Jeff, Gary F., List: The footnote quoted by Jeff about "the quaternion of metals" is from "Prolegomena" (1906), while the comment quoted by Gary F. about "different dimensions of the logical Universe" is from "Bedrock" (1908). (This is not obvious in the electronic version of CP, where all of the footnotes--both Peirce's own and those provided by the editors--are jumbled together; I had to look at the original published version <http://www.pragmaticism.net/works/csp_ms/P01128.pdf> of "Prolegomena" in order to disentangle them.) Consequently, it seems quite tenuous to link the two concepts, such that "the quaternion of metals" somehow corresponds to "four dimensions--one real and three imaginary."
In fact, Peirce wrote that *all four* Metal tinctures correspond to Actuality, while Color is used for Possibility and Fur is used for Intention (CP 4.554). However, he did not spell out in "Prolegomena" *why *there are *exactly four* tinctures for each Mode of Being. On the other hand, in "Bedrock" he referred to "the different tints representing different kinds of possibility" (CP 4.578). Can we perhaps infer from this that there are *exactly four* different kinds of Possibility, as well as *exactly four *different kinds of Actuality and *exactly four* different kinds of Intention? If so, what are they? As for the "different dimensions of the logical Universe," Peirce explicitly attributed this concept to his former student, O. H. Mitchell; and in two alternative drafts of "Bedrock," he stated the following. CSP: Yet since the Universe, which force[s] upon us all those enduring thoughts that we call truths, makes its power felt in three ways so utterly different that we may well liken them to a set of three mutually perpendicular directions from which any object may be viewed, we must distinguish, Firstly ... the *Universe of Real Capacities*; then, Secondly ... *the Universe of Actual Fact*; and Thirdly ... the *Universe of Tendencies *... I have suggested [in "Prolegomena"] that we resort to the heraldic tinctures; to wit, to *color *for the Universe of Capacities, to *metal *for the Universe of actuality, and to *fur *for the universe of tendencies ... (R 300:72-75[37-40]) CSP: As to the Mitchellian Dimensions of the Universe, it is easy to see that their mutual relations,--imaged by perpendicularities in sets of three,--are relations between different Modalities. (R 300:76[38]) Hence there are *only three* such dimensions, not four; and they correspond *directly *to the three Universes of Capacities, Actualities, and Tendencies--i.e., the three Modes of Tincture, not the four different tinctures *within *any one of them. It thus seems clear to me that in the "Prolegomena" footnote, Peirce intended *only *the first sense of "quaternion," and *not *also the third sense as Jeff conjectured. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Wed, Apr 3, 2019 at 7:30 AM <[email protected]> wrote: > Jeff, list, > > First, a correction: In the “Bedrock” (MS 300), Peirce makes *many* > comments about the “Prolegomena” — indeed, the “Bedrock” was drafted to be > the “next paper” which Peirce mentions at the very end of the “Prolegomena” > — but of course the reverse is impossible, given the order of composition, > which Peirce tells us explicitly at the beginning of the “Bedrock.” Anyway, > the CP editors inserted another footnote into CP 4.553 which *they* took > from MS 300, in which Peirce mentions “different dimensions of the logical > Universe.” For that reason I would answer “Yes” to your question regarding > quaternions, “would it also make sense to say that the representation of > these modes in the gamma system can be interpreted in the third sense of > the term as well, where we employ a mathematical system of numbers that are > understood to be in four dimensions--one real and three imaginary?” > > But having said that much, I’m not prepared to go into further detail > because I am not yet familiar enough with Peirce’s writings on quaternions. > For the time being, then, I’ll have to leave the further exploration of > that to you (and others who may be better prepared than I to do the > exploring). > > I’ve been devoting my free time over the past two days to reading through > Ahti Pietarinen’s full transcription of the talk Peirce gave at the > National Academy of Science meeting in April 1906. I must thank Jon A.S. > for posting the link to that ( here > <https://www.researchgate.net/profile/AHTI_Pietarinen2/publication/271419583_Two_Papers_on_Existential_Graphs_by_Charles_Peirce/links/54c753d30cf289f0ceccf607.pdf> > ), as I think it is at least as informative as the other texts I’ve been > posting here, and anyone who’s been following this thread with interest > should read it, in my opinion. After I’ve finished reading through it > myself, I’ll try to pick out some highlights from it and tie up some “loose > ends” of the thought process Peirce was going through in drafting all of > these documents. After that I’ll be ready to dig deeper into the matter of > quaternions (with your help of course). > > Gary f. > > *From:* Jeffrey Brian Downard <[email protected]> > *Sent:* 2-Apr-19 20:16 > *To:* 'Peirce List' <[email protected]>; [email protected] > *Subject:* Re: [PEIRCE-L] Phaneroscopy and logic > > Gary F, List, > > The texts to which you are drawing our attention are fascinating. Let me > ask a question that we should be able answer in a yes or no way, even if we > don't see all of the implications of the competing answers. > > In "Prolegomena to an Apology to for Pragmaticism," Peirce makes some > comments about "The Bedrock beneath Pragmaticism." The remarks are found in > the CP in footnote 1 to 4.553 (on page 443 of Vol. 4). He says: "It is > chiefly for the sake of these convenient and familiar modes of > representation of Petrosancta, that a modification of heraldic tinctures > has been adopted. Vair and Potent here receive less decorative and > pictorial Symbols. Fer and Plomb are selected to fill out the quaternion of > metals on account of their monosyllabic names." > > When he refers to the "quaternion" of the metals, it is clear that he > means to use the term in the first of the sense that he articulates in the > Century Dictionary, which is something that belongs to a group of four. In > making the point, would it also make sense to say that the representation > of these modes in the gamma system can be interpreted in the third sense of > the term as well, where we employ a mathematical system of numbers that are > understood to be in four dimensions--one real and three imaginary? In a > number of places, both in the earlier writings on the symbolic systems of > logic and the later writings on the existential graphs, Peirce applies the > mathematical system of the quaternions for sake of thinking about the > values of the variables where the values are (1) continuous in their > variation (and not merely binary T or F), and (2) related as part of a > system having more than three dimensions. As such, I think that the answer > may be "yes", that we might interpret the relations between the tinctures > that are used to designate the boundaries around different sheets as > related in manner that is analogous to a four dimensional system of > quaternions. > > The reason I point this out is that it has a direct bearing on the way we > might interpret the improvement offered on the gamma graphs where the > relation between the recto and verso is taken to represent a relation > between existential facts and possibilities of different kinds (depending > on the tint of the outer boundary on the verso side)--where a cut in a page > is conceived to go down through subsequent pages in a book that represents > other kinds of possibilities depending upon the tint of the recto and verso > of each of those pages. > > In the system of the quaternions, the relations between the dimensions is > different in a number of respects from that which is represented in an > algebra of multiple dimensions where all of the dimensions are understood > in terms of rational or real systems of number. One of the big differences > is that in the system of quaternions, the multiplication of values in two > of three imaginary dimensions (say *i* and *j*) takes you directly to a > value in the other dimension (say *k*). > > Why possible basis might I have for suggesting that Peirce may drawing on > the Hamiltonian system of quaternions as a possible model for interpreting > the relations between what is asserted on different pages have different > tinctures? The straightforward reason is that Peirce is well aware that, in > systems of number that are not complex (e.g., the integers, rationals or > reals), there is no closure over the inverse operation of multiplying > something by itself (i.e., raising it to a power). The inverse of this > operation (e.g., taking the square root), requires the use of a system of > complex numbers in order to have closure for the system. One of the things > that the system of gamma graphs allows--which the alpha and beta systems do > not--is the representation of the operation of hypostatic abstraction. In > logical terms, this allows the introduction of objects that are formed on > the basis of abstractions of predicates--such as with a lambda operator in > logics of Church or a Hilbert operator in the systems of Hilbert. As such, > I think that Peirce sees that a modal logic--such as he is exploring in the > gamma graphs--may need something that has the formal properties of the > quaternions as a basis for interpreting the possible values of the > variables. That, at least, is the guess I'd like to explore. > > --Jeff >
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