Gary R., List: GR: If the vector 3ns ->1ns ->2ns "matches up with. . .the nature of semiosis," what is that vector's relation to that of the vector of determination (2ns ->1ns ->3ns) in consideration of semiosis, that the Object 2ns determines the Sign 1ns for the Interpretent 3ns then?
The first vector applies to semiosis *itself* as a real and continuous process (3ns) such that *individual* signs with their objects and interpretants are indefinite material parts (1ns) until they are deliberately marked off as actual parts (2ns). The second vector applies to the *directionality* of semiosis, always from the object (2ns) through the sign (1ns) toward the interpretant (3ns). In the former case, the categories are assigned directly to the semiosic whole and its parts. In the latter case, the categories are assigned to the three correlates of the genuine triadic relation of representing or (more generally) meditating, *after* they have been prescinded from the real and continuous process in accordance with the first vector. GR: I wonder, how does this match up with Times as a continuum as Peirce's discussion of Time elsewhere in the manuscript takes it up as such? Likewise, the first vector applies to time *itself *as a true continuum (3ns) whose parts are indefinite moments (1ns) until distinct lapses with finite durations are deliberately marked off with durationless instants (2ns); and the second vector applies to the *directionality* of time, always from the past (2ns) through the present (1ns) toward the future (3ns). As you might recall, I apply all six categorical vectors to time in my "Temporal Synechism" paper (https://rdcu.be/b9xVm). CSP: The idea of time must be employed in arriving at the conception of logical consecution; but the idea once obtained, the time-element may be omitted, thus leaving the logical sequence free from time. That done, time appears as an existential analogue of the logical flow. (CP 1.490-1, c. 1896) GR: Here Time would appear to follow the vector of order (1ns -> 2ns -> 3ns). How so? Peirce's point is that temporal and logical flow are *isomorphic*--just as time runs from the past through the present toward the future, reasoning in accordance with a conditional proposition or argumentation runs from the antecedent/premisses through an implication/inference to the consequent/conclusion. GR: Indeed, it was "The Mathematics of Logic, An attempt at developing my categories from within" which first got me thinking about the possibility of there being movement through the categories, that is, categorial vectors, and further research led me to find all six in many, various places throughout Peirce's oeuvre. I have long considered this to be an important and valuable insight with great potential to help Peirce scholars break free of the natural but mistaken tendency to treat the categories that he named 1ns/2ns/3ns as if they designate a *sequence *(vector of order). As I have said before, the categories are instead properly applied to phenomena in accordance with their *different *relations when viewed from *different *perspectives, as demonstrated by my own example of finding all six vectors when analyzing time. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Mon, Oct 13, 2025 at 5:26 PM Gary Richmond <[email protected]> wrote: > Jon, List, > > I apologize for the length of this post and even after paring down some of > the Peirce quotations. > > JAS: "In my view, this vector (3ns → 1ns → 2ns) matches up with not only > the constitution of being in cosmology, but also the nature of semiosis in > general (prescinding individual signs with their objects and interpretants) > and perception in particular (prescinding predicates and hypostasizing some > of them into subjects). Hence, "vector of continuity" would reflect its > applicability across mathematics, phaneroscopy, the normative sciences > (especially semeiotic), and metaphysics." > > > If the vector 3ns ->1ns ->2ns "matches up with. . .the nature of > semiosis," what is that vector's relation to that of the vector of > determination (2ns ->1ns ->3ns) in consideration of semiosis, that the > Object 2ns determines the Sign 1ns for the Interpretent 3ns then? > > The material just before the snippet I quoted (beginning "It will be very > difficult for many minds") is quite interesting especially as the subtitle > of the manuscript, "The Mathematics of Logic" from which the quotation was > taken, is "An attempt at developing my categories from within" so that most > all that is analyzed, even Time, is done so to show some categorial > character or relation. > > So, as you wrote: > > JAS: As he states a few paragraphs later, "temporal succession is a > mirror of, or framework for, logical sequence" (CP 1.496). Here we have a > different categorial vector, that of *determination*--the accomplished > past (2ns) determines the nascent present (1ns) to determine the contingent > future (3ns), just as the object (2ns) determines the sign (1ns) to > determine the interpretant (3ns). > > > I wonder, how does this match up with Times as a continuum as Peirce's > discussion of Time elsewhere in the manuscript takes it up as such? > > CSP: It will be very difficult for many minds--and for the very best and > clearest minds, more difficult than for others--to comprehend the logical > correctness of a view which does not put the assumption of time before > either metaphysics or logic instead of after those kinds of necessity, as > here arranged. (CP 1.490) > > > I must admit that when I first quoted that snippet I was thinking > especially in terms of the *vector of *(Hegelian)* orde*r being the > inverse of the *vector of involution*. > > > In a number places in the manuscript Peirce discusses categorial > *involution* (following the *vector of involution*, 3ns --> 2ns -> > 1ns) as the inverse of Hegel's 'evolution' (alternatively,* dialectical > method* following the *vector of order*, 1ns -> 2ns -> 3ns). (By the way, > in places in "The Mathematics of Logic" Peirce refers to categorial > involution as "analysis," and so at first I was tempted to call it the *vector > of analysis*. But very soon I decided that by 'analysis' what Peirce > meant wasn't analysis more generally speaking but, specifically,the > analysis of *categorial involution*; and so I settled on the *vector or > involution*.) > > It is also important to note that what Peirce refers to as "the method of > dilemma" i.e. *dialectical method*, Hegelian 'evolution' (1ns -> 2ns > ->3ns) is not at all Peirce's understanding of evolution in his -- the > modern -- usual sense (such as when we refer to biological evolution) which > follows the *vector of process* (1ns -> 3ns >- 2ns). > > But continuing, immediately following the quoted snipped above, Peirce > writes: > > But that is an objection, not to this particular item of the development, > but to the general plan of it. To admit the force of the objection and > carry it out to its consequences would simply result in r*eversing the > whole order of development, making it begin with polyads, analyzing these > into triads, and then finding dyads in triads, and monads in dyads **[which, > of course, is exactly involution]**. There is not only nothing erroneous > in such an arrangement, but the conceptions cannot be thoroughly grasped > until it has been carried out*. But this is only one of two sides of the > shield, *both of which must be examined, and which have to be synthesized* in > the really philosophical view. The reason of this is, that although *the > view which takes the triad first is necessary to the understanding of any > given point*, yet it cannot, from the very nature of the case, be carried > out in an entirely thoroughgoing manner. How, for instance, would you > begin? *By taking the triad first. You thus do, in spite of yourself, > introduce the monadic idea of "first" at the very outset.* T*o get at the > idea of a monad, and especially to make it an accurate and clear > conception, it is necessary to begin with the idea of a triad and find the > monad-idea involved in it.* But this is only a scaffolding necessary > during the process of constructing the conception. When the conception has > been constructed, the scaffolding may be removed, and the monad-idea will > be there in all its abstract perfection. *According to the path here > pursued from monad to triad, from monadic triads to triadic triads, etc., > we do not progress by logical involution -- we do not say the monad > involves a dyad - - but we pursue a path of evolution **[Hegelian > dialectic; for Peirce evolution as we generally think of it follows the > vector of process: 1ns -*>* 3ns -*>*2ns]* [. . .] > > > So far Hegel is quite right. But he formulates the general procedure in > too narrow a way, making it use no higher method than dilemma *[i.e., > dialectic]*, instead of giving it an observational essence. [. . .] The > great danger of the evolutionary *[i.e, Hegelian]* procedure lies in > forcing steps that are not inevitable, in consequence of not having a > sufficiently distinct apprehension of the features of the conception in > hand to see what it is that must immediately succeed it. *The idea of > time must be employed in arriving at the conception of logical consecution; > but the idea once obtained, the time-element may be omitted, thus leaving > the logical sequence free from time. That done, time appears as an > existential analogue of the logical flow*. CP 1.490-491 > > Here Time would appear to follow the vector of order (1ns -> 2ns -> 3ns). > Returning > to a consideration of involution, in an earlier comment in the > manuscript Peirce writes: > > The general law of quality [. . .] has *three clauses*, relating > respectively to single qualities, to pairs of qualities, and to triads of > qualities. The first clause is that every quality is perfect and in itself > such as it is. The second more complex law is that two qualities have one > or other of two sorts of relations to one another; namely, they may be, > first, independent of one another, somewhat resembling and somewhat > differing from one another, or secondly, one of them may be merely a > further determination of the other, *this latter being essentially the > first of the pair in the order of evolution, or synthesis **[vector of > order]**, while it is the second of the pair in the order of involution > or analysis **[vector of involution]*. CP 1.484 > > > And a bit later bit later. . . > > The triadic clause of the law of logic recognizes three elements in truth, > the idea, or predicate, the fact or subject, the thought which originally > put them together and recognizes they are together; from whence many things > result, especially a* threefold inferential process which either first > follows the order of involution from living thought or ruling law, and > existential case under the condition of the law to the predication of the > idea of the law in that case **[involution]*; or second, proceeds from > the living law and the inherence of the idea of that law in an existential > case, to the subsumption of that case and to the condition of the law; or > third, proceeds from the subsumption of an existential case under the > condition of a living law, and the inherence of the idea of that law in > that case to the living law itself. Thus the law of logic governs the > relations of different predicates of one subject. CP 1.485 > > > Also: > > But the event may, on the other hand, consist in the coming into existence > of something that did not exist, or the reverse. There is still a > contradiction here; but instead of consisting in the material, or purely > monadic, repugnance of two qualities, it is *an incompatibility between > two forms of triadic relation*, as we shall better understand later [. . > . ]CP 1.493 > > The dyadic requirement of the law of time is that if a subject > existentially receives contrary attributes, of the two contrary states an > existentially determinate *one is first in the existential order of * > *[Hegelian]** evolution and second in the existential order of > involution, while the other is second in the existential order of evolution > and first in the existential order of involution*; and of any two events > whatever, a determinate one is related to the other in this same way > (although the two events are not joined, as the two states are joined in > the event), unless they are independent of one another. . . CP 1.495 > > > So, within "The Mathematics of Logic" Peirce describes several vectors > without naming them as such (they are 'orders of')' and since the Hegelian > order (the *vector of order*) being the inverse of categorial involution > (the vector of involution), it was suggested to me that there might be > relations between the six vectors, something which proved to be the case. > Indeed, I've hypothesized cycles of the six vectors, on which I need to do > more research and well as regards other relations among the vectors. > > Indeed, it was "The Mathematics of Logic, An attempt at developing my > categories from within" which first got me thinking about the possibility > of there being movement through the categories, that is, categorial > vectors, and further research led me to find all six in many, various > places throughout Peirce's oeuvre. > > Best, > > Gary R >
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