of Raymond Smullyan which are more focused on logic.

The only really nice non-textbook taster of university-level mathematics that I have found in Alice in Numberland by Baylis and Haggarty. However it's out of print so you might have problems getting hold of a copy. It is a lovely book though if you can get your hands on it.

Otherwise you are looking at textbooks. I'd recommend maybe 'Introductory Mathematics: Algebra and Analysis' by Geoff Smith as a gentle but rigorous intro to the basics that I'd expect every maths student to learn at the start of their degree course. There are lots of alternatives out there too though. I taught myself lots from Herstein before going to university but that's pretty heavy going and there are better books out there these days. If you look at other books, I'd probably suggest getting one on abstract algebra maybe, covering things like sets/functions and group theory rather than say analysis or linear algebra to start off with, as it's easier to get into the right mathematical mindset if you're not distracted by content which you already have intuitions about.


It may be helpful if you state your mathematical background, though I'm guessing that, if you had taken any proof-based math course, you wouldn't be asking this.

I recommend The Art of Problem Solving I and II. On the one hand, they're intended for (mathletic) middle and high-schoolers. On the other hand, some of their problems are quite challenging, and much of the material therein is what my school teaches in its intro discrete math courses since very few students learned it in middle and high school.

http://www.artofproblemsolving.com/Store/contests.php


I'm currently a junior in high school. I am doing some summer work in math and considering doing it as a degree in college, but so far I have only taken basic Algebra, Geometry, Trigonometry, and currently in Statistics (as well as in a math-focused Computer Science track). I'm looking at Calculus I next year.


I love Calculus Made Easy by Silvanus Thompson (updated by Martin Gardner).


This is an absolute must. You can get it free (the original version) here : http://www.gutenberg.org/ebooks/33283


Related to this, I've found most text books aren't the best way to learn the material since they don't actually provide answers to solutions. You get the theory but little followup practice to apply your understanding of that theory.

Are there any good texts that are more problem workbook style? One example that comes to mind is "Exercises in Probability", or some of the 3000 Problems books as published by Schaum.


I'm working through Calculus by Michael Spivak. If you want a thorough knowledge of calculus(who doesn't?) then I wholeheartedly recommend it. While it's more deep than broad, it will surely help you learn to start thinking more like a mathematician.

http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098896


In my view, the best place to start for a good grounding in rigorous mathematics is Velleman's 'How to Prove It'

As for a good broad overview of many areas, the title that springs to mind is 'the nature of mathematical modelling' by Gershenfeld, though you'd better have some decent maths experience before tackling that one - it can be tough-going, but is refreshing in its breadth and clarity.


Here's my list:

Basic

1. Chapter Zero (Carol Schumacher) 2. Naive Set Theory (Paul Halmos)

Linear Algebra

1. Finite-Dimensional Vector Space (Paul Halmos) 2. Linear Algebra Done Right (Steven Axler)

Real Analysis

1. Real Mathematical Analysis (Charles Pugh) 2. Introduction to Analysis (Maxwell Rosenlicht)

Algebra

1. A first course in abstract algebra (John Fraleigh)

These books are better for self-directed reading compared to some of the classics like Rudin or Herstein. These should keep you busy for a while.


If youre looking to learn mathematics as a tool rather than an end goal the list above seems far too abstract. A good foundation in analysis is probably as abstract as you'll need for a majority of applied fields. For computational science and learning you need to know (albeit very well):

linear algebra (strang, trefethen, golub and van loan) optimization (nocedal, bertsekas) probability (rice, casella & berger, grimmett) statistical learning (tibshirani, bishop)

A good free online book was recently an HN topic: http://news.ycombinator.com/item?id=1738670


You might want to have a look at the "Princeton Companion to Mathematics". It's goals seem close to your "broad but not too deep" objective.


Check out The Unapologetic Mathematician "blath":

http://unapologetic.wordpress.com/


Linear Algebra Done Right, Axler. Don't learn Linear Algebra without it.


http://topics.nytimes.com/top/opinion/series/steven_strogatz...


I really like this one: http://academicearth.org/subjects/mathematics


I'd definitely recommend Godel, Escher, Bach by Doug Hofstadter. Not just about mathematics, but fascinating.

Also, anything by Martin Gardner.


GEB is excellent as a pop-science/pop-philosophy book, but (like most pop-sci books) it's terrible for actually learning any subject. You need more than one proof every 200 pages.


Yup. It gave me a nice intro to formal systems / number theory iirc. Intro only though, and it was only a small part of the book.




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