Writing a research paper in mathematics requires developing new ways of thinking about mathematical concepts and problems. Unlike other subjects, mathematical research papers rely more heavily on formal proofs, rigorous logic, and quantitative or algebraic evaluations rather than qualitative analysis. The structure and style of a math research paper is also quite different from papers in other fields.
A math research paper generally follows the IMRAD structure of Introduction, Methods, Results, and Discussion. The Methods and Results sections may look quite different than those in other disciplines. The focus is on developing new theorems, proving theorems, exploring counterexamples, and rigorously evaluating problems through mathematical logic rather than experimentation. References should also involve citing seminal works in the field to provide proper context and attribution for new contributions to mathematical knowledge.
Introduction
The introduction lays the groundwork and provides necessary context or background information for understanding the problem or topic being addressed. It should clearly define any terms, functions, variables, or concepts that will be used throughout the paper. The introduction also establishes the significance and motivation for exploring the problem by discussing how it relates to other open questions in the field or potential applications. The specific research questions or hypotheses being explored should be explicitly stated at the end of the introductory section.
Methods
While experimental methods are generally not applicable to pure math research, the methods section is still important for explaining the theoretical and logical approaches being taken. This includes defining any assumptions being made and outlining the systematic process of exploring the problem through derivation of theorems, lemmas, proofs, or other logical arguments. Any mathematical definitions, theorems, or previous results from the literature that will serve as a framework should be cited and summarized. The methods explain how rigorous logical reasoning and formal mathematical proofs will be used to develop new results and address the research questions.
Results
The results section presents the key outcomes and findings of the mathematical problem exploration. This generally involves stating any new theoretical results, theorems, corollaries, or lemmas that were derived through the methods. Each result should be explicitly stated and then proven through formal mathematical proofs, working through a logical chain of steps and equations. Counterexamples used to disprove related conjectures may also be presented. Graphs, tables, or diagrams can be used if helpful for illustrating relationships but are not generally the primary means of communication as in other disciplines. The focus is on rigorous logical arguments and proofs rather than experimental data.
Discussion
The discussion section analyzes and interprets the results in the context of the original research questions or problem. It elaborates on the significance and implications of the new theoretical findings for the field of mathematics. Any limitations or edge cases that were discovered should also be addressed. The discussion then ties back to other relevant literature, comparing and contrasting the new results. It explores the potential applications or directions for future work stemming from the research. The conclusion explicitly states how the paper has contributed new theoretical foundations to mathematics.
In addition to the typical IMRAD structure, a mathematical research paper may also include the specific formatting of stating definitions and theorems as propositions. Proofs are always presented with clear logical steps separated by lines or numerals. Equations and mathematical expressions are also prominent features throughout the text. Figures, graphs and diagrams can help visualize relationships when applicable but are secondary to the theoretical definitions, statements and rigorous proofs. References should be cited both inline within sentences as well as in a references list using numbered citations in order of appearance.
A strong mathematical research paper contributes new theoretical results to understanding open problems through rigorous logical arguments rather than empirical findings. With practice, researchers can learn to communicate complex mathematical ideas with precision using the specialized style and structure expected in this field. Achieving complete logical consistency and formality while maintaining clarity of communication are hallmarks of high quality math research writing.