Jeffrey wrote: "In Sung’s response to you, he seems to conflate the orientability of a space with (012515-1) the Euclidean or non-Euclidean character of the space."
Thank you for pointing out my ignorance. I have two related questions. "Do you think that my argument for using the Moebius band as a geometric representation of (012515-2) the principle of complementarity (as I defined in my previous email) can still stand, if I associate the principle with the orientability of the Moebius surface rather than with its curvedness ?" "Do you think that the orientation of the Moebius surface can be reversed without curving its surface, (012515-3) i.e, without destroying the parallel line axiom of Euclid?" All the best. Sung On Sun, Jan 25, 2015 at 7:02 PM, Jeffrey Brian Downard < jeffrey.down...@nau.edu> wrote: > Hello Ben, Gary F., and Mary, > > First of all, a quick response to Mary's last post. I think there is no > need to bow out of the conversation. The purpose of the Peirce list, as I > understand it, is for people with different backgrounds to work together to > sort through the texts with the aim of improving our shared understanding > of the ideas and arguments. We can, of course, go beyond the texts and > talk about our own ideas or the ideas of people other than Peirce. It will > be helpful, of course, if we tell the others on the list when we are trying > to interpret Peirce and when we are really trying to do something very > different. My assumption is that, whatever particular ends might be > guiding each of us as we make an entry or give a reply, we are all trying > to find the truth about the kinds of questions that Peirce was also trying > to answer. As such, the different particular ends we might have—including > that of making our own points or of interpreting Peirce’s texts—are all > subservient to this larger goal of engaging in honest inquiry in the search > for truth. > > If protocol demands that we wait in silence for the experts on the graphs > to chime in, then I'll need to bow out as well. Despite the fact that I've > been working on these texts for a while, the Gamma Graphs are still quite a > puzzle to me. These graphical systems of logic may have been Peirce's chef > d'oevre, but I'm afraid that I lack the cooking skills needed to set foot > in his kitchen. > > The underlying topological ideas that Peirce is drawing on are also a > source of puzzlement, so it is often hard for me to see what the > implications are of introducing a many of the topological conceptions into > the graphical system of logic. In my own studies, I've found that > introductory texts on topology could only take me so far. There were just > too many unstated assumptions that were being left out of the explanations, > and too many of the texts moved too quickly to algebraic explanations--so > it was difficult for me to improve my intuitive grasp of the basic ideas. > Peirce's own writings in topology are helpful (see the New Elements of > Mathematics, vol. 4), but often it is difficult to see what he is doing > when he imports the topological conceptions into his work on the logical > graphs. > > So, if you want to study some topology for the purposes of understanding > how Peirce is using these mathematical ideas in the development of the > existential graphs, here are two recommendations of resources that have > been helpful for me. > > 1. Jeffrey Weeks, The Shape of Space > 2. Norman Wildberger, Online Course on Algebraic Topology, > https://www.youtube.com/playlist?list=EC41FDABC6AA085E78 > > Let me add a few suggestions for thinking about Peirce's work in the > graphs--and the relevance of questions about such things as Mobius bands > and Klein bottles. It is possible that what I have to say is only so much > review for you and many others. Regardless, I'll provide a short summary > of some key ideas to see if we are on the same page--and to give others who > have a better understanding of geometry a opportunity to spell out where I > might be confused or mistaken. > > Peirce firmly believes that one of the great advances that Cayley and then > Klein made in geometry was to show how projective geometry supplies us with > a basis for understanding the relations between metrical geometries and how > topology provides us with a basis for understanding the relationships > between and ordering of the other two main branches of geometry. In effect, > topology studies the properties of connectivity and dis-connectivity of > spaces, but there are no postulates that enable us to work with straight > lines or with lengths of lines and degrees of angle. Given the > difficulties of clarifying what is involved in relations of continuous > connectivity in space, the mathematicians have made an executive decision > to treat these matters separately under the heading of topology. For a > number of reasons, Peirce thinks that this executive decision on the part > of the mathematicians was wise because it enables us to see how all of > hypotheses that lie of the foundations of geometry are related one to > another. Projective geometry takes up those properties of connectivity and > studies the properties of proportion that hold when straightness is > introduced, and metrical geometry adds the study properties such as length > and degrees of degree of angle. One of the great advances in the 19th > century was to see with greater clarity that the kind of metrical geometry > one is working with depends on the system of homoloids that one takes to be > dominant in setting up the postulates for a metrical system. > > Historically, mathematicians came at these issues by thinking about > different ways of stating what might obtain when one tries to draw lines > through a point that are parallel to a given line. In a system of metrical > geometry, there are three possibilities: (1) there might be only one line > that can be drawn, or (2) there might be no lines that can be drawn that do > no intersect with the given line, or (3) there might be more than one line > that can be drawn that do not intersect. Mathematicians like Cayley and > Klein thought about these matters by working with the complex plane that > has both real and imaginary dimensions, and Klein then used group theory to > spell out the relations between the geometries in more general terms. The > idea of the imaginary dimensions of the complex plane adds a further > complication that Peirce shows us how to set to the side when he provides > an "introductory" explanation of these different kinds of metrical spaces: > see “The Non-Euclidean Geometry Made Easy” in vol. 8 of the Chronological > Writings. > > What does a Mobius strip or a Klein bottle have to do with any of this > discussion about the relationship between parabolic, hyperbolic and > elliptical systems of metrical geometries? The short answer is that the > questions are not directly related. Each of these kinds of geometries can > be orientable, and each can be non-orientable. The orientable character of > a space is determined by whether or not it has a cross-cap. We can figure > out where or not a given surface has such a twist in it by decomposing the > space and seeing if the parts match up in the way that a Mobius band is > connected. > > The “curvature” (or non-Euclidean character) of a surface is something we > can figure out by putting a polyhedron over the space and counting up the > number of vertices, sides and edges. The Euler equation enables us to see > whether the space has a value for the Euler characteristic that is > negative, 0, or positive. In Sung’s response to you, he seems to conflate > the orientability of a space with the Euclidean or non-Euclidean character > of the space. For the sake of simplicity, we can think of the > orientability and the “curvature” of surfaces as separate matters. Look at > the tables that Weeks and Wildberger each give, where they categorize > surfaces by the number of cross caps and/or holes that are found in them. > Here is an abbreviated version of their tables: > > Euler # Orientable Nonorientable > 2 Sphere > 1 Disc Projective plane > 0 Torus Klein bottle > -1 Double cross cap > -2 Double Torus Triple cross cap > > (and so on, with higher negative values) > > We can see that a torus and the Klein bottle have same Euler > characteristic of 0; as such, they are both homeomorphic with the Euclidean > plane. This is counterintuitive. After all, how can a Euclidean plane be > topologically homeomorphic with a torus? One is clearly flat, and the > other is clearly curved. The answer comes when we remember that > straightness and curvedness are something that we are setting aside in > topology. In topology, we have not settled on a dominant system of > homoloids. We’re only looking at how one part of a surface is connected or > disconnected with another part of the surface. Topologically speaking, the > parts of a one-holed torus are connected to the other parts of the surface > in the same ways that the parts of the Euclidean plane are connected to > each other—and the same holds for a space that is shaped like a Klein > bottle. > > Having said this much about the Euclidean or non-Euclidean character of > two dimensional surfaces, what more can we say about the orientability and > nonorientability of such two dimensional spaces? Once again, to simplify > matters, orientability and two-sidedness are related but separate matters. > If you run around a space like the Mobius strip—but one that has two twists > and not just one—then you end up with the same orientation that you started > with once you return back to your starting point. That is another way of > saying that a strip with two twists is really two-sided, but a strip with > one twist is really one-sided. > > So far, our discussion of these ideas has been confined to two-dimensional > surfaces, and all of the points are about the intrinsic properties of those > surfaces. We could ask questions about how those parts of the surface are > related to parts of a three or higher dimensional space of which the > surface is just a part, but then we would be looking at the extrinsic > properties of the surface. The division between intrinsic and extrinsic > properties is something that applies to any space we might study. We > could, for instance, study the intrinsic properties of three dimensional > spaces, and then later look at the extrinsic properties involved in > thinking about the three dimensional space as part of a larger four > dimensional space. > > With this bit of mathematical prelude, we could draw on some of these > ideas from topology—including the ideas of orientability and two-sidedness > of a surface--to try to sort out what Peirce might be doing by introducing > the idea of distinguishing between the recto and verso sides of the sheet > of assertion. I add the qualification “might” to emphasize the fact that I > take it to be an open question, at least as far as my understanding goes, > and my aim is to try to figure it out with the help of others on the list. > > When it comes to interpreting the suggestion that he has made a “new > discovery”, I think a great place to start is with the particularities of > his remarks in the essay on “The Improvement on Gamma Graphs.” Let me > start with the first thing he infers from the new discovery of how we might > work with a two-sided sheet, where the recto side represents actual facts, > and the verso side represents one or another of various kinds of > possibility. We should note that the first kind of possibility he > considers is subjective possibility. This is the kind of possibility that > involves ignorance on the part of the inquirer. It is being used to deal > with the kind of assertion we have when a person asserts “The stove is not > hot” and then adds, “but I’m not a cook, so it is possible that I am in > error.” Starting with this very familiar sort of possibility, where we > admit that there are a number of qualifications we need to add to our > assertions in order honestly to admit it is possible we might be in error > due to our ignorance of certain relevant matters, what is involved in the > very first inference that he draws from the new discovery? > > He says: “First, the cut may be imagined to extend down to one or another > depth into the paper, so that the overturning of the piece cut out may > expose one stratum or another, these being distinguished by their tints; > the different tints representing different kinds of possibility.” (CP, > 4.578) > > What is implied in this sentence? We should note that there are a number > of ideas in play. > > First, the paper has a depth such that a cut through it will expose > multiple stratum. Don Roberts interprets the passage in the following > way: “The cut would retain its function of negation, but the effect of > scribing a graph on the verso would be to exclude a possibility (not simply > an actuality) from the universe.” (88) Having made this point, he adds > that “Peirce arrived at this analysis while considering the ‘anolomy’ by > which in EG the relation ‘other than’ is expressed differently from any > other relation. (LN 265r). It is the only relation requiring that a graph > be partly in one area and partly in another; it requires that a line of > identity cross a cut.” Ben has tried to explain some of the implications > of seeing the anomoloy of having to treat ‘other than’ in this unique way. > > Second, the cut goes through several sheets, and it exposes one or another > of the stratum as it penetrates into the depths of the book. As such, each > sheet in the book has two sides, and the strata of the layers are exposed > by cutting through sheets having different tints of color. Each of these > tints represents different kinds of possibility—such as the practical > possibility that I am in error in thinking that I can carry out a > resolution, as well as the deeper metaphysical possibility that my > assumptions about the nature of the real are in error. > > Let me interpret the significance of this in light of what Peirce says in > the Prolegomena about the multiple sheets in the book. In this essay, the > different sheets in the book represent stages in a process of inquiry, > where the Graphist and the Interpreter are engaged in a dialogue animated > by the purpose of articulating the truth. As a starting point, the > Graphist and the Interpreter are taken to have a number of shared > assumptions. As Roberts points out, “the Phemic sheet, before anything is > scribed on it, represents whatever is taken for granted at the outset by > the Graphist and Interpreter.” > > The Sheet of Assertion is devoted to the expression of propositions held > to be true, the Sheet of Interrogation is devoted to questions one might > raise about those propositions, while the Sheet of Destination is devoted > to resolutions. How might we understand Roberts’s suggestion that the > “tinctures were designed with more than formal logic in mind; they were > meant to provide a structure in terms of which Peirce could apply his > categories to propositions and inferences, to hypotheses, questions, and > commands.” (Roberts, 100) That is, the graphs were designed to represent > “all that ever could ever be present to the mind in any way or any sense.” > (Ms, 499(s)). > > Consider a simple example, such as the case of a child who is learning > about a stove. Suppose the child already has associated words such as > ‘stove’ with objects of this kind, and the word ‘hot’ with the experience > of heat. What is involved in the child moving beyond such psychological > associations and gaining logical self-control over the conduct of his > thought? From early in his career in “Questions Concerning Certain > Faculties” Peirce is trying to explain what is involved in learning the > logical meaning of falsity, and thereby learning the meaning of error and > the logical relationships involved in negation. Let us suppose that the > child has already formed a habit of belief to the effect that all mid-sized > objects that capture his attention are suitable for being touched. What is > involved in learning that this assertion is false? In the case of the > child, the parents will try to stop him from touching the stove and will > say “The stove is hot, don’t touch!” Logically speaking, what is involved > in the child forming hypotheses to the effect that the representations > about the stove being suitable for being touched are false, and that there > is a real difference between what he has represented to be the case and > what his parents have represented to be the case? In learning this, he > gains a conception of his own mind as being distinct from the minds of > others. > > In his discussion of the logic of relatives of second intention, Peirce > says the following: > “The general method of graphical representation of propositions has now > been given in all its essential elements, except, of course, that we have > not, as yet, studied any truths concerning special relatives; for to do so > would seem, at first, to be ‘extralogical.’ Logic in this stage of its > development may be called paradisiacal logic, because it represents the > state of Man's cognition before the Fall. For although, with this > apparatus, it is easy to write propositions necessarily true, it is > absolutely impossible to write any which is necessarily false, or, in any > way which that stage of logic affords, to find out that anything is false. > The mind has not as yet eaten of the fruit of the Tree of Knowledge of > Truth and Falsity. Probably it will not be doubted that every child in its > mental development necessarily passes through a stage in which he has some > ideas, but yet has never recognised that an idea may be erroneous; and a > stage that every child necessarily passes through must have been formerly > passed through by the race in its adult development. It may be doubted > whether many of the lower animals have any clear and steady conception of > falsehood; for their instincts work so unerringly that there is little to > force it upon their attention. Yet plainly without a knowledge of falsehood > no development of discursive reason can take place.” (CP, 3.488) > > Let’s see if we can draw on some ideas from topology to try and clarify > what might be going on in his claims about the method of graphical > representation that he is exploring in the Prolegomena and the essay on the > improvement in the Gamma Graphs. For the sake of our discussion, let us > suppose that the child’s way of experiencing such things as the blackness > and hotness of the stove are similar to our own, but that they have not yet > been sorted out in the way we’ve sorted things out. It is as if the child > has wandered into a cave filled with experiences of various odors, warmth > and cold, and textures of smoothness and roughness, and has not yet figured > out what is up and what is down with respect to the feelings of such things > as hot and cold. Putting the matter in less metaphorical terms, let us > suppose the child does not yet have a conception of where to draw the line > between hotter and colder so that he can understand the real difference > between saying “the stove is suitable for being touched because it is not > hot,” and “the stove is too hot, don’t touch it!” In the child’s way of > seeing the world, the relationship between hot and cold might be a matter > of one feeling shading into the other, but he has not yet established a > suitable set of breaks in this relationship. His sense of what is possible > in the way of such feelings is like a Mobius band because he has not yet > established a break in the continuity between one and the other kind of > experience. > > The key to explaining the break in the continuity between these > experiences is that he discovers, contrary to what he had supposed, that he > is unable to hold his hand to the stove despite his resolution to do so. > As such, the general supposition made in the child’s initial hypothesis now > runs up against a hard fact in the child’s experience. The discovery that > he is unable to hold his hand to the stove connects the facticity of the > balking of his will with the possibilities that were represented under the > hypothesis about stoves generally being well suited for being touched. How > might we represent what is being learned here in a graphical system of > logic? Peirce is pointing out that a system like Euler’s graphs is unable > adequately to represent the connection that is being made between the > possibilities covered under the general concepts represented in the child’s > hypothesis and the actual fact that is learned through the experience of > being unable to carry out the practical resolution. The same is true, he > seems to be saying, about the Alpha and Beta graphs. They do not contain > the kinds of logical relations needed adequately to represent what the > child is learning about the relationship between the general conceptions > represented in the hypothesis and the particularities of his experience of > being unable to carry out his resolution. > > How is this represented in the Gamma Graphs? That is, how do the Gamma > Graphs represent the logical relations in a way that will clarify the > connections between asking questions, carrying out resolutions, and > revising our assertions in the light of experience? We start by taking all > that is shared in the child’s and parents’ understanding to be represented > on the blank Phemic sheet. After that, the child scribes his hypothesis on > the Sheet of Assertion, and then the parent’s challenge what is ascribed on > that sheet by interpreting things differently. At this point, there is a > dispute between the Graphist and the Interpreter, and we need to understand > how this dispute might be resolved through inquiry. Despite the parent’s > warnings, the child persists in trying to touch the stove, and “I will > touch the stove” is scribed on the Sheet of Resolution. These are all > separate sheets in a book. When the practical resolution proves to be > something the child cannot do, this creates a cut through the sheets in the > book exposing the strata of the layers. > > At this point, it is difficult to sort out what, logically speaking, might > going on when the child has conducted the test of his hypothesis and has > discovered that it is false. In this Gamma system of graphical > logic--where we are trying out different ways of trying to picture the > relations as iconically as possible for the sake of developing better > philosophical explanations of these fundamental logical conceptions--we are > supposing that the child’s ability to use his actual experiences of hot and > cold as signs that refer to the possible experiences he might have in the > future are only relations of references. > > That is, these experiences stand in a dyadic relation of reference to one > another—but this is merely a matter of similarity. As such, the child can > understand that the feeling of the hotness of the cookie when he eats it is > a particular experience that is contained in the more general experience of > hotness, and the same holds for the experience of touching the hot stove. > Both experiences are similar because they are both contained in the same > general representation. But the relation is only one of particular feeling > that is contained in the more general representation that serves as > container. What is needed to establish a referential relation, and then to > establish even richer relations that are materially and formally ordered? > > In order to sort out the moves that Peirce is making in the development of > the Gamma Graphs, we’d need to try to picture how he is thinking about the > kinds of dyadic relations that are needed to get transitivity in this > ordering of hotter and colder tested against actual experiences, and we > would need to see how these dyadic relations are being conceived of as part > of larger triadic relationship under general rules. So, we’ve got a lot to > sort out. Having said that, I’m hoping that we can see that the > introduction of the idea that we can distinguish between the recto and > verso sides of the sheets is something that is being applied to all of the > sheets in the book. So, we picture the child beginning to understand the > differences between what is on the recto side of the Sheet of Resolution > and the Verso side of that same sheet. And, what the child sees is that > the fact that he is unable to touch the stove establishes that this > resolution was not possible for him to carry out in this case, and he sees > that this has a number of logical implications for what he has scribed on > the sheet of assertion. > > Let me make something of a leap at this point and offer the following > suggestion in the hopes of holding a number of threads together in my > hand—even if they are only being held together quite loosely. In learning > that his hypothesis about the stove being suitable for being touched is > false, the child is learning how to order his possible experiences of > hotter and colder and connect them up to what he can and can’t do. On this > line of thought, the balking of the will is the ground for setting up an > order so that he might, for instance, treat colder as a positive values and > hotter as negative values, and the line between the two as a kind of zero > point. (note: what matters is the break between positive and negative, > and not that one is positive and the other is negative) > > It should be clear that the assignment of the quantitative differences of > positive and negative values to the qualities of his experience must be > something of an achievement for the child. Lacking something like a > thermometer, how does he sort out the feelings of hotter and colder? We > can picture the connections that are made in terms of one two-dimensional > surface of assertion intersecting with another two-dimensional surface of > resolution. The possibility of having these kinds of intersections may > require that we conceive of each of the sheets as both two-sided and > orientable, and not as non-orientable single-sided sheets. And this may be > what is taking place when the cut that is established on the sheet of > resolution is connected to the sheet of assertion, so that the child is > able sort his actual and possible experiences of hot and cold into a system > that is both materially and formally ordered under a rule. > > In this case, the rule that is connecting them is a logical rule of > negation. And, we could try to picture how this sorting is grounded by > trying to develop a graphical system of logic where the recto and verso on > the sheet of resolution is connected to the recto and verso on the sheet of > assertion by a cut, and where the lines that connect one experience of a > stove to another later experience of the stove establish both the dyadic > and triadic relations necessary for the child to understand the > implications of saying to his parents “My assertion was false, and I was in > error.” > Ok, I admit that I am groping my way around in something like a dark cave > trying to get a handle on a knotted set of problems. How can one make any > progress if each of the problems seems to be a thread that is firmly > knotted together with the others? How can we begin to sort things out so > that we might grasp one clear question with a firmer grip and then follow > this thread through the knotted maze? Peirce tries to explain the question > I’m struggling with in the following way in the Critic of Arguments: > > Diagrams and diagrammatoidal figures are intended to be applied to the > better understanding of states of things, whether experienced, or read of, > or imagined. Such a figure cannot, however, show what it is to which it is > intended to be applied; nor can any other diagram avail for that purpose. > The where and the when of the particular experience, or the occasion or > other identifying circumstance of the particular fiction to which the > diagram is to be applied, are things not capable of being diagrammatically > exhibited. Describe and describe and describe, and you never can describe a > date, a position, or any homaloidal quantity. You may object that a map is > a diagram showing localities; undoubtedly, but not until the law of the > projection is understood, nor even then unless at least two points on the > map are somehow previously identified with points in nature. Now, how is > any diagram ever to perform that identification? If a diagram cannot do it, > algebra cannot: for algebra is but a sort of diagram; and if algebra cannot > do it, language cannot: for language is but a kind of algebra. It would, > certainly, in one sense be extravagant to say that we can never tell what > we are talking about; yet, in another sense, it is quite true. … It is > requisite then, in order to show what we are talking or writing about, to > put the hearer's or reader's mind into real, active connection with the > concatenation of experience or of fiction with which we are dealing, and, > further, to draw his attention to, and identify, a certain number of > particular points in such concatenation. CP, 3.419) > > So, to end with a question: how should we conceive of the cut of negation > cutting through the Sheet of Resolution and the Sheet of Assertion--thereby > exposing a strata of layers? > > My suggestion for interpreting the Gamma Graphs in light of the goals > established at 3.419 for developing graphical systems of logic is that the > cut forms a kind of boundary that exposes the relations between the strata > in the different sheets, and thereby establishes lines of intersection > between those sheets. The intersection is something that causes us to > re-interpret the connections between a given resolution (e.g., the child > says: “I can touch the stove”) with the discovery of what he practically > is unable to do (e.g., represented in his mind as “I couldn’t do it”). > This cut through the sheet of Resolution is based on a point where the > child has discovered a practical impossibility. Many such discoveries can > be connected together by a rule—such as a rule of induction--thereby > forming connected lines. In this way, each of the points of practical > impossibility are connecting to the sheet of assertion—thereby establishing > something like a homoloidal relationship between what is asserted actually > to be the case and what might possibly follow from such an assertion. > > Thanks for your patience with this overly long post, but I wanted to try > my hand at connecting some dots, > > Jeff > > > Jeff Downard > Associate Professor > Department of Philosophy > > --------------------------------------------------------- > From: Libertin, Mary [mli...@ship.edu] > Sent: Sunday, January 25, 2015 6:42 AM > To: biosemiot...@lists.ut.ee > Subject: [biosemiotics:7983] Re: NP 8.3 and the > > Dear Ben, Jeff, Gary F., > > Your comments make much sense. Upon rereading Peirce I realized one of my > mistakes. I had been reading the verso as being on the underside of the > recto. It is rather on the same side. I did notice Peirce briefly > referring to Klein in his discussion of existential graphs. Klein bottles > work in the fourth dimension and operate can be seen as consisting of two > mobius strips in some fashion. I do not wish to pursue this except to > mention that my reference to the mobius strip in the context of boundaries > and continuity was not totally off the wall, but it was a miss. > > I will bow out of the conversation and listen to the experts. Thanks for > your explanations and tactfulness with regards to my mistake. > > Best, > Mary Libertin > > > On 1/22/15 3:39 PM, "Jeffrey Brian Downard" <jeffrey.down...@nau.edu> > wrote: > > >Lists, > > > >Ben has made a quick remark offlist, and I wanted to respond to the > >Lists. He says, "A surprising thing to me is that Peirce in the Gamma > >graphs treats possibility, necessity, etc. without mentionng that he is > >not starting like in probability theory from a set of given data > >parameters like in probability theory, but instead (somewhat like > >contemporary modal logic) supposing, for instance, unspecified > >conditions, or an unspecified state of information, in virtue of which > >which g is possible. Of course if one does it like probability theory, > >then the possibilities and necessities are merely logical possibilities > >and necessities and don't belong to a separate province within logic. The > >approach of leaving unspecified the data parameters, the states of > >information, etc., that one might like to specifically know, suggests to > >me the idea of devising deductive formalisms with special utility for > >inductive inquiries. But that's just an initial impression." > > > >Here is my response: interesting remarks, Ben, especially the "idea of > >devising deductive formalisms with special utility for inductive > >inquiries." One of the moves Peirce makes as he transitions from the > >Beta to the Gamma graphs is to think of the lines of identity as being > >really composed of branching relations--at least potentially. In the > >essay on the improvement of the Gamma Graphs, he says: > > > >"The truth is that concepts are nothing but indefinite problematic > >judgments. The concept of man necessarily involves the thought of the > >possible being of a man; and thus it is precisely the judgment, "There > >may be a man." Since no perfectly determinate proposition is possible, > >there is one more reform that needs to be made in the system of > >existential graphs. Namely, the line of identity must be totally > >abolished, or rather must be understood quite differently. We must > >hereafter understand it to be potentially the graph of teridentity by > >which means there always will virtually be at least one loose end in > >every graph. In fact, it will not be truly a graph of teridentity but a > >graph of indefinitely multiple identity." (CP, 4.583) > > > >This shouldn't be too surprising, I think, because the lines of identity > >in the Beta system are thought of extensionally as existing objects that > >are joined by actually having or not having specific qualities. As such, > >the lines are an iconic representation of the dyadic relation of an > >actual matter of fact. As Peirce says in his discussion of the > >nomenclature and division of dyadic relations: "The author's writings on > >the logic of relations were substantially restricted to existential > >relations; and the same restriction will be continued in the body of what > >here follows." (CP, 3.574) > > > >Once we move from the Beta to the Gamma system, we are connecting things > >with different modal characteristics, and we are connecting things across > >different universes of discourse. As such, the character of the > >connection between qualities that are present in an existing thing are > >being connected to the possible qualities that possible things might have > >--including the possible changes that might occur to this object if > >certain conditions were to obtain. Peirce sees that the specification of > >such possibilities is governed by some rule (i.e., either one in our > >understanding or one that is in the world). Connecting qualities, > >individuals and objects under rules requires some way of dealing with the > >generality of the rule itself and the way that it holds across different > >possible states of affairs. > > > >So, here is a suggestion for Gary F., as he thinks about the character of > >the sheet of assertion in the Beta and Gamma systems. In effect, the > >movement from the Beta to the Gamma graphs forces us to reinterpret the > >meaning of the empty spaces found on the recto side of the sheet of > >assertion, and the relationships between those empty spaces and those > >that are occupied on the verso side of that sheet. This gives new > >meaning to the boundaries between spaces and the connections between > >those spaces. Instead of thinking of the relationship between recto and > >verso extensionally as, "it is actually the case that this object x has > >this property F," and "it is not the case that this object x does not > >have this property F," we are thinking differently about how the two > >sides of the sheet are related one to the other. As I mentioned in an > >earlier email, Peirce is classifying the referential relation as a > >species of dyadic relation proper (i.e., one that is genuine and not > >degenerat > > > > e--as a reference happens to be). In the movement to the Gamma Graphs, > >Peirce is trying to find a way to represent--as iconically as > >possible--the introduction of a triadic relation between reference to > >ground, reference to object and reference to interpretant. All legisigns > >bring these three functions together and binding them together--under a > >rule--as it were. > > > >One reason I find your last remark especially interesting, Ben, is that > >the leading principles of induction and abduction are rules of a special > >sort, and Peirce is trying to understand how we might clarify the > >relationship between the rules of synthetic inference and the leading > >rule that governs deductive inference. As a side remark, it is really > >interesting to see him explore the limits of what could and couldn't be > >done with Euler graphs in his entry on that subject. A comparison > >between Peirce's remarks on the limitations of the Euler system of > >diagrams and what is introduced--piece by piece--in the development of > >the Alpha, Beta and Gamma systems, is really quite instructive for > >thinking about these big questions about the leading principles of > >synthetic inference--and the grounds of the validity of these principles. > > > > > >Like you, I think that Peirce was very much motivated by these kinds of > >philosophical questions--and that they are helping him clarify many of > >the goals that are guiding him in the development of the existential > >graphs generally, and especially in the development of gamma graphs. > >Peirce is focusing on the question of how to understand the nature of > >different kinds of conditionals (and not just those that are conditional > >propositions de in esse) because we want to gain greater insight into > >what is involved in the illative transformation when the reasoning is > >synthetic and not just when the transformation is deductive. Consider, > >for instance, Peirce's remarks in the "Apology for Pragmatism" when he > >explains why he chose the scroll as an iconic representation that enables > >us to see what is going on when we draw inferences from a conditional > >proposition de inesse. (CP 4,564) Very quickly, he clarifies the > >permissions (one might label them postulates, if you are thinking like > > > > a geometer) called "the rule of deletion and insertion,"the rule of > >iteration and reiteration,"and " the rule of the double cut," etc. > > > >Let me close by saying that I place great weight on Peirce's conclusion > >that, ultimately, there are only three such permissions in the > >existential graphs that are needed to understand the nature of the > >illative transformation. Those are colligation, iteration and erasure. > >(CP, 5.579) My assumption is that he is making a point about any kind of > >illative transformation when he says this, and not just the > >transformation involved in a deductive inference. After all, his main > >point in this passage is that these three permissions are precisely what > >is needed in order to gain a deeper understanding of the self correcting > >character of any kind of inference--including inferences by induction and > >abduction. > > > >--Jeff > > > >Jeff Downard > >Associate Professor > >Department of Philosophy > >NAU > >(o) 523-8354 > >________________________________________ > >From: Benjamin Udell [bud...@nyc.rr.com] > >Sent: Thursday, January 22, 2015 11:19 AM > >To: Gary Fuhrman; Jeffrey Brian Downard > >Subject: Re: OFF-LIST Re: Contradictories, contraries, etc. WAS Re: > >[PEIRCE-L] Natural Propositions : Chapter 8 - On the philosophical > >nature of semiosis? > > > > > >________________________________________ > >From: Benjamin Udell [bud...@nyc.rr.com] > >Sent: Tuesday, January 20, 2015 10:52 AM > >To: biosemiot...@lists.ut.ee; Peirce List > >Subject: Re: [PEIRCE-L] RE: NP 8.3 and the Improvement on the Gamma Graphs > > > >Mary, Gary F., > > > >Gary F., thanks for changing the subject title. I had renamed it > >'Contradictories, Contraries [etc]' and then it unexpectedly veered back > >toward the original subject, I should have changed the title when that > >happened. > > > >Mary, you did indeed write the starting post in this subthread. (You sent > >it only to the biosemiotics list, but Gary Richmond forwarded it peirce-l > >(I provide these links so everybody can peruse) > >(gmane) http://thread.gmane.org/gmane.science.philosophy.peirce/15394 > >(IUPUI) https://list.iupui.edu/sympa/arc/peirce-l/2015-01/msg00102.html > >and replied to it: > >(gmane) http://thread.gmane.org/gmane.science.philosophy.peirce/15404 > >(IUPUI) https://list.iupui.edu/sympa/arc/peirce-l/2015-01/msg00112.html > > and your text (originally in reply to Jeffrey Brian Downard) follows > >Gary R.'s reply.) > > > >I wasn't active in the subthread till a bit later but I did read your > >original post. As regards the questions that you posed there: > > > >1. From your original post: > > > >> For example I, like many readers, relate the dicisign overall as > >>Stjernfelt has presented it to his far-reaching cpt. 8: "Operational and > >>Optimal Iconicity in Peirce¹s Diagrammatology.² How do the two kinds of > >>iconicity (chapter 8) Optimal and Operational Icons), make sense when I > >>relate them to or place them in dialogue with the dynamic and immediate > >>objects of the index? I wonder, does a dicisign posit or ³say² that > >>there exists (may exist, hypothetically exists) a written or spoken > >>proposition SRO (Subject Relation Object)? Š that the whole proposition > >>(seen completed after the fact or seen hypothetically completed before > >>the fact of writing or utterance or action) is made up of two parts? To > >>distinguish the object as optimal and operational in relation to the > >>dicisign, I consider the index as it operates in an icon and the index > >>as it operates as an index. (The node between the two, the index and > >>icon, as they reach out and for that moment exist. Is Stjernfelt saying, > > > > in other words, that there (1) exists an object, undistributed in > >relation to the subject and that there (2) exists an object of this > >specific subject under discussion that is distributed (that are under > >discussion,that are being thought, that are coming into a realer or more > >iconic existence)? What and who have or will have placed these in > >discussion may be the Grapheus and the Graphist, the realist and the > >doubter, but the Universe. > >[End quote] > > > >I confess that I didn't understand it! I admit that I was feeling kind of > >obtuse. I was hoping that others' subsequent discussion would clarify it. > > > >2. From your original post: > > > >> I find some loose ends in my thinking about Peirce, amplified somewhat > >>by NP. Is the recto/verso Sheet of Discourse, the ³leaf² pointed to by > >>Stjernfelt, boundless, and in what dimension? I always imagine it as a > >>mobius strip when the sign is in process, but the boundaries of the > >>Universe of Discourse that are discussed by linguists and others are > >>raised. Just now I continue with the leaf (sheet of assertion) analogy > >>and consider the node of life at the stem as it grows. I will continue > >>to think through these icons. > >[End qote] > > > >I thought about the Mobius strip idea but I stopped because I was > >uncertain about whether existential graphs have chirality, but I think > >that they don't, and anyway it doesn't matter (I was wondering about a > >graph that locally seems on the verso, what happens to it when one moves > >it around the Mobius strip to what locally seems the recto). Shaping a > >sheet into a Mobius strip makes it all recto and no verso, as Gary F. > >said, and eliminates the ability to negate a graph. I think Peirce > >somewhere talks about logic without negation. Anyway it can have only > >particular affirmatives and conjunctive compounds of particular > >affirmatives, a one-sided logic so the Mobius strip is actually perfect > >for it. If you want it to be unbounded, the surface of a Klein bottle > >would do that > >https://people.math.osu.edu/fiedorowicz.1/math655/Klein2.html . Anyway I > >guessed that you were trying to think of a way for there to be a > >referential relation between a recto graph and a verso 'possibility' grap > > > > h. I remember once trying to think of some topological trick for that. > > > >Best, Ben > > > >On 1/20/2015 10:36 AM, Gary Fuhrman wrote: > > > >Mary, > > > >The subject line got truncated in your post so I made up a new and > >shorter one to continue the thread. > > > >I can only speak for myself — I read your post carefully more than once, > >but left it to others to reply to it (which Gary R had already done, > >actually) because I had no answers to the questions you raised in it. I > >couldn't make a connection between your suggestion of “would-be > >hypothetical situations, such as the mobius strip” and Peirce's idea of > >using the verso of the sheet of assertion as the area inside a cut. In > >fact I still don't see a connection. A mobius strip, being a bounded > >surface with only one side, doesn't have a verso, and I don't see how it > >relates to Peirce's “discovery” that the verso of the sheet represents “a > >kind of possibility” and not just the negation of the graph within the > >cut. I also couldn't get a handle on your question “Would boundedness > >exist in a mobius strip?” or its relevance to the issue we’ve been > >discussing yesterday and today. > > > >Maybe it’s just my obtuseness, but you’ll need to explain what you were > >driving at before I can see its relevance to Jeff’s post that I did reply > >to. (I assume you want to be given credit for more than just mentioning > >the “verso” in your post, but I don’t yet see what else in it anticipates > >Jeff’s post). > > > >gary f. > > > >-----Original Message----- > >From: Libertin, Mary > >Sent: 20-Jan-15 8:36 AM > >To: biosemiot...@lists.ut.ee<mailto:biosemiot...@lists.ut.ee> > >Subject: [biosemiotics:7975] Re: Contradictories, contraries, etc. WAS > > > >Jeffrey, Gary R, Lists, > > > >I brought up significance of the verso side of the existential graphs in > >my most recent post last week, but have not been acknowledged as the > >initiator of this thread. Gary R. responded to my comment on the > >distributed and undistributed significance of the immediate and direct > >objects of the dicisign. I wrote: > > > >"I do think we should go on. Stjernfelt places his discussion of the > >dicisign in as large a Universe of Discourse as is practical for his > >audience. We need to be more tolerant of interdisciplinary analogies. I > >also think we need some instruction when we find it necessary, which > >means we should ask. Here are some of the questions that came to mind > >after the third time reading NP: how is the sheet of assertion, recto and > >verso sides, to be understood in various ³would¹be² hypothetical > >situations, such as the mobius strip. Would boundedness exist in a mobius > >strip? The concepts of in/out, the whole or the part of the universe of > >discourse are in chapter 8, along with many other important thoughts, > >juxtapositions, questions, and musings. . . ” > > > >I have been researching this area and find it surprising that my initial > >discussion has been overlooked. If this is the first time the issue has > >been discussed I wish to be given credit or acknowledged in the > >discussion. > > > >Mary Libertin > > > >Mary Libertin, PhD > >Professor of English > >Shippensburg University of PA > >Shippensburg PA 17257 > >mli...@ship.edu<mailto:mli...@ship.edu> > > > >On 1/19/15, 11:16 PM, "Jeffrey Brian Downard" wrote: > > > > > > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to > peirce-L@list.iupui.edu . To UNSUBSCRIBE, send a message not to PEIRCE-L > but to l...@list.iupui.edu with the line "UNSubscribe PEIRCE-L" in the > BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm > . > > > > > > -- Sungchul Ji, Ph.D. Associate Professor of Pharmacology and Toxicology Department of Pharmacology and Toxicology Ernest Mario School of Pharmacy Rutgers University Piscataway, N.J. 08855 732-445-4701 www.conformon.net
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