Writing Proofs

Issues dealing with the writing of mathematical proofs should be addressed throughout an introduction to proofs course.  Completing a proof does bring personal satisfaction, but mathematicians also have the responsibility to be able to communicate their results to others.  For a long time now, I have emphasized the importance of writing in courses I teach for mathematics majors, and I have been actively involved in helping students develop their writing abilities in the introduction to proofs course.  

This is also important to others in the Department of Mathematics at Grand Valley State University as is shown the following two paragraphs, which are based on a document developed by the Department of Mathematics at Grand Valley State University for use in the Writing Center at the university.

At the risk of oversimplification, doing mathematics has two distinct stages.  The first stage is to convince yourself that you have solved the problem or proved a conjecture.  This stage is a creative one and is quite often how mathematics is actually done.  The second equally important stage is to convince other people that you have solved the problem or proved the conjecture.  This second stage often has little in common with the first stage in the sense that it does not really communicate the process by which you solved the problem or proved the conjecture.   However, it is an important part of the process of communicating mathematical results to a wider audience.

A mathematical proof is a convincing argument (within the accepted standards of the mathematical community) that a certain mathematical statement is necessarily true.  A proof generally uses deductive reasoning and logic but also contains some amount of ordinary language (such as English).  A mathematical proof that you write should demonstrate that you have gained a deep understanding of the mathematical concepts involved and should convince an appropriate audience that the result you are proving is in fact true. 

It should be clear from this that we consider writing to be an important part of the mathematical process.  This is one reason the title of our introduction to proofs course is "Communicating in Mathematics."  Issues dealing with the writing of mathematical proofs are addressed throughout the course.  This was also one of the reasons that I wrote the book Mathematical Reasoning: Writing and Proof.   As I search the Internet now, it is great to find many documents dealing with the writing of proofs in mathematics course.  The Department of Mathematics at GVSU has also developed such a document, which can be accessed at this link.  Because of such documents, I am sure that having a textbook dealing with these issues is not as important as it used to be.  However, I still like the idea of having a text that supports what I try to have my students learn.

Feel free to make comments describing what you feel should be part of the writing guidelines for the introduction to proofs course.  One nice things about having an open-access text is that I can include revisions much more quickly than I could before.