In mathematics, proofs are all the rage. I’m currently TAing a course on theory of computation, which is the most math many students have seen in the last four years, and definitely not high school style mathematics (i.e. ‘shut up and calculate’). So I’ve been asked some tips on how to cope with proof writing and I ended up writing quite a long ‘guide’, which I like enough to publish here, for the public to ridicule.
Quick anatomy of a theorem
Theorems are mathematical statements of the form:
Bear in mind that the above is not how theorems are usually written down. Mathematicians speak with words, not symbols (that’s mathematics). So you rarely encounter a theorem which is expressed exactly as an implication as above. Often mathematicians prefer to abbreviate them in more succinct statements, and there could be considerable work to be done to unpack them into a clear implication, or to track down what’s exactly the setting a theorem is being formulated in. Mathematicians love to disseminate terminology around so that they can say things like ‘all blorbs on a smooth zib are 3-sgurz’. Indeed, one could say mathematics is all about this.
Theorems beget proofs: a theorem without a proof is simply a statement, and could be true, false or even neither until a proof is found. If you have a statement but not a proof, you got yourself a conjecture, a claim or an hypothesis if your name is Bernhard Riemann.
Writing a proof
It is crucial to realize proofs are nothing but explanations of why something is true. We do this all the time in real life, so do not let the formality of the context hold you back. Proofs are mathematicians explaining things to one another. They have some lingo that helps, and they are very exacting on the correctness of the argument (sometimes we do not always do in real life, admittedly). Pause to internalize this and then keep reading.
It follows from the point above that the first step in writing a proof is to understand yourself why something is true. This might seem trivial, but it’s definitely the hardest part. Sometimes it takes minutes, sometimes hours, occasionally months or years. But when you do, you crossed the river and can’t come back.
The proof is a shallow manifestation of a deeper phenomenon happening in your brain, which is developing intuition for a new realm. Mathematics is about concrete manipulations of abstract concept. You find yourself dealing with a world whose rules are very unfamiliar to your brain at first, like they were when you were a child. Slowly, you learn your way around them. You start to develop that intuition that you develop for the physical world (‘if I let this object go, it falls’, ‘poo smells’, ‘mom is just behind the door, she didn’t stop existing’, etc.) [0].
So before you write anything, be free and aim at intuit. Play with the assumptions, recall facts you know about the objects at hand, break things, experiment, challange the new abstract world around you [1]. At this step, you don’t need to be formal. Refrain from be formal, actually. You need to feel the theorem in your guts to move to the second step, and oftentimes rigid formal statements do not sit well with handwavy feelings.
Bear in mind: proof writing is recursive. Every proof is made of small proofs chained together, and each of those is made of even smaller proofs, and so on. Think of a proof as a rock wall to climb. You’re not gonna jump to the top in one step. Instead, you break down the wall in smaller parts to climb individually. Each of these small parts is, in the end, comprised of single holds that you can actually switch between in one go. So don’t feel daunted in front of a big proof: it’s actually many small proofs in a trenchcoat.
Corollary: it really helps to work backwards from the conclusions, since it is equivalent to position yourself closer to the end of the climb. It’s not always the definite strategy, but it’s a strategy. Seeing what’s ahead improves your chances of finding a step, and from there you can proceed backwards again.
Applying this at least one time is often necessary, in order to unpack what proving the thesis actually amounts to (e.g. ‘prove 
When you feel why the theorem you need to prove is true, go ahead and sketch an explanation. It’s useful to pretend you’re explaining this to a very skeptical friend of yours. They’re going to challenge everything you say (it takes time to internalize what can be not challenged, what mathematicians call ‘trivial’). Your job is to keep yourself true to your intuition and explain it in detail. Perhaps your imaginary friend will actually poke a hole in your intuition. That’s actually the best thing that can happen: often this brings you to realize a deeper truth, and you end up with a deeper understanding of the problem.
Indeed, it’s important to realize the key to problem solving is continuous back and forth between being wrong and being correct. The worst thing you can do is freeze and not try anything: you’re depriving yourself of useful errors. ‘You learn from your mistakes’ is a very deep truth.
Sometimes this means going ahead and sketching to your imaginary skeptical friend a solution you know is wrong and explain to them why you believe so. Making something explicit to yourself (‘rubber ducking’) is an unreasonably powerful technique. This is because reasoning unravels thoughts in potentiality: as you explain something to yourself, your brain can explore its ramifications. Like in those (old?) games where the map is hidden until you walk through it, you need to move through the dull parts to glance the interesting one.
Finally, use ‘solved’ proofs to test yourself. Sipser’s book [we’re using this in our ‘Theory of computation’ course, ndr] is full of proofs, most of them are quite straightforward (one could call them ‘constructions’). This is great, it means you can exercise your skills by recreating these proofs.
You should not read and repeat them, that’s useless. You should tackle the task as if it was a theorem you found in the wild and try to prove it yourself. If you get stuck, the solution can give you a hint. Be parsimonious with hints though, or you’ll never force your brain to learn. If you are successful and get to the end, you can now compare your proof with Sipser’s. It doesn’t have to be the same! Many theorems have multiple proofs. It has to be valid, though. Try to understand what Sipser’s proof tells you about yours and viceversa.
Also, this is a crucial step to equip yourself with intuition about the objects the course is about. The ultimate goal of studying a proof is to internalize the intuition conveyed by it. You also learn techniques to deal with specific problems (indeed, in this course most proofs are formulaic, and follow a pattern you can familiarize yourself with in advance by reading the book proofs), e.g. reduction techniques.
On a minor note, writing proofs means writing in a certain literary style, so reading and imitating ‘the masters’ is how you learn the correct style of prose [2]. Keep in mind that your goal is to explain why something is true to someone. Providing intuition on why you do something is the main goal. That’s what you need to convey. The rest is to be extra-convincing and plug all the holes someone might poke in the argument.
TL;DR
The three most important take-aways to learn how to write proofs/approach proof writing:
- Develop intuition for the material at hand, clarify to yoursel what the proof actually requires (what am I given as hypotheses? What do I actually need to prove?),
- Rubber duck and don’t be afraid of being wrong,
- Iterate on your ideas, even the wrong ones, and divide your goal in subgoals,
- Learn from the examples.
To conclude, the best quality a mathematician can have is perseverance. That something I noticed when I started my bachelor: finally, I wasn’t the only one who was capture by a problem even after class was dismissed. My fellow mathematicians wouldn’t abandon a puzzle after the first hurdle. They derived joy from the challenge.
Perhaps you don’t want to be a mathematician, but keep in mind success often comes through insistence! So don’t give up!
Footnotes
[0] I clearly remember this process when I first learned commutative algebra and (the rudiments of) algebraic geometry. It felt like forcing my brain to restructure itself. Nothing worked out, and I flunked my exams the first time. I was like a 1yo barely able to balance themselves.
So I spent more time on it, reviewed the concepts (learned more concepts actually, I took an homological algebra course in the meantime) and finally things started to click. I familiarized myself with rings, modules and their behaviours. I internalized the examples, and learned what to expect from my interactions with them. When I tried my exams a second time, they felt like asking a 6yo to jump, clap their hands and pour a glass a water without spilling it. All things I was familiar with by now. I aced them.
[1] I believe proof by contradictions are exactly this: challenge the conclusions and try to understand why it breaks down. This so unreasonably powerful that many people do it even when unnecessary, leading to ‘fake’ proofs by contradiction, proof by negation. So one of my favourite techniques is reason by contradiction and them remove the scaffolding to get a perfectly fine (i.e., constructive) proof by negation.
[2] Yeah, some ‘masters’ are really bad at writing. Some aren’t. Imitate the proofs you found clear, and strive to keep that clarity.
General abstract nonsense. 
