by Peter Francis Cassels
Many people whose formal mathematical education ended in high school or early college would describe mathematics as the study of numbers and shapes. If asked further, perhaps they would spend some time recalling the tedium of performing calculations, or of solving contrived word problems which are far removed from any sort of reality. This description, however, is woefully inaccurate at describing the actual practice of (pure or abstract) mathematics as practiced by mathematicians and students at the late undergraduate and beyond levels. A better description would be that mathematics is the study of reasoning through the use of proof. Mathematicians create and describe complicated objects which they reason about with in precise language, and when successful they are able to solve problems with. I have often believed that there is an apt comparison of a mathematician to a lawyer—the mathematician must make an airtight case on some proposition to the jury who is their reader. They must be sure to consider every possibility and case to ensure that they have provided sufficient reasoning for their proof. It is this reason why, perhaps counterintuitively, that learning writing is so important for students of mathematics.
It is first worth asking oneself, what is a proof? Put simply, in mathematics, a proof is a justification for a certain proposition which starts with a set of axioms—smaller statements which are assumed to be true—and finishes with showing how if one accepts these axioms then one is forced to accept the conclusion of the given proposition. They can range from very simple and short to extraordinarily complicated. In length, a correct proof can be a few sentences, (or no words at all in some rare cases where a picture can suffice) or it can take up hundreds of pages. In some sense these are the building block of mathematics; if mathematics is the practice of precise reasoning, then to communicate one’s ideas, you must use proofs. Herein lies the true beauty of mathematics. In most cases, it is not the final result which possess the beauty, it is instead the construction of the proof, with the beautiful ideas flowing through the words. The feedback which is constantly applied to early math students’ work, that one must always “show your work” is indeed true—mathematical facts without justification is close to worthless.
There are a number of practices which mathematicians generally agree upon with the purpose of maximizing clarity in writing proofs. First, it is always of the utmost importance to always be as clear as is reasonable. Different audiences and different situations demand different levels of clarity, but mathematical writing (and in particular the writing of proofs) should always have the reader in mind. One only writes in math with the intention that someone will read it, so it is of the utmost importance to ensure that this can be done for the intended audience. For example, when writing a more involved proof, it is best practice to sketch out the steps that you will be taking, so your reader can easily follow along.
While many believe math to be communicated through symbols and calculations, in reality clear communication through writing is equally important.
The field and study of mathematics is indeed a beautiful one, but in order to see its true beauty and power, one must engage with the writing of mathematics. The way to expose others to mathematics is through writing proofs. It is imperative for anyone who wishes to share their mathematical work with others. While many believe math to be communicated through symbols and calculations, in reality clear communication through writing is equally important.
Peter Cassels is a 2021–22 WIP TA for mathematics.