It starts by ranting about uppity mathematicians and academics while showing how simply you can get your head around basic calculus.
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 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."
-- The Pleasure of Finding Things Out (http://books.google.com/books?id=Md0IirlFUfEC&lpg=PA194&...)
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.
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.
IMHO math as a language should be taught first.
Proofs and Refutations by Lakatos also touches on this topic, though it is not the primary or sole discussion there.
"The Pleasure of Counting" by Tom W. Körner. See a review at http://plus.maths.org/content/pleasures-counting and another one here http://www.maa.org/reviews/counting.html
Quote from the second review:
>> 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.
While this is probably true everywhere, it's especially true in any hard science.
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.
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.
“[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. :)
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.