Ben, List You are confronted with the mathematical notion of the composition of morphisms. This notion appears as an axiom in the definition of a category. Category theory is the study of mathematical structures and their relationships. It's a unifying notion that began with the observation that many properties of formalized systems can be unified and simplified by point-and-arrow diagrams. Their formal definition is not gratuitous; it aims to capture these observations in exact, well-defined terms. Admittedly, they first concerned mathematical objects, but points and arrows are commonly used in all fields of knowledge. I don't think I'll be denied on this list.
We only need to see the definition of a category to be convinced that it will give us the means to think of these unifications with exactitude (this is Peirce's "exact thinking" at the source of an "exact philosophy"): *The doctrine of exact philosophy, as I understand that phrase, is, that all danger of error in philosophy will be reduced to a minimum by treating the problems as mathematically as possible,** that is, by construction some sort of a diagram representing that which is supposed to be open to the observation of every scientific intelligence, and thereupon mathematically,--that is, intuitionally,--deducing the consequences of that hypothesis. *(NEM IV:12, unidentified fragment) *But in order completely to exhibit the analogue of the conditions of the argument under examination, it will be necessary to use signs or symbols repeated in different places and in different juxtapositions, these signs being subject to certain "rules," that is, certain general relations associated with them by the mind. Such a method of forming a diagram is called algebra. All speech is but such an algebra, the repeated signs being the words, which have relations by virtue of the meanings associated with them. What is commonly called logical algebra differs from other formal logic only in using the same formal method with greater freedom. I may mention that unpublished studies have shown me that a far more powerful method of diagrammatisation than algebra is possible, being an extension at once of algebra and of Clifford's method of graphs; but I am not in a situation to draw up a statement of my researches*. (CP: 3.418). [emphasize mine] NB: Clifford's algebras remain in current scientific research (see https://en.wikipedia.org/wiki/Clifford_algebra). These quotations are self-explanatory regarding the adequacy of category theory to realize Peirce's program. I am aware, of course, that he tried to realize it himself with his existential graphs. But this should not block the path of research since mathematicians, after Peirce, have forged potent tools suitable for achieving the same goal. I'm now giving the most "literary" definition of a category possible; it requires only a common mathematical habitus to be apprehended. A category is defined by three kinds of entities: objects, relations between these objects (called morphisms or arrows), and an operation of composition of these morphisms noted "o" thus defined: for three objects X, Y, and Z, for any morphism f between X and Y and any morphism g between Y and Z, there exists a morphism between X and Z noted g ∘ f which is the compound of f and g. There are other axioms, in particular that of associativity, of which everyone can take note Let me come back to g ∘ f. This notion appears in the last year of the high school science sections. To understand it better, I place myself in set theory; sets are made up of elements; these will be the objects of the category, and relations will be the applications of one set in another. To verify that sets do indeed constitute a category, the operation "o" y needs to be well defined: for each x of X, we have f (x)= y in Y, and for each y of Y, we have g(y)=z in Z. To compose the two is to apply g to the result of f (x)=y by f, hence the new application (g ∘ f ) of X in Z well defined by : (g ∘ f )(x)= g(f(x)) = g(y) = z. I now answer your question by literally reversing your doubt into belief (I hope!): not only does "O → S → I" not break down into the dyads "O → S" and "S → I," but it is the composite of the two dyads that constitute the triad in any mind. As a result, it gives I the power to represent O and thus to be a First (i.e., a First correlate of a new triad - otherwise, it's the third correlate that becomes a First as soon as it's present in mind since it's the one that attention is now focused on, thus initiating semiosis by simple iteration). Such a thing is only possible if the mind has previously internalized the relationship between O and S through previous "collateral" experience in the external world. All this Peirce writes here : *I define a Sign as anything which on the one hand is so determined by an Object and on the other hand so determines an idea in a person's mind, that this latter determination, which I term the Interpretant of the Sign, is thereby mediately determined by that Object. A sign, therefore, has a triadic relation to its Object and to its Interpretant. *(n° 47 bis – 1908 - Letter to Lady Welby in CP 8.343 ). *Every object of experience excites an idea of some sort; but if that idea is not associated sufficiently and in the right way so with some previous experience so as to narrow the attention, it will not be a sign.* (from n°56 - 1911 - MS 849) This is the latest stage in his reflection on the triadic sign. He extended it to the hexadic sign, defined using a sequence of five determinations between 6 elements everyone knows. The question of the determinations of the decadic sign is still open. I challenge anyone to master the combinatorial explosion of the number of classes of signs without these determinations. Regards, Robert Marty Honorary Professor ; PhD Mathematics ; PhD Philosophy fr.wikipedia.org/wiki/Robert_Marty *https://martyrobert.academia.edu/ <https://martyrobert.academia.edu/>* Le dim. 7 janv. 2024 à 18:54, Ben Udell <baud...@gmail.com> a écrit : > Hi, Robert, all, > > I wish a whole lot of us 15 or 20 years ago had noticed a paragraph that > you quote in your message, > > *The conceptions of a First, improperly called an "object," and of a > Second should be carefully distinguished from those of Firstness or > Secondness, both of which are involved in the conceptions of First and > Second. A First is something to which (or, more accurately, to some > substitute for which, thus introducing Thirdness) attention may be > directed. It thus involves Secondness as well as Firstness; while a Second > is a First considered as (here comes Thirdness) a subject of a Secondness. > An object in the proper sense is a Second.* (EP 2: 271) > > We had some long arguments many years ago at peirce-l about what Peirce > meant by "First" etc., when he wasn't explicitly tying those adjectives to > the categories. Joe Ransdell, Gary Richmond, I, and probably others, > argued that, yes, Peirce was alluding to his categories. > > I also remember a whole lot of discussion about Peirce's shift to viewing > the sign as not only determining the interpretant but also being determined > by the object. At the time, a 1906 quote was the earliest that I could > find (I happened to find it at Commens.org I think), and Joe came up with a > quote that prefigured Peirce's shift, from 1905 or 1904, I wish I could > remember (and I tried years ago without success to find Joe's message about > it), but I don't want send anybody on a wild goose chase. > > Folks, here, by the way, is a link to Robert's "76 DEFINITIONS OF THE SIGN > BY C.S. PEIRCE" > http://perso.numericable.fr/robert.marty/semiotique/76defeng.htm > > It includes added quotes absent from the Arisbe version. > > Robert, you wrote below that "*O → S → I*" reads: > > "*O determines S, which determines I*." > > I haven't tried to learn any category theory, since I got intimidated by > its being reputedly based in very high or abstract algebra. > > Generally I recall people saying that — > > an object determines a sign to determine an interpretant > > — rather than that — > > an object determines a sign, which determines an interpretant > > — a phrasing which makes the sign's determination of an interpretant seem > possibly coincidental to the sign's being determined by an object, like > dominoes toppling, each one the next, though the earlier dominoes are not > finally-caused to topple the later ones (except if they are literal > dominoes that some person set up to fall that way). I remember (though not > in detail) a whole lot of discussion of this at peirce-l. Does the > category-theoretical understanding of "O determines S, which determines I" > avoid that seeming problem? To put it another way, how does "*O → S → I*" > keep from breaking down into dyads "*O → S*" and "*S → I*"? I'm not > trying to be argumentative, I'm actually wondering. > > Best, Ben > > On 1/7/2024 10:10 AM, robert marty wrote: > > Cécile, List > > I present here, in the most condensed form possible, the merits of a > purely algebraic formalization of Peirce's semiotics, entirely indexed to > the history of its development. > > > *How do we distinguish the correlates of a triadic sign? How do we > formalize the triadic sign?* > > This question arises because the definition of a triad, strictly speaking, > implies no a priori distinction between the elements it links together. If > you represent them by letters, you're surreptitiously introducing > lexicographical order and by numbers, the order of natural integers. This > is why I draw attention to an important warning Peirce gives about "First" > and "Second" in a footnote to the Syllabus in Part III (EP 2, selection 20): > > **The conceptions of a First, improperly called an "object," and of a > Second should be carefully distinguished from those of Firstness or > Secondness, both of which are involved in the conceptions of First and > Second. A First is something to which (or, more accurately, to some > substitute for which, thus introducing Thirdness) attention may be > directed. It thus involves Secondness as well as Firstness; while a Second > is a First considered as (here comes Thirdness) a subject of a Secondness. > An object in the proper sense is a Second.* (EP 2: 271) > > This warning should shed light on the following definition of the Sign > (which, in my opinion, is far from the best) on page 272: > > *A Sign, or Representamen, is a First which stands in such a genuine > triadic relation to a Second, called its Object, as to be capable of > determining a Third, called its Interpretant, to assume the same triadic > relation to its Object in which it stands itself to the same Object.* (EP > 2: 272) > > On the other hand, the definition given in the fifth version of the > Syllabus (EP 2, selection 21), which is much more precise, will avoid > confusion: > > *Representamen is the First Correlate of a triadic relation, the Second > Correlate being termed its Object, and the possible Third Correlate being > termed its Interpretant, by which triadic relation the possible > Interpretant is determined to be the First Correlate of the same triadic > relation to the same Object, and for some possible Interpretant. A Sign is > a representamen of which some interpretant is a cognition of a mind.* (EP > 2: 290 or CP 2.242 ) > > This is the version Jon Alan Schmidt has chosen to formalize the sign and > semiosis as a spiral. This definition is not that of an ordinary triad > since one correlate, the First, has the power to determine another, the > Third, to make it the new First of a new triad, conferring on it the same > power, and so on. Hence, the image of the spiral, of which Jon Alan Schmidt > provides a projection on a plane, requires raising this projection in the > reader's mind. The repeated addition of the notation S ïI is intended to > capture this "power" and, in fine, "*to capture the idea that the sign > mediates between the object and interpretant*." > > But Peirce himself did much better to achieve this. Once again, I must > point out that Peirce modified his definitions of the Sign by introducing, > around 1904-1905, the determination of the sign S by the object O. Anybody > can consult the list of 76 definitions I published in 1990: it's available > on Peirce.org. There's no need to mention Existential Graphs, which require > a considerable intellectual investment, especially for non-expert readers. > > Perhaps the most straightforward is this one (emphasis added): > > *A sign may be defined as something (not necessarily existent) which is so > determined by a second something called its Object that it will tend in its > turn to determine a third something called its Interpretant in such a way > that in respect to the accomplishment of some end consisting in an effect > made upon the interpretant the action of Sign is (more or less) equivalent > to what that of the object might have been had the circumstances been > different.* (n° 36 - v. 1906 - MS 292. Prolegomena to an Apology for > Pragmaticism) > > The most explicit about the creation of the triadic relation due to the > concatenation of the two determinations is the following: > > *I define a Sign as anything which on the one hand is so determined by an > Object and on the other hand so determines an idea in a person's mind, that > this latter determination, which I term the Interpretant of the Sign, is > thereby mediately determined by that Object. A sign, therefore, has a > triadic relation to its Object and to its Interpretant.* (n° 47 bis – > 1908 - Letter to Lady Welby in CP 8.343 ). > > It's clear, then, that the composition of the two determinations gives > rise to the triadic relation for Peirce. That's why I've underlined > "therefore." Consequently, the formalization is simplified considerably, > without any loss of information, by : > > O → S → I > > The arrows represent determinations, and this diagram reads: > > O determines S, which determines I > > Referring to the Peircean conception of a determination: > > *We thus learn that the Object determines (i.e. renders definitely to be > such as it will be) the Sign in a particular manner.* (CP 8.361, 1908) > > We can see that O determines I by transitivity. Peirce verified this in MS > 611 (Nov. 1908). > > This diagram has the considerable advantage of being equivalent to the > mathematical object below: > > Schematic representation of a category with objects *X*, *Y*, *Z* and > morphisms *f*, *g*, *g* ∘ *f*. > <https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/200px-Commutative_diagram_for_morphism.svg.png> > (click) > > It's an algebraic category, the simplest there is (non-trivial). This one > is the archetypal example of a category on the Wikipedia site devoted to > this part of mathematics, which emerged in the second half of the 20th > century (Category theory - Wikipedia > <https://en.wikipedia.org/wiki/Category_theory>). In 1977 (in French) and > 1982 (in English), I was able to use it to generate, in just a few pages, > not only classes of triadic signs but also, above all, to show that these > classes are naturally organized in a lattice structure (which Peirce had > intuited in the form of affinities). I've verified that Peirce knew about > this type of structure, but limited by set theory, he couldn't obtain it > formally. In his classification of the Sciences, this lattice occupies the > place of the *Grammatica Speculativa*. It's his ultimate form. > > Robert Marty > Honorary Professor ; PhD Mathematics ; PhD Philosophy > fr.wikipedia.org/wiki/Robert_Marty > *https://martyrobert.academia.edu/ <https://martyrobert.academia.edu/>* > _ _ _ _ _ _ _ _ _ _ > ARISBE: THE PEIRCE GATEWAY is now at > https://cspeirce.com and, just as well, at > https://www.cspeirce.com . It'll take a while to repair / update all the > links! > ► 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 UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the > message and nothing in the body. More at > https://list.iupui.edu/sympa/help/user-signoff.html . > ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and > co-managed by him and Ben Udell.
_ _ _ _ _ _ _ _ _ _ ARISBE: THE PEIRCE GATEWAY is now at https://cspeirce.com and, just as well, at https://www.cspeirce.com . It'll take a while to repair / update all the links! ► 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 UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
