Dear Jon Alan Schmidt and Frederik Stjernfelt, Thank you both very much for your helpful replies and for sharing these valuable resources.
Jon Alan, the chapter "Peirce on Abduction and Diagrams in Mathematical Reasoning" sounds highly relevant to my project. Frederik, I appreciate you attaching your work on Peirce's corollarial-theorematic distinction, which is central to my inquiry. I am already familiar with your important contributions, and I am particularly eager to now dedicate time to reading these specific works you've shared. Thank you again for your time and generosity. Best regards, Matías Saracho El vie, 11 jul 2025 a las 16:44, Frederik Stjernfelt (<[email protected]>) escribió: > Dear Matias – > > I wrote a bit about Peirce’s corollarial-theorematic distinction some > years ago, attached. > > Good luck! > > Best > > Frederik > > > > *Fra: *[email protected] <[email protected]> på > vegne af Matias <[email protected]> > *Dato: *fredag, 11. juli 2025 kl. 19.39 > *Til: *Jerry LR Chandler <[email protected]> > *Cc: *[email protected] <[email protected]> > *Emne: *Re: [PEIRCE-L] Inquiry: Constraints on Diagram Transformations in > Peircean Theorematic Reasoning (Logical Possibility & Campos) > > 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 > > > > El jue, 10 jul 2025 a las 19:00, Jerry LR Chandler (< > [email protected]>) escribió: > > > > > > On Jul 9, 2025, at 10:23 AM, Matias <[email protected]> wrote: > > > > Dear Mr. Chandler, > > Thank you for your insightful question and for your comments on my > original post. I appreciate you prompting me to clarify my intentions. > > My primary interest is to understand the process of creative mathematical > reasoning. My interest in Peirce derives from this, as he proposed that > mathematical reasoning is intrinsically creative. Similarly, while I > maintain a broader interest in AI and other scientific fields for their own > sake, for the purposes of this specific discussion, my interest in them is > only insofar as they may help us to understand this process. > > What you say about chemical structure reminds me of what Emily Grosholz > says about "intelligible objects." Moreover, she explicitly compares the > intelligible objects of mathematics and the object of study of chemistry. > > > > The poet turn of Ms Grosholz is very close to CSP’s own words: > > W:6, p.35 > > “The chemist sets up… > > > > … The procedure of the mathematician is closely analogous to this.” > > > But all this is with respect to the object, and my focus is on the side of > the inquiry. In this respect, I suspect that what Peirce describes as > theorematic reasoning is similar to the way Margaret Boden describes > Kekulé's discovery of the benzene ring structure. He was working within a > conceptual space that excluded rings as possible structures, and Kekulé's > breakthrough was to propose a structure not readily conceivable within this > conceptual space. Here, the constraints have a role: first, guiding the > investigation; and subsequently, serving as a background to appreciate > Kekulé's creative idea. > > > > The necessity for a cognitive source of referential symbols / words / > experimental images is needed in both cases. > > > > Note that the chemical reference system was in its infancy during CSP > early years and co-matured with CSP himself and that his writings reflect > this shifting foundations of meanings. > > Mathematical notations were well established - arithmetic as associative > and distributive operations grounded computations. Chemical arithmetics > was to be developed after CSP past; it was in primitive stages of > development during his lifespan. > > > > > What is disconcerting about Campos's view is that he seems to restrict the > experimentation upon diagrams to what is possible within a mathematical > system, and seems to be collapsing the object of inquiry with the > particular frames or representations from which we study it. Of course, it > is very possible that I am misunderstanding Campos. This potential > misunderstanding on the matter, particularly given my acknowledged lack of > expertise in Peirce's broader philosophy, is precisely what motivated my > original questions: first, to confirm my interpretation of his argument, > and second, to gauge the extent to which his view is representative within > the scholarship. > > Thank you again for your time and consideration. > > > > You are welcome. > > > > May I pose questions? > > > > Of what use do you seek this information? Poetry? > > > > What thoughts could be stipulated from these esoteric notions? > > > > How would semiosis be influenced by the distinction between externally > grounded symbols and the internal playground of symbolic manipulations? > > > > Are you seeking syntagmata? > > > > Cheers > > > > Jerry > > > Best regards, > > Matías Saracho > > > > > > El mar, 8 jul 2025 a las 20:57, Jerry LR Chandler (< > [email protected]>) escribió: > > Hi > > > > First, are you interested serious science or merely probing the groundings > and soundings of transformer theory? > > > > I have inserted some comments in the text that address your questions at a > highly superficial level. > > On Jul 8, 2025, at 10:20 AM, Matias <[email protected]> wrote: > > > > Dear List Members, > > I am interested in determining the kinds of creativity involved in > theorematical reasoning. For this inquiry, I am adopting Margaret Boden's > typology, which identifies three distinct creative processes: > combinatorial, exploratory, and transformational. This framework has led me > to consider the nature of the constraints imposed upon the transformations > allowed within the diagram of the premises during a theorematical > deduction, particularly as expressed in what Campos terms "experimental > hypothesis-making." In this regard, I would greatly appreciate your > insights on two specific questions: > > Is Campos accurate in equating "logical possibility" with "possible > within the mathematical system" as a constraint on the formulation of > experimental hypothesis-making in the following paragraph? > > > > I find this to be a bit strange. Yes, but… this is rather naive. > > > > > Is it reasonable to assume that the views expressed by Campos in this > paragraph represent the standard interpretation of this matter within > Peircean scholarship? > > > > Even more strange! > > The question of “the form of the icon” allows combinatorial explosions…I > suspect that CSP language here is addressing the very troubling issues of > “chemical isomers", such as the mention of tartaric acids, Pasteur’s famous > quanta objects with multiple crystalline forms/ > > > Here is the citation from Campos (2010): > > "The logical possibility of the experimental diagram, within the assumed > mathematical framework, is a necessary condition that constrains what > experimental signs the imagination may submit for observation. As Peirce > put it generally in 1906, “that which is displayed before the mind’s > gaze—the Form of the Icon, which is also its object—must be logically > possible” (Peirce 1906b:C4.531). Logical possibility is in part a matter of > mathematical diagrams being subject to the normative rules of good > reasoning that Peirce’s logical critic prescribes, including for example, > for mathematical deduction, the law of the excluded middle and the law of > noncontradiction. But it is also a matter of what is possible to create > within a general hypothetical framework. As we have seen, for example, in > Euclidean geometry postulates affirm possibilities while axioms deny them. > Along with definitions, they are in fact the more general hypotheses that > frame a realm of possibility for mathematical investigation." (Campos, > 2010, The Imagination and Hypothesis-Making in Mathematics: A Peircean > Account, p. 337) > > > > "As we have seen, for example, in Euclidean geometry postulates affirm > possibilities while axioms deny them." > > > > This is also a bit strange language. OK. Many would introduce the > notions of axiological reasoning or some other abstractive terminology. > > > > Frankly, I can not imagine a “standard” for CSP terminology. > > At this point in history, more than a 100 years after he past, CSP’s > writings are mostly a playground for philosophers; the scientists, > logicians and linguistics have moved on decades ago. > > > > Just my opinions… > > > > Cheers > > > > Jerry > > > > > > > I very much appreciate your invaluable comments. Thank you for your time > and consideration. > > Best regards, > > 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. 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_ _ _ _ _ _ _ _ _ _ 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.
