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Recommend math books for a language geek

screw_dog said:
The only way you could possibly not need a calculus book is if you already have a complete mastery of the subject. It's that vital (IMO).
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I'm afraid I have to disagree here. If I were to suggest any fields of study to the non-mathematician, calculus would be among the last, precisely because it has so little use.

I would suggest instead logic, probability/statistics, "discrete mathematics," and abstract algebra.
 
drkitten said:
I'm afraid I have to disagree here. If I were to suggest any fields of study to the non-mathematician, calculus would be among the last, precisely because it has so little use.

I would suggest instead logic, probability/statistics, "discrete mathematics," and abstract algebra.
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I thoroughly agree with drkitten. Personally, I'd avoid statistics textbooks, because they almost always either assume more mathematics than a nonmathematician would want to deal with or state miracles without explanations. There are a number of books which combine logic, probability, and discrete mathematics in a nice unified picture. I have not perused many, but your best bet is probably a well-written undergrad introduction to discrete mathematics text. The well-written is the kicker, isn't it?
 
T'ai Chi said:
As far as logic goes, I'd recommend some books by Smullyan, or GEB by Hofstadder(sp).
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I'd like to second the recommendation for Godel, Escher, Bach by Douglas Hofstadter. It's really excellent for learning logic (and a little number theory) from the ground up... and it's also a great read.

Going further, I think GEB is the single greatest work of non-fiction I've ever read, and I know I'm not the only one who has said that.

Read the customer reviews on Amazon to get some idea of what kind of an effect this book has on people:

http://www.amazon.com/gp/product/04...104-8253465-0823940?s=books&v=glance&n=283155

One person wrote:
It's without a doubt the most challenging thing I've ever read, but also the most rewarding.
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That about sums it up.
 
Vorticity said:
I'd like to second the recommendation for Godel, Escher, Bach by Douglas Hofstadter.
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Hmmm.... What's the consensus on that, folks? I'd heard that GEB was to math/logic what The Dancing Wu Li Masters is to physics -- a pop read that's not of much value if you really want to understand the field. (Actually, I'd go much farther to say DWLM is an outright fraud, but haven't heard that bad a pan of GEB.)
 
Piggy said:
Hmmm.... What's the consensus on that, folks? I'd heard that GEB was to math/logic what The Dancing Wu Li Masters is to physics -- a pop read that's not of much value if you really want to understand the field.
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I would disagree.

GEB is a popularization of some rather deep philosophy and rather deep mathematics. But to my knowledge there's not a single statement in there that's actually a misrepresentation.

The final part of WLM is nothing but misrepresentation.

You're not going to achieve lots and lots of knowledge by reading GEB, but you won't come away with any wrong ideas. And it provides a good intuitive basis if you want to go in for further study. (In which case I recommend the collected works of Raymond Smullyan, particularly What is the Name of this Book?).
 
drkitten said:
You're not going to achieve lots and lots of knowledge by reading GEB, but you won't come away with any wrong ideas.
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And it is a beautiful book, delightful to read. Check what Martin Gardner said about it:

Martin Gardner said:
Every few decades and unknown author brings out a book of such depth, clarity, range, wit, beauty, and originality that it is recognized at once as a major literary achievement. Gödel, Escher, Bach[...] is such a work.
[...]
For laymen I know of no better explanation [...] of what Gödel achieved and of the implications of his revolutionary discovery.
[...]
By the end of GEB Hofstadter has introduced his readers to modern mathematical logic, non-Euclidean geometries, computability theory, isomorphisms, Henkin sentences, Peano postulates, Feynman diagrams for particles that travel backward in time, Fermat's Last Theorem, transfinite numbers, Golbach's conjecture, Turing machines, computer chess, computer music, computer laguages, molecular biology, artificial intelligence, free will, holism vs. reductionism, and a kind of sentence philosophers call counterfactual.
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Still, be advised it's not a 'real' math book. If you really want to learn math, Calculus is the place to start. Spivak's book is a wonder, a standard freshman text, so is Apostol's. However, both are for people who already know some math. A good introduction could be:

Silvanus P. Thompson, Calculus Made Easy, St. Martin's Press, 1998.
 
Nancarrow said:
Piggy, our recommended text for applied maths at uni was Riley, Hobson and Bence. I notice all the reviews there are from Cambridge folks... go figure! I agree that it's the bible of applied maths. It has a refresher chapter on calculus at the beginning, but if you've done no calculus at all before, you may want to find a book on that specifically.

ETA: just noticed that in the newer edition they stuck in a chapter on 'preliminary algebra' which I think is a very good idea. Also they've expanded the statistical stuff at the end. Sod it, I might have to buy a new copy now! Even though I'm an accounts assistant!
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Yikes! That's the book I had to buy for my graduate course in math methods. It certainly can get advanced! Although, now that I look at the first few chapters, it does cover algebra and calculus (rather quickly). I've found it useful, but then, I've been using it for graduate physics courses.
 
drkitten said:
You're not going to achieve lots and lots of knowledge by reading GEB, but you won't come away with any wrong ideas.
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The main thing that I learned from GEB was that I finally understood the difference of primitive- and mu-recursive functions. [I still use the Chapters 5.3-5.5 of the first edition of Lewis&Papadimitriou as the example on how you shouldn't write undergrad texbooks on mathematics. I don't know anyone who would have figured out recursive functions from that. They fixed the section for the second edition.]

(In which case I recommend the collected works of Raymond Smullyan, particularly What is the Name of this Book?).
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Seconded. Particularly that book.
 
Glad to have my reservations about GEB removed... b/c it really sounds fascinating!
 
Piggy said:
Not to seem stupid here... but rather to seem ignorant... because I am ignorant....

Y'all still haven't answered my question: What is done with calculus?
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What's done with calculus is solving differential and integral equations. Which are equations you set up to model physical situations. As was said above, it's fundamental to science.

My experience with calculus is that it was almost impossible (for me, anyway) to learn it from reading textbooks. The summer before I took my first calculus course, I got a calc textbook and tried to learn it, and failed miserably. When I took the course, for the first few weeks I couldn't get what they were driving at, until finally there was an "aha" moment and it all became clear. You might be better off taking courses rather than reading books. But you probably also should first get a handle on trig and particularly analytical geometry, which I found has more practical use in everyday application than most of the higher math I took.
 
joe87 said:
What's done with calculus is solving differential and integral equations. Which are equations you set up to model physical situations. As was said above, it's fundamental to science.
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Expanding upon joe87's answer -- calculus is about solving problems where values change in interrelated ways over time. For example, you fire a bullet up in the air, and it goes up for a while, then it goes up more slowly, then it starts coming down, and eventually lands (hopefully not in you).
You eat a candy bar, and your level of blood sugar starts going up, which triggers a rise in your blood insulin levels, which in turn causes your blood sugar level to go down,.... Since time is continuous, the changes have to be smooth.

It's also useful for optimization tasks in continuous domains, such as figuring out what the very highest point of a curve is or the way to shape a bridge for maximum load.

However, I still disagree that calculus per se is the most important part of mathematics; although it's fundamental to "science" (and specifically to the hard sciences like physics and chemistry), it's got rather limited application to most aspects of "real life." Most of the questions that people have tend to be discrete -- do I take an extra card in blackjack, or not? (I can't take 2/3 of a card) Is it faster to drive straight through the city or around the beltway? (I only have a few choices, not a range) Does my car give better gas mileage with the radio on or off? (there's not really an intermediate state) Which calling plan will save me the most money? (even calculus won't help here)
 
Thanks for the low-down on calc, etc.

I probably won't have the opportunity to buy more books for about another month. And I dunno how soon I'll finish Gould's The Structure of Evolutionary Theory. Trying to read that one simultaneously with math would probably be too tall an order.

All this is very useful.

FWIW, my main interests are probability, mathematical logic (rather than rhetorical logic, which is what I'm accustomed to), astronomy, and physics.
 
Ok, it's been over a month now. Get cracking. :D

Another recommendation: the four-volume World of Mathematics, James R. Newman (editor), Simon and Schuster, 1956. It's been reprinted since then, by Microsoft Press I think, and also by Dover. Absolutely wonderful.

One of the many selections it contains is by Alfred Tarski on symbolic logic, taken from the beginning of his book, Introduction to Logic and to the Methodology of Deductive Sciences, which Newman describes as "the best all-round primer in the field." I don't have that book, but if the rest of it is as good as the first two chapters, I can heartily recommend it too.
 
Thanks again. I'm not finished w/ The Structure of Evolutionary Theory yet. It's slow going b/c I'm relandscaping the yard, painting the inside of the house, and restructuring my finances, too.

But when I do complete Gould, I'm on to math!

I'm actually very excited about finally tackling the subject. I'm such a geek.
 
You might want to check out a book by Ken Amdahl titled Calculus for Cats and possibly his other book Algebra Unplugged. Calculus for Cats will help you understand what calculus is and "what you do with it." Both of these are good books, but his best is There Are No Electrons, if you have an interest in learning about electicity.
 
Piggy:

I think it's admirable that you're going to "tackle" mathematics, and don't want to be negative ... but mathematics is a gigantic (really, really, really big!) field. I spent four years as a math undergrad, and 35 years as a deeply interested math hobbyist (working in a technical field) ... and consider myself woefully under-educated mathematically. The size and depth of mathematical knowledge is staggering and far beyond any one individual's grasp in a single lifetime of intense study. And it's very tough going on your own, even with the best books.

So I strongly suggest you narrow your focus a bit and decide what you hope to get out of your efforts: possibilities include: a broad high level picture of the power and/or beauty of math; some specific tools for application to a particular problem or type of problem; a deeper understanding of one branch of mathematics that intrigues you; an introduction to mathematical methods and 'rules'; or something else entirely. And once you decide that (or maybe even to help you decide what you want out of it) I recommend taking a class rather than starting on your own with books.

Given all that ... if you have an interest in mathematical logic, and in recursion (mathematical, musical, or artisitic) reading 'Godel, Escher, Bach' is a wonderful experience. Take the time to read it slowly.

Good luck!
 
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