Matias, List: In light of your helpful elaboration on what you have in mind, a book chapter that I coauthored with Joseph Dauben and List moderator Gary Richmond a few years ago might be of interest--"Peirce on Abduction and Diagrams in Mathematical Reasoning" ( https://philpapers.org/archive/DAUPOA.pdf), which appears in *Handbook of Cognitive Mathematics* ( https://rd.springer.com/referencework/10.1007/978-3-030-44982-7). Some of the other chapters in that volume might likewise be relevant to your project.
Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Fri, Jul 11, 2025 at 12:38 PM Matias <[email protected]> wrote: > Dear Mr. Chandler, > > I truly appreciate your answer. It challenges me to make explicit what I > am seeking. Here, I will explain what I am pursuing. > > First, as I stated in my previous message, my primary focus is to > understand creative mathematical reasoning. This led me to wonder whether > Peirce's notion of theorematic reasoning can offer a good starting point. I > first encountered that notion while reading an article by Daniel Campos, at > a time when I was working in a research group dedicated to the philosophy > of mathematical practice based in Argentina, specifically at the National > University of Córdoba. At that time, I had done little research on Peirce's > thought around this notion, but it was not the central concern of the > group's activities at that moment. For several reasons, we had to pause the > activities of our group some time ago, and I am taking advantage of this > pause to return to Peirce's notion and to explore its applicability to > understand the actual practice of mathematical inquiry. In a sense, I am > proceeding from scratch. > > Allow me now to briefly describe how I currently conceive a research > project built upon Peirce's ideas. This is a general framework within which > more limited objectives could be inserted. From my perspective, this > project would ideally comprise three steps: 1) to clarify Peirce's notion > of theorematic reasoning; 2) to explore its applicability to mathematical > inquiry at a suitable level of granularity; and 3) to focus on the > underlying cognitive mechanisms behind theorematic reasoning. This final > step could potentially be tackled in an interdisciplinary way. > > Currently, I am focused on the clarification of Peirce's notion, and this > is what motivated my original message. As far as I can see, there is > consensus that Peirce's notion of theorematic reasoning is underdeveloped > in his remaining works, but a salient fact, pointed out by some > specialists, is that through this notion, Peirce characterizes mathematical > reasoning as a creative process of thought. The central trait is the > creative experimentation with a diagram that embodies the relations stated > in the condition of the theorem being sought to prove. This experimentation > consists in a continuous interplay between experimental hypothesis-making > and judicious observation (terms I borrow from Campos). Of course, > theorematic reasoning can comprise other operations beyond experimentation, > such as hypostatic abstraction, but the need to perform experiments with a > diagram seems to me the common trait of all forms of theorematic reasoning. > Also, theorematic reasoning is a type of inquiry into hypothetical worlds > created by the mathematician. > > My reading of Margaret Boden's The Creative Mind and her distinction > between three creative outcomes—the surprising combination of known ideas, > the exploration of conceptual spaces, and the transformation of these same > conceptual spaces—leads me to ask what kind of creativity is implicit in > diagrammatic experiments. I think that such experiments comprise all three > types identified by Boden, as Peirce conceived the notion. > > First, there is at least one example of theorematic reasoning that > exhibits the combinatorial kind of creativity, namely, the demonstration of > Desargues' theorem (or the ten-point theorem) produced by von Staudt. In > this case, a two-dimensional array of lines is related to a > three-dimensional disposition of figures, which reveals the truth of the > theorem almost directly. > > The second type of creativity is represented in the much more frequently > cited example of the Pons Asinorum, where the construction is the product > of the exploration of what is possible within the conceptual space defined > by Euclidean axioms, postulates, and definitions. > > I do not know if Peirce provides an example that directly illustrates the > third type of creativity, but I suspect that the notion of theorematic > reasoning comprises this type as well, and that this makes the notion > particularly interesting as a model for the reasoning involved in > mathematical inquiry. In this respect, I think that we can encounter a > parallel between theorematic reasoning and Kekulé's discovery of the > benzene molecular structure. The experimentation with a diagram, in such > cases, helps us first to detect the limits of the conceptual space within > which we are working, and subsequently to break these limits by > transforming this space. > > In all this project, I think that a profound understanding of Peirce's > notion is fundamental, but if necessary, we may need to modify his original > ideas somewhat to apply them to the understanding of mathematical reasoning. > > I apologize if I did not introduce myself before. I hope this helps you to > understand the context of my first two questions. I thank you very much > again for the time you took to read and answer my messages. > > Sincerely, > > Matías Saracho >
_ _ _ _ _ _ _ _ _ _ ARISBE: THE PEIRCE GATEWAY is now at https://cspeirce.com and, just as well, at https://www.cspeirce.com . It'll take a while to repair / update all the links! ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to [email protected] with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iu.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
