" In short, it wanted to put the meaning back into mathematics. But it was meaning of a very dif- ferent kind from physical reality, for the meaning of mathematical ob- jects states "only the relationships between mathematically 'nndefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. "
Other introductory books I've found very useful are the 'Dover Books on Mathematics' introductions series, I've found their graph theory and topology books rather concise and clear to read -- to my knowledge they're availible at archive.org in the collection 'folkscanomy mathematics'.
Are you referring to the book authored by Richard Courant (http://www.amazon.com/dp/0195105192)?
> Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving.
I find the book to be a great introduction to topics that are (actually) explored at the upper division level. The vast majority of what I've learned from lower division mathematics courses was a list of methods rather than understanding (which is understandable as it's really geared for engineers, who arguably care more about results rather than a deep understanding), with the exception of linear algebra and perhaps calculus 1.
A couple of recommendations (not specific to just Calculus):
- What is Mathematics? (Courant http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...)
- Calculus (Apostle http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...).
- Mathematics from the Birth of Numbers (http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg...) This book was written by a Swedish surgeon without any background in Mathematics. He started working on this when his son started attending university. A recommended read.
- The Calculus Lifesaver (Adrian Banner). This book is supposed to be a guide for students to crack their exams. But I found the book surprisingly informative. http://press.princeton.edu/titles/8351.html
- Godel Escher Bach. I've read only the first couple of chapters. My interest in mathematics was rekindled to a great degree by Godel and the Incompleteness Theorem. (http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#The_Incompleten...)
- http://us.metamath.org/. The concept alone makes me happy! Metamath is a collection of machine verifiable proofs. It uses ZFG to use prove complicated proofs by breaking it down to the most basic axioms. The fundamental idea is substitution - take a complicated proof, substitute it with valid expressions from a lower level and keep at it. It introduced me to ZFG and after wondering why 'Sets' were being taught repeatedly over the course of years when the only useful thing I found was Venn diagrams and calculating intersection and union counts, I finally understood that Set theory underpins Mathematical logic and vaguely how.
- The Philosophy of Mathematics. From the wiki: studies the philosophical assumptions, foundations, and implications of mathematics. It helped me understand how Mathematics is a science of abstractions. It finally validated the science as something that could be interesting and creative. http://plato.stanford.edu/entries/philosophy-mathematics/
I think the Philosophy of Mathematics should be taught during undergraduate courses that has Maths. It helps the students understand the nature of mathematics (at least the debates about it), which is usually pretty fuzzy for everyone.
Mathematics: A Very Short Introduction (Tim Gowers): http://www.amazon.com/Mathematics-Short-Introduction-Timothy...
The Language of Mathematics: Making the Invisible Visible (Keith Devlin): http://www.amazon.com/Language-Mathematics-Making-Invisible-...
What is Mathematics?: An Elementary Approach to Ideas and Methods (Richard Courant): http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...
They give brief but interesting glimpses into several areas of higher math, so they could help you decide what you want to pursue in more depth.
So, only two recommendations for you, one book, one website:
1) If I'm only allowed to recommend one math book to beginners, It'll definitely be:
What is Mathematics - by Richard Courant
and Herbert Robbins
Take a look at the review by Albert Einstein. Yes, Albert Einstein!
This site explains math intuitively unlike the traditional formal approach.
There's also the Princeton Companion for Mathematics, which isn't out yet but is available online. It's a wonderful book.
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