Gary F, John S, List, In the second Lecture, Peirce is providing relatively informal explanations of the starting points for the EG. If these explanations were to be cast in a more rigorous way, how might we characterize each of the starting assumptions?
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. 4.396-7. 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. Peirce thinks of the EG as a part of pure mathematics. As such, we should be able to characterize each of the starting points as a definition, postulate or axiom. Peirce provides the following explanations of each: 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 [CP 2.219-26; NEM 2.7]. John S has suggested that the first common notion concerning the character of the sheet of assertion is the only axiom in the system. In common parlance, we sometimes use the word “axiom” to refer to any sort of starting point for mathematical deduction. Peirce points out that this loose way of speaking can be misleading. It wasn’t clear to me whether John was using “axiom” in the broader or the stricter sense. Does the first common notion fit the following criteria? a) it is a self-evident truth; b) the statement of the axiom is superfluous to the conclusiveness of the reasoning; c) it only serves to show a principle involved in the reasoning; d) it is generally a truth of observation, such as the assertion that something is true; e) drawing on the Century Dictionary definition, we could add that an axiom is a matter of common knowledge—such as an item of knowledge drawn from another area of mathematics that is, relatively speaking, beyond doubt. Let us ask the same question about the other common notions. If any do not fit these criteria for a common notion, then are they definitions or postulates? Yours, Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: g...@gnusystems.ca <g...@gnusystems.ca> Sent: Saturday, October 21, 2017 6:18:49 AM To: 'Peirce-L' Subject: RE: [PEIRCE-L] Existence and Reality (was Lowell Lecture 1: overview) Gary R, list, I’ve just caught up with yesterday’s flurry of posts and would like to thank you for this one, along with Jon Alan Schmidt and Jeff Downard for their excellent contributions. (Jeff, I do agree with your revision of my hasty remark on the constitution of mathematical objects.) I don’t really have anything to add to the thread, except to say that we’ll be looking deeper into Peirce’s phenomenological “categories” when we reach the third of the Lowell lectures, and that should help to clarify the concepts of Firstness, Secondness and Thirdness. As Peirce remarked in that lecture, “you must have patience, for long time is required to ripen the fruit” of phenomenological inquiry. In the meantime, though, we’re looking into Lecture 2, where Peirce takes up “the subject of necessary reasoning, mathematical reasoning, with a view to making out what its elementary steps are and how they are put together.” In doing this, he is following up on his promise to show that the “three great classes of argument, Deductions, Inductions, and Abductions … profess to tend toward the truth in very different senses, as we shall see.” I’ve been reading Peirce on this subject for years, but when I get a chance to study, in its original context, a Peirce text that I’m not familiar with, it always challenges and deepens my prior understanding of what Peirce was talking about. That’s why I’m so grateful to the SPIN project for making many of Peirce’s unpublished manuscripts, including those of the Lowell Lectures, available to us all, and providing a platform for open-source scholarship. As I continue to post pieces of Lowell 2, I’ll start including links to the online manuscript pages themselves, so readers can get a better idea of what they look like. And again, my whole transcription of Lowell 1 and 2 are on my website if you want to read them without interruption. Where I find Peirce’s train of thought hard to follow, I’ll post a comment on that section, and I hope others will do the same. Gary f. 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 From: Gary Richmond [mailto:gary.richm...@gmail.com] Sent: 20-Oct-17 17:45 To: Peirce-L <peirce-l@list.iupui.edu> Subject: Re: [PEIRCE-L] Existence and Reality (was Lowell Lecture 1: overview) Gary f, Mike, Jon S, Edwina, John, Jeff, List, Gary f wrote: Gf: I think Jon’s post should clarify what is meant by a “real possibility.” But I’d like to add a point about the “universal categories”: they are not watertight compartments, or separate bins into which phenomena can be sorted. Any given phenomenon, such as an argument or a blueprint, can have its Firstness, its Secondness and its Thirdness. In fact you can’t have Thirdness that doesn’t involve Secondness, or Secondness that doesn’t involve Firstness. I agree that in offering the complete quotation and his succinct and cogent comments, that Jon went very far in clarifying the notion of a "real possibility." And it would seem from his response that Mike is satisfied with that clarification. On the other hand, he seemingly completely disagrees with your take on the "universal categories," Gary, while I tend to strongly agree with you. Indeed, I think that your comments on the categories help clarify especially what they can and cannot be expected to do. Peirce calls them mere hints and suggestions, and I think that is just about right. To make them "compartments" or "bins" has the effect of completely separating them from each other, something which in my opinion one can only do for the purposes of certain kinds of analysis. Peirce will say that in some facet of some phenomenon that 1ns, say, is predominant. While,, as you noted, when there is 3ns present at all (say, in consideration of Reality) that it will involve 2ns, and that 2ns will involve 1ns. One might speak of a quality, a 1ns, say, some particular hue of red, but it is a mere abstraction until it is embodied in say a rose or an apple, and once it is embodied the other categories will come into play. Elucidating the example I offered Mike of a blueprint, you wrote: Gf: A blueprint is a First relative to the universe of real buildings, i.e. it is the mere idea of a building. A physically instantiated blueprint, like a “replica” of an existential “graph,” is a Second in the universe of representations, a token of a type. And it is a Third in its function as an iconic sign interpretable by the builders. So, one might say that all three are involved in considering the categoriality of a blueprint. So, except that in my understanding every Sign in having an Interpretant aspect always involves Thirdness merely because it involves an Interpretant, in the example to follow, a Rheme (something which I know Edwina does not agree with, but which I'd rather not get into a discussion of in this thread), I would tend to agree with her recent comment: ET [A] blueprint could be analyzed as a 'rhematic iconic sinsign' 2/255 "An Iconic Sinsign [e.g., an individual diagram] is an object of experience in so far as some quality if it makes it determine the idea of an object. Being an Icon, and thus a sign by likeness purely, of whatever it may be like, it can only be interpreted as a sign of essence, or Rheme. It will embocy a qualisign".2.255 The nature of a Sign is not its isolate nature [as a blueprint, as a building] but its Relationship as a Sign with other Signs. . . So, while I'm all for considering the categories phenomenologically apart from semiotics, such as analyzing their distinctive characters, or how they interact with each other, if one asks for examples, as Mike did, then one is thrown quasi-necessarily into semiotic considerations. Jon S, who wrote that he would tend to agree with Mike "that, in consideration of the categories one ought not be "looking at 'blueprint' [semiotically but] in terms of the nature of the object [in categorizing] the reality of our real world" added: Jon S: "[It] seems to me that Peirce ultimately maintained a subtle distinction between the phenomenological/phaernoscopic Categories that describe three different kinds of predicates and the metaphysical Universes that contain three different kinds of subjects. Of course, in accordance with his overall architectonic, the latter would in some sense depend on the former." I have tended to call this less "a subtle distinction" than an application of the Categories to Metaphysics. Getting back to Gary f's comments, he concluded: Gf: We certainly can’t define these categories as arguments. An argument is a phenomenon, and so is a process such as an inquiry; both are phenomena in which Thirdness is predominant. But the categories are elements of any and all phenomena that can be “before the mind” (any kind of “mind”) in any way. That includes mathematical and other imaginary objects, which may be intelligible without being perceptible by the senses. Indeed it is only in the mathematical realm that necessary reasoning can be done, because the objects of pure mathematics have no being except what they are defined to have. I think that the last comment just above, that "it is only in the mathematical realm that necessary reasoning can be done," will be important to keep in mind as we explore the Lowell lectures in other threads here, while I would strongly support Jeff's amendment offered today. JD: An amendment might take the following form: the objects of pure mathematics have a character that is determined by the definitions, postulates, common notions and diagrams in which the various conceptions are framed. But this notion that the categories are but hints and suggestions holds in the case of categorial vectors as well. Late in his career, in the Neglected Argument, Peirce comments on the three stages of a complete inquiry. My present abstract will divide itself into three unequal parts. The first shall give the headings of the different steps of every well-conducted and complete inquiry, without noticing possible divergencies from the norm. I shall have to mention some steps which have nothing to do with the Neglected Argument in order to show that they add no jot nor tittle to the truth which is invariably brought just as the Neglected Argument brings it. The second part shall very briefly state, without argument (for which there is no room), just wherein lies the logical validity of the reasoning characteristic of each of the main stages of inquiry. . . [The third places the NA in the context of the three stages.](CP 6.468). Peirce immediately goes on to "give the headings" of "the main stages of inquiry," and they are, in order, Retroduction (Abduction), Deduction, and Induction, and he discusses each of them and their characteristic logical validity in the following couple of pages. I gave these three in my trikonic form (which puts 1st, 1ns, abduction (a hypothesis is formed) |> 2nd, 3ns, deduction (there is an analysis of the implications of the hypothesis were it valid in the interest of constructing tests of it 3rd, 2ns, induction (the actual experiment testing of the hypothesis occurs) But John S found Peirce's tripartite diagram of the "main stages" of an inquiry inadequate and offered his own well-known cyclical diagram as a corrective. While I would agree that in some sense Peirce's simplified outline (and the diagram above) represent a very partial, even skeletal view of inquiry, and that there is certainly a cyclical character to inquiry when one gets much beyond its main stages; and while I would suggest that the above diagram of these stages represent but one analytical categorial pathway through inquiry, and that many others could be posited as concurrently in effect, that this was certainly not what Peirce was attempting to do in offering this simplified model (which, for my purposes, was also intended to show the categorial ordering of the stages). So, to conclude: I am brought back to Gary F's comment about the "universal categories," that "they are not watertight compartments, or separate bins into which phenomena can be sorted," and I would suggest this is so not only for the categories taken separately or as triads, but for the 6 possible tricategorial paths (see the "Mathematics of Logic," CP 1.471) which they may take, that these too should be seen as but "hints" and "suggestions." Best, Gary R [Gary Richmond] Gary Richmond Philosophy and Critical Thinking Communication Studies LaGuardia College of the City University of New York 718 482-5690<tel:(718)%20482-5690>
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