Gary R., Robert, List: I take no exception to anything in Gary R.'s reply, and in light of his and Robert's comments along with Jon A.'s remark in the other thread, I am now persuaded to embrace the formulation that 3ns involves 2ns and 1ns, while 2ns involves 1ns. However, I would welcome some further discussion of whether involution (or presupposition) in this context is really a *genuine *triadic relation in the case of 3ns, rather than a *degenerate* triadic relation that can be reduced to transitive dyadic relations (3ns involves 2ns, which involves 1ns).
Unfortunately I am not adept enough with mathematical category theory to make heads or tails of Robert's exposition below. It still seems to me that "category" means something quite different in that context than it does for Peirce when he is writing about 1ns, 2ns, and 3ns. Am I wrong? If so, I would appreciate some further explanation of how they relate to each other. I will also point out once again my disagreement regarding the logical order of the three interpretants in the hexad. I believe that it should be If→Id→Ii, and it seems to me that this is *more *consistent with "the immutable suite of the three 3ns→2ns→1ns" as 3→2/3→1/2/3, just as the two objects are Od→Oi as 2→1/2. Am I overlooking something in the underlying logic that *requires* the sequence of the interpretants to be Ii→Id→If? Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Wed, May 6, 2020 at 6:10 AM robert marty <[email protected]> wrote: > Gary, Jon Alan, Jon Awbrey, List > > *1 *-First I note that the formulation "3ns involves 2ns, which involves > 1ns" is very dangerous car it forgets that 2ns has its autonomy and 1ns > too. If you look at the podium on remains in the inner cylinder. It seems > to me that Peirce's reproach to Hegel is: > > "*He has usually overlooked external Secondness, altogether. In other > words, he has committed the trifling oversight of forgetting that there is > a real world with real actions and reactions. **Rather a serious > oversight that".* > > It is therefore important to prefer"3ns involves 2ns and 1ns, while 2ns > involves 1ns" which preserves the autonomy of the Peircian categories so > as not to encourage the idea of a possible peircean hegelianism. " > > 2 – On the specific question *"about **the conceptual relationship > between Peirce's trichotomic category theory and contemporary mathematical > category theory if any"*, I will mainly have a limited response to the > field at hand, that is, the classification of signs. Then I can give some > personal reflections on the general scope of mathematical category theory > in the humanities. > > I was just preparing a text on the comparison of the ways in which classes > of signs are generated by different authors and I chose the most > interesting and successful in my eyes, i.e. Gary's trikônics, the triangles > of Priscila Farias and Joao Queiroz and the signtree of Priscila Borges. It > can be said at first glance that they are equivalent since they generate > the same classes of signs that can be characterized by sequences of numbers > of length n = 3, 6 or 10 taken in the set {1,2,3} and verifying that each > number must be less or equal to the previous number. But the question is, > since these are classes based on different graphic metaphors, what is the > common formal structure - if there is one - of which they are the graphic > inscriptions. For this it is necessary to go in the field of posets and > more precisely totally ordered sets the simplest that are the chains: > > " A set with a partial order is called a *partially ordered set* (also > called a *poset*). The term *ordered set* is sometimes also used, as long > as it is clear from the context that no other kind of order is meant. In > particular, totally ordered sets > <https://en.m.wikipedia.org/wiki/Total_order> can also be referred to as > "ordered sets", especially in areas where these structures are more common > than posets. > > For *a, b*, elements of a partially ordered set *P*, if *a* ≤ *b* or *b* > ≤ *a*, then *a* and *b* are *comparable > <https://en.m.wikipedia.org/wiki/Comparability>*. A partial order under > which every pair of elements is comparable is called a *total order > <https://en.m.wikipedia.org/wiki/Totally_ordered_set>* or *linear order*; > a totally ordered set is also called a *chain* (e.g., the natural numbers > with their standard order)". > > ( https://en.wikipedia.org/wiki/Partially_ordered_set) > > > > Clearly the abstract diagram *3ns**à**2ns**à**1ns* (let's call the P) is > a chain which is common to all three approaches. > > > > We have also the maps between partially ordered sets > > > > " Definition 6: A function f : P → Q between partially ordered sets is > order-preserving if x ≤P y ⇒ f(x) ≤Q f(y). > > > > Definition 7: Two partially ordered sets P and Q are isomorphic if there > exists a bijective, order-preserving map between them whose inverse is also > order-preserving" > > (http://www-math.mit.edu/~levine/18.312/alg-comb-lecture-7.pdf ) > > > > To use this notion of the preservation of order, it is necessary to > identify in each graphic metaphor a Q chain . > > > > *I claim that these Q chains are materialized at the moment when everyone > chooses the convention that consists of locating the sign, the object of > the sign and its interpretant on the graphic icon he has chosen.* > > > > *As for Gary*: starting from the object at the lower corner of his > trikône he goes up to the sign at the top corner following the "vertical" > side and then from there he should go to interpretant it from the top side > but if his graph indicates a direct relationship between the object and the > interpretant, which is the same because it is the concatenation of the > first two paths. What is thus traced is an OàS àI chain. > > > > *As for **Priscila Farias et Joao Queiroz :* ( > https://www.researchgate.net/publication/249933979_On_diagrams_for_Peirces_10_28_and_66_classes_of_signs > ) > > it is the same ; the way is taken from Peirce for which the categories > are assigned thereby : the object to the upper left corner to go to the > sign at the bottom corner and from there to interpretant it in the upper > right corner. They create what they call "triangular coordinates" but it's > the same OàS àI chain and it's do the same it when they process > graphically n = 6 and n = 10 which has the effect of multiplying the > triangles. > > > > *Priscila Borges* uses the graphic metaphor of the developing tree: > > > https://www.researchgate.net/publication/263463845_THE_SIGNTREE_FROM_SIGN_STRUCTURE_TO_PEIRCE'S_PHILOSOPHY_THROUGH_READING_A_VISUAL_MODEL_OF_THE_66_CLASSES_OF_SIGNS > > > > "So, the diagram construction begins by the idea of tree rings. They are > used in dendrocronology to count the age of trees. As years go by rings > grow in trees, but they are also affected by climate factors. More than > sign of time, tree rings show interaction between systems. All these > concepts are welcome in semiotic process. Each ring corresponds to one > trichotomy: the first trichotomy comes in the centre, the second > trichotomy in the second ring and so on. " > > > > In this text she does it for n =10: > > > > "Consequently, since the object determines the sign, and not the sign > determines > the object, it was necessary to put the dynamical object in the central > ring, followed by the immediate object and the ground of sign. Given the > first three correlates, comes the first relation: between sign and > dynamical object. This relation determines the possible interpretants, > called immediate interpretants that when are existent become dynamical > interpretants. So, the elements that compose the second relation are > given: between sign and dynamical interpretant." > > > > Moving from a growth ring of its tree to the next it builds the de facto > chain for n = 6: > > > > Od àOiàSàIiàIdàIf > > > > And for n =1 0 she obtient very beautiful diagrams intelligently colored. > > > > *My conclusion is that all these iconographic constructions are > isomorphic; they are produced in the same way using the applications of the > immutable suite of the three **3ns**à**2ns* *à**1ns **and the f > application in n-length chains similar to the protosigns I defined in the > article on the trichotomic machine. They all lead - we just saw - to sets > (in the sense of set theory). The results: classes of signs without > explicit relations between them. * > > > > *Now here's the jump in the category theory :* > > > > *"Every poset (and every **preordered set > <https://en.m.wikipedia.org/wiki/Preorder>**) may be considered as a > **category > <https://en.m.wikipedia.org/wiki/Category_(mathematics)>** where, for > objects x and y, there is at most one **morphism > <https://en.m.wikipedia.org/wiki/Morphism>** from x to y. More > explicitly, let hom(x, y) = {(x, y)} if x ≤ y(and otherwise the empty set) > and (y, z)**∘**(x, y) = (x, z). Such categories are sometimes called **posetal > <https://en.m.wikipedia.org/wiki/Posetal_category>**."* > > > > "Posets are equivalent > <https://en.m.wikipedia.org/wiki/Equivalence_of_categories> to one > another if and only if they are isomorphic > <https://en.m.wikipedia.org/wiki/Isomorphism_of_categories>. In a poset, > the smallest element, if it exists, is an initial object > <https://en.m.wikipedia.org/wiki/Initial_object>, and the largest > element, if it exists, is a terminal object > <https://en.m.wikipedia.org/wiki/Terminal_object>. Also, every preordered > set is equivalent to a poset. Finally, every subcategory of a poset is > isomorphism-closed <https://en.m.wikipedia.org/wiki/Isomorphism-closed>." > ( > https://en.wikipedia.org/wiki/Partially_ordered_set#Mappings_between_partially_ordered_sets > ) > > > > so the same mathematical objects that are involved in the ensemblist > mathematical models that I have just listed can be looked at differently; > they are now algebraic categories. On can use all the conceptual apparatus > of the categories and first the functors and especially the bonus of > natural transformations of functors which brings us back to the trichotomic > machine. This machine naturally produces the same classes of signs of > course but with the order of a lattice revealed by the natural > transformations of functors that we will be able to exploit to increase our > knowledge of the signs and especially to create a methodology as an example > I did in the case of nicotine. > > > > The general idea that has guided me for a long time is that Peirce's > thought is "functorial" and that his universe of thought is above all > relational. This is the reason for the fact that I continued work started > in my book "The Algebra of Signs". I try to express all its semiotics by > starting with a formalization of the "percipuum" in the category of > relational structures. But that's another story... > > > > For now I am sorry to find that I submitted my nicotine analysis to the > criticism on May 3rd and that I did not get any reaction. I believe that I > show and demonstrate how a positive image of semiotics is formed and how it > gains in "semioticity" until it becomes able to compete with the negative > image of nicotine installed in a Dicent Symbol, at the top of the lattice. > > See > https://www.academia.edu/42930701/Nicotine_a_semiotic_confrontation_between_life_and_death > > > > Best regards, > > Robert Marty > > Le mer. 6 mai 2020 à 06:47, Gary Richmond <[email protected]> a > écrit : > >> Jon, Robert, List, >> >> JAS: Overall, we seem to be more or less on the same page. >> GR: I think that's so. >> >> JAS: I understand the impetus for using "presupposition" rather than >> "involution," since the former term is more familiar to modern >> mathematicians and logicians than the latter. >> GR: I too understand the impetus for Robert's using "presupposition" as >> being more familiar to modern mathematicians than "involution." But how >> many of them are familiar with Peirce's three category theory at all? I >> continue to believe that in consideration of Peirce's semeiotic (and all >> that follows from it) that "involution" is the more accurate and evocative >> term. >> >> JAS: I have no objection to saying that 3ns involves 2ns and 1ns, while >> 2ns involves 1ns. I just find it more succinct and equally accurate to say >> that 3ns involves 2ns, which involves 1ns; this already entails that 3ns >> also involves 1ns. >> GR: Logically, of course, you are correct and your more succinct version >> is equivalent. But saying that "3ns involves 2ns and 1ns" brings the >> fundamental trichotomy into high relief immediately. But it is a minor >> point, perhaps one merely of emphasis. >> >> JAS: I am not wedded to Peirce's adaptation of Aristotelian terminology >> (1ns/2ns/3ns = form/matter/entelechy), which is most prevalent in his >> writings around 1904--e.g., in "New Elements" (EP 2:303-305) and "Sketch of >> Dichotomic Mathematics" (NEM 4:292-300)--but I find it helpful in certain >> contexts. >> GR: I suppose it is helpful in certain contexts to employ Peirce's >> tricategorial adaptation of Aristotelian terminology. But is it possible >> that the movement from one equivalent terminology to another -- especially, >> but not only, within a single analysis -- has impeded the more general >> acceptance of some core Peircean ideas? I don't think there's an easy >> solution to this or any of the terminological questions we've taken up on >> the list over the last few years, but I think that there may be a >> communicational problematic here worth considering. >> >> JAS: I share Gary R.'s interest in learning more about "the conceptual >> relationship between Peirce's trichotomic category theory and contemporary >> mathematical category theory if any." I am more familiar with Fernando >> Zalamea's opinion (which I share) that Peirce's mathematical conception of >> continuity is more consistent with category theory (synthetic/top-down) >> than set theory (analytic/bottom-up). >> GR: Zalamea is, in my estimation, one of, if not the leading contemporary >> expert writing on mathematical continuity today. Again, I would be most >> interested in your thoughts, Robert, about "the conceptual relationship >> between Peirce's trichotomic category theory and contemporary mathematical >> category theory if any." >> >> Best. >> >> Gary R >> >> "Time is not a renewable resource." gnox >> *Gary Richmond* >> *Philosophy and Critical Thinking* >> *Communication Studies* >> *LaGuardia College of the City University of New York* >> >>>
----------------------------- PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
