Gary F, John S, List,
The EGs are being developed as a mathematical system of logic. Peirce conceives of the framework within which the system is being developed as a topological system. As such, I'm working on the assumption that it will be helpful to think of the EG as a system of mathematical logic that draws on ideas from topology--e.g., Euler's work on logic diagrams, Euler's work on graph theory, Euler's formula for the topological characteristic of a figure and a surface and Listing's later development of this formula--as so many sources that Peirce has available at his fingertips. Setting cosmological matters to the side, I do think that it will be helpful to consider these topological ideas as we try to understand Peirce's motivations and inspirations as he makes decisions about how to set up the diagrammatic relations that are important for using and understanding the EG as a formal system. Mathematical inquiry often is driven by some real problem that could not be solved by the other sciences. As such, the mathematician is called in to help, and the first step is to build a simplified and idealized system that can be used to address the most important parts of the problems at hand. If many of the problems that are motivating these inquiries are coming from the side of the normative sciences--i.e., theory of semiotics including the critical logic--then I believe it is natural to suppose that it is a set of perceived shortcomings in the algebraic systems of logic that were developed in the latter part of the 19th century--including his own algebraic systems--that are motivating and inspiring the development of this topological system of logic. Making the comparison between the two will be helpful precisely because it will clarify where the EGs have analytical advantages over the algebraic systems. For the purposes of analyzing the elemental steps in both common and in more specialized forms of reasoning, the algebraic systems are falling short. Instead of spending more time trying to improve those systems, he is moving to a more diagrammatic approach that takes recent diagrammatic work in topology as a starting point. Having said that, I admit it is an open question as to how much we can learn from looking at topological figures and conceptions as we try to understand what Peirce is doing in the development of the EG. It is possible that we may go too far and import ideas that are out of place. The only way to figure out where it will be helpful to draw on these topological ideas and where it won't is to push the boundaries, so to speak. --Jeff --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: g...@gnusystems.ca <g...@gnusystems.ca> Sent: Monday, October 30, 2017 11:05:43 AM To: 'Peirce-L' Subject: RE: [PEIRCE-L] Lowell Lecture 2.6 Jeff, I share your interest in Peirce’s topological ideas — mostly because they are significant for his cosmology. But EGs are not designed to represent the cosmos, and I’m reluctant to apply topological theories to EGs if they’re going to complicate the issues instead of simplifying them. Peirce illustrated his second Lowell lecture by drawing the diagrams on a blackboard, which itself represents the sheet of assertion, and it would be physically impossible to draw a line on the blackboard around the blackboard. John hasn’t said what he actually had in mind, but I’m guessing that it was a line or double line drawn around a part of the sheet of assertion which has a graph on it. JD: We shouldn't lose sight of the fact that, for the SA, a cut is not simply a path. Rather, the cut takes what is inside the boundary and moves the that part of the surface to a different surface--one that represents what is negated. GF: I don’t think so. Inside the cut is another surface, another “area,” but the surface in itself does not represent what is negated. The blank sheet of assertion is a graph, and does represent everything implicitly understood to be true (between graphist and interpreter); but the blank area inside a cut is not a graph. It does represent a universe of discourse different from the one represented by the sheet of assertion, and any graph scribed in that area is read as false of the universe outside the cut. That to me is a very different idea from the surface itself representing what is negated. John’s idea seems still more different from Peirce’s idea in Lowell 2: John appears to regard all graphs, all partial graphs and all areas as being on the sheet of assertion. But Peirce says explicitly that neither the antecedent nor the consequent of a conditional can be scribed on the sheet of assertion, because neither one is being asserted! Hence the need for other areas, other universes, to be separated (by cuts) from the places on which the enclosures are drawn. Maybe Peirce was never satisfied with his EGs; maybe he abandoned the gamma graphs because he concluded that what he was trying to represent with them could not be visually represented. But if that’s the case, and I’m quite willing to believe it is, I want to understand why it can’t be done. And I think the best way of understanding that is to thoroughly investigate Peirce’s attempts to do it, from the ground up. Gary f. From: Jeffrey Brian Downard [mailto:jeffrey.down...@nau.edu] Sent: 30-Oct-17 12:19 Cc: Peirce-L <peirce-l@list.iupui.edu> Subject: Re: [PEIRCE-L] Lowell Lecture 2.6 Hello Gary F, John S, List, Gary F has raised the following question about a remark John made concerning the SA in the EG: "Having said that, I have to say also that some of the statements in your post are even more confusing than Peirce’s presentation in Lowell 2. You wrote, “Peirce said that a blank sheet of assertion is a graph. Since it's a graph, you can draw a double negation around it.” — Eh? How can you draw anything around the sheet of assertion, which (by Peirce’s definition) is unbounded?? Can you show us a replica? It would be usual for those of us who learned Euclidean geometry in middle and high school to think of the SA as a surface that is akin to the Euclidean plane. Under the postulates that govern this system, parallel lines never meet, so we picture the plane as stretching out in all directions endlessly. In topology, we think of an unbounded surface differently. After all, the figures constructed in a 2-dimensional topological surface can be stretched and twisted indefinitely without changing any of the continuous connections between the parts of such figures. Leaving aside the homoloidal character of lines taken to be straight and the metrical properties of such a surface, the underlying topology of the Euclidean plane is that of a parabolic surface. Such a surface is unbounded, but lines return to themselves. The reason is that the parabola surface has the global structure of a torus. It is clear that Peirce is reflecting on the topological character of the SA itself as he explains the starting assumptions for the alpha and beta system of graphs. Such reflections are prominent in the NEM, the 9th Lecture in Reasoning and the Logic of Things, etc. The global character of the SA will be determined by the assumptions that govern the construction of figures in this 2-dimensional surface. We can study this surface the same that that we would study any 2-dimensional surface in topology using the Euler characteristic, and we can study its global properties more carefully by reflecting on the additional features that Listing and Peirce added to Euler's version of the equation. For a classification of types of surfaces based on the Euler characteristic, see: https://en.wikipedia.org/wiki/Euler_characteristic Euler characteristic<https://en.wikipedia.org/wiki/Euler_characteristic> en.wikipedia.org In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré... For richer explanations of Peirce's understanding and development of this formula, see: Havenel, Jérôme. "Peirce’s topological concepts." New essays on Peirce’s mathematical philosophy (2010): 283-322. Let's consider an example taken from the table that classifies the different kinds of surfaces. The projective plane is an non-orientable unbounded surface, and it has an Euler characteristic of 1. The global properties of this surface are quite interesting. Within the system of postulates that govern the generation of the surface, all parallel lines converge. This is something we can picture in a more familiar way by considering a perspective drawing in which all parallel lines converge on the infinitely distant horizon. In effect, the projective space is a generalization on this idea from perspective geometry. What we should note is that the absolute in a projective surface is effectively a generalization of the infinitely distant horizon within the perspective geometry. My reason for picking this example of a topological surface is that it provides us with an example of a 2 dimensional space in which a path can be drawn all of the way "around" the surface--i.e., as the line that serves as the absolute. Is there any restriction on doing the same kind of thing in other sorts of topological surfaces? That is can we draw a path all of the way around a spherical (i.e., elliptical) or toroidal surface? If such a path can be drawn "all the way around" these sorts of unbounded surfaces, is there a restriction on making a cut "all the way around" the SA? We shouldn't lose sight of the fact that, for the SA, a cut is not simply a path. Rather, the cut takes what is inside the boundary and moves the that part of the surface to a different surface--one that represents what is negated. As such, already in the Alpha graphs, the SA is not a simple 2 dimensional surface. Rather, the SA can be used to represent is all that can be positively asserted, and this surface appears to be related--in some fashion--to another surface that represents all that can be denied. Spending some effort on the question of how those surfaces are related within the Alpha and Beta systems might be worth our time. My hunch is that there is a significant difference between the way they are related in these two systems, and and even more significant difference when we consider, as John S suggests, what they represent when we are using these systems--especially gamma--to analyze synthetic forms of inference such as induction. The first step in approaching this sort of question here in the context of our discussion of the 2nd Lowell Lecture is to consider how the part of the surface that is inside the inner portion of the scroll is related to the part of the surface that is inside the outer part of the scroll. With a clearer idea of what that figure represents as a relation between those two parts of the surface, we can consider the import of the drawing the pseudograph as a cut that is entirely filled in as black. With this diagram, we see Peirce exploring a way to represent the relation between what is possible as a positive assertion and what is impossible--at least within the context of the development of the alpha system that he is explaining here. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: John F Sowa <s...@bestweb.net<mailto:s...@bestweb.net>> Sent: Monday, October 30, 2017 7:20:48 AM To: peirce-l@list.iupui.edu<mailto:peirce-l@list.iupui.edu> Cc: Dau, Frithjof Subject: Re: [PEIRCE-L] Lowell Lecture 2.6 Gary F, The issues are far deeper than notation or computer processing. 1903 was a critical year in which Peirce began his correspondence with Lady Welby. That led him to address fundamental semiotic issues. > I’ll have to confess at this point that I have no interest in learning > EGs for the sake of learning a new notation system, or for the sake > of knowledge representation in automated systems. Last week, Ontolog Forum sponsored a telecon, in which I presented slides on "Context in Language and Logic". It addressed complex semiotic issues, and I mentioned Peirce at various points. Following are the slides. Slide 2 also has the URL for the audio: http://jfsowa.com/ikl/contexts/contexts.pdf > the elementary phenomena of reasoning, that I’d like to understand better. I agree that's important, and I also agree that Peirce was seeking the most fundamental methods he could discover. But I also believe that he abandoned the recto/verso system because (a) the questions raised by Lady Welby led him to more significant problems, and (b) those low-level ideas paled in comparison to his goal of representing "a moving picture of the action of the mind in thought." > The three pairs of rules you attached (from NEM) are essentially > the same as the three pairs he gives later on in Lowell 2, except > for the shading and the absence of lines of identity. For his EGs of 1903, they are logically equivalent. In fact, that is why his recto/verso description and his "magic blot" have no real meaning: they have no implications on the use of the graphs in perception, learning, reasoning, or action. But the 1911 system can be generalized to modal logic, 3-valued logic, and probability. And by the way, that letter of 1911 was addressed to Mr. Kehler, one of Lady Welby's correspondents, and the main topic was probability and induction. That's also significant. Implications of his 1911 system: 1. The rules come in 3 symmetric pairs, and each pair consists of an insertion rule (i) and an erasure rule (e), each of which is the inverse of the other. This feature supports some important theorems, which are difficult or impossible to prove with other rules of inference. 2. The rules are *notation independent*: with minor adaptations to the syntax, they can be used for reasoning in a very wide range of notations: the algebraic notation for predicate calculus (Peirce, Peano, or Polish notations); Kamp's discourse representation structures; many kinds of diagrams and networks, and even natural languages. 3. They can be adapted to theorem proving with arbitrary icons inside an EG. I demonstrated that with Euclid's diagrams inside the ovals of EGs. But they can also be used with icons of any complexity -- far beyond Euclidean-style diagrams. 4. The psycholinguist Philip Johnson-Laird observed that Peirce's notation and rules are sufficiently simple to make them a promising candidate for a logic that could be supported by the neural mechanisms of the human brain. That is true of his later system, but not the recto/verso system. For an overview of these issues, see my slides on visualization: http://jfsowa.com/talks/visual.pdf To show that Kamp's DRS notation is isomorphic to a subset of EGs, see slides 20 to 27 of visual.pdf. To see the application to English, see slides 28 to 30. (But this is true only for that subset of English or other NLs that can be translated to or from Kamp's DRS notation.) For the option of including icons inside the areas of EGs, see slides 31 to 42 of visual.pdf. For more detail about Euclid, see slides 19 to 39 of http://www.jfsowa.com/talks/ppe.pdf Note: There is considerable overlap between visual.pdf and ppe.pdf, but slides 19 to 39 of ppe.pdf go into more detail about Euclid. For theoretical issues, see slides 43 to 53 of visual.pdf. For the theoretical details, see http://jfsowa.com/pubs/egtut.pdf I'm working on another paper that goes into more detail about Peirce's "magic lantern of thought". The 1911 system can support it. But the recto/verso system cannot. John
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