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 > >
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