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 .
----------------------------- 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 .
