# Mathematics Proofs

Any good links for someone stuck with writing these? I don’t mean examples, but rather something of a step by step outlining the thought process involved.

This is a bit tricky to answer as a each problem is different and there is a good deal of inspiration involved in finding a proof. Basically the one thing that I learned in my first few years of college mathematics what how to fiddle with things, even if you don’t know where it will lead. In general mathematicians don’t see the whole proof from beginning to end at the start, but instead work it out by trying different things.

Start with your axioms and manipulate them to see what other conclusions you get even if you don’t know how it will eventually help. Start with your desired end result and what conditions would be required to show that was true, and what would then be required to show that those conditions were true. Working back and forth from each one in hope that they somehow meet in the middle. Finally go back to the start and write down all of the steps you took to get from A to B checking that you can justify each step, and throwing out all of the dead ends you hit in your first run through.

I also found onebook that might be helpful, although I haven’t read it so I can’t vouch for it.

Have you read “How to Solve It” by George Polya? If not, get it out of your nearest library.

Oh one final hint. If you get stuck in a proof, one thing to try is to look for a counter example. Sometimes in the process of constructing your counter example you encounter an impassible stumbling block which ends up being the core of a proof that not such example exists.

If this is for a class, then you have the added advantage of knowing which techniques were just covered in class. If you’ve spent the past week on the method of runcibilation, then it’s a good idea to start by asking yourself how you can runcibilate the problem at hand.

Yeah, you should familiarise yourself with different proof techniques. Conjectures amenable to proof by induction are usually easy to spot, though choosing what to induct on is not always obvious. You just have to try it, and see what works.

You might find some really basic help on this site.

There isn’t really a step-by-step recipe for writing proofs in general, but there kinda sorta is a step-by-step way to go about doing certain kinds of proofs, or using particular proof techniques (like the mathematical induction that Capt. Ridley’s Shooting Party mentioned). The Wikipedia article on proofs lists and describes several methods of proof.

We might be able to help you more if you gave us more context. Is this for a particular class or in a particular subject?

There are two different questions that the OP might have meant. The easier one is how do you find a proof when the result is known to be true (maybe it is assigned or on an exam) or you have a conjecture and are trying to verify it. If the first, then what you have been doing in class is the crucial thing. The second is harder. One of the hard parts is seeing what conjecture seems likely.

Someone said to start with the axioms. I have never done that and I don’t think man mathematicians do. One general piece of advice, though is to start with the definitions. I have found that it is like pulling teeth to get students to start with the definitions and apply them.

For the second question, I guess the main technique I use is to ask from what simpler thing the conclusion might follow. If that simpler thing looks plausible, then try to prove that. Of course, it will often happen that that simpler thing is false, which doesn’t tell you a thing about the main conclusion. But try try again, look for counter-examples, play a computer game, who knows. YMMV

My first reaction was that you have to learn how to do proofs by doing proofs. The best way is to see how others prove things, then try to do them yourself. I’d suggest narrowing yourself to a particular math area like algebra, analysis, or geometry because there are specific types of proofs in these areas – from straight-forward proofs that follow from axioms or definitions, proofs by contradiction, induction, constructive, etc. I don’t think there’s a single approach that always works. So pick a subject area, get a good book on the subject, and read the proofs. I have a bias to algebra (groups, rings, etc.) since that’s where I cut my eye teeth mastering proofs.

Mathematical proofs are merely arguments. They have certain conventions, and there is a higher standard of rigor than in other fields, but they are just arguments.

Your goal in writing proofs is to convince the reader of the truth of the theorem. That’s it. Read a draft of your proof before turning it in. If you didn’t already believe the theorem, would your proof convince you? Why or why not?