I, of course, completely agree with this assessment, Frederik, and think your final point is a good one: after all, in the dizzying ascent through the alephs, how can one decide justifiably that aleph-16 is or is not sufficient to answer CSPs metaphysical desideratum?
-----Original Message----- From: Frederik Stjernfelt [mailto:[email protected]] Sent: Monday, December 01, 2014 10:25 AM To: [email protected]; Peirce List Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. Dear Jeff, Michael, Gary R, Lists, Peirce's engagement with continuity is a huge issue (see books by Kelly Parker and Matthew Moore addressing this). One comment - when writing Diagrammatology (2007) I spent considerable time grappling with this issue for the evident reason that many diagrams from continuous representations and, what is more, that continuous interpretations and manipulations of them seem to be the key to the generality of claims inferable from them (a large appendix of the book addresses Peirce's discussions of Cantor etc.). My conclusion, however, was that Ps interest in the continuum was not only mathematically but also metaphysically motivated. Peirce wanted a concept of the continuum sufficiently strong to support his scotist realism - the latter restatable as the claim that the reference of a true universal is a set of cases which exceeds any extension in terms of sets of past, actual, and future single cases. It is an open question, however, if that desideratum may not be satisfied with a weaker notion of the continuum than Peirce's strong wish for a "supermultitudinous" collection transgressing all Cantorian Alephs Best F > It was not my intention to raise hard questions about the later stages in development of Peirce's account of continuity. Rather, I was merely suggesting that our inquiries concerning such difficult conceptions should be guided by a clear understanding of the methods we'll use. As such, I asked: What might we do to arrive at greater clarity...about continuity and the continuum? > > My aim in the email sent as a follow-up was to offer two suggestions by way of an answer about what we could do. > > 1. First, we can model the efforts to gain greater clarity about the conception of continuity on Peirce's earlier efforts to deal with similarly difficult conceptions--such as those of of relative, relation and relationship. (CP, 3.456) Such efforts should start with an attempt to improve our concepts at the level of the first grade of clarity. Towards this end, we can focus on the things we typically say using the conceptions and the things we typically say about the conceptions. So, the points I was making about the way we use the terms 'continuous', 'continuity' and 'continuum' were just initial efforts to make some progress in improving our clarity about the familiar ways we already use the terms. The points might seem trivial, or obvious, but this is where we should start. Once we've arrived at greater clarity at the first grade, we can then turn to the second and then the third grade of clarity--where we will use the pragmatic maxim to move beyond abstract definitions to explanations that can be put to the test. > > 2. As we work our way to better abstract definitions and then towards more fruitful hypotheses concerning the nature of continuity, it will likely be worth our while to draw on the clarifications Peirce has provided of the conceptions of relative, relation and relationship in order to refine our explanations. So, we might ask, what kind of relationship is involved in saying that the experience of things changing from one time to the next is an experience of continuous change? What kind of relation does the memory of something that happened in the past have to that which is taking place in experience right now? Drawing on the clarifications that he has provided, the memory of something happening in the past is one relative, and the experience of what is happening now is another relative, and we can ask what kind of relationship holds between these two relatives. What kinds of relatives are these? Are they monadic, dyadic or triadic relatives? If they are dyadic or triadic, in what ways might they be degenerate or genuine? Considered in separation from one another (by abstracting from the relation that holds between them), the relatives are just an icon or image without "a local habitation and a name." Moving to questions about how we might arrive at more clarity at the second grade, let us ask who kind of relationship (or fundamentum relationis) holds between these relatives? The relation is the relationship considered as something that may be said to be true of one of the objects in the relationship. So, we could ask how some qualities of experience in a larger continuum of possible feelings are located at one time in a larger continuum of times in our experience, and how that relative stands in relation to the relative that is experience when we notice that things have later changed. At this grade of clarity, we are treating the relatives a nominal relatives. How might we make our way to a third and higher grade of clarity? As we examine these experiences, we need to ask what suppositions need to be made about the continuum of feeling and the continuum of time in order to explain the connections that are formed between memory of past and experience of present. At this point in the inquiry, it would probably be helpful to turn from the abstract definitions of relation, relative and relationship we've been drawing on, and move to something that we can manipulate and test--like a diagram. In this way, we might be able to move from those ideas which, from lack of contrast, "ordinary experience fails to bring into prominence." (CP, 3.490) In doing so, we can make observations of our thoughts as we find them in their expressions and form ideas as second intentions. The idea of "incipient cohesiveness," which serves as a germini of continuity, seems to be an idea that we arrive at by reflecting on such observations and asking what we are able to do with the diagrams that embody the ideas. I'll stop here, but I think this might be a fruitful line of inquiry if our aim is to arrive at greater clarity about the conceptions that we are using to formulate and test hypotheses about the conception of continuity. > > Hope this helps to clarify why I asked that question. > > --Jeff > > Jeff Downard > Associate Professor > Department of Philosophy > NAU > (o) 523-8354 > ________________________________________ > From: Michael DeLaurentis [[email protected]] > Sent: Tuesday, November 18, 2014 10:38 AM > To: Jeffrey Brian Downard; 'Peirce List' > Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. > > Jeff I apologize for not having the time to pursue this fully or to reproduce the complete quotes that are relevant, but I dont think the distinction is very elusive. Specifically, I would point you to CSPs 1897 manuscript, generally published under the title On Multitudes [selection 24 in Matthew E Moores fine collection, Philosophy of Mathematics -- Selected Writings], where CSP discusses incipient cohesiveness, a germin of continuity, with specific reference to the first abnumeral multitude, Cantors imperfect continuum, many further imperfect continua away from CSPs true continuum, where the paradox of uncountably many individual [individ-uable] dimensionless points with specifiable loci on a line is resolved via ascent to generality via possibility, and a very different meaning for uncountable the inexhaustible plenitude which sparked this sub-string days ago [leading unfortunately to distracting talk of a continuity somehow not exhibitable in a corresponding continuum]. > > > > -----Original Message----- > From: Jeffrey Brian Downard [ <mailto:[email protected]> mailto:[email protected]] > Sent: Tuesday, November 18, 2014 12:26 AM > To: 'Peirce List' > Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. > > > > Michael, List, > > > > That passage is quite a gem. Having said that, I find the discussion difficult to make out. What might we do to arrive at greater clarity about what the distinction between the conceptions of a perfect and an imperfect continuum consists in and what the basis of the distinction rests on? > > > > --Jeff > > > > > > > > Jeff Downard > > Associate Professor > > Department of Philosophy > > NAU > > (o) 523-8354 > > ________________________________________ > > From: Michael DeLaurentis [[email protected]] > > Sent: Monday, November 17, 2014 9:53 PM > > To: 'Jerry LR Chandler'; 'Peirce List' > > Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. > > > > Lest you lost sight of my sample citations -- and for your further musement -- I part with this very late gem [ca. 1908], where I think the continuity-continuum seamlessness is quite evident continuous, if you will. > > > > In going over the proofs of this paper, written nearly a year ago [1907], I can announce that I have, in the interval, taken a considerable stride toward the solution of the question of continuity, having at length clearly and minutely analyzed my own conception of a perfect continuum as well as that of an imperfect continuum, that is, a continuum having topical singularities, or places of lower dimensionality where it is interrupted or divides . If in an otherwise unoccupied continuum a figure of lower dimensionality be constructed __ such as an oval line on a spheroidal or anchor ring surface __ either that figure is a part of the continuum or it is not. If it is, it is a topical singularity, and according to my concept of continuity, is a breach of continuity. If it is not, it constitutes no objection to my view that all the parts of a perfect continuum have the same dimensionality as the whole. (Strictly, all the material, or actual parts, but I cannot now take the space that minute accuracy would require, which would be many pages.) That being the case, my notion of the essential character of a perfect continuum is the absolute generality with which two rules hold good, first, that every part has parts; and second, that every sufficiently small part has the same mode of immediate connection with others as every other has. (CP 4.642) > > > > With that, I will rest my case. > > > > From: Michael DeLaurentis [ <mailto:[email protected]> mailto:[email protected]] > > Sent: Monday, November 17, 2014 7:56 PM > > To: 'Jerry LR Chandler'; 'Peirce List' > > Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. > > > > Jerry A lot of words, but no explication whatsoever of any distinction CSP makes between continuity and its instantiation in continua. After some irrelevancy re the Continuum Hypothesis, you make some statements about continuity and the philosophy of the natural sciences, continuity and chemistry, and continuity in term of units, individuals and collections -- three sets of comments about continuity, none involving the continuum. Wheres the purported distinction between continuity and the continuum i.e., between the continuity exhibited in continua and any other alleged continuity? Where are CSPs words that indicate such a distinction? What has this claim One notion of continuity was constructed by CSP from units, individuals and collections -- contrasted with this claim A second notion of usage of continuity emerges from Cantor's view of the number line as a closed interval that could be separated into two notions of distance got to do with a purported distinction PEIRCE makes between continuity and the continuum, other than that the former is exhibited in, and only in, continua? And how do these meandering musings confirm that Kirstis intuition was spot on? Nothing you say even begins to addresses this. > > > > I cited the late articles and passages where, to the contrary, CPS, as even Kirsti has now acknowledged, moves seamlessly between the two, in just the manner I have described. You have splashed a bunch of disconnected comments around, but have cited nothing in Peirce to the contrary. > > > > I dont see the point of continuing this thread if youre just going to toss a hodge-podge of unrelated statements around. And with this post, I will therefore close end my responses to these aimless meanderings. > > > > > > From: Jerry LR Chandler [ <mailto:[email protected]> mailto:[email protected]] > > Sent: Monday, November 17, 2014 7:26 PM > > To: Peirce List > > Cc: Michael DeLaurentis > > Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. > > > > List, Michael, John, Kirsti: > > > > On Nov 14, 2014, at 11:41 AM, Michael DeLaurentis wrote: > > > > Jerry All due respect, but my post concerned the distinction Kirsti claimed to find, not anything in your post. So I dont see the relevance.. > > > > The immediate relevance of my post is that this is a listserve for CSP writings and the recent correspondence relate to his writings, although I do not think that is what was of concern to you. > > > > Your post leave me puzzled about your penultimate post which in turn was puzzling, so I reviewed the tread from its beginning and read more widely. > > > > The immediate motivations for my contributions and, as I understand it, yours also, was the issue raised by Kirsti with respect to the possible distinction the continuum and continuity in philosophy, mathematics and CSP's writings. > > > > Your post (Nov 12) expresses this perspective: > > Continuity is simply the unique quality which continua, and only continua, exhibit. > > > > Does this assertion close the philosophical issue that Kirsti raised? > > > > In response to John Deely's questions (post 18 in the listing) "Deely, John N." [[email protected]< <mailto:[email protected]> mailto:[email protected]>] kirjoitti: > > Kirsti, would you mind clarifying for me, if possible (but not necessarily) with some specific ref. to a Peirce text(s), your remark re the difference between "continuity" and "continuum": > > > > I believe that it is relevant to cite the specific texts 4.172-176 from CSP in part of the answer to John's and Kristi's questions. > > These paragraphs brought to mind the Cantor's famous "Continuum Hypothesis" . A couple of citation from the web place the concept of the Continuum is a completely different context that that of mere continuity. > > > > [Introduction. Arguably the most famous formally unsolvable problem of mathematics is Hilbert's first problem: Cantor's Continuum Hypothesis: > > {The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number< <http://mathworld.wolfram.com/CardinalNumber.html> http://mathworld.wolfram.com/CardinalNumber.html> between that of the "small" infinite set of integers< <http://mathworld.wolfram.com/Integer.html> http://mathworld.wolfram.com/Integer.html> [aleph_0] and the "large" infinite set of real numbers< <http://mathworld.wolfram.com/RealNumber.html> http://mathworld.wolfram.com/RealNumber.html> (the "continuum< <http://mathworld.wolfram.com/Continuum.html> http://mathworld.wolfram.com/Continuum.html>"). Symbolically, the continuum hypothesis is that [aleph_1=c] . Problem 1a of Hilbert's problems< <http://mathworld.wolfram.com/HilbertsProblems.html> http://mathworld.wolfram.com/HilbertsProblems.html> asks if the continuum hypothesis is true.] > > > > Another aspect of this issue was how did CSP relate his views on continuity to the philosophy of the natural sciences? > > And these to synechism? > > 4.584 (1906) It is that synthesis of tychism and of pragmatism for which I long ago proposed the name, Synechism > > > > Yet, with respect to chemistry and continuity, he writes CP1.62 (1896?) Now it enters into every fundamental and exact law of physics or of psychics that is known. The few laws of chemistry which do not involve continuity seem for the most part to be very roughly true. It seems not unlikely that if the veritable laws were known continuity would be found to be involved in them > > > > This is to be contrasted with his statements in 4. 173 where he justifies the origin of continuity in term of units, individuals and collections, strongly implies a consistency with the legisigns of chemistry with atoms as units, individuals as proper names of elements and collections becoming continuous. > > > > Thus, my conclusion from these readings is that Kristi's intuition was spot on. > > One notion of continuity was constructed by CSP from units, individuals and collections. > > A second notion of usage of continuity emerges from Cantor's view of the number line as a closed interval that could be separated into two notions of distance, as shown in his well know "removal of the middle third" argument to construct infinite numbers of continuous closed intervals from a line of UNIT length. The "Continuum Hypothesis" is a proposition about Cantor's mathematical philosophy. It is not an extension of CSP's notion of continuity. > > > > On another topic, I think it is important to support Stefan's quote of CP 5.131: > > "Man makes the word, and the word means nothing which the man has not made it mean, and that only to some man. But since man can think only by means of words or other external symbols, these might turn round and say: You mean nothing which we have not taught you, and then only so far as you address some word as the interpretant of your thought. In fact, therefore, men and words reciprocally educate each other; each increase of a mans information involves and is involved by, a corresponding increase of a words information." > > > > I was not aware of this quote, but have had a similar thought in mind for decades from my sensory experiences in the world. The observation that meaning is individualized is true for all individuals as a consequence of their antecedent sensory experiences. It is also true of language usage among disciplines. It is particularly important for those who love knowledge. > > > > Cheers > > > > Jerry > > > > > > > > > > > > > > > > > > From: Jerry LR Chandler [ <mailto:[email protected]> mailto:[email protected]] > > Sent: Friday, November 14, 2014 12:33 PM > > To: Peirce List > > Cc: Michael DeLaurentis; John N. Deely; Määttänen Kirsti > > Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. > > > > List, Michael, Kirsti, John: > > On Nov 12, 2014, at 11:47 AM, Michael DeLaurentis wrote: > > I dont find any such distinction, implicit or explicit, in Peirces late writings. > > Motivated by your assertions, I re-read 4.172 and later paragraphs, searching for distinctions between CSP logic and set theory logic. > > In contrast to your assertion, I certainly find numerous critical philosophic distinctions between CSP logic and Cantorian/Russellian logic with respect to inquiry into the mathematics/logic of the continuum. > > Although a large number of texts could be cited, availability of time and energy restrict my rhetoric principally to 4.172 to 4.176. > > 1. 4.173 introduces with the notion of a collection. > > A collection is a consequence of "bring or gather together", parts of a whole. > > CSP bases his notion of relation on collections as parts of a whole. It requires activity to bring together a collection. > > Thus, CSP is grounding his argument, among other mathematical concepts, on the theory of numbers, the collectability of numbers, and the antecedent parts being brought together to construct a whole. > > This is clearly distinct from Cantor / Russell views which pre-supposes a geometric line. > > 2. 4.174 (and 4.172) introduces the notion of a relative of a part versus the relative of a whole, drawing on the statistical example in 4.172 and the concept of a unit of a partition of a role of a pair of dice. Each role of the pair of die generates a relative value among all possible roles of the pair of six-sided die, exactly 36. > > This is clearly distinct from Cantor / Russell views. > > 3. 4.175 "But when the units lose there individual identity because the collection exceeds every positive existence of the universe, the word multitude ceases to be applicable. I will take the word multiplicity to mean the greatness of any collection discrete or continuous." > > I infer from this, in light of 4.172-175, that individual identity is related to parts of a whole such that parts, as units, can be collected into whole, generating the NOUN, collection. The "bringing together" of a collection is of the nature of a sublation. The quality of the collection, is, presumable for CSP, a matter of sensory experience, as one perceives from the usage of the term "because" in this sentence, inferring causality. > > (And qualities are an aspect of sensory experiences, are they not?) This is clearly distinct from Cantor / Russell views of memberships and classes. > > Yes, set theory, as a dominant force in modern mathematics, has ignored the logical basis of CSP notion of multitude and his terminology for distinguishing between parts and wholes, points and lines, and sensory experiences. > > But, CSPs philosophy expressed in 4.172-4.175 is consistent with many aspects of chemical logic; modern mathematics is not consistent with chemical logic for very specific reasons of the non-transitivity of the mathematics of chemical sublations of individual identities. Non-transitivity is illustrated, for example, by the handedness of chemical isomers.) I conclude that although many many aspects of CSP logic and set theory logic are consistent with one another, the distinction between them (modes of constructions) at the rhetorical and semantic levels differ in mathematically profound ways. > > The basic conundrum of the nature of distinction between discrete and continuous mathematics remains alive and open. Indeed, a very active subfield of mathematics is the Brouwer School of intuitionism. > > ( <http://en.wikipedia.org/wiki/Intuitionistic_logic> http://en.wikipedia.org/wiki/Intuitionistic_logic ) Parenthetically (or perhaps metaphorically) I conclude that studying CSP texts without an in-depth knowledge of the state of the science in the 2nd half of the 19 Th Century is like attempting to solve a crossword puzzle with only the superficial "across" clues. The depth of his thought corresponds with knowledge of mathematics and the natural sciences and the natural propositions in his time, that is, the "down" clues. > > Extending the metaphor, the sensory experiences of the American cultural milieu of the late 19 Th Century are interwoven into the very fabric of CSP's text. > > Cheers, Jerry > > (BTW, Thanks to Gary F. for suggesting a puzzle analogy for hermeneutics.) > > > > > > > > > > > > > > No virus found in this message. > > Checked by AVG - <http://www.avg.com%3chttp:/www.avg.com%3chttp:/www.avg.com%3chttp:/www.avg. com> www.avg.com<http://www.avg.com<http://www.avg.com%3chttp:/www.avg.com>> > > Version: 2015.0.5577 / Virus Database: 4213/8570 - Release Date: 11/14/14 > > > > ----------------------------- > > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. 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