What an awesome book - for my first couple of years at University I struggled with maths for a number of reasons, but one of which was that I expected it to be difficult and complicated and therefore refused to see that actually in most cases the concepts and processes involved are actually quite straightforward but presented in complex ways. Once I realized this I never really had a problem with maths again (and I went on to do six years of post graduate work in an engineering related field).
I wish I had something like that book while I was at school - something that emphasized that maths can be easy if you approach it the right way.
I think Feynman may have used this book. He mentioned having a calculus book that had the same quotation that this one begins with:
"I don't believe in the idea that there are a few peculiar people capable of understanding math, and the rest of the world is normal. Math is a human discovery, and it's no more complicated than humans can understand. I had a calculus book once that said, 'What one fool can do, another can.' What we've been able to work out about nature may look abstract and threatening to someone who hasn't studied it, but it was fools who did it, and in the next generation, all the fools will understand it. There's a tendency to pomposity in all this, to make it deep and profound."
This is one of my all time favorite technical books - I've been searching for a dead-tree copy on and off for years. You've reminded me that times have changed, the technology has improved, and I should search again.
Thank you.
ADDED IN EDIT: A quick search shows that the updated version is co-authored by one of my personal heroes - Martin Gardner. Now I'll need to get two copies.
I must confess that I always read mathematics like a novel skipping over the theorem proofs. I was mostly reading example and quick summary of the theorem.
IMHO, proofs as their names imply, are mostly used to prove correctness with the most rigorousness as possible.
In this perspective, I don't mind trading trust (of the manual/person/web site) for rigorousness. I can get the big picture by reading the summary and, usually, a couple of well chosen examples help me understanding it in depth.
Sadly, it is now a burden for me to read any deep mathematical books. For instance, the dragon book was really annoying to read since it was more written in a formal tone instead of a example/quick explanation.
Anyone know of a good book on mathematics as a language? What I mean is a book that spends most of its time simply introducing the notation and the concepts that it refers to rather than dealing excessively with the mechanics of doing math.
I do not know if it is exactly what you want, but Chapter Zero ( http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Math... )comes close. It talks about a lot of the foundational mathematics that and symbolism that is often never taught to non-math majors, but it does so in an easily understood way.
Proofs and Refutations by Lakatos also touches on this topic, though it is not the primary or sole discussion there.
>> Körner says, in his preface, that this book is "meant, first of all, for able school children of 14 and over and first-year undergraduates who are interested in mathematics and would like to learn something of what it looks like at a higher level." I don't know about English school children; there won't be many American 14-year-olds who will read this book. But that doesn't matter: for those who do read it (and I hope many potential math majors do read it), it offers a unique look at what mathematics, especially applied mathematics, is like.
>> Rather than attempt a book on "mathematics for poets", Körner explains that he decided to talk as if he were speaking to another mathematician. Almost all of his topics involve only elementary mathematics (though here and there an occasional remark or exercise goes quite a bit deeper than that), but the attitude is quite sophisticated. As he says in the preface, this means that most of the book's intended readers will find at least a few points where they are in over their heads. Körner urges them to skip over such parts or, even better, to find some professor willing to discuss them. Read this way, Körner's The Pleasures of Counting is really a pleasure, and may well attract many students to mathematics. It strikes me, in many ways, as the ideal book for independent reading or for a first-year seminar.
"The best way to understand what you are reading is to make the idea your own. This means following the idea back to its origin, and rediscovering it for yourself."
While this is probably true everywhere, it's especially true in any hard science.
>> "Reading Mathematics is not at all a linear experience ...Understanding the text requires cross references, scanning, pausing and revisiting"
The key problem I face is when cross-references go outside of the current article, since almost certainly then those cited articles would cite some more themselves and you can never get to the bottom of it. Even worse is when the author references something from outside without stating it, an example of which could be using some variable without defining it.
Stephen Hawking believes what Euclid did in culminating mathematical knowledge (into self-contained 13 volumes) is something that is needed again but has never been done in the modern times. His "God Created The Integers" I suppose is a small attempt in that direction.
I would have imagined Eric Wittgenstein's mathematics online encyclopedia (http://mathworld.wolfram.com/) could be such a book. But it has too much backward and "forward" cross-referencing.
I am currently reading Bertrand Russell's Principia Mathematica in my free time.
_The Princeton Companion to Mathematics_ is the closest thing to what you're looking for that I know of.
I have a copy, and it's wonderful for looking up common topics in mathematics. The problem is that math is so enormous as a field that you'd essentially have to print and bind every journal article ever written to truly encompass all of the current mathematical knowledge.
Thanks for the link to the great article. This sentence in particular caught my attention:
“[The phrase] 'It follows easily that...' does not mean if you can’t see this at once, you’re a dope, neither does it mean this shouldn’t take more than two minutes, but a person who doesn’t know the lingo might interpret the phrase in the wrong way, and feel frustrated."
I encounter this phrase frequently and it always leaves me feeling like a dope. :)
Regarding "Reading Mathematics is not a linear experience": You have to say, compared with what. I remember that compared with theoretical physics, mathematics seemed emphatically like a linear experience. A friend of mine tried once to express in gestures the difference between learning mathematics and learning physics. He said: "Mathematics is like this:" Then he opened a book, moved his finger slowly along the first line on the page, left to right, humming like a computer that's processing something. Once he got to the end of the line, he tapped the middle of the line and said "Beep beep", then he moved to the second line, moved his finger from left to right humming, tapped in the middle and said "Beep beep" and so on. The he said: "Physics is like this:" And moved the finger along the first line, then the second line, etc., and only after he got to the bottom of the page he tapped the center of the whole page and said "Beep beep". I remember there was a bunch of us there and we all agreed that he was onto something.
This has gone straight into my "Great Articles" collection. Thank you.
If anyone is interested in helping me debug and test the site I'm working on (ad hoc) to collect, browse and retrieve great articles, email me and I'll send you an access code.
I submitted this article after watching his Theory of Computation lecture course on YouTube. It's an introduction to finite state machines, grammars, Turing machines and other related theoretical ideas. I'm writing a program for a problem traditionally tackled by Natural Language Processing and after stumbling through a bunch of academic papers about parse trees and context free grammars, it was obvious I'd need to take the plunge and learn this stuff from the beginning. Anyway it's a good course, tough going but he's a sympathetic teacher. I recommend the videos (search Youtube for Pumping Lemma and that should get you lecture 3, then browse the 'More From' list).
Great article, but the "imaginary reader" at the end is very condescending and stubborn. A purely curious imaginary reader would have served the purpose better.
It starts by ranting about uppity mathematicians and academics while showing how simply you can get your head around basic calculus.