How to Develop a Mindset for Math

Yes, math -- like every other discipline that can take itself seriously -- is about models and relationships. But given these models, we ought not necessarily "apply them to real-world objects." Rigorous math is powerful precisely because it is abstract.

One of this guy's prized "deep insights" is that negative numbers refer to deficits of cows. Granted, that's a slick (if not wholly obvious) way to represent an intuitively awkward concept, but it engenders a false confidence. Negative numbers don't exist to represent bookkeeping deficits. They are there because some formal mathematical argument grounded in pure logic said they had to be there. The fact that we use negatives to refer to cows reveals more about primitive functions of the human brain (distinguishing among objects, counting) than it does the deep algebraic structure of the numbers themselves.

I admire attempts to ground difficult abstractions in physical reality as much as the next guy, but we shouldn't confuse these simplistic observations with mathematical insight.