Edwina wrote: "I am glad you have accepted the possibility of mathematical mapping to explain semiosis."
I have done nothing of the sort and quite the opposite: I strongly maintain that mathematical mapping does *not* "explain semiosis." In fact, I wrote: "The difficulty arises. . . when a notation like *f(x)=y* is taken as a general schema for semiosis itself," that I reject "an input-output schema to stand for [semiosis.]" Since I've made my position clear enough, at least I hope to others, I have nothing more to offer this discussion at present. Best, Gary R On Thu, Apr 30, 2026 at 8:07 AM Edwina Taborsky <[email protected]> wrote: > Gary R, List > > I am glad you have accepted the possibility of mathematical mapping to > explain semiosis. > > In your focus on continuity,{Thirdness] , however, you ignore Secondness > - the unique reality of this time and this space. I refer to Peirce’s > example in 8,314 of the weather, which refers to the semiosic > understanding of the weather only in THIS time and THIS space.The world > operates not merely within continuity but vitally, within the hard > closures of finite entities within Secondness. > > What does the function map? It maps, as I said, the unique interpretation > of an object [x] within this particular time and space, according to the > general habits/rules [f] and provides a specific - to that time and place- > interpretation [y]. > > As I said already > "the function is a* rule* of mediation - and I said that quite clearly in > my outline - when I referred to and gave examples of the HABITS or rules or > laws of mediation - and mediation is of course, a ‘mode of connection - > but, more, it is a mode of transformation, where it transforms, via its > rules, habits, knowledge base... input data into a viable interpretation. . > > And yes, this semiosis triad, which I compare to the function [ and I am > hardly the only person to do so!!!] does lead to , at a specific time and > place, *one interpretation*. I repeat - It has to -otherwise the world is > chaos. Again, the world operates also within Secondness, which refers to > finite entities in time and space. And since this function is a triad, with > not only each node containing vital data but a different process for each > node - - it’s not a dyadic process - > > Again, I emphasize that the function with its *multiplication action*, is > not a linear dyadic set. The mediate action,* f,* [which in semiosis is > habits/rules/laws [there’s your continuity! ] transforms the input data of > the object in ways that no addition dyad could do. > > . This triad/function/ is a vital action- I must interpret the input > sounds or images to be: one interpretation at this specific time and > space.Not multiple interpretations,but ONE. > Again- I refer to the weather example in 8,314. > > As for the concept of continuous semeiois - that’s basic to Peirce - but > my outline of ONE specific action of semiosis hardly nullifies the concept > of the world as a continuous semiosic process. > > Edwina > > On Apr 30, 2026, at 1:52 AM, Gary Richmond <[email protected]> wrote: > > Jeff, Jon, List, > > Let me restate the issue as I see it as simply as I can. I do not deny -- > indeed, I agree -- that Peirce freely employed mathematical constructions > , that mapping can indeed illuminate certain aspects of sign activity, > especially within formal logic. The difficulty arises, however, when a > notation like *f(x)=y* is taken as a general schema for semiosis itself. > That basic form presents an input, an operator, and a *determinate* output > linked by, as Jon noted, equality, that is, *a completed relation* (however > flexible such notation may be, Jeff, in advanced mathematics). That structure > is *not* neutral in consideration of semiosis, which is most definitely > *not* an operation performed on an object to yield an interpretant, but > an irreducibly triadic mediation in which object, sign, and interpretant > co-constitute one another, so to speak, within a continuous process. C > hains of dyadic mappings cannot capture that continuity. > > Further, the form of the function suggests closure while Peirce insists > on growth. A function is oriented toward producing values. Semiosis, on > the other hand, and as previously noted, generates further signs > indefinitely. What is especially at stake here concerns, in my view, > *category*: the difference between *a mechanism *terminating in outputs vs > a* process *governed by evolving habits. Again, as I earlier suggested, if > one wanted to retain the language of 'function' at all, it would have to > be understood as a habit of mediation formed for a specific time and > purpose, and open-ended -- certainly not as an operator mapping inputs to > outputs. > > So my concern is not with mathematical modeling per se, but with the risk > of allowing a dyadic, input-output schema to stand in for what is > fundamentally 3ns: mediation, continuity, and the living growth of > meaning. > > Best. > > Gary R > > > On Wed, Apr 29, 2026 at 8:56 PM Jeffrey Brian Downard < > [email protected]> wrote: > >> Jon, all, >> >> You say: "This is precisely why I agree with Gary R. that it is *highly >> misleading* to use the mathematical function notation f(x)=y to describe >> semiosis--the equal sign clearly implies a dyadic relation in which a given >> input x always and only produces the same output y, which is brute >> necessitation in accordance with determinism, not the irreducibly triadic >> process by which "Symbols grow" (CP 2.302, EP 2:10, 1894)." >> I want to push back a bit on the suggestion that mathematical function >> notation is inherently misleading when applied to semiosis. >> I agree that semiosis, for Peirce, is irreducibly triadic, and I also >> agree that it would be a mistake to reduce it entirely to a merely dyadic >> input-output relation. But it does not follow that the use of function >> notation is therefore inappropriate. The notation (f(x)=y) does not, by >> itself, commit one to metaphysical determinism, brute necessitation, or a >> denial of triadicity. It simply expresses that, under a given rule or >> mapping, one term is related to another in a mathematically specifiable way. >> That seems especially important in Peirce’s case, because the growth of >> symbols is bound up with processes of inference. If, for the sake of >> argument, we set aside abduction and induction and focus just on deduction, >> Peirce’s logical algebras were designed precisely to analyze formal >> transformations of signs. In that context, mathematical mappings are not >> alien to semiosis; they are one of the tools by which semiotic processes >> can be rendered exact. >> So I do not think the real issue is whether one may use functions at all. >> The real issue is whether one mistakes a formal model for an exhaustive >> metaphysical account. One can use a function, or a family of functions, to >> model some aspect of semiosis without claiming that semiosis is therefore >> nothing but a dyadic mechanism. >> It is also worth noting that mathematical models need not be confined to >> simple linear or mechanically necessitated relations. Modern mathematics >> gives us nonlinear dynamics, iterative mappings, stochastic operators, >> probabilistic transition structures, and other formal systems in which >> branching, instability, and sensitivity to conditions play a central role. >> If the objection is to an overly rigid or simplistic model, I share that >> concern. But that is not an objection to mathematical modeling as such, nor >> to functional representation in particular. >> In a hypercomplex algebra, such as the quaternions, octonions and >> nonions, the branching patterns of such complex functions can have several >> results—and the results are often highly sensitive to initial conditions. >> These are, as Peirce points out, the hallmarks of living systems. One could >> replace the equality sign in the expression of a functional relationship >> with other types of signs (e.g., >, <, *≈*). One might point out that >> "strict" functions take the relation of equalty as fundamental, but this is >> not true of functions generally. >> To cite an example, Euler studied approximation functions in his work on >> cartography. Peirce draws on this tradition in his own work on map making. >> Indeed, Peirce himself seems quite willing to think in terms of mappings, >> transformations, and exact formal relations when doing logic, mathematics, >> and cartography. The relation between a map and the terrain is not one of >> simple equality, yet it is still a rigorously intelligible representational >> relation. >> >> I think something similar can be said of sign systems more generally: a >> system of signs used by a person or community may stand in a mapping >> relation to features of the world without that relation being reducible to >> mere identity. >> So my hesitation is with the claim that function notation must be avoided >> in semiotics. That seems to me too strong. The suggestion that we should >> avoid using mathematical functions to build exact models in semiotics runs >> counter to Peirce's practice. Having said that, there is reason to be >> cautious when using models that systematically ignore or suppress the >> temporal development of sign-action, or the irreducibly triadic character >> of semiosis. But mathematical functions, broadly understood, are perfectly >> compatible with formalizing--and thereby clarifying--various aspects of >> those processes. >> Best, >> Jeff >> >> ------------------------------ >> *From:* [email protected] <[email protected]> on >> behalf of Jon Alan Schmidt <[email protected]> >> *Sent:* Tuesday, April 28, 2026 5:59 PM >> *To:* Peirce-L <[email protected]> >> *Subject:* Re: [PEIRCE-L] Peirce and critical thinking >> >> List: >> >> ET: And yes, this semiosis triad, which I compare to the function [and I >> am hardly the only person to do so!!!] does lead to, at a specific time and >> place, *one interpretation*. >> >> >> To clarify, a sign token that occurs at one specific time and place can >> determine multiple dynamical interpretants--effects that the sign token >> *actually >> does* have--each of which occurs at a specific time and place, and all >> of which are not necessarily identical. That is why *misi*nterpretation >> happens, to the extent that an individual dynamical interpretant deviates >> from the final interpretant of the sign itself--the effect that it *ideally >> would* have. >> >> ET: And since this function is a triad, with each node containing vital >> data - it’s not a dyadic process - and you are quite incorrect to claim >> such - ie, the function is not the same as a a mathematical equal sign - >> which IS dyadic!!! >> >> >> This is precisely why I agree with Gary R. that it is *highly misleading* >> to use the mathematical function notation f(x)=y to describe semiosis--the >> equal sign clearly implies a dyadic relation in which a given input x >> always and only produces the same output y, which is brute necessitation in >> accordance with determinism, not the irreducibly triadic process by which >> "Symbols grow" (CP 2.302, EP 2:10, 1894). >> >> Regards, >> >> Jon Alan Schmidt - Olathe, Kansas, USA >> Structural Engineer, Synechist Philosopher, Lutheran Christian >> www.LinkedIn.com/in/JonAlanSchmidt >> <http://www.linkedin.com/in/JonAlanSchmidt> / twitter.com/JonAlanSchmidt >> >> On Tue, Apr 28, 2026 at 6:12 PM Edwina Taborsky <[email protected]> >> wrote: >> >> >> Gary R, List >> >> Yes, you are quite correct- the function is a* rule* of mediation - and >> I said that quite clearly in my outline - when I referred to and gave >> examples of the HABITS or rules or laws of mediation - and mediation is of >> course, a ‘mode of connection - but, more, it is a mode of transformation, >> where it transforms, via its rules, habits, knowledge base... input data >> into a viable interpretation. . >> >> And yes, this semiosis triad, which I compare to the function [ and I am >> hardly the only person to do so!!!] does lead to , at a specific time and >> place, *one interpretation*. It has to -otherwise the world is chaos. >> And since this function is a triad, with each node containing vital data - >> it’s not a dyadic process - and you are quite incorrect to claim such - ie, >> the function is not the same as a a mathematical equal sign - which IS >> dyadic!!! >> >> . This triad/function/ is a vital action- I must interpret the input >> sounds or images to be: one interpretation at this specific time and >> space.Not multiple interpretations,but ONE. My neighbour is saying Hello - >> and is NOT an evil witch putting a curse on me. …I must interpret the image >> I see to be Rain- and not gold coins…or harmful chemicals or angels >> dropping from the sky or... >> >> As for the concept of continuous semeiois - that’s basic to Peirce - but >> my outline of ONE specific action of semiosis hardly nullifies the concept >> of the world as a continuous semiosic process. >> >> Edwina >> >> On Apr 28, 2026, at 6:40 PM, Gary Richmond <[email protected]> wrote: >> >> Doug, List, >> >> Since she mentioned it in her response to you, I must remark that Edwina >> Taborsky's interpretation of Peirce’s semiosis as a function, *f(x) = y*, >> where x is the object, f is the sign, and y is the interpretant, is >> exceedingly problematic in my view. In her model the sign 'operates' on >> the object to 'produce' the interpretant. This would mean that the >> interpretant is generated through a* transformation *comparable to a >> mathematical mapping*.* While this model may seem appealing to those >> trained in analytic philosophy or, say, Shannon's classical information >> theory -- since it offers a clean *input-output *picture of semiosis -- >> nonetheless, it replaces Peirce’s open and continuous semiotic mediation >> with a closed system of transformation. >> >> Further, a function implies a dyadic and determinate relation: one value >> of x leads to one value of y. Peirce’s semiosis, however, is at its very >> heart triadic and relational. Yes, the object determines [in Peirce's >> sense of 'constrains'] the sign which determines [constrains] the >> interpretant, but *none of these determinations is mechanical*. For >> Peirce the sign does not * transform *the object into an interpretant; >> rather, it *mediates *between them within a process that remains open >> and continuous. In reducing semiosis to transformation,Taborsky's model >> ignores the potential, and the creative aspects which animate Peirce’s >> semeiotics, including as regards his conception of meaning. >> >> In short, the formula f(x) = y collapses Peirce’s open-ended process of >> mediation (3ns) into a fixed act of transformation (2ns) in that* it >> substitutes function for a law of habit*. Whereas a function ends with >> an output, Peirce’s semiosis generates new interpretants as new signs >> indefinitely, each interpretant, in turn, becoming a sign. This process >> has no definite closure. Some have called this 'infinite semiosis': the >> process where the interpretant of a sign becomes a new sign for the same >> object, and so initiates an endless chain of meaning. It implies that >> meaning-making is never static or fully realized, but is always open to >> further interpretation and unfolding. >> >> And there is one additional danger: that of over-formalizing Peirce’s >> ideas. While it is true that Peirce himself sometimes used mathematical >> analogies (although, and for prime example, he strongly argued against >> using his system of Existential Graphs as a calculus), his semeiotic is >> anything >> but computational. By turning semiosis into a neat function, one is >> treating signs as static conversions rather than as *evolving acts of >> interpretation* rooted in culture, experience, and inquiry (there are >> biological analogies of these). In addition, it's important to note that >> Peirce’s >> semeiotic is deeply entangled with his phenomenology and metaphysics (see, >> for example,“An Outline Classification of the Sciences” in the 1903 >> *Syllabus*) and is, in my view, not fully detachable from them. >> >> There is, perhaps, a limited sense in which Taborsky’s metaphor might be >> helpful. But if so, she would have to treat the 'function' *not as a >> deterministic formula but as a rule of mediation*; that is to say as a >> pattern of relational continuity rather than causal mapping. As such it >> might serve as a shorthand for the way the sign organizes its reference. >> But then Taborsky would have to acknowledge that* f* here is *not an >> operator but a habit*: a mode of connection among object, sign, and >> interpretant within a continuum of semiosis, something I think that she >> is unlikely to do. >> >> Perhaps the greatest flaw in Taborsky’s model as I see it is that it >> doesn’t capture the continuity -- the evolutionary continuity -- the >> *vitality* in Peirce’s thought. His semeiotic came out of his own >> evolving ideas about synechism and habit-formation, stressing the >> continuity and growth of interpretation. A fixed function like *f(x)=y* >> freezes what Peirce understood as a living, evolutionary process. For these >> reasons, in my view such essentially dyadic models as Taborsky's >> misunderstand and misinterpret Peirce’s semiosis. >> >> Best, >> >> Gary Richmond >> >> _ _ _ _ _ _ _ _ _ _ >> ► 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] . >> ► <a href="mailto:[email protected]";>UNSUBSCRIBE FROM >> PEIRCE-L</a> . But, if your subscribed email account is not your default >> email account, then go to >> https://list.iu.edu/sympa/signoff/peirce-l . >> ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; >> and co-managed by him and Ben Udell. >> > _ _ _ _ _ _ _ _ _ _ > ► 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] > . > ► <a href="mailto:[email protected]";>UNSUBSCRIBE FROM > PEIRCE-L</a> . But, if your subscribed email account is not your default > email account, then go to > https://list.iu.edu/sympa/signoff/peirce-l . > ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and > co-managed by him and Ben Udell. > > >
_ _ _ _ _ _ _ _ _ _ ► 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] . ► <a href="mailto:[email protected]";>UNSUBSCRIBE FROM PEIRCE-L</a> . But, if your subscribed email account is not your default email account, then go to https://list.iu.edu/sympa/signoff/peirce-l . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
