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 / twitter.com/JonAlanSchmidt On Mon, Aug 11, 2025 at 1:06 AM Gary Richmond <[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 >
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