1 Jérôme Havenel Peirce’s Topological Concepts, Draft version of the book chapter published in New Essays on Peirce’s Philosophy of Mathematics, Matthew E. Moore ed, Open Court, 2010. Abstract This study deals with Peirce’s writings on topology by explaining key aspects of their historical contexts and philosophical significances. The first part provides an historical background. The second part offers considerations about Peirce’s topological vocabulary and provides a comparison between Peirce’s topology and contemporary topology. The third part deals with various relations between Peirce’s philosophy and topology, such as the importance of topology for Peirce’s philosophy of space, time, cosmology, continuity, and logic. Finally, a lexicon of Peirce’s main topological concepts is provided1. Introduction In Peirce’s mature thought, topology is a major concern. Besides its intrinsic interest for mathematics, Peirce thinks that the study of topology could help to solve many philosophical questions. For example, the idea of continuity is of prime importance for Peirce’s synechism, and topology is “what the philosopher must study who seeks to learn anything about continuity from geometry” (NEM 3.105)2. In Peirce’s reasonings concerning the nature of time and space, topological concepts are essentials. Moreover, in his logic of existential graphs and in his cosmology, the influence of topological concepts is very apparent. However, Peirce’s topological vocabulary is difficult to understand for three reasons. First, in Peirce’s time topology was at its beginnings, therefore both terminology and concepts have evolved a lot in the 20th century, and we find Peirce struggling to find light in a quite unexplored environment. In particular, to the best of my knowledge, Peirce has never studied - nor mentioned -, Poincaré’s major contribution to algebraic topology, his “Analysis situs” (Poincaré, 1895), followed by a series of five addenda from 1899 to 1904. Indeed, Poincaré’s papers have provided some of the key concepts of modern topology. Secondly, in almost all contemporary introductions since Lefschetz, topology seems to be grounded in the notion of set, and topological spaces are defined as sets3. But to ground topology in the theory of sets is historically misleading. If the notion of continuous transformation can be characterized within the framework of set theory, the main intuitions of the founders of topology were related to general geometry. As Gauss, Listing, Klein and Poincaré, Peirce considers 1 I offer my grateful thanks to André De Tienne, Marc Guastavino, Matthew Moore, Marco Panza, and Fernando Zalamea for their very helpful comments. 2 RLT, p. 246, MS 948, 1898. 3 Lefschetz, 1949, p. 3. 1 2 topology as a general geometry. Indeed, what Peirce calls topology corresponds roughly to algebraic topology and some aspects of differential topology4. If Peirce is concerned with point-set topology when he characterizes the continuum as being supermultitudinal, point-set topology is not topology for Peirce. Thirdly, Peirce frequently changed the names he used to refer to topological notions. It is therefore useful in introducing Peirce’s topological vocabulary to give the main definitions, their historical context, and philosophical significance. Hence, there is a glossary at the end of this study. As a forerunner in topological researches, Peirce offers various definitions of topology, but we can start with the following: “topology, or topical geometry, is the study of the manner in which places are intrinsically connected, irrespective of their optical and metrical relations” (NEM 2.165). One can notice Peirce’s terminological hesitation between topology and topical geometry. In the Cambridge Lectures of 1898, he explains that what Listing calls “topology”, he prefers to call it “topic”, to “rhyme with metric and optic” (NEM 3.105). One should remember that for Peirce, there is no distinction of meaning between topology, topical geometry, geometrical topics, topics and topic. Likewise, there is no distinction of meaning between graphics, optic and projective geometry5. Metrics, metric and metrical geometry also mean the same thing for Peirce. 1) Historical Background As the son of Benjamin Peirce, considered by many to have been America’s first truly creative mathematician, and being himself an important figure in the field, Charles Sanders Peirce was well informed of the works of many prominent mathematicians of his day. In particular, we know that Peirce was deeply interested by mathematicians such as Cantor, Cayley, Clifford, Dedekind, Gauss, Helmholtz, Klein, Lobatchewski, Riemann, Story, and Sylvester6. For example, Clifford was a friend of Benjamin Peirce and was one of the first to recognize the importance of the paper on the founding of geometry published by Riemann in 1867. In 1873, Clifford translated Riemann’s work in English, and Charles Sanders Peirce has read this translation7. Peirce attended several meetings at the New York Mathematical Society, before it became the American Mathematical Society in 1894. However, from the 1890s to the end of his life, Peirce’s contacts with the mathematical community shrunk dramatically. Peirce met several mathematicians when he was a member of the Johns Hopkins University from 1879 until 1884. Among his colleagues were Sylvester and Kempe, and it is likely under Kempe’s influence that Peirce became very interested in the topological question of the Four Colour Problem8. “On November 5, 1879, Dr. Story presented a communication by Mr. A. B. Kempe … 4 This question will be elaborated later on. 5 NEM 2.625, MS 145: “… Projective Geometry, which I prefer to call, with Clifford, Graphics”. 6 NEM 3.979-980; NEM 3.883. 7 Riemann, 1873. 8 The Four Color Problem is a topological question since the sizes and shapes of the map regions do not matter; what matters is only the way they join together. 2 3 ‘On Geographical Problem of the Four Colors.’ … Mr Peirce is recorded as then discussing…” (Eisele, 1979, p. 217). Sylvester’s close friend Arthur Cayley was invited in 1882 to give a course of lectures at Johns Hopkins University9. During his stay Cayley undoubtedly influenced Peirce. Cayley also influenced both Sylvester and Clifford in their idea of using chemical diagrams to represent algebraic invariants. In a letter to Oscar Howard Mitchell, on December 21, 1882, Peirce explained that he was greatly impressed by this idea10. Unsurprisingly, since Peirce was both a professional chemist and logician, deeply interested by new mathematical ideas. Cayley was one of the few mathematicians of his time to be interested in Listing’s Census-theorem, published in 1861, which represents the first important mathematical step in topology since Euler. Cayley’s interest in Listing’s work was probably related to his work on the theory of the combinatorial machinery for obtaining various chemical graphs11. The same year Listing published his Census- theorem, Cayley published a paper “On the Partitions of a Close”, which he thereafter recognized to be “a first step toward the theory developed in Listing’s memoir”12. Cayley reported on Listing’s Census-theorem to the London Mathematical Society in 1868, and he published a summary of it in the Messenger of Mathematics, in 1873. It is very likely under Cayley’s influence that Peirce read Lisiting for the first time, but as shown by his “Notes on Listing” in MS 159, Peirce was at first disappointed and not convinced by Listing’s work. As quoted by Murphey (1993, p. 197), in MS 159, Peirce says that Listing’s work is “Talkee, Talkee” and that he does not “understand this so far, at all”. The dating of MS 159 is difficult; Robin says: “1897?”; whereas Murphey seems to think that it is rather 1882-188413. In his article “theorem” written for the Century Dictionary, so in 1884-1889, Peirce gives a definition of Listing’s theorem, but a short one: “Listing’s theorem, an equation between the numbers of points, lines, surfaces, and spaces, the cyclosis, and the periphraxis of a figure in space: given in 1847 by J. B. Listing. Also called the Census theorem” (CD, “theorem”); and Peirce does not define the Census theorem in the CD. However, in the SV, such a definition is given which was likely written by Peirce, as it is suggested by the reading of MS 1597. Whatever the case may be, in MS 137, written in 1904, Peirce speaks with great respect of Listing’s Census-theorem as being a masterpiece. Around the same time of Listing’s first works in topology, the theory of invariants was invented by George Boole who is best remembered as one of the creators of mathematical logic. The theory of invariants was subsequently developed by mathematicians such as Arthur Cayley and James Sylvester. In his 1859 “Sixth Memoir on Quantics”, which is widely quoted by Peirce, Cayley developed the geometrical application of the theory of invariants. It was already known that for projective geometry, in its legitimate transformations, lengths, 9 Parshall and Rowe, 1994, p. 79-80. 10 L 294. 11 Murphey, 1993, p. 196. 12 Cayley, quoted in Murphey, 1993, p. 196. 13 Murphey,1993, p. 197. 3 4 angles, areas and volumes are not always preserved. The distance between two points is a numerical value that is invariant in metrical geometry. However, in the realm of projective geometry, this quantity does not remain invariant and is therefore useless. But in projective geometry there is one numerical invariant which is called the cross-ratio. Because it is invariant under projective transformations, the cross-ratio, which is a ratio of ratios of distances, is a fundamental quantity for projective geometry. In 1859, Cayley set himself the task of establishing the metrical properties of Euclidean geometry on the basis of projective concepts. Cayley generalized Laguerre’s idea of defining the size of an angle in terms of the projective concept of cross-ratio.