| What is the logic at work in the inquiring activity of mathematicians? I address this question, which pertains to the philosophical debate over whether, in addition to a logic of justification of mathematical knowledge, there is a logic of inquiry and discovery at work in actual mathematical research. Based on the philosophy of Charles Sanders Peirce (1839-1914), I propose that there is a logic of mathematical inquiry, and I expound its form. I argue that even though there are not rules or algorithms that will lead to breakthrough discoveries and successful inquiry with absolute certainty, Peirce's philosophy provides a way to describe (i) the conditions for the possibility of mathematical discovery; (ii) the actual method of inquiry in mathematics and its associated heuristic techniques; and (iii) the logical form of reasoning that warrants the application of mathematical theories to the study of actual scientific problems in nature. With regard to (i), I discuss the role of the problem-context of discovery and describe the epistemic conditions necessary for carrying out mathematical reasoning. With respect to (ii), I argue that experimental hypothesis-making in the course of analytical problem-solving, and not deduction from axioms, is the actual method of mathematical research. Regarding (iii), I argue that abduction and analogy warrant the application of mathematical theories to the study of actual scientific problems. The discovery and early development of mathematical probability, culminating with Jacob Bernoulli's Ars Conjectandi (1713), serves as the historical case study to examine critically the proposed logic of mathematical inquiry. I discuss the practical implications of the proposed logic of inquiry for a philosophy of mathematical education. |
