Of the many similarities between mathematics and art, one stands out above the rest. The layman can feel they understand the concepts, but find modern practitioners completely incomprehensible. When snooty art critics pontificate about carefully arranged poop, or mathematicians berate students about the true meaning of multiplication, one naturally concludes that these people are either way smarter than the rest of us, or they’re insane.
I was 21 when I bought my first abstract math book, “Abstract Algebra An Introduction” by Thomas Hungerford. I started into the first chapter confident that I would burn through the content, just like every math class I’d taken to that point. That didn’t happen. I spent weeks reading and rereading early sections, desperately searching for the significance in the text; it just didn’t click. Everything I had studied before was progressive, and clearly applicable; there were slots in my brain practically waiting for each new piece of information. This abstract algebra was just a bunch of new names for parts of the algebra I learned in the eighth grade. I mean, it’s nice to know what an Abelian group is, and go through the exercise of creating one, but why should I care? These nineteenth century Germans were messing with me from beyond the grave.
Many moons have turned since then. In the intervening years I’ve done more reading and explored many of my own ideas. And, while I’m far from understanding the intentions of nearly any mathematician, I can see value in abstraction. By breaking familiar concepts down to their bare mechanics you can see what the ideas have in common, and where the proofs in one are applicable to another. Abstraction allows rapid exploration and creation of new mathematical ideas by trivializing the content and focusing on the structure. Unfortunately, in school, we’re taught to focus on the content, and the structure is just something to take for granted…it just works, ya’ know.