Gary F, List,
Thank you for pointing out the source of the footnote I quoted. For those interested in seeing how Peirce applies ideas drawn from the mathematics of quaternions to algebraic systems of logic, see section 3 in "The Simplest Mathematics". (CP 4.250-306). At 4.258, Peirce provides an interpretation of truth and falsity in a system of dichotomic logic. He offers a diagram in which the values of what is true and false are arranged along horizontal lines, with an origin in the middle. Peirce says that a value, x, which is in the upper right quadrant of truth can be considered to have a transformation by rotation into the lower left quadrant of falsity. Here is a modification of his diagram showing the rotation. [cid:1aebdaf2-178f-4d5a-ad5a-a95117375331] This type of rotation is precisely the kind of transformation that we find in the complex plane. After developing the system of logic and providing some definitions of key relations, including the logic connectives along with the relations of quantity, aggregation and composition, he supplies an interpretation of how the values of the variables in such a system will be transformed under different functions. In order to explain how logical multiplication (functional and relative) works, he interprets the system in terms of a multiplication table of the quaternions. In the next section, Peirce is using a set of chemical diagrams as a model for explaining how he intends to develop a trichotomic system of logic. In the runup to the discussion, he points out in the first paragraph of the section that he has already, in the previous discussion of logical multiplication, being employing a conception of multiplication that is "purely triadic". One reason the account of operational multiplication is purely triadic, I take it, is that it is being interpreted in terms of a multidimensional system of quaternions. I would be interested to see how the account of relations that he develops on the chemical model in the 1902 explanation of the trichotomic system of algebraic logic compares to what he is doing in 1906 development of the existential graphs. In particular, I would like to better understand how the account of multiplication that is developed in sections 3 and 4 of "The Simplest Mathematics"--which is explained in terms of the composition of relations--compares to the account logical multiplication that is developed in the 1906 account of the gamma graphs in the "Apology" and "Bedrock" essays. If anyone has thoughts about this comparison, I would be interested in hearing suggestions about how multiplication is being modeled in this modal system--topologically conceived. Yours, Jeff P.S. For those who might like to see a gentle introduction to the complex plane (including a very accessible illustration of the Riemann surface) and the system of quaternions, I recommend the following videos which provide very helpful moving diagrams. 1. See the 12 videos in the series "Imaginary numbers are real: " https://www.youtube.com/watch?v=T647CGsuOVU [http://img.youtube.com/vi/T647CGsuOVU/0.jpg]<https://www.youtube.com/watch?v=T647CGsuOVU> Imaginary Numbers Are Real [Part 1: Introduction] - YouTube<https://www.youtube.com/watch?v=T647CGsuOVU> www.youtube.com For early access to new videos and other perks: https://www.patreon.com/welchlabs More information and resources: http://www.welchlabs.com Imaginary numbers ... 2. For an explanation and visual illustration of the quaternions, see: <https://www.youtube.com/watch?v=d4EgbgTm0B> https://www.youtube.com/watch?v=d4EgbgTm0Bg [http://img.youtube.com/vi/d4EgbgTm0Bg/0.jpg]<https://www.youtube.com/watch?v=d4EgbgTm0Bg> What are quaternions, and how do you visualize them? A story of four dimensions. - YouTube<https://www.youtube.com/watch?v=d4EgbgTm0Bg> www.youtube.com How to think about this 4d number system in our 3d space. Home page: https://www.3blue1brown.com Thanks to supporters: http://3b1b.co/quaternion-thanks Quant... Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: g...@gnusystems.ca <g...@gnusystems.ca> Sent: Wednesday, April 3, 2019 5:29:43 AM To: 'Peirce List' Subject: RE: [PEIRCE-L] Phaneroscopy and logic 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 <jeffrey.down...@nau.edu> Sent: 2-Apr-19 20:16 To: 'Peirce List' <peirce-l@list.iupui.edu>; g...@gnusystems.ca 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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