Hi Charles, List, You are expressing the same general idea, but in the terms of a Cartesian geometry. Euclidean geometry does not involve hypotheses (e.g., postulates, substantive definitions, precepts of construction) having the character of a system of coordinates. As Descartes shows, there are a number of different ways of set up such coordinate systems that result in a variety of different sorts of analytical geometries (e.g., polar, cylindrical, spherical, etc.)
So, to respond your question about mapping from one point on a figure to another, there are only three points of incidence that are determinate in a triangle in a Euclidean geometry (i.e., the three vertices). The precepts that guide the mathematician in reasoning about matters of congruence require that certain steps be carried out--utilizing a straightedge but not a ruler. One might think that the process of reflecting the triangle on the piece of paper using a pencil and straightedge is a transformation of a token sinsign. But, it is clear that the reasoning about the congruence of the figures is reasoning about an idealized triangle. After all, the token sinsign actually drawn on the page has lines of some thickness, while the idealized mathematical triangle does not consist of lines having such properties. My sense (not well substantiated, but mine nonetheless) is that Kant is explicitly drawing on perspective geometry when he makes a number of points about the transcendental unity of apperception and the role of this representation in giving direction to our inquiries. This idea is expressed in his lectures on logic in terms of the ways that our present self makes an appeal to our future self that is attempting to correct the errors of our ways (see the Jäsche Logic on standards of correctness and error). Peirce clearly states that our conception of self is born from a realization that my own representations have been in error--and that more may turn out to have the same sorts of defects (Questions Concerning Certain Faculties). If we consider what Peirce says about the transcendental unity of apperception (see CP 5.71 and 6.378, for instance), we see him working through some of Kant's ideas about the special character of this representation. At present, I'm looking fairly closely at what Peirce says in MS 612 for the sake of trying to get a bit clearer on how he tries to clarify the conception of "determination". In that MS, he engages in a dialogue with himself. It is really quite a fascinating piece which he reworked in a number of ways in other MS (610-615). In that MS, he says this about the self: The first point of the first principle is that, when a man meditates, he does not, as my master Kant (my attitude toward) whom is substantially such as I would have my disciples take toward me, namely, a critical attitude,) says he does, incessantly repeat: "I think," although it is true that, when he reaches a decided belief, he may perform an exertion of the kind called a Resolution of the will, with a view to producing in his constitution a Determination, i.e., a tendency to conduct himself in harmony with that belief. Otherwise, he only thinks of himself as being ignorant, or has having fallen into error, or as having comported himself ill. When he gathers his attention upon an Idea, it is not of himself that he thinks; it is rather to the Idea that he addresses a command, "Come on, now; play your part in this situation." The second point of the first principle is that although the object of which he is thinking is not himself, nevertheless what he thinks is addressed to himself. By that I mean that he is all along appealing to his subsequent self, the self who shall have thought the matter out and come to a definitive belief on the subject. When two people are in heart to heart conversation, each is aware of what is passing in the other mind by substantially the same means by which he is aware of what is passing in his own, though I do not say he is as completely cognizant of the one as of the other. He no more thinks about the other's mind than he does of his own. (MS 612, Nov. 11, 1908) See: http://fromthepage.com/display/read_work?page=9&work_id=149 How might the notion of a self that is in the future serve to give direction to our inquiries? I believe that, as for Kant, Peirce holds that the representation of the self serves a crucial function in providing a kind of "point of perspectivity" out there on the horizon where we imagine that our many lines of inquiry might converge on stable beliefs about what is true and what is false concerning some question. (CP, 8.94) With that much said, let me offer a grand gesture of something that I find enticing in Peirce's remarks about the way we might represent the processes of self-control that are essential for the success of experimental inquiry in a graphical logic. This is meant only as a gesture, but I'm hoping it might provide some orientation as to where I'm hoping to head. I'd like to see how Peirce's notion of the self might help us better understand what is involved in moving from a system gamma graphs that is sufficient for representing deductive inferences to a system of gamma graphs that will be sufficient for representing synthetic inferences. (See, for example, CP 3.488-491) --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________________ From: charles murray <[email protected]> Sent: Tuesday, May 3, 2016 5:29 AM To: Jeffrey Brian Downard Cc: Peirce List Subject: Re: [PEIRCE-L] Topology, the Gamma Graphs, and representations of self 3 May 2016 Jeff, all: In my March 24 post I expressed interest in your remarks about Peirce's mathematics and theory of the self. With a hint from your March 29 response I have located what I'm curious about in your post dated 4 July 2014 from the De Waal Seminar. You write: "My first suggestion is that we start by thinking about the self as a relationship between representations. In Euclidean geometry, for instance, we say that one figure is congruent with another if we can move one around and superimpose it on top of another. The basic ways in which figures can be moved in a diagram is by translation, rotation and reflection. It turns out that reflection is the fundamental kind of symmetry on a Euclidean surface because we can achieve the same effect as translation by reflecting over two lines. The same is true of rotation (but the lines cross). [quotation continued:] Before we introduce any ideas from projective geometry, how might we use this geometric notion of reflection to think about the symmetries that hold between one representation and another representation that is, in some sense, 'congruent' with it? Or, if you prefer a weaker relation, how is one representation similar to another? In what ways might one representation mirror another?" In my rudimentary understanding of reflection as an operation upon graphs on a Euclidean surface, certain facts about points on the reflected graph necessitate certain facts about points on the reflecting graph. For example, given a coordinate system with the graph A of a 1-1 function, there is a reflecting graph B such that for any point on A specified by the x coordinate _a_ and the y coordinate _b_, there is a point on B specified by the x coordinate _b_ and the y coordinate _a_. This gives reflection about the graph of the function y = x. Graphs of other relations give other reflections, e.g., reflections about the x axis, y axis or the origin. There are conditions on whether there is a reflection and if so what it is like. First, there are conditions on the algebraic properties of that which is being graphed. It matters whether it is a relation, whether the relation is a function, what results from negating the function, or negating the function's argument, and whether the function is many to one or one to one. There are corresponding conditions on graphic representation, which include provision for a coordinate system with an x and y axis, their intersection at a point of origin, points specifiable by x and y coordinates, and the possibility of various symmetries between graphs with respect to this system. Extending these rudimentary remarks, I suppose reflection _might_ be best understood as conditioned by the movement of a point from one place to another, represented graphically by having a point on the graph maintain its identity through changes in facts about it, such as having one or both its coordinates change value from positive to negative or vice versa, or having its x coordinate become its y coordinate and vice versa. More needs to be said to account for preservation of identity through such changes. It also seems important to be more careful than I am above in keeping straight whether one is speaking of a graph as a legisign or a sinsign; it should be recognized that reference to "points" as part of a sinsign may be misleading. Are we on the same page so far? Best, Charles
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