Sung, List, The arguments Peirce makes against Kempe are rather involved, so I won't try to reconstruct them here. If you'd like to take a look at the primary texts, you might start with the references in Grattan-Guinness, Ivor. "Re-Interpreting'λ': Kempe on Multisets and Peirce on Graphs, 1886-1905." Transactions of the Charles S. Peirce Society (2002): 327-350. The pieces by Kempe are available online.
--Jeff Jeff Downard Associate Professor Department of Philosophy NAU (o) 523-8354 ________________________________________ From: Sungchul Ji [s...@rci.rutgers.edu] Sent: Monday, April 07, 2014 1:09 PM To: Jeffrey Brian Downard Cc: Sungchul Ji; Peirce List Subject: RE: [PEIRCE-L] de Waal Seminar: Chapter 5, Semeiotics, or the doctrine of signs Jeff asked: "Given your way of putting things, let me ask the (040714-3) following question: is there anything that is obscured if you assume that there is an "internal node" at the heart of every elementary relation?" To the best of my knowledge, I have not found any problem represetning Perice's sign triad in terms of the 4-node network at all. Instead, it gives a graphical basis for differentiating his "represetnamen" and "sign", as I discussed in my recent email to which you reponded. " . . . are you able to articulate the basic points (040714-4) Peirce is making against Kempe's analysis of mathematical form in your terms?" I am ignorant of the basic points that Peirce is making agaisnt Kemp's analysis of mathemtical form. Can you explain them to me or direct me to appropriate links ? Thanks. Sung > Sung, Jerry, List, > > I understand that we are using somewhat different language and graph > systems as bases for understanding Peirce's arguments in his phenomenology > and semiotics. Jon has suggested that Peirce's different ways of > diagramming these relations are hardly more than hints of ideas, so we > should be careful about how much we derive from one or another manner of > representing these things graphically. I want to resist Jon's suggestion. > As such, let me try to restate the question. Given your way of putting > things, let me ask the following question: is there anything that is > obscured if you assume that there is an "internal node" at the heart of > every elementary relation? I am suggesting that any graphical system, > including the one that you have formulated or a bipartite system of graphs > that allows nodes to be connected to several lines (i.e. edges, relations, > etc.) will run the risk of obscuring things that Peirce wanted to analyze > further. > > I recognize that you and Jerry seem to have different dispositions towards > the way Peirce is using graph theory to analyze these things. You are > saying that your system is equivalent to Peirce's and that it expresses > the heart of what he is really trying to say. Jerry, on the other hand, > seems to suggest that Peirce is hamstrung by limitations that have their > source in outmoded 19th century ways of thinking about and graphing > chemical relations. > > So, let me try to state the question I've been trying to push as a > challenge. For those who think that the most elemental relations can be > understood as a node with one, two or three relations jutting out from it > (such as you have characterized them in your diagrams, and as the nodes > and edges are characterized in the example of the bipartite graph on the > WikiPedia site I referred to earlier), are you able to articulate the > basic points Peirce is making against Kempe's analysis of mathematical > form in your terms? My hunch is that Peirce's arguments are valid, that > they can be used against alternate analyses of the reasonings--and that > these alternate analyses fail because they tend to obscure points about > the elemental character of a triadic relation that Peirce wanted to make > as clear as possible. > > --Jeff > > Jeff Downard > Associate Professor > Department of Philosophy > NAU > (o) 523-8354 > ________________________________________ > From: Sungchul Ji [s...@rci.rutgers.edu] > Sent: Monday, April 07, 2014 11:23 AM > To: Jeffrey Brian Downard > Cc: Peirce List > Subject: RE: [PEIRCE-L] de Waal Seminar: Chapter 5, Semeiotics, > or the doctrine of signs > > Jeff wrote (040714-1) and (040714-2): > > ". . . I don't believe that I'm using monad, dyad and (040714-1) > triad to refer to the number of nodes in a network." > > If my classification of the networks shown in Table 1 is correct, you are > indeed not using the terms “monad”, “dyad”, and “triad” to refer to the > number of nodes in a network but rather the number of edges (which is > synonymous with your “binding sites”) of a network (see the last two > columns in Table 1). > > _____________________________________________________________ > Table 1. A network representation of the Peircean “medad”, > “monad”, “dyad”, and “triad”. > _____________________________________________________________ > > Number of > ________________________________ > > Name (N) network nodes* binding edges > sites (x) > _____________________________________________________________ > > medad o 1 0 0 > (1-node N, > or a point) > _____________________________________________________________ > > monad o – x 2 1 1 > (2-node N) > _____________________________________________________________ > > Dyad x – o – x 3 2 2 > (3-node N) > _____________________________________________________________ > > x > | > triad o 4 3 3 > (4-node N) / \ > x x > _____________________________________________________________ > > *There are two kinds of nodes – the focal (denoted by “o”) and the > peripheral (denoted by “x”). The peripheral nodes are also called > “binding sites” and can be occupied by n-node network where n can be 0, > 1, 2, 3 or 4. > > “As far as I can tell, it appears that your (040714-2) > characterization of the four node network might > be doing the same thing.” > > In what sense do you think the 4-node network representation of the > Peircean sign (i.e., the 4-node network in Table 1 with three x’s replaced > by representamen, object, and interpretant) is like a “bipartite system of > graphs” that “obscure relations that Peirce wanted to analyze” ? > > > With all the best. > > Sung > ___________________________________________________ > 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 > >> Sung, Jerry, list, >> >> I am trying to use the terms and the relations in a manner that fairly >> represents what Peirce says in the texts. For the sake of clarity I >> don't >> believe that I'm using monad, dyad and triad to refer to the number of >> nodes in a network. Rather, I'm using it to refer to the number of >> unsaturated bonding sites on three kinds of relations that Peirce claims >> are elementary. One of those, the monad, functions like a terminal node >> with with only one unsaturated bonding site. The others are not nodes. >> Rather, they are relations with two or three unsaturated bonding sites. >> The difference between them is not merely a matter of the number of >> sites >> available, it is also a matter of one being a relation with no branches >> and the other having a branch. Topologically, those are fundamentally >> different kinds of relations. The dyadic and triadic relations can, of >> course, be saturated by being bound to two or three monadic nodes. Once >> that has been done, there are no more bonding sites available and either >> can be treated as a medad. Having said that, medadic complexes can >> always >> be altered by the insertion of triadic relation--creating new bonding >> sites within the complex. >> >> There are several different ways that graphical systems could be set up >> with different systems permitting or forbidding different kinds of >> combinations. I'm not claiming that the graph theoretical system that >> I'm >> describing--which seems to fit Peirce's two primary descriptions of the >> elementary kinds of relations in his phenomenology best--is the only >> kind >> of graph theoretical system that could be used to analyze mathematical >> diagrams and inferences. I'm simply saying that a bipartite system of >> graphs seems to obscure relations that Peirce wanted to analyze further >> and thereby make clearer. As far as I can tell, it appears that your >> characterization of the four node network might be doing the same thing. >> If that is the case, they both obscure the very same things that Peirce >> found problematic in Kempe's graphical system. >> >> I haven't laid out the arguments. Thus far, I've only pointed to them. >> I >> do think, however, that his arguments against Kempe's analyses of >> different kinds of mathematical diagrams and inferences might be >> instructive for understanding aspects of the kinds of relations are >> taken >> to elemental in his semiotic theory. >> >> --Jeff >> >> Jeff Downard >> Associate Professor >> Department of Philosophy >> NAU >> (o) 523-8354 >> ________________________________________ >> From: Sungchul Ji [s...@rci.rutgers.edu] >> Sent: Saturday, April 05, 2014 3:04 PM >> To: Jeffrey Brian Downard >> Cc: Jerry LR Chandler; Peirce List; Vinicius Romanini >> Subject: RE: [PEIRCE-L] de Waal Seminar: Chapter 5, Semeiotics, or >> the doctrine of signs >> >> (For undistorted Fgirue 1, see the attached.) >> >> Jeff wrote: >> >> ". . . the fundamental elements of experience (040514-1) >> studied in the phenomenological theory are >> diagrammed as a node with a single bonding site >> (e.g., a monad), a straight line with two bonding >> sites (e.g., a dyad), and a branching line with >> three bonding sites (e.g., a triad). I think >> there is much to be gained by using these kinds >> of figures in the graphs we construct to analyze >> the objects, relations and inferences in mathematical >> (or any other) kind of reasoning." >> >> It seems that you are using the terms "monad", "dyad", and "triad" in >> (040514-1) to refer to the number of the nodes in a network. Thus, the >> “branching line”, i.e., -<, is a triad because it has three nodes or >> “binding sites” as you called them, a line segment is “dyad” because it >> has two ends (or nodes), and a point is “monad” because it is just one >> node all by itself. This way of interpreting the terms is also employed >> by John and Edwina. >> >> But these same terms can be used to refer to “edges” of a network as >> well >> as I have been advocating against the opinions of John and Edwina. For >> example, the word “triad” can refer to “a triadic relation” as >> represented >> by the three edges of the “branching” line, -<. But when it comes to >> the >> word “dyad”, things get somewhat complicated, because “dyad” can mean >> “two nodes” of a line segment as you explained above or “one edge” >> (i.e., one relation) which a line segment is. In other words, “dyad” >> can >> mean, confusingly, “two nodes” or “one edge”, i.e., “two relata” or “one >> relation”. Similarly, the word “monad” can mean one “node” (or >> relatum) >> with no edge (or relation). So, depending on whether these terms refer >> to >> nodes or edges of a network, their meanings change. >> >> The 4-node network representation of the Peircean sign (see Figure 1 in >> [biosemiotics:5631]) seems able to provide a convenient visual tool to >> analyze Statement (040514-2) as shown in Figure 1. Focusing on the >> components of the Peircean sign only, Figure 1 can be read as >> >> “A representamen is determined by its object and determines >> (040514-2) >> its interpretant in such a way that the interpretant is indirectly >> determined by (or is consistent with) the object.” >> >> One of the main points of Figure 1 is that the sign, ‘n-ad’, has at >> least >> two choices, denoted as (. . .) and [. . .], as its representamen, >> object, and interpretant. Hence, “n-ad” can be interpreted in two (I >> would say “complementary”) ways, depending on which aspect of the sign >> is >> prescinded in one’s mind, just as light can be “interpreted” in two >> ways, >> as particle or as wave, depending on which aspects of light is measured >> by an instrument, leading to the following dictum: >> >> “The word ‘n-ad, as represented by a network, can >> (040514-3) >> be viewed as the complementary union of nodes and >> edges, just as light is the complementary union of >> waves and particles.” >> >> >> (node-adicity) >> [edge-adicity] >> REPRESENTAMEN >> | >> | >> | >> “n-ad” >> SIGN >> / \ >> / \ >> / \ >> (node) (relata) >> [edge] [relation] >> OBJECT INTERPRETANT >> >> Figure 1. The application of the 4-node network representation of a >> sign >> to clarifying the ambiguity of the word ‘n-ad’ such as monad, dyad, >> triad, >> etc. >> >> Statement (040514-3) applies to Piece’s tripod, -<, leading to the >> conclusion that >> >> “Peirce’s tripod network, -<, can be interpreted (040514-4) >> in two complementary ways -- as a node-triad >> or as an edge-triad. That is, as a set of three >> relata/nodes or as a set of three relations/edges.” >> >> The significance of (040514-4) lies in the fact that, if the three nodes >> of the tripod are arranged linearly, and not “triangularly”, the >> node-adicity would be 3 and the edge-adicity two. >> >> With all the best. >> >> Sung >> ___________________________________________________ >> 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 >> >>> Jerry, List, >>> >>> You make the following claim: "In today's mathematics, a chemical icon >>> is >>> an exact mathematical object, a labelled bipartite graph." I'd like to >>> ask: if we understand the chemical icon to be a formal diagram >>> involving >>> vertices and lines like the ones that Peirce uses in his discussions of >>> possible diagrams of chemical molecules, then what might be missed if >>> we >>> analyze the diagrams using bipartite graphs as a logical tool? >>> >>> For now, I'd like to set to the side a number of points you make that I >>> would consider matters of metaphysics. The question of whether >>> Peirce's >>> phenomenology and semiotics provides the conceptual tools needed to >>> establish a metaphysics that will be adequate to explain the real >>> nature >>> of molecules and molecular relations is a difficult question, and it's >>> one >>> that I wait until the discussion of the chapter on metaphysics to >>> broach. >>> >>> Having set such issues to the side, I'd like to focus on a set of >>> points >>> you make that starts with the mathematics of formal graphs, runs into >>> the >>> phenomenological account of the categories, and then proceeds into >>> Peirce's critical grammar. Starting with the math, you say that >>> Peirce's >>> understanding of graph theory is based on diagrams used to study >>> chemical >>> relations. I'll grant that much. Sylvester explored these kinds of >>> diagrams to examine character of algebraic invariants, and Peirce drew >>> on >>> the same kinds of chemical diagrams for a number of mathematical and >>> logical purposes. One of Peirce's purposes was to to challenge a claim >>> Kempe makes in his essay on mathematical form. Kempe asserts that all >>> mathematical objects, relations and inferences can be analyzed in terms >>> of >>> a simple set of graphs that he developed for the purpose. Peirce >>> denies >>> the claim. He argues that the analysis of the objects, relations and >>> inferences in any part of mathematics requires triadic relations. In >>> effect, he is claiming that the proper analysis of the relations >>> between >>> monads, dyads and triads, is obscured in Kempe's account. >>> >>> Let me see if I can turn one of the points Peirce makes in his argument >>> against Kempe against a 20th or 21st century analysis of mathematical >>> form >>> in terms of bipartite graphs. Like a modern bipartite graph, the main >>> elements of Kempe's system are nodes and edges. In this system we >>> treat >>> every vertex in a diagram as a node, and every line that connects >>> vertices >>> as edges. I'm no expert in graph theory, but my understanding is that >>> a >>> graph is bipartite if all of the nodes can be grouped into two sets in >>> such a fashion that every line connecting vertices in a given formal >>> diagram are represented by edges connecting nodes in one or the other >>> of >>> two sets. See, for instance, the WikiPedia entry on bipartite graphs >>> for >>> a simple explanation and a set of examples. In order to have an >>> example >>> before us, let's consider the graph at the top of that webpage: >>> http://en.wikipedia.org/wiki/File:Simple-bipartite-graph.svg >>> >>> What concerns does Peirce have about the use of these kinds of graphs >>> as >>> a >>> tool for analyzing the objects, relations and inferences used in one or >>> another area of mathematical inquiry? Well, it would help to have a >>> clear >>> example of a mathematical diagram. That way, we have an example of a >>> diagram that is being analyzed and an example of the kind of graph that >>> can be used to analyzed it. The diagram that Peirce draws on in the >>> Harvard Lectures of 1903 in his argument against Kempe is Pappus' proof >>> of >>> the 9-ray theorem in projective geometry. >>> >>> You seem to be saying that Peirce's analysis of mathematical form is >>> inadequate because it fails to take into account the kinds of >>> developments >>> that were made in the 20th century as work in graph theory marched >>> forward. I beg to differ. My hunch is that these bipartite graphs >>> obscure the very same points that Kempe obscured. As such, we should >>> be >>> careful if we intend to use such mathematical systems to explore the >>> adequacies or inadequacies of Peirce's approach to analyzing the >>> possible >>> systems of hypotheses that might lie at the foundations of any area of >>> mathematics and the inferences that can be drawn from such hypotheses. >>> >>> What is being obscured? In short, the bipartite graphs allow several >>> edges to meet on one node. What Peirce analyzes the form of such >>> intersections, he says that allowing this kind of combination fails to >>> bring out the dyadic or triadic character of the relations being >>> analyzed. >>> As such, he fundamental elements of experience studied in the >>> phenomenological theory are diagrammed as a node with a single bonding >>> site (e.g., a monad), a straight line with two bonding sites (e.g., a >>> dyad), and a branching line with three bonding sites (e.g., a triad). >>> I >>> think there is much to be gained by using these kinds of figures in the >>> graphs we construct to analyze the objects, relations and inferences in >>> mathematical (or any other) kind of reasoning. To press one of >>> Peirce's >>> points, what is needed in the way of a formal graph if we're going to >>> analyze the character of the projective space in which the diagram of >>> Pappus' theorem is constructed. It is a two-dimensional surface, and >>> it >>> is different from other surfaces in that there is a peculiar twist in >>> the >>> space. One of the things that Pappus's proof enables us to see--and I >>> mean literally "see"--is the commutivity of the mathematical space that >>> contains the lines, points and rays of the diagrams used it the proof. >>> The key thing that we see is that it wouldn't matter how the lines are >>> moved in this space. Any movement would result in the rays >>> intersecting >>> in fashion that produces three points that are collinear. Peirce >>> insists >>> that seeing this relationship is crucial to the proof, and that the >>> analysis of what it is that we're seeing is obscured if we think of the >>> formal relations as nothing more than nodes and edges where more >>> several >>> edges meet at one node. It's not just a matter of not seeing what is >>> being packed into the meeting of those several edges at this one node. >>> Rather, we don't see the order involved in constructing the rays and >>> the >>> intersection of those rays. In effect, the act of constructing those >>> rays >>> and intersections is what defines the character of the surface as a two >>> dimensional projective space. >>> >>> --Jeff >>> >>> >>> >>> Jeff Downard >>> Associate Professor >>> Department of Philosophy >>> NAU >>> (o) 523-8354 >>> ________________________________________ >>> From: Jerry LR Chandler [jerry_lr_chand...@mac.com] >>> Sent: Friday, April 04, 2014 10:48 AM >>> To: Peirce List >>> Cc: Vinicius Romanini; Jeffrey Brian Downard >>> Subject: Re: [PEIRCE-L] de Waal Seminar: Chapter 5, Semeiotics, or the >>> doctrine of signs >>> >>> Vinicius, Jeff, Ben: >>> >>> (This post is a bit on the technical side. Do not have time today to >>> make >>> it simpler with longer explanations of the categories of exact >>> relations >>> mentioned in this text.) >>> >>> A simple interpretation of the Peircian distinction between the meaning >>> associated with the grounding terms "icon" and "index" is possible if >>> one >>> recognizes that both terms are consequences of his knowledge of >>> chemistry >>> as it stood in his day. >>> >>> An chemical icon, as a visual form, either internal to a mind or >>> external >>> as a form of an existent object, is only one form of a material object. >>> >>> An index, as a set of marks or as a listing of multiple terms or >>> objects >>> in some form or another, is a necessary concept for chemical >>> representations. Furthermore, this index is essential to GENERATING or >>> CREATING the chemical icon. In today's mathematics, a chemical icon is >>> an >>> exact mathematical object, a labelled bipartite graph. >>> >>> Both the chemical icon and the chemical index are absolutely necessary >>> to >>> create the semantic notion of a symbol (which CSP defines a symbol as >>> either a word OR a concept.) [This is one possible understanding of >>> the >>> conundrum of why CSP used the "or" conjunction here.] >>> >>> Thus, this interpretation is congruent with the philosophical >>> categories >>> of Quality, Representation and Relations. >>> It is also consistent with the more pragmatic view of CSP's thoughts >>> about: Thing, Representation and Form. >>> >>> Thus, this chemical interpretation of the three terms, icon, index and >>> symbol are congruent with the ontology of matter as CSP understood it >>> in >>> the late 19 th Century. >>> >>> Finally, this interpretation is necessary for the completeness of the >>> medad as a sentence representing a complete thought. In the case of >>> chemical logic, as it stood in CSP's day, the medad can be thought of >>> as >>> a >>> sentence describing the binding of chemical elements into a "radical". >>> >>> With these insights as part of the ground, I would like to extend my >>> remarks to CSP's motivations in general. >>> My recent posts are parts of a broad thesis of CSP's motivations for >>> his >>> inquiry into semiotics and its meaningfulness in terms of today's usage >>> of >>> symbols in scientific notations. (Jeff's posts raise comparable >>> issues.) >>> >>> The philosophical hypothesis is simple: >>> >>> Peircian semiotics is grounded in the 19th Century view of chemical >>> semiotics as well as the logic of 19th C. mathematical terms. >>> >>> Unfortunately for this world view, 21 st Century chemical semiotics are >>> both numerically, logically, and symbolically remote from 19th Century >>> chemical semiotics. Chemical terms have been given new meaning in order >>> to >>> be congruent with the electrical structures of atoms as a atomic >>> numbers. >>> Consequently, the meaningfulness of Peircian semiotics is problematic >>> as >>> it severely restricts the conceptualization of information as the >>> breadth >>> and depth of the intentions embedded in a symbol by a speaker or >>> writer. >>> >>> The reasoning for the emergence of modern chemical semiotics from the >>> 19 >>> th century Peircian semiotics is also simple. In the 19th Century, the >>> chemical table of elements was based on the relative masses of each >>> element as a means of explaining correlates. >>> >>> In the first half of the 20 th Century, modern chemical semiotics >>> emerged >>> from the Peircian forms by the inclusion of electrical logic into the >>> iconization of chemical objects. Consequently, a chemical icon of >>> today >>> is an exact representation of BOTH mass and electricity (as illustrated >>> by >>> physical quantum mechanics.) This representation of a chemical object >>> as >>> a binary object necessarily infers that it can not be represented as a >>> geometric point. >>> >>> The gradual shift of representation of matter as weight (19 th C., a >>> singular point, a measure of "beingness") to the representation of >>> matter >>> as both mass and electricity (21 st C., a form comparable to an >>> extension >>> of a beta-existential graph) forced the change in meaning of chemical >>> and >>> biological and medical symbols. >>> >>> By simple extension, a biological icon of today represents both mass >>> and >>> electricity and iconic correlates (in addition to many other predicates >>> and copulas.) >>> >>> Cheers >>> >>> Jerry >>> >>> >>> >>> >>> >>> >>> >>> >>> On Apr 3, 2014, at 7:50 PM, Vinicius Romanini wrote: >>> >>> Jeff, list >>> >>> Jeff said: Having taken a look at MS 7, I'd like to ask a quick >>> question >>> about the first assertion. What is Vinicius claiming when he says >>> that >>> icons don't *enter* our concepts as such? Looking at page 15 of the >>> MS, >>> I >>> see Peirce saying the following: "An icon cannot be a complete sign; >>> but >>> it is the only sign which directly brings the interpretant to close >>> quarters with the meaning; and for that reason it is the kind of sign >>> with >>> which the mathematician works." Shortly after making this point, he >>> develops the examples of the weather vane and the photograph. >>> >>> >>> V: A pure icon would be an immediate intuition of the form of the >>> object, >>> which Peirce denies. Every cognition is based on previous ones. But we >>> can use abstract concepts to diagram an idea iconically, as when we use >>> mathematical symbols to express the truth of a theorem. We then >>> contemplate the icon represented in the symbol. The quote bellow might >>> help: >>> >>> "The third case is where the dual relation between the sign and its >>> object >>> is degenerate and consists in a mere resemblance between them. I call a >>> sign which stands for something merely because it resembles it, an >>> icon. >>> Icons are so completely substituted for their objects as hardly to be >>> distinguished from them. Such are the diagrams of geometry. A diagram, >>> indeed, so far as it has a general signification, is not a pure icon; >>> but >>> in the middle part of our reasonings we forget that abstractness in >>> great >>> measure, and the diagram is for us the very thing. So in contemplating >>> a >>> painting, there is a moment when we lose the consciousness that it is >>> not >>> the thing, the distinction of the real and the copy disappears, and it >>> is >>> for the moment a pure dream -- not any particular existence, and yet >>> not >>> general. At that moment we are contemplating an icon." (CP 3.362) >>> >>> Vinicius >>> >>> >>> ----------------------------- >>> 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<mailto:peirce-L@list.iupui.edu> . To >>> UNSUBSCRIBE, >>> send a message not to PEIRCE-L but to >>> l...@list.iupui.edu<mailto: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 . >>> >>> >>> >>> >
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