Well, Obviously…

So there was a math professor lecturing to their class.  The class had just started but they were already getting into some heavy-duty stuff.

No more than five minutes into the class, some smart-ass pipes up, interrupting the professor’s unfurling grandiosity: “Wait, why is what you just wrote there true?”

The professor, interrupted from their unveiling of knowledge, scoffs back: “Well, that’s obvious!  You just have to…  Well, you just… Huh, well if you… Mmmm….”

The professor scratched their head.  The professor twiddled the chalk.  The professor scribbled some scratch into the corner of the board, hiding it from onlookers with their body.  The professor forgot the students in the class and paced back and forth muttering trains of thoughts.  The professor spent the entire class period trying to figure out why what they wrote was true.  The professor was very bothered.

At the end of the class the professor mumbled: “I can’t think of why this is true.  We’ll meet again on Thursday.”  And then hobbled out the door leaving the students sitting there.

So finally, come Thursday, all the students shuffled in for their next class.  Then the professor burst into the room with a wide gait.

The professor swoops up the chalk and exclaims: “I’ve thought about why that statement is true.  I was right!  It’s obvious.”

The professor then picks up where they left off with the obvious statement now “settled”, unfurling their grandiosity onto the passive students…

Russell Impagliazzo told this joke last semester during a course he was teaching for the Fine-Grained Complexity seminar at the Simons Institute.  While it was certainly a funny and somewhat relatable commentary on the culture of mathematics and academia, it also reminded me of my philosophy on the nature of mathematics.

To aphorism-ify a thought by another friend of mine who is named Russell: Math is the art of turning the non-obvious into the obvious.

Now, this isn’t necessarily a new thought.  In fact, it seems that most mathematicians are already Platonists.  That is, they view math as an act of discovery rather than creation.  It’s actually the reason I fell in love with math: I’m not just hacking together some ad hoc creation that’s limited by my imagination, but am discovering pieces of something that is pre-existing.  Something that is universal and immutable and, best of all, bigger than me!

But even with all this Platonism-awareness, it may be easy to forget (or it may not be obvious) that the notion of obviousness is relative.  In fact, the notion of obviousness is all there is, to me, in mathematics.

For a very simple example, let me ask you this easy question: What are the roots to x^2 -5x+6=0?

Some of you may say, Oh, well, obviously the roots are 2 and 3.  You’d of course be right.  So my next question to you is Why was that obvious?

Did you simply just look at the equation and the numbers flew at you from the screen?  Or did you do some remodeling first?

I’d bet the majority of you (almost subconsciously) factored the equation in your head to say x^2 -5x+6=(x-3)(x-2) reaching way back into the junior high or even elementary school days.  And, for some reason, (x-3)(x-2)=0 makes it more obvious that the roots are 2 and 3.  I mean, hell, they’re right there!

But what did we actually do.  We rewrote the same equation in a different way.  How silly and time-wasting is that?!  x^2 -5x+6 was always equal to (x-2)(x-3), so we didn’t create anything new.  We just rewrote it (maybe just in our heads and maybe not even fully out).  But then, BAM!, it was obvious to us what the answer should be.

The point is, that is all we ever do in math.  We take something and just rewrite it a bunch of times until we finally get it into a form that’s obvious for our little mortal brains!

This says to me that there are two possible philosophies for math if you really understand its essence: Either it is the most pointless and futile of all things a human can do, or it is the most pure and profound of all things a human can do.

Pointless and futile because anything that we ever prove was always and will always be true.  Our “discovery” of it changes absolutely nothing.  We were just simply able to bang our heads on the wall enough to make something that was always just utterly and completely obvious to the universe fit into our human heads.

But pure and profound because there can be no feigning at human grandiosity of creation or superiority over the universe.  Our discovery of a proof changes nothing except our human connection with mathematics.  It means that math is nothing except the human experience that we get out of it.  The awe, the wonder, the purpose.  The pursuit!

Ah, the pursuit.  When you realize that all a proof is is taking a true statement that is not obvious to us and simply rewriting it until we (finally) get to a point where the concluding rewriting is obvious to our little brains (Q.E.D.), is when you realize that math is all about the human endeavor and pursuit of the beast.  We simultaneously become humble and shuffle through the towering majesty of math while becoming boisterous and with purpose trying to hunt down and pursue the beast, pridefully winning a match against the Supreme Fascist.

The only problem is that this truth doesn’t stop people from feigning grandiosity of creation or superiority over the universe.  While most mathematicians may be Platonist, it’s not always obvious to follow Platonism’s logical conclusion to the relativity of obviousness.

In fact, I think most of math’s problems of “academic dryness” or pedagogy boil down to forgetting that there is no such thing as obvious.  That “obviousness” is not just entirely relative to humans, but is a completely individual experience.

Recognizing that there is no such thing as obvious (or that everything is obvious in the sense that it’s either true or not) let’s us realize that math is all about finding ways, that are as human as possible, to fit it into our mortal context.  To fit it into our historical context, to fit it into our poetic allusions, to make it visual, audible, and spacial simply because that’s how we operate as humans.  To make analogies to events in our life and childhood. To pinch the air as we think about topology.  To tilt your head to see hallucinatory linear transformations on the wall.  To talk about mesh frames and soap, and donuts and coffee mugs, and peas turning into suns, and cats turning to static and then back again, and ham sandwiches, and snakes swallowing themselves whole.

Math is inextricably tied to our human bodies and histories and literature and politics and hobbies and fears and food.  The more we realize this, the more of this human endeavor we can explore and exalt in.

So when we say that it’s obvious what the roots to x^2 -5x+6=0 are, we should recognize that we can only do so because we memorized that we should factor a quadratic equation to get the zeroes.  And that’s because we memorized that (x-a)(x-b)=0 implies x-a=0 or x-b=0 (and maybe we learned much later that this works since \mathbb{C} is an integral domain and so has no zero-divisors).  Just looking at x^2 -5x+6=0 doesn’t make it obvious what the roots are, even though we consider it an exceedingly simple problem; it’s the fact that we have techniques built up over the centuries that fit it into our human context and let’s us have a place for it in our small brains.

It’s only obvious to us because we stand on the shoulders of giants!

Although, in retrospect, I don’t know how we ever thought they were giants.  They’re obviously windmills…

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