Gary F, List,
Gary F: In Lowell 2, though, Peirce makes no mention of axioms or postulates; instead, he derives all his “permissions” from definitions, making use of both “branches.” And that comes later in the lecture. Jeff: I'll agree that he draws on the definitions when setting forth the permissions, but it is not obvious that the latter are "derived" in the ordinary sense of that term from the definitions. In the Syllabus of 1903, for example, Peirce articulates the conventions for the EG first and then moves to the rules of transformation. In the explanation of the rules of transformation, he spells out a set of definitions that hold "regardless of their interpretation." With those definitions in hand, he then articulates the "code of permissions (CP 4.394-17) How does this terminology of conventions, definitions and permissions compare to the more typical mathematical terminology of definitions, postulates and common notions (or axioms)? It seems clear to me that the permissions are functioning as a set of precepts. That is, they tell us what we are permitted to do. This certainly fits with Peirce's account of a mathematical postulate. Here is an observation about Peirce's use of definitions and postulates that might be worth sharing. One thing that strikes me as odd about the discussion of topology and projective geometry in the NEM is that most of the key ideas are introduced as definitions. Just where I expect him to set forth a set of postulates, he sticks with definitions. Then, late in the game, he introduces some postulates when there is no obvious reason why he might not have treated them as definitions. What is Peirce doing in the NEM by relying so heavily on definitions? My hunch is that his aim in the NEM is to introduce these mathematical ideas to students. Peirce sees that, for pedagogical purposes, he can take what has been postulated by others in more original research oriented works and, given the aims of the NEM, he can convert the ideas into definitions because the meanings of the conceptions was already well established when he was writing the textbook. It is only when he is departing from the usual way of explaining things that he needs to introduce postulates of his own. Peirce might, at times, resort to similar strategies when introducing and explaining conceptions in his different essays and lectures on the EG. --Jeff Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: [email protected] <[email protected]> Sent: Sunday, October 22, 2017 5:40:17 AM To: [email protected] Subject: RE: [PEIRCE-L] Lowell Lecture 2.3 Jeff, As we’ll see later in Lowell 2, Peirce does work toward more formal definitions of the EG conventions in that lecture. But he starts “from scratch,” appealing to the imagination of his audience to follow what he’s doing and thus get a sense of what these diagrams are supposed to represent (“the course of thought.”) It is interesting to compare what he says in Lowell 2 with his more detailed presentations of EGs, several of which are in CP 4. The quote you supply here is from “New Elements” (EP2:302 as well as NEM), which is closely related to the Lowell Lectures in several ways. In Lowell 2, though, Peirce makes no mention of axioms or postulates; instead, he derives all his “permissions” from definitions, making use of both “branches.” And that comes later in the lecture. Don Roberts’ book is a valuable resource, indispensable really, for studying EGs, but it’s eclectic in its use of sources, and I think it’s equally valuable to follow Peirce’s lead as he introduces these concepts to an ‘innocent’ audience, assuming as little as possible about their prior knowledge of logic and mathematics. From the manuscripts I get the impression that Peirce is thinking through (or rethinking) these concepts as he writes, and (for me anyway) this brings EGs to life in a way that textbook summaries rarely do. I find it rewarding to follow the course of Peirce’s thought instead of trying to summarize it all the time. By the way, his use of the word “word” to explain the graph/replica distinction (i.e. the type/token distinction) reappears in the 1906 “Prolegomena” (CP 4.537). Gary f. From: Jeffrey Brian Downard [mailto:[email protected]] Sent: 22-Oct-17 01:42 To: [email protected] Subject: Re: [PEIRCE-L] Lowell Lecture 2.3 Gary F, John S, List, In this second Lecture, Peirce is providing informal explanations of the starting points for setting up the EG. If these explanations were to be cast in a more rigorous way, how might we characterize each of these assumptions or conventions? In his monograph, Don Roberts characterizes the first three conventions in the following way: Conventions CI. The sheet of assertion in all of its parts is a graph. 4.396, 397. C2. Whatever is scribed on the sheet of assertion is asserted to be true of the universe represented by that sheet. 4.397. C3. Graphs scribed on different parts of the sheet of assertion are all asserted to be true. 4.433. The logical system is being developed as a part of pure mathematics. As such, we should be able to characterize each as a definition, postulate or axiom. Peirce provides the following [CP 2.219-26]. 1. A definition is the logical analysis of a predicate in general terms. It has two branches, the one asserting that the definitum is applicable to whatever there may be to which the definition is applicable, the other (which ordinarily has several clauses), that the definition is applicable to whatever there may be to which the definitum is applicable. A definition does not assert that anything exists. 2. A postulate is an initial hypothesis in general terms. It may be arbitrarily assumed provided that (the definitions being accepted) it does not conflict with any principle of substantive possibility or with any already adopted postulate. By a principle of substantive possibility, I mean, for example, that it would not be admissible to postulate that there was no relation whatever between two points, or to lay down the proposition that nothing whatever shall be true without exception. For though what this means involves no contradiction it is in contradiction with the fact that it is itself asserted. 3. An axiom is a self-evident truth, the statement of which is superfluous to the conclusiveness of the reasoning, and which only serves to show a principle involved in the reasoning. It is generally a truth of observation, such as the assertion that something is true. 4. A diagram is an icon or schematic image embodying the meaning of a general predicate; and from the observation of this icon we are supposed to construct a new general predicate [NEM 2.7]. 1. 2. Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: [email protected]<mailto:[email protected]> <[email protected]<mailto:[email protected]>> Sent: Saturday, October 21, 2017 6:45:12 AM To: 'Peirce-L' Subject: [PEIRCE-L] Lowell Lecture 2.3 Continuing from Lowell 2.2: …The board itself is a graph, since it represents the universe as consisting of single imaginary things. The board and what is written on it together make up another graph. It is quite important, however, to distinguish between a graph and a graph replica. Suppose an editor writes to me and asks for an article of 4000 words. You know what he means by words in that case. In what I write the single word “the” may occur twenty times on every page. Every time it will count as a separate word. Yet in another sense, it is the same word. In this latter sense, the word the consists in the sum total of general conditions to which ink-marks or voice-sounds must conform in order to be understood in a certain way. In this sense the word the is, strictly speaking, never written; but what is written conforms to it, or, as we say, embodies it. In the other sense a written word is written once and only once, since every act of writing makes a new word. Now when I speak of a graph, I mean the general type of whatever means the same thing and expresses that meaning in the same way so far as the conventions of this system take cognizance of the ways; while that which is scribed once and only once and embodies the graph, I call a graph-replica. For brevity, however, I speak of “scribing a graph” just as we speak of writing the word the. The phrase may be defended as employing the word “scribe” in a special sense. I will now put an additional replica upon the board, or the sheet of assertion [cid:[email protected]] Let us agree to understand that each of those replicas has the same meaning as if it stood alone; and that it shall be the same with any other two replicas on the sheet, so that I now assert not only that there is a ripe pear, but also that it rains. By calling this system a system of existential graphs, my meaning is that two graphs at different points of the board, whether far or near, are both asserted, each just as much as if the other were not there. Manuscript and transcription: https://fromthepage.com/jeffdown1/c-s-peirce-manuscripts/ms-455-456-1903-lowell-lecture-ii/display/13600 7 (C. S. Peirce Manuscripts, MS 455-456 (1903) - Lowell Lecture II) | FromThePage<https://fromthepage.com/jeffdown1/c-s-peirce-manuscripts/ms-455-456-1903-lowell-lecture-ii/display/13600> fromthepage.com 7 (C. S. Peirce Manuscripts, MS 455-456 (1903) - Lowell Lecture II) - page overview. 5 since existential graphs will be the only ones dealt with, I shall call it a graph, simply. For example, what I have just written scribed is ... http://gnusystems.ca/Lowell2.htm<http://gnusystems.ca/Lowells.htm> }{ Peirce’s Lowell Lectures of 1903 https://fromthepage.com/jeffdown1/c-s-peirce-manuscripts/ms-455-456-1903-lowell-lecture-ii
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