Mathematical Creativity: A Peircean Abductive Proposal

This paper explores mathematical creativity through the lens of Charles Sanders Peirce's concept of abductive reasoning. Peirce, who defined mathematics as the study of pure hypotheses and their consequences, saw abduction as the process of forming explanatory hypotheses, the only logical operation that introduces new ideas. We argue that mathematical creativity, with its focus on generating novel and meaningful ideas, patterns, and solutions, is intrinsically linked to abduction and takes place through diagrammatic reasoning. Mathematicians, faced with surprising facts or observations, engage in abductive reasoning to propose new theorems or conjectures that explain these phenomena. This creative leap, going beyond deductive and inductive reasoning, lies at the heart of mathematical discovery. We illustrate this process with examples from contemporary mathematics, including Wiles' proof of Fermat's Last Theorem and Perelman's resolution of the Poincaré Conjecture. Furthermore, we connect Peirce's notion of diagrammatic reasoning to mathematical creativity, highlighting the role of experimentation and observation in this process. Finally, we link our approach to Fernando Zalamea's advocacy for a "synthetic philosophy of contemporary mathematics" that engages directly with the creative practices of mathematicians.