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
----------------------------- 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] . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
