Looking to attain fluency in mathematics, not academic mastery

I'm a business/international relations person and a lot of my job is flying around. I have had a lot of downtime recently, and couldn't find a sustainable hobby to fill in that time.

Until I found Michael Spivak's Calculus and decided that it was legitimately a very fun book to read. I didn't actually do the problem sets, but I read the book carefully, and can say that while I by no means mastered the material, I'm generally conversant in it. I might actually end up doing the problem sets at some point, but that's another thing...

In a similar vein to my previous endeavors, becoming "fluent" in undergraduate biology and philosophy through self-study during my downtime, I'd like to do the same thing with mathematics and statistics.

Can someone help me plan out and structure what books I should read and in what order? Let's try to avoid popular science books. I liked the level of technicality in Spivak's book. Again, I'm not trying to reach any sort of academic mastery, just technical "conversational" fluency.

There are plenty of "what should I read" questions around, but I think mine is slightly different, by virtue of asking for a structure, and specifying what I want to achieve. Also, I like the proof-based approach used by Spivak, and would like to see something similar for statistics.

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To clarify, when I read Spivak's book Calculus, I didn't skip the dense parts. I read and understood the proofs. Whether I could replicate them on my own is another issue--I attribute this to the lack of problem sets completed--but I enjoyed the dense parts of Spivak's books. So, I am absolutely looking for something more technical than A Brief History of Time, etc, etc.

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Any equation? Let's see. Here's an equation for you: $s^2=g_{\mu\nu}v^\mu v^\nu$. What does it mean, and what can you do with it? – celtschk Aug 17 '12 at 15:02
I meant I want to learn to the point where I can understand any equation. The last comment lacked context. – nanana Aug 17 '12 at 15:07
It seems you've read Spivak's Calculus as one might pick up and read any coffee table-type book. But to be honest, I can't imagine one can simply read such an active textbook and enculture a deep understanding of the material without truly engaging with it, i.e., completing exercises, re-producing proofs, asking questions, etc. That is, I'm not necessarily sure if one can can claim being conversant in the material and having understood all the proofs without at least a vague idea for how to reconstruct them. – Dustin Tran Aug 17 '12 at 16:19
The OP is not looking to "enculture a deep understanding" nor to attain "academic mastery," but simply to acquire "conversational fluency" as a hobby. Now we can argue about what being mathematically conversant actually entails, but I think it's pretty clear what the OP is after. I think it's great what the OP is doing, and wish more non-mathematicians would have a similar interest -- with exercises completed now, later, or never. Not everyone is looking to become a professional mathematician. – Jesse Madnick Aug 17 '12 at 19:14
@nanana: Hehe. No judgements here. Based on your question, I recommend looking into graph theory and combinatorics. These fields seem like ones that would be right up your alley. – Joel Cornett Aug 18 '12 at 0:14

Given what you've said, I recommend looking at the following.

1. The MAA's New Mathematical Library books

Note that #2 above, a well known English translation that first appeared in 1965 (the date of the 3-volume hardback edition I have), is now available as a relatively cheap Dover paperback. Incidentally, all 12 of the present amazon.com reviews give this book a 5-star rating (the maximum).

3. English translations of the Russian Popular Lectures in Mathematics Series. Besides the google search I embedded in the previous sentence, see also David Singmaster's list.

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When I first came into possession of "Fearless Symmetry" by Avner Ash and Robert Gross, I had no idea who the target audience was. On the one hand, this book is very far from being a textbook; it is not adequate reading for a mathematician seeking technical expertise in the area of Galois representations. On the other hand, it is certainly not a "popular science" book. When I initially heard about this book, I had hopes of sharing it with family members who wondered what I studied. Once I flipped through it, such hopes were dashed. Surely, no ordinary lay reader would want to read this book.

Judging from your question, it seems like you may actually be the target audience.

This book attempts to build up an explanation of what Galois representations are for those people who haven't seen math since high school or their first year of college. This is a Herculean task, and the reviews on Amazon suggest that the authors probably did not totally succeed, but they also didn't totally fail.

You mentioned that you like to do problems sparingly; you'll be happy to know, then, that problems are sprinkled sparingly throughout the book to help you judge whether you're keeping up.

This book will give you a whirlwind introduction to many aspects of modern algebra. By the end, you will have glimpsed the cutting edge.

If you decide to read this book, please check back and let me know what you thought of it!

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Visual Complex Analysis by Tristan Needham is a highly readable book for someone that understands Calculus, delves at least a bit into the history, and also at least mentions applications along with the theory.

I do not think many people would be able to really understand what it is saying without doing some problems in Complex Analysis. But I read it as a companion book when I was taking a Complex Analysis course without doing the problems in that book (I did the ones in the assigned textbook) and it helped me tremendously with undersanding and was pleasant to read.

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For your kind of cultural interest, I would recommend Imre Lakatos:"Proofs and Refutations: The Logic of Mathematical Discovery".

This is a kind of "mathematical philospohy", but by extended example! And it does have b oth proofs and technical content, in the appendix it ends up with introducing Poincare's ideas about homology ...

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It looks a bit like you'd like some advice on texts for topics. I think you would get maximum benifit from those if you did the problems. If you aren't going to do problems, I'm thinking I should recommend some general reading on math rather than texts.

If you want a big general reference, I imagine that the Princeton Companion to Mathematics would suit you well.

If you wanted to get a kind of casual blanket knowledge about what different flavors exist in mathematics, there are quite a few "popular" reading books you could look at.

If you have any interest in physics, I could recommend Penrose's "The Road To Reality". The first half is chock full of a lot of mathematical ideas and general discussion of them. Dense parts can be skipped over without any problem.

I also remembered Paulos' "Innumeracy" being good (but I didn't really enjoy his " Mathematician reads a newspaper" book.)

While I haven't had time to read it, I imagine Hofstadter's "Godel, Escher, Bach" would have an interesting perspective of some math ideas.

There are a lot of things like "50 math ideas you need to know" that would make good plane reading too, I imagine (without having read them.)

It sounds like you aren't as interested in problem solving, but if you ever did want to try your hand at mathematical puzzles you would pick up any of Martin Gardner's books on math puzzles. Great for planes!

Edit It sounds like you were serious about doing problems, so I'll have to add some texts!

1. Kelley's General Topology

2. Stein and Shakarchi's Analysis book

3. Anderson and Fuller's Rings and Categories of Modules

By the way, the Road to Reality mentioned far above contains lots of exercises!

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Hmm... to clarify, I am planning, at some point, to actually do the problems. I did the same with biology. I could understand from reading alone, but I couldn't "integrate" the knowledge without doing a few problems. But, well, a hobby is a hobby, and I want to indulge myself in a bit of speedy reading first. – nanana Aug 17 '12 at 13:36
OK, if you are serious about doing problems, then I will add a few texts to this list above :) – rschwieb Aug 17 '12 at 13:36
Thanks for taking the time, @rschwieb! – nanana Aug 17 '12 at 13:43
Ah, I'd like to note that, while I could finish Spivak's Calculus, I'm sure that's more because of Spivak's rigor than my proficiency in math. I think I may have skipped plenty of introductory or higher texts, with things I might want to read? Or is it unnecessary? The books you list seem very close to graduate level mathematics... – nanana Aug 17 '12 at 13:49
@nanana Mainly I listed them because I thought they had good exercises. I wish I could list more introductory books, but I kind of skipped them. Real analysis is a good gateway to topology, and linear algebra and geometry are good gateways to algebra. Algebra and topology are good gateways to category theory. This tourist map certainly doesn't do the whole of mathematics justice, but it's the one I'm most familiar with! – rschwieb Aug 17 '12 at 13:54

Here are some books I enjoyed reading:

• Aigner and Ziegler. Proofs from the book.
• Milnor. Topology from the differentiable viewpoint.
• Bau and Trefethen. Numerical linear algebra.
• Remmert. Theory of complex functions.
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You might like "Algebraic Topology: An Intuitive Approach" by Hajime Sato.

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I would add "How to solve it" by G. Polya to the list. It is a nice read about approaching and solving mathematical problems. You may also want to refer to books in Schaum's series. They cover a variety of topics in mathematics.

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