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 misinterpretation 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<http://twitter.com/JonAlanSchmidt> On Tue, Apr 28, 2026 at 6:12 PM Edwina Taborsky <[email protected]<mailto:[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]<mailto:[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
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