List, I just noticed that the message below, from Jeff Downard, did not go to the list, although he clearly intended it to. I think the question he poses at the end is a fascinating one, but hardly know where to start in working toward an answer. (Perhaps I should start with Peirce’s writings on topology in NEM?) Anyway I hope Jeff or John Sowa or others will take up this topic (perhaps with a change of subject line), as it’s something I’ve often wondered about.
One passage that might be relevant is the topological cave-exploration passage in Peirce’s 1898 Cambridge lecture on the Logic of Continuity, which I’ll append at the bottom of this message. I find it both challenging and fascinating to (try to) follow. Gary f. From: Jeffrey Brian Downard <jeffrey.down...@nau.edu> Sent: 20-Feb-19 10:58 To: g...@gnusystems.ca Subject: Re: [PEIRCE-L] Analyzing Propositions (was EGs and Phaneroscopy) Gary F, Jon S, John S, List, I, too, agree with Gary F's remarks on the points John S has made about the EG's. That is, they seem on track to me so long as we constrain our attention to Peirce's treatment of lines connecting dots to relations of identity--conceived of dyadically and triadically (i.e., as teridentity). Having said that, I find it difficult to see how Jon S's suggestions with respect to the MEG's are supposed to provide an account which treats lines as more general relations and not as lines of identity. After all, it is obvious that some relations are not well captured by lines connecting spots, including the relation of the conditional de inesse and the relation between affirmation and negation--which are fundamental to all versions of the EG. Having said that, it would help me better understand what Jon S is trying to accomplish with the MEG's if some comparison were made between the way he is using and interpreting lines and branches to represent various kinds of relations and the way Peirce uses lines and branches when he develops the conceptions of the potentials and the selectives. Having read and listened to the presentation John S gave at the APA Pacific Division meetings a little while back on the use of EG's in the Alpha and Beta forms to represent some of Eulid's reasonings in the Elements, I am curious about a diagram in the gamma system that Peirce offers to represent the 5th postulate in Book I. Don Roberts provides some helpful commentary on Peirce's graph of that postulate on pages 76-7 of his monograph. I'd be curious to hear from John S (and others) how such representations in the gamma system might be used to analyze examples of more complex reasoning in geometry. As test cases of more complex inferences, consider the reasoning of Riemann to the postulates that lie at the basis of elliptical geometry and of Lobachevsky to the postulates that lie at the basis of hyperbolic geometries. Furthermore, consider the proofs they give of the theorems that follow from those alternate systems of hypotheses. In both cases, a key move was a reconsideration of the conceptions that are involved in the 5th postulate--and Peirce is clearly thinking this through in a number of places, including the last lecture in RLT, the EM and NEM, and his later remarks on the relations that hold between topology, projective geometry and metrical geometries. Let me try to offer a first response to Gary's comments about the discussion that we had on the points that Peirce explores concerning the "EGs and Phaneroscopy" in the Lowell Lectures (thanks again to Gary for providing transcriptions of those lectures). I think the example above might help to supply us with a case to consider. Instead of looking at the postulates that lie at the basis of metrical or projective geometries, consider the simpler set of postulates, definitions and axioms that lie at the bases of 19th-century topology. Peirce's articulates a number of the key postulates in the EM and NEM. With a list of those hypotheses in hand, we could try to answer the same kinds of question that Peirce tried to answer in "The Logic of Mathematics, an attempt to develop my categories from within" for the case of number theory. I've tried to reconstruct the main moves in Peirce's phenomenological analysis of the formal categories and the role that each plays in the hypotheses that lie at the basis of discrete and finite systems of number in an essay that appeared in the Cuadernos. The same kind of analysis could be given, I think, for the key conceptions in the conventions that lie at the bases of the EG--including the generation of a surface as a sheet of assertion, and the generation of the scroll as a line that creates a boundary between areas on that surface. Interpreting the diagrammatic representation of these logical conceptions will, I believe, require us to consider the topological postulates and conceptions (i.e., the generation of surfaces and boundaries) that seem to be informing the way Peirce is developing each system of the EGs. So, let me frame a question: how might we draw on the phenomenological account of the formal and material categories to analyze what is involved in our experience of generating topological surfaces and boundaries--especially a boundary like a scroll that is interpreted as a representation of the relation of "if ____, then ___"? --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 Appendix: The “cave” passage from RLT 251-3: [[ If I were to attempt to tell you much about the different shapes which unbounded three dimensional spaces could take, I fear I might seem to talk gibberish to you, so different is your state of mental training and mine. Yet I must endeavor to make some things plain, or at least not leave them quite dark. Suppose that you were acquainted with no surface except the surface of the earth, and I were to endeavor to make the shape of the surface of a double ring clear to you. I should say, you can imagine in the first place a disk with an outer boundary. Then you can imagine that this has a hole or holes cut through it. Then you can imagine a second disk just like this and imagine the two to be pasted together at all their edges, so that there are no longer any edges. Thus I should give you some glimmer of an idea of a double ring. Now I am going in a similar way to describe an unbounded three-dimensional space, having a different shape from the space we know. Begin if you please by imagining a closed cave bounded on all sides. In order not to complicate the subject with optical ideas which are not necessary, I will suppose that this cave is pitch dark. I will also suppose that you can swim about in the air regardless of gravity. I will suppose that you have learned this cave thoroughly; that you know it is pretty cool, but warmer in some places, you know just where, than others, and that the different parts have different odors by which they are known. I will suppose that these odors are those of neroli, portugal, limette, lemon, bergamot, and lemongrass,— all of them generically alike. I will further suppose that you feel floating in this cave two great balloons entirely separated from the walls and from each other, yet perfectly stationary. With the feeling of each of them and with its precise locality I suppose you to be familiarly acquainted. I will further suppose that you formerly inhabited a cave exactly like this one, except it was rather warm, that the distribution of temperature was entirely different, and that [the] odors in different localities in it with which you are equally familiar, were those of frankincense, benzoin, camphor, sandal-wood, cinnamon, and coffee, thus contrasting strongly with those of the other cave. I will further suppose the texture-feeling of the walls and of the two balloons to be widely different in the two caves. Now, let us suppose that you, being as familiar with both caves as with your pocket, learn that works are in progress to open them into one another. At length, you are informed that the wall of one of the balloons has been reduced to a mere film which you can feel with your hand but through which you can pass. You being all this time in the cool cave swim up to that balloon and try it. You pass through it readily; only in doing so you feel a strange twist, such as you never have felt, and you find by feeling with your hand that you are just passing out through one of the corresponding balloons of the warm cave. You recognize the warmth of that cave[,] its perfume, and the texture of the walls. After you have passed backward and forward often enough to become familiar with the fact that the passage may be made through every part of the surface of the balloon, you are told that the other balloon is now in the same state. You try it and find it to be so, passing round and round in every way. Finally, you are told that the outer walls have been removed. You swim to where they were. You feel the queer twist and you find yourself in the other cave. You ascertain by trial that it is so with every part of the walls, the floor, and the roof. They do not exist any longer. There is no outer boundary at all. Now all this is quite contrary to the geometry of our actual space. Yet it is not altogether inconceivable even sensuously. A man would accustom himself to it. On the mathematical side, the conception presents no particular difficulty. In fact mathematically our own shaped space is by no means the easiest to comprehend. That will give you an idea of what is meant by a space shaped differently from our space. The shape may be further complicated by supposing the two balloons to have the shape of anchor-rings and to be interlinked with one another. After what I have said, you cannot have much difficulty in imagining that in passing through one of the balloons you have a choice of twisting yourself in either of two opposite ways, one way carrying you into the second cave and the other way into a third cave. That balloon surface is then a singular surface. ] RLT 251-3 ]
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