An Overview of the Discipline

Professor Dennis DeTurck points out that "mathematics as a discipline is not the study of numbers and shape. It's the study of structure, and structures can occur in many different types and contexts. But structure is the underlying theme." He adds, "The function that you use to figure out where to put a weight on a beam to make it balance is the same function you use to figure out the mean waiting time for a bus. It's structure; it's the form without the content." Professor David Harbater similarly describes the discipline: "Math is the study of structure, of shapes and functions. We define math more by what it does. It provides a language of science, a very precise language that helps people convey complicated science and complicated ideas."

Math is a way of understanding the world in terms of formal logical systems. Often these consist of quantitative models where one attempts to understand the relationship between variables. Some divide mathematics into two fields, applied and theoretical. Applied mathematics concerns itself with the practice use of math in industries such as engineering, science, and business. Theoretical or pure mathematics is mathematics developed for its own sake. As many point out, there is no clear divide between these two approaches. Research may originate on the theoretical side and make its way to the applied side, and vice versa. The two fields are significantly interdependent.

Writing in the Discipline

Reasoning

Mathematics generally focuses on explanatory rather than justificatory/argumentative reasoning. A proof needs to be logically airtight to be persuasive, including the outline and corollaries. Facts do the convincing, rather than arguments. A proof is typically accompanied by an explanation of the steps taken to arrive at its conclusion. A minor amount of justification occurs when the scholar is obliged to persuade the reader of the significance of his or her findings.

Key terms include:

• Theorem: mathematical statement that has been proven and is thus held to be true. May be used to prove further theorems.
• Proof: convincing logical demonstration of the conclusion. "Proofs establish the assertions that you study," notes Dr. Harbater.
• Conjecture: proposition that is still unproven, but yet held to a high standard of truth (Riemann Hypothesis). Believed to be true but not yet a theorem.
• Corollary: side note that typically follows a theorem. "Serve as little sign posts not meant to disrupt logical thought or an idea (in writing), but follow logical thought," explains Dr. DeTurck.
• Lemma: "precedes a theorem, and serves as a logical building block in establishing the validity of the theorem," describes Dr. Harbater.
• Axiom: proposition or fact considered to be self-evident; universally held truth that requires no proof.

Evidence

The accepted evidence for mathematical articles is the formal proof. "You start with what you know and have an airtight argument that deduces each step from the previous step until you reach a conclusion," observes Dr. Harbater. Proofs are based on axioms, which are considered to be inherently true, and theorems, which have previously been proven. A good proof should be indisputable. The peer review process helps to assure that incorrect proofs are not published.

Computers have been very helpful to mathematicians as a tool for creating proofs. First, it is possible to discover a phenomenon through computer simulations. Once discovering a phenomenon, a formal proof must follow to support the experimental findings. Second, a computer can aid in the actual proof by using it to check thousands of cases after a mathematician has reduced the number of possible cases to a finite number. Despite the powerful uses of a computer, it cannot replace a rigorous proof.

Authorship

While in the past math scholarship has been mainly single-authored, in recent years research and publishing has grown more collaborative, with multiple-authored texts no longer uncommon. Many scholars in the field of mathematics share their work on arXiv.org, a preprint repository, before submitting it to a journal, as a means of getting feedback from other mathematicians.

Goal

The main goal of mathematics is problem definition and solution. A theorem is put forth and then proved. Mathematicians may aim to prove a longstanding problem or to develop a new theorem. They may explore a new field or look at an old field in a new way. While most scholarly work in mathematics presents correct, indisputable proofs, the work may not be interesting or useful. Thus, a substantial rhetorical demand for math scholarship is to convince the audience that the findings are significant.

Writing Tips

Important Criteria for Student Writing

Professors place having original ideas at the top of the list of desired qualities in student writing in the field of mathematics. This originality is, in turn, contingent upon the student having demonstrated mastery of other’s ideas. Mathematicians can further a field only after understanding what has already been done. Second, reasoning and evidence in the form of proofs are vital to all writing assignments in this discipline. A good proof shows mastery of the formal reasoning of mathematics. Finally, organization is important to any piece as it will help the reader understand the ideas presented in the paper.

Common Student Errors

A very common error in papers is that students do not explain their reasoning in proofs very clearly. Many times, this can be fixed by going through the proof again and making sure it makes sense. Another error is that students do not always use complete sentences when using mathematical symbols. Note that in math, "Let f = 4x + 3" can be considered a sentence.

Dr. Harbater points out that common error is for students to jump right into the reasoning. An explicit introduction is required. "Mathematical writing needs to let you know where you're going, and it's the same idea in lecture," explains Dr. Harbater. "I always start by forecasting what I'm going to teach that day, how it relates to what we've done, and how it relates to where we’re going."

Student writers may also err by using circular reasoning. Circular reasoning attempts to prove a statement by using the statement itself as part of the proof, or by using a fact that logically depends upon the statement you are setting out to prove.

Finally, novice writers may provide too many unnecessary details in their proofs, presenting every minute step rather than understanding that one's readers can fill in such details themselves.

Style

Because clarity of content is so important, mathematical writing should be clear and concise. A writer needs to be very sensitive to the audience, explaining only as much as needed to get the point across. Too little explanation will inhibit understanding, while too many details will overwhelm and obscure the proof. Grasping how much the audience already knows or takes for granted, and how much it needs to know, are critical to the writing process in mathematics.

Math texts are typically written in the present tense and try to limit the vocabulary as much as possible to ensure precision and stability of meaning. Concision is highly valued, as is clarity. The writing typically has a very specific audience with specialized knowledge in the field; the reader needs to understand the terminology to engage meaningfully with math scholarship.

Genre

Student Assignments

Undergraduate writing assignments in math mainly consist of problem sets with written solutions. Students are typically asked to write proofs, which are about two paragraphs each. In advanced math classes, the homework problems are "prove the following," and half or more of the course may focus on mathematical writing, which is tightly correlated with mathematical thinking. A goal of the upper level math courses is to learn how to write mathematically.

Professional Writing

Scholarly articles published in academic journals are the main genre in this field. Other genres include books, book reviews, reports, and expository pieces about another mathematician’s research.

Additional Resources

Wayne Booth, Gregory Colomb, and Joseph Williams' The Craft of Research

© 2013-2014 The University of Pennsylvania

Meet the Professors

Dr. Dennis DeTurck

Dr. DeTurck has authored more than 50 papers. More...

Dr. Jonathan Block

Dr. Block is a topologist and geometer. When he begins writing an article, he goes to a cafe so that when he needs to take a break he can just look around and relax for a bit. More...

Dr. Charles Epstein

Dr. Epstein works in mathematical analysis with partial differential equations and in applied fields with image analysis, population biology, and magnetic resonance imaging. Writing is a large part of what he does. He writes books, journal articles, and an "endless number of reports." More...

Dr. David Harbater

Dr. Harbater believes "good mathematical writing is good mathematical thinking." More...