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 On Sun, Aug 10, 2025 at 6:28 PM Jon Alan Schmidt <[email protected]> wrote: > Jeff, List: > > Peirce indeed identified topology--which was a relatively new field in his > day, often called "topical geometry" or "geometrical topics" instead--as > the branch of mathematics most directly concerned with continuity. However, > his three types of cosmologies seem to be grounded instead in *projective > *geometry, where a straight line represents infinity and a closed curve > represents all three types of conic sections ( > https://cre8math.com/2017/07/10/what-is-projective-geometry/). They are > distinguished only by the number of places where the curve intersects that > line at infinity--zero for an ellipse, one for a parabola, and two for a > hyperbola. That is why they translate into three different types of > cosmologies--elliptic, which has no real starting or stopping point; > parabolic, which eventually ends up at the same point where it began; and > hyperbolic, with initial and final points that are distinct. > > Peirce presumably describes an elliptic cosmology as "Epicurean" because > it "cannot consistently regard mind as primordial, must rather take mind to > be a specialization of matter," i.e., it requires materialism; a parabolic > cosmology as "pessimistic" because it holds that "nature develops itself > according to one universal formula," but only toward "the very same > nothingness from which it advances"; and a hyperbolic cosmology as > "evolutionist" because it begins with something "neither requiring > explanation nor admitting derivation," and proceeds in accordance with "a > principle of growth of principles, a tendency to generalization" (CP > 6.582-5, 1890). He also summarizes the differences as follows. > > CSP: If you think the measurable is all there is, and deny it any definite > tendency whence or whither, then you are considering the pair of points > that makes the absolute to be imaginary and are an Epicurean. If you hold > that there is a definite drift to the course of nature as a whole, but yet > believe its absolute end is nothing but the Nirvana from which it set out, > you make the two points of the absolute to be coincident, and are a > pessimist. But if your creed is that the whole universe is approaching in > the infinitely distant future a state having a general character different > from that toward which we look back in the infinitely distant past, you > make the absolute to consist in two distinct real points and are an > evolutionist. (CP 1.362, EP 1:251, 1887-8) > > > Personally, I have a hard time seeing much relevance of these cosmological > concepts to Peirce's later development of Existential Graphs, but I am open > to being persuaded otherwise. He left the Gamma part incomplete, a > collection of various fragments, in contrast to the Alpha and Beta parts > that he spelled out in considerable detail. He also ultimately saw the need > "to add a *Delta *part in order to deal with modals" (R 500:3, 1911 Dec > 6), likely because by then he had abandoned cuts altogether--including the > broken cuts of Gamma for "possibly false" subgraphs--in favor of shading > oddly enclosed areas. > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Structural Engineer, Synechist Philosopher, Lutheran Christian > www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt > > On Sun, Aug 10, 2025 at 10:23 AM Jeffrey Brian Downard < > [email protected]> wrote: > >> Hello Jon, Gary F., Gary R., and all, >> How should we interpret Peirce’s claims about the *topological* distinction >> between three types of cosmologies? >> Jon has assembled an extensive set of textual references spanning roughly >> twenty years of Peirce’s work (1886–1906). My general approach is to read >> Peirce as engaged in a cycle of inquiry, with his initial focus on the >> *abductive* phase—framing questions and developing competing hypotheses. >> He claims that all such hypotheses can be grouped into one of three types: >> >> - *Hyperbolic* or *evolutionist* cosmology (CP 1.409, EP 1:277, >> 1887–8; CP 6.581, 1890; CP 8.317, 1891) >> - *Parabolic* or *pessimistic* cosmology >> - *Elliptical* or *Epicurean* cosmology (CP 1.362, EP 1:251, 1887–8; >> CP 6.582–5, 1890; R 953, c. 1897) >> >> My assumption is that Peirce is asking: *What kinds of models can be >> applied to these competing hypotheses?* It is striking that as early as >> 1886, he is applying the mathematics of topology to classify possible >> cosmological models. From his study of mathematical inquiry, he concludes >> that topology provides the fundamental set of hypotheses for the study of >> continuous systems. >> This raises a question: to what extent did this tripartite division of >> explanatory models shape his later work, particularly the development of >> the existential graphs—especially the gamma system? My own inclination is >> to think that Peirce devised the existential graphs, at least in part, to >> clarify such philosophical and cosmological hypotheses. If so, we might >> explore how the gamma system could be used to make these competing >> hypotheses more precise and to frame them in a way that allows for >> empirical or logical testing. We can also ask, to what extent is he guided >> by hypotheses that lay at the bases of topology to the development of the >> existential graphs. >> Great cosmologists, such as Einstein and Penrose look to various >> geometries for guidance in their physical--and philosophical-- inquiries. >> Peirce is suggesting that we should look, first, to topology before turning >> to the metrical questions of geometry. >> Yours, >> Jeff >> >> _ _ _ _ _ _ _ _ _ _ > ► 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.
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