I’ve been reading Zvi Mowshowitz’s criticism of a contest, and one of his statements got me to ponder. I’ll quote stripped of most of the context, I think appropriately for my purposes:
My claim here is that #1 is very different from #2, but that #2 and #3 are very similar.
I thought to myself “could #3 be similar to #1 in this case, or does it have to be pretty different?”. Then I thought that the triangle inequality would imply the latter. But could we be in a space where the triangle inequality doesn’t hold? We could define “difference” and “similarity” based on a non-metric function on pairs of points. This sort of feels wrong, though. Is there a reason that the English words “difference” and “similarity” should be interpreted as different ranges of numerical values of a metric? In which contexts is a metric interpretation of those words more or less useful?
waltonmath 