Jon S, Gary R, Gary F, all, It is worth noting that Peirce argues topology supplies the key principles (i.e., postulates and theorem) that are foundational for projective and metrical geometries. See the discussion of topology and theses geometries in the NEM. The distinction between elliptical, parabolic and hyperbolic surfaces and higher dimensional spaces is rooted in topology. See the helpful discussion in Jeffrey Weeks, The Shape of Space, for instance.
The reason I raised the topological questions about Peirce's distinction between three kinds of philosophies is that they are, at root, different accounts of the beginning and ending points of inquiry, considered as some kind of limit. Those beginning and ending points are taken to mirror an analogous account of the beginning and ending points of cosmological evolution, also considered as some kind of limit. Jon S claims that Peirce's claims should be understood in terms of projective geometry and not topology. I am suggesting that the claims can be understood more deeply if we consider both, and especially the way the postulates of topology provide a foundation for projective and metrical systems of geometry. I take Peirce seriously when he claims that the EG's were designed to have the richness needed to represent any kind of proposition, and any kind of inference, regardless of whether it is deductive, inductive or abductive. As such, I am interested in various ways we might draw on the existential graphs, especially in their gamma form, to clarify the claims he is making about elliptical, parabolic and hyperbolic philosophies. Without getting into the details, the differences in the assumptions will likely show up in the different ways philosophers might interpret the different universes of discourse, and the manner in which the conceptions of truth and reality are represented in a logical system such as the EG's. The differences will show up prominently when we consider the ways in which abductive and inductive inferences might be represented--especially when we consider the ways the modal features of are treated--such as a the modal features of the claims that a hypothesis is plausible or that the conclusion of an inductive inference is probable. The claims of probability involved in induction need to be interpreted in terms of metrics for degrees of confidence, margins of error, and the like. I'm curious about how we might represent these conceptions in propositions expressed in terms of the gamma graphs. --Jeff ________________________________ From: [email protected] <[email protected]> on behalf of Jon Alan Schmidt <[email protected]> Sent: Monday, August 11, 2025 10:27 AM To: Peirce-L <[email protected]> Subject: Re: [PEIRCE-L] Topology, Cosmology, and Existential Graphs (was Semiosic Ontology and Evolution) Gary R., List: As far as I know, the best introduction to Peirce's work in mathematical topology is Jérôme Havenel's paper, "Peirce's Topological Concepts" (https://www.academia.edu/25930950/Peirces_Topological_Concepts_Jerome_Havenel). The linked version is his draft of a chapter that appeared in the 2010 book, New Essays on Peirce's Mathematical Philosophy, edited by Matthew E. Moore. As Havenel notes in his introduction, it is important to recognize that modern "point-set topology is not topology for Peirce." That is because it is an extension of Cantor's arithmetical (bottom-up) conception of the real numbers as constituting a "continuum," which Peirce considered instead to be a "pseudo-continuum" as opposed to his own geometrical (top-down) conception of a "true continuum." You are exactly right to think that the sheet of assertion functions as a topological surface. In fact, Peirce begins one of his very first manuscripts about Entitative Graphs (R 482, 1896-7) by talking about Listing numbers (LF 1:212), and he applies them directly to Existential Graphs (EGs) a decade later (R 293; NEM 4:323-4, LF 3/1:327-8, 1906-7). As Ahti-Veikko Pietarinen notes in his introduction to R 484 (1898), "Peirce even ventures a topological possibility that a double cut is a coarsened form of one continuous line infolded within itself 'so as to produce cells and cells within cells'" (LF 1:331), although Peirce later explains it as iteration (when added to any area) or deiteration (when removed from any area) of the blank. His manuscript entitled "Topical Geometry" (R 145, 1905) includes a brief introduction to the Beta part of EGs (LF 1:510-1), noting that "the Alpha part has no line of identity" and "The Gamma part has been a good deal considered, but has not been settled." In the "Logical Tracts" that Peirce prepared for the 1903 Lowell Lectures, he characterizes topology as "a branch of geometry which not only leaves out of consideration the proportions of the different dimensions of figures and the magnitudes of angles (as does also graphics, or projective geometry--perspective, etc.) but also leaves out of account the straightness or mode of curvature of lines and the flatness or mode of bending of surfaces, and confines itself entirely to the connexions of the parts of figures" (LF 2/1:134-5). This description likewise applies to EGs, where "the straightness or mode of curvature" of both the heavily drawn lines of identity and the lightly drawn lines for cuts--ultimately replaced by boundaries of shaded areas--are likewise irrelevant, the flatness of the sheet's surface is a convenience rather than a requirement, and what matters from a logical standpoint are "the connexions of the parts of figures"; the spatial arrangement of the subgraphs within each area makes no difference at all in the interpretation. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt<http://www.LinkedIn.com/in/JonAlanSchmidt> / twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt> On Mon, Aug 11, 2025 at 1:06 AM Gary Richmond <[email protected]<mailto:[email protected]>> wrote: Jeff, Jon, List, Not being a mathematician, yet I've been interested enough in Peirce’s work in topology (especially after reading his 1898 Lectures) as to have taken up a self-study of it many years ago. I can't say I made much headway. But because Peirce never systematized his work in topology in a paper, and because his ideas appeared decades before topology became a formal mathematical field, it appears that his work in this area hasn't been fully explored in any direction. We know Peirce followed European, and especially German mathematics, was aware of and influenced by the work of Gauss, his student Riemann, and Listing (who, btw., coined the term topology in 1847 in his book Vorstudien zur Topologie (Introductory Studies in Topology). In any case, and as already noted, Peirce's work in topology shows an astonishing prescience about ideas that wouldn’t even be named, and certainly not rigorously developed until well after his death. Although this may not have direct relevance to the three cosmological concepts being discussed and Peirce's development of EGs, I'd like to ask, as existential graphs are laid out on a sheet of assertion, am I wrong to think that the sheet of assertion may itself function as a topological surface? I am glad, Jeff, that you brought this neglected topic of topology to the List; and thanks, Jon, for your thoughtful response to Jeff. Best, Gary R
_ _ _ _ _ _ _ _ _ _ ► 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] . ► <a href="mailto:[email protected]?subject=SIG%20peirce-l";>UNSUBSCRIBE FROM PEIRCE-L</a> . But, if your subscribed email account is not your default email account, then go to https://list.iu.edu/sympa/signoff/peirce-l . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
