Jeff, some responses inserted .
Gary f. From: Jeffrey Brian Downard <jeffrey.down...@nau.edu> Sent: 15-May-18 12:54 To: g...@gnusystems.ca Subject: [PEIRCE-L] Lowell Lecture 5 Hi Gary F, List, Thanks for your initial hints about Lecture 5. Up to this point in my reading and re-reading of the Lowell Lectures, I've had a feeling that some parts of the lectures read something more like "ideas on detached topics" and less like a clear path of inquiry. In particular, it was difficult for me to see the connections between the discussion of self control in lecture 1, the discussion of the first two parts of the existential graphs, alpha and beta, and then an interlude on phenomenology and semiotics before turning to the gamma graphs--only to turn to a discussion of the mathematics of multitudes. To be honest, I found it rather bewildering. [gf: I think that for Peirce, these subjects are all so intertwined in his system that he finds it difficult to talk about any one of them without at least mentioning the others. That doesn't bother me as a reader, but then I think readers have the same problem with my book that you have with these lectures, so maybe I think like Peirce in that respect. (Though very differently in other respects!) I quoted Wittgenstein in my book saying that "the very nature of the investigation . compels us to travel over a wide field of thought criss-cross in every direction." Maybe that applies to Peirce too.] The table of contents that you've provided at the start of your transcription of lecture 1 is helping me see the forest for the trees. The lectures are organized in the following way: 1. What Makes a Reasoning Sound? 2. Existential Graphs, Alpha and Beta 3. General Explanations, Phenomenology and Speculative Grammar 4. Existential graphs, Gamma Part 5. Multitude 6. Chance 7. Induction 8. Abduction [Gf: That list is actually from one of the pages of MS 470, i.e. Lecture 5, which suggests that it was only halfway through the drafting process that Peirce decided on this ordering of his material.] The topics of the last two chapters helps to clarify the purpose of starting with the question about what makes a reasoning sound at the beginning. His main aim, I am supposing, is to use the EG to put ourselves in a position to better understand what makes inductive and abductive reasoning sound. [gf: That could be, but I haven't read 7 or 8 yet. We'll see. But the same should apply to multitude, which is one way of using the logic of mathematics to develop the related concepts of infinity, continuity, and generality. It all comes down to the nature of argument as the "complete sign" par excellence.] If I take chapter 1 and chapters 7 and 8 as bookends, it helps me get a better grasp of what Peirce hopes to accomplish in lectures 2-6. If the goal is to explain--better than before--what makes inductive and abductive reasoning sound, then we can see some strands of thought that are being wound together into a cable that represents a fairly clear path of development. That is, Peirce is offering a relatively simple and abbreviated explanation of the alpha system of diagrammatic logic, and how the beta system is a development of the ideas in the alpha system--and he is explaining in the third chapter how the formulation of those formal systems is informed and guided by the observations being analyzed in the phenomenology and the logical problems and questions that he is trying to address in the semiotics. The general strategy is--as with any area of science--to make the theory of semiotics more rigorous by making is more mathematical. In this case, he is developing and drawing on systems of formal logic that are primarily topological and diagrammatic in character and not those that are algebraic and symbolic. [gf: I think I see what you mean about making it more rigorous, but I'm sure Peirce would not want to sacrifice experiential factuality for the sake of rigor. Logic/semiotic has to be a positive science, in other words.] If we look at what he is doing in lectures 2-6 from the vantage point of a larger historical perspective, we can see that the attempts to develop and employ topological systems of formal logic represents a rather important development in the history of modern philosophy. Let us recall that one of Boole's main aims in his formal logic was to draw on algebraic approaches for the sake of analyzing reasoning involving probabilities. This was a particularly pressing task because mathematical reasoning about probabilities and the application of those ideas in statistics were giving rise to a number of philosophical problems and questions that required better logical tools. Having participated in the development of these systems of logic, Peirce could see that the algebraic systems of logic of Boole, DeMorgan, Schroder, etc. were subject to a number of limitations--especially when it comes to analyzing the more basic mathematical conceptions involved in reasoning about probabilities and the application of those ideas in statistics. As such, we can understand Peirce's larger strategy in lectures 2-6 as an attempt to improve on what Boole sought to accomplish--but using logical tools that are better suited for the job at hand, which is to analyze the conceptions, propositions and arguments more minutely and with greater rigor. The central conception Peirce is focusing on in lecture 5 is that of a multitude, and he is attempting to understand how that mathematical conception works within the context of different systems of numbers (e.g., the counting numbers, the integers, the rationals, etc.). He is focusing on this conception because he sees that logical confusions were arising in the work of the mathematicians in the theory of probability--and a number of these confusions could be traced based to longstanding issues in number theory. Assuming that this overview of the argumentative strategy is on track, then I am keen to better understand how the arguments in lecture 5 might be reconstructed, and especially how the gamma graphs are being used as tools in those inquiries. In particular, I'd like to gain more clarity about how Peirce is using the newly coined terms "sam" and "gath" to distinguish, as Gary F says, between the essential and the existential aspects of collections in the context of his analysis of mathematical collections and multitudes. With that much said, let me pose a question for the sake of trying to start a discussion of lecture 5: how does the use of the terms "sam" and "gath" in his analyses and arguments help us better understand the character of really important forms of mathematical inference, such as the Syllogism of Transposed Quantity and the Fermatian form of inference? [gf: I don't have a good answer to that, but I notice that early on in the lecture, Peirce says that "the doctrine of multitude is nothing but a special application of the doctrine of ordinal numbers. But the special objects of its series have a special character which permits them to be studied from a special point of view; and that point of view is a logical point of view. It is not the pure mathematical forms that we study in the doctrine of multitude. It is on the contrary a branch of logic which, like all logic, is directly dependent upon mathematics." Now, to me, the sam/gath distinction is a phenomenological one, and that brings the "doctrine of multitude" out of the realm of pure mathematics and into the realm of positive science, specifically logic. But that's all I'm prepared to say at the moment.] --Jeff http://gnusystems.ca/Lowell5.htm }{ Peirce's Lowell Lectures of 1903
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