the lack of intuitive explanations for math is a major pet peeve of mine
I'm a very visual thinker, and that is one reason I enjoy the new Art of Problem Solving textbook Prealgebra by Richard Rusczyk, David Patrick, and Ravi Boppana--
it is full of interesting visual "explanations" and substitute for proofs in a book intended for a young audience.
That said, I finally realized that I was limiting my mathematical development by insisting that every mathematical idea must appeal to my visual intuition. Some mathematical ideas are proven even if they don't appeal to visual intuition. In the words attributed to John von Neumann, "in mathematics you don't understand things. You just get used to them."
Thanks for the pointer! I found Needham's book awesome, I've barely made a dent in it but love the visualizations.
I don't think visualization is the only intuitive method -- you can have a general "sense", not sure how to put it more specifically -- I have a "sense" about growth of e without a specific diagram.
Agreed that not every concept can be understood... yet. There's a quote I love to rail on, in reference to Euler's formula:
"It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth." (Benjamin Peirce, 19th-century Mathematician)
Really? Yes, it may be baffling at first, but we can _never_ understand it? Only if that's our attitude :).
I'm a very visual thinker, and that is one reason I enjoy the new Art of Problem Solving textbook Prealgebra by Richard Rusczyk, David Patrick, and Ravi Boppana--
https://www.artofproblemsolving.com/Store/viewitem.php?item=...
it is full of interesting visual "explanations" and substitute for proofs in a book intended for a young audience.
That said, I finally realized that I was limiting my mathematical development by insisting that every mathematical idea must appeal to my visual intuition. Some mathematical ideas are proven even if they don't appeal to visual intuition. In the words attributed to John von Neumann, "in mathematics you don't understand things. You just get used to them."
http://en.wikiquote.org/wiki/John_von_Neumann
That point of view makes a lot of sense to many of the best mathematicians.
One more example of really interesting visual explanations of mathematical concepts is Visual Complex Analysis
http://usf.usfca.edu/vca/
by Tristan Needham. The book is delightful, and well reviewed, but it is not the sole path toward getting used to complex analysis.