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. Chains 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 / 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.
