by Isnayni Hadi
If I were to ask you, “What is mathematical writing?” would you think of an equation or maybe a sequence of symbolic statements? Years of grade-school mathematics has trained us to make that association: math is just a series of solving equations. The curriculum requires us to take algebra, geometry, and a little bit of calculus for some. From those courses, we often assume that mathematical research involves a long equation in which we are “solving for x”, but mathematical writing consists of writing proofs. A proof is a logical argument that convinces the reader that the statement they are proving is true. As you transition from lower-division math classes like calculus or differential equation to proof-based classes, it can be hard to shake off the equation problem solving way of reasoning to a more logical-based reasoning. However, writing a mathematical proof has the same processes as any other kind of writing.
An important consideration for writing any kind of piece is asking ourselves, “Who will the audience be?” In other words, who will we be writing for. The same thought process applies for writing proofs. Initially, we learn to write proofs to a professor in our homework or exams, and we model proof-writing based on the professor. Knowing that the audience in our coursework is the professor, we must be detailed with our proof so that the professor can gauge what we learned and grade accordingly. Any lack of detail may result in points being deducted. However, writing a proof for class work can differ from writing a proof for a publication. For example, the audience of a math textbook is likely a student, so it will often omit details to “leave it to the reader” to force the student to critically think about a proof. Understanding who you are writing proofs to is important to gauge what background material the audience knows. If you want to reference a theorem in a homework assignment that was not covered in class, you might need to prove it before you can use it. However, if the professor already proved it in class, you could simply reference it. The structure of your proof relies on the background material you are assuming.
An important consideration for writing any kind of piece is asking ourselves, “Who will the audience be?” In other words, who will we be writing for.
Beyond knowing who your audience is, it is important to keep in mind that mathematical writing has the same standard and expectations as any other writing. Just like in English class, this means writing a proof in full sentences and paragraphs with correct grammar. Math ideas can be too abstract and complex to express so providing structure to your writing makes it easier to read. It helps to include motivation and intuition for the proof to present a clear argument to the reader. Proofs need clear, concise communication and rigor to be able to convince the reader that your proof is correct.
Everyone has different styles of writing proofs. It is a form of expressing your creativity and presenting the argument the way you want. There are certainly different ways to argue the same claim as it is one of the many beauties of math to witness the different ways of approaching and thinking of a problem. This is the reason mathematics is seen as a collaborative field because many people come together to contribute their approach to a problem so singular pronouns are often avoided in proofs for homework, textbooks, papers, and from my knowledge, any kind of mathematical writing.
Finally, just like any other writing, you need to proofread your proof. It is important to review your proof and analyze your argument since otherwise your proof would be considered wrong. Since proof writing is intended for the math community to read, proofs need to be reviewed and revised before it is reviewed. Although proof writing can seem very rigorous at first, it is a way to showcase beautiful mathematical ideas.