Why Proofs Are Hard
For some reason you want to learn, or are forced to learn, proof-based mathematics. Algebra clicked for you at some point, you conquered calculus, and you then went on to summit the mountains of multivariable calculus, linear algebra, and differential equations of both the ordinary and partial kind. But now you've been tasked with proving something. Something horribly simple. Something obvious. Something so obvious, in fact, that you don't even know how to prove it. It's just...true. Take, as my favorite example, the fact that the square root of 2 is irrational. Suddenly you can't help but want to hide in a hole so that nobody can see you. You were blasting through the wave equation and had all of those sexy partial differential symbols dripping off your page, but now you've been knee-capped by some kind of peasant math that was supposed to stay in middle school!
This is, in my own estimation, a typical first interaction with proofs. And just to make it feel like this horoscope was specially designed for you today, here's how it continues:
You aren't a quitter, so you decide to pay attention. Maybe it seems simple, but it's proof-based math, which, after all, comes with some kind of bragging rights. Yet despite the ease with which all the example proofs you have seen are written, they seem almost like declarations from high upon the mountain-top. Someone makes some conjecture, cobbles together a bunch of reasonable sounding ideas as you nod along in agreement, and then triumphantly declares that the result has been demonstrated. But it gets worse when you try to write your own proofs. Where do you begin? How did the guy who wrote the example proof know where to start or what properties to reference or what arguments to make? Frustration blows a swift breeze along these shores. Perhaps you find a clever argument, only to be told "that's circular reasoning." What? Not it's not! The swift breeze becomes a gale, and rain begins to make craters on the sand.
Fear not. While you've just discovered that everything you've learned so far is little more than glorified algebra (yes, even the wave equation), the knowledge of proof-writing is not restricted to only the most esteemed druids of the intellectual vanguard. This statement is not some saccharine inspirational bromide, either. It's true. It's really true. And not in some saccharine inspirational bromide kind of way like I just said, because I know you're thinking that anyway. Proofs are made artificially difficult for a two main reasons, an understanding of which I believe should be immensely helpful to anyone trying to learn the ways of the Force.
The first reason is the fact that many proof-based topics in math cover areas that are already very familiar to students. Take again the proof of the irrationality of the square root of 2. Students already know so much about numbers that it's hard to un-think all of it. Which concepts depend on which? Which properties can you use or not use? When enough ideas get batted away with the hammer of "We Haven't Proven That Yet," it's no surprise that paralysis can set in. The fact that you can't use some knowledge that you already know to be true seems absurd. It's like being asked to forget your native tongue and then relearn it in the order it had evolved since the beginning of time - a maddening and seemingly useless Sisyphean effort.
The second reason is that new proof students critically fail to appreciate their total lack of relevant experience. But how can this be so for successful math students? Because algebra involves using theorems and properties to solve for numerical results, whereas proofs involve using theorems and properties to derive other theorems and properties. The idea of this kind of dependence in the rules of math is totally new. As a result, a student who knows algebra and is learning proofs is in the same position as a student who knows arithmetic and is learning algebra. There is a fundamental intellectual leap required to "get" algebra, as well as experience needed to be conversant in a familiar world of algebraic operations such as factoring polynomials, using the laws of exponents, and manipulating trigonometric identities. Similarly, there is another fundamental intellectual leap required to "get" proofs, as well as experience needed to be conversant in a familiar world of proof methods such as induction, contradiction, and counting in two ways.
The problem is that new proof students have high expectations of themselves based on accumulated experience in an unrelated field. Up until now, new proof students only needed to keep expanding their algebraic toolbox in order to succeed, since all math problems were algebraic. But all math problems are not algebraic, and neither is all mathematical thinking, so students invariably hit a wall when they try to use their useless algebraic tools to solve proofs. In reality, new proof students have, almost by definition, no relevant experience in this realm of thinking. It is the failure to properly understand, and therefore adequately appreciate, the major difference between proofs and algebra that leads to false expectations and unnecessarily damaged self-images.
There is a third reason that ties into the previous two. Despite the self-congratulatory nature of many math classes and textbooks, their lecturing and writing are rarely unambiguous enough to satisfy what are now millions of what-if alarm bells ringing in students' minds. When everything is questioned, it's impossible to know which facts remain in the realm of the known and usable. Words themselves become totally suspicious. What is a "set" or, god forbid, an "object," anyway? Each noun and verb becomes a rabbit hole of ultra pedantic confusion, and textbooks' and teachers' princely use of the adjective "clearly" to arbitrarily make "obvious" leaps without argumentation is positively maddening (because isn't it all clear and obvious?!). Books and teachers that even notice these problems, much less overcome them, are sadly the exception. Books and teachers that attempt to (much less manage to) explain the whole mindset of proofs and its difference from the mindset of algebra are also similarly exceptional.
The foregoing three reasons combine to make learning proofs unnecessarily hard: false familiarity induces incorrectly high expectations, and poor resources make everything worse. However, I hope that in trying to diagnose why proofs are hard, I have opened the door to making them easier to understand. I also hope that this diagnosis can provide some genuine encouragement for new students in proofs. Knowing both that you're not alone and that your problems are well-understood can provide tremendous solace when you feel like you're the only one suffering from a disease you can't even describe.
But fear not, I don't intend to leave this disease with just a diagnosis. It's one thing to know that the bubonic plague comes from bacteria and not demons, but it's another thing entirely to use penicillin to actually cure it. In a series of follow-up posts, I will explain how to avoid the problems discussed here and in turn think about proofs clearly and productively.