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(4) as a matrix function which (a) rescales when you rescale a column, (b) isn't changed by addition of one column to another, and (c) changes sign when columns are swapped. Fun to think through why all these are equivalent! Many other ways of thinking of the determinant, too.
Some considerable fraction of mathematics seems to be keeping multiple ways of thinking about an object in your head, and moving back and forth between them.
Learning new math concepts often comes down to identifying the "primitive" object and then figuring out how the new concept extends this primitive (credit to @geomblog for impressing this idea). Having multiple interpretations makes it much easier to see extensions
I'm not sure what you mean - can you give an example?
You've probably seen Thurston's "37" ways to think about derivatives (and much more): http://arxiv.org/abs/math/9404236 … pg 3
That's a terrific paper.
Flashbacks to the definitions of "open sets" in real analysis....
Helpful to keep the types annotated. They are objects in categories. You can apply forgetful transformations, same object but with different structures allowed to be used. For example, integers as a CRing but can forget to just as a Set. Different ways to work with them.
I have found this especially true for linear algebra. Think of all the ways to describe invertability.
Is that process itself part of mathematics? Is there an optimal number, development, and moving back and forth between ways of thinking about an object for maximally efficient problem solving or creative extension?
Mathematicians are SO VERY VERY much smarter than economists, of which I am one.
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