Great comment -- chiming in because intuitive math (or the lack thereof) is a hot-button issue for me :).
I really dislike the von Neumann quote "Young man, in mathematics you don't understand things. You just get used to them."
I know what he's saying (there are some concepts you just need to internalize), but taken at face value it implies you stop looking for insights once you've "gotten used to it" (vs really grokking it).
I think learning is a spiral of theory & practice, i.e., present some principles, explain with examples, present more principles, explain with deeper examples, and so on.
Shameless plug, but check out this article on imaginary numbers:
I try to explain imaginaries by starting with negative numbers (something we're familiar with, but was counter intuitive at the time) and building up with examples (3 cows - 4 cows is "absurd", right? sqrt(-1) is "absurd", right?).
I don't think you can just define "i" as sqrt(-1) and give a bunch of problems, or talk about abstract visualizations for pages with any meaty examples: it's an interleaving spiral of both.
Square root is a fundamentally geometric idea: you need to have a concept of area before you can find the ratio of an area to the side of a square. Much better than talking about what is or isn’t absurd in terms of pure symbols (like √-1 or whatever) is to give some geometric motivation of the definition of area in terms of orthogonality, which brings in the link to rotation, and finally results in complex numbers. By embedding complex numbers in so-called “geometric algebra”, you put them in their proper coherent and comprehensible context. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
Can you recommend an introduction to geometric algebra & calculus? (preferably online) I've quickly read some but they were vague and in general quite unlike other mathematical texts. For example the document you linked to doesn't actually define the operations it uses. You can sort of conjure up a definition by piecing various statements together (by using the laws plus the operations' behavior on basis elements), but surely there must be a clearer introduction available? Other introductions were just plain wrong, for example one claimed that the curl of the curl is always 0. All that I've seen were strongly trying to persuade the reader of the awesomeness of GA rather than teaching it.
I don't think von Neumann was saying "some concepts you just need to internalize" at all; his idea was more along the lines of
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" (Jerry Bona)
In other words, intuitive notions of "understanding" aren't necessarily useful when dealing with "pure" mathematical ideas that lack real-world antecedents. (N.B.: the axiom of choice, the well-ordering principle, and Zorn's lemma are equivalent)
Thank you, I really love this kind of thing (former physics major here who sadly, I suppose, went through college never really peering deep into the meaning of tools I used on a daily basis).
I really dislike the von Neumann quote "Young man, in mathematics you don't understand things. You just get used to them."
I know what he's saying (there are some concepts you just need to internalize), but taken at face value it implies you stop looking for insights once you've "gotten used to it" (vs really grokking it).
I think learning is a spiral of theory & practice, i.e., present some principles, explain with examples, present more principles, explain with deeper examples, and so on.
Shameless plug, but check out this article on imaginary numbers:
http://betterexplained.com/articles/a-visual-intuitive-guide... HN discussion: http://news.ycombinator.com/item?id=2712575
I try to explain imaginaries by starting with negative numbers (something we're familiar with, but was counter intuitive at the time) and building up with examples (3 cows - 4 cows is "absurd", right? sqrt(-1) is "absurd", right?).
I don't think you can just define "i" as sqrt(-1) and give a bunch of problems, or talk about abstract visualizations for pages with any meaty examples: it's an interleaving spiral of both.