Writing Mathematical Proofs
Beginning your Proof
The very first step is to know what it is you are assuming, whatever that may be (see “Types of Proofs”). A reliable next step is to, perhaps on a piece of scrap paper, write out every detail of your assumptions and various results that follow immediately from them. Another good strategy is to look through textbooks and lecture notes for proofs of similar claims to the one you are hypothetically working on at the moment. In many cases, professors may ask of students to solve problems similar to or generalized from examples worked out in class. In these cases, some elements of the proofs given can be useful in solving the new problem, but in every case simply rewriting the proof with new terms will likely be insufficient.
From there, because there is such a wide variety in problems and proofs even within some subdisciplines of mathematics, there is no advice that applies and can be useful in general.
A strategy recommended often by UVM Math professor Jeff Dinitz is to look at the conclusion of a given statement and think about how to get there. In a sense, to try and write the proof “backwards.”
Language
Language in proofs should be formal and impersonal. In seeking to communicate a point concisely and objectively, writing very dryly is preferable. This does not preclude all possibility of unique voice and dry humor in mathematical writing, but it is far from a priority.
A balance of mathematical symbolic writing and plain English is paramount. As a general rule of thumb, anything that can be explained without using a dense line of esoteric math should be. A proof consisting of a paragraph of the written word is much more readable than two lines of symbols. There are settings where that is appropriate, but these are generally the academic circles that look to find the most concise version of a proof. These are, of course, illegible to anyone outside the field (and, likely, outside the subdiscipline within mathematics), but accessibility is not really the goal of this genre of proof, nor is this the ideal student’s writing in mathematics should reach for.
Conventions and Requirements
Completeness
A proof must be complete. This means that your proof does not have any logical holes and works in any scenario given statement A. This also means that something a lot of beginning math students will want to gravitate towards, a “proof by example,” does not exist and that such a thing will not service a substantial proof. This is not to say that someone should never try to work out specific examples of a problem-that can be very helpful-but the final draft of a proof must work for any instance.
Clarity
Every statement made within a proof must have a clear logical connection to the previous parts of the proof. As a check, perhaps go through each logical step and think “this is true because.” If that because is not very obvious (and be careful with the threshold of obviousness), it is advisable to add another step in between.
Conciseness
In short, no statement should be made in a proof that does not contribute directly to the statement you are trying to prove. This is a similar tenet to writing in many different disciplines, but it is especially true in mathematics, particularly when statements and proof start getting more and more complicated and each sentence carries more and more weight. This ties in with the note above about not including examples in proofs. If the example is covered by the more general version of the proof, it is redundant to include the example in the proof.