I took a fundamental concepts of math course in Boston. It dealt a lot with proofs and how to formally write a proof and different ways of proving different statements. One of the things I found most interesting was when we covered what are called distance functions. In 95% of mathematics only one distance function is used, the existence of other distance functions isn’t even mentioned! So here’s what I’m talking about. A right triangle exemplifies the classic distance function which most students use, to the exclusion of even being aware that others exist. Say the two sides of equal length are length 1, if one uses the classic distance function (which declares length = the shortest distance between two points), then the length of the hypotenuse = the square root of 2. As it turns out, you can create as many distance functions as you want, as long as they satisfy these three rules:

1. the dist from point a to point a = 0
2. the dist from point a to point b = the dist from point b to point a
3. if you have three points a,b,c; the dist from point a to point b + the dist from point b to point c is > or = the dist from point a to point c.

One of the most well known of the very unknown alternate distance functions has been nicknamed “taxi driver” because diagonals aren’t allowed. If you were to apply this distance function to the right triangle, the length of the hypotenuse would be the sum of the lengths of both equal sides, or, 2. The punch line to all this is that you can perform mathematical operations, like differentiation and integration, with respect to these alternate distance functions, but it gets wicked complicated.