How to Think About Proofs

In a previous post, I explained why proofs are so difficult to understand. In this post, I'd like to explain how to think about them productively.

A proof takes given axioms, theorems, and other mathematical properties and combines them in a way that demonstrates the truth of a new mathematical property. The first step in conquering proofs is to acquire a new mindset about solving problems, one that is better suited to writing proofs rather than solving equations. The next step is to understand exactly which logical moves are permitted by the given knowledge, and just as importantly those moves that are not. The final step is to acquire a toolbox of proof techniques that enable you to more easily solve an ever-growing suite of problems.

Switching from an equation-oriented mindset to a proof-oriented mindset is the first step you'll need to take. You may have an excellent skill set in the realm of equation crunching, but this skill is only marginally helpful in the world of proofs. By way of analogy, your expert aim with a rifle, honed over years of shooting, is of no use when you are unarmed and wrestling on the ground. So, what exactly goes into this proof-oriented mindset? You must transition from wanting to solve equations to get numerical results to wanting to combine theorems and axioms to get further theoretical results. The fact that you "already know" that the Fundamental Theorem of Calculus is true, for example, is no more important than the fact that you "already know" the value of a definite integral because it's in the back of the book. The focus in writing proofs is on the logical process by which abstract properties are combined to arrive at new properties. Free your mind from wanting to jump straight to the result because "it's obvious," and you'll have taken a great leap towards enlightenment. 

Understanding exactly what kind of information you are and are not allowed to use is often the trickiest part of learning how to do proofs, especially when beginning in subjects like real analysis or abstract algebra. For example, consider using the field axioms to prove that $a\cdot(b-c)=ab-ac$. The field axioms list a number of very basic algebraic operations, but it is easy to accidentally extend the power of these axioms beyond what they truly are. You may think that $a(b-c)=ab-ac$ follows directly from the distributive property, but that is incorrect. The distributive property says that $a\cdot(b-c)= a\cdot b + a\cdot (-c)$. There is no axiom that says that $a\cdot (-c)=-a\cdot c$, so you need to prove it. All of these tiny algebraic manipulations that are by now second nature to you must be derived from the axioms before you can use them in subsequent proofs. If you can clearly focus on manipulating the axioms to this end, and you can ignore the strong impulse to just "do algebra" since "it's obviously true," you've made it over half way the way up the mountain. You will hone this skill over time as you get practice with more and more examples and counterexamples.

The final step is to expand your toolbox of proof techniques. Plenty of things can be proven by diligently deducing things from axioms, such as the tidbits derived from the field axioms above, but many techniques require more thought. For example, some proofs may require that you show that A implies B, but showing this directly may be very hard. However, the laws of logic hold that proving the contrapositive of a statement is equivalent to proving the original statement. Armed with this knowledge, you could try to instead prove that not B implies not A, and then conclude that A therefore implies B. Proof by contraposition is hardly the only technique available - others include proof by contradiction, proof by induction, and counting in two ways. The first time you see these techniques, it may seem like they're being pulled from a magician's hat. Over time, however, you will see them reused time and again. The more techniques you acquire, the more ways you'll be able to tackle tricky proofs. Eventually you will develop an eye for which techniques can be used in which circumstances.

Learning how to do proofs is hard, but it is completely within the reach of mortals. The preliminary step is to acquire an understanding for why proofs are hard, both in their own right and artificially as an unavoidable byproduct of mathematical education up until their introduction. From here, you need to shift your focus from crunching algebraic formulas to combining theoretical properties. Once you understand this new task, you need to be able to execute it well. This means knowing how to make sound logical steps and avoiding unsound ones. Understanding why a particular step is sound or unsound is the name of the game. From there, you just need to build up your toolbox of proof techniques, and at that point, the sky's the limit.

I hope that this advice has made proofs both easier to think about and easier to write. However, talking about doing hard things can only go so far. Knowing how one might theoretical understand proofs is like knowing how one might theoretically use muay thai to win a cage fight - the only meaningful next step is to just start writing proofs (or sparring in gym). In a follow-up post, I'll explain why graph theory is one of the best places to get started.