Just to give one more example of the integral inseparable relation between continuity [and the continuous] and continua, heres CSP from a 1908 Monist supp [after laying out Cantors aleph-1 notion]: I proceed to define what I think it is that constitutes a true continuum [CSP emphasis], or continuous object [my emphasis].
Again I ask: where is the language IN PEIRCE that separates these patently integrated concepts? From: Michael DeLaurentis [mailto:[email protected]] Sent: Tuesday, November 18, 2014 9:33 AM To: 'Jeffrey Brian Downard'; 'Peirce-L' Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. Jeff -- If I may briefly intercede, a few too-brief comments since my time is very limited. (1) This is indeed a very different discussion from the one I was trying to have with Jerry, with a quite different set of questions. (2) I think I made pretty clear through implication, if not expressly (though I think I was sufficiently express), that, as you state in very similar terms below, the new term [continuum is used] to refer the thing that has the property of being continuous. Thats an elementary point of diction, whatever else it is. Its not just that the function of the monadic predicate: -is continuous tend[s] to [function] like an adjective: continuous IS an adjective. So these are at the first level merely usage points. (3) One of my points in the prior exchanges was that Peirce was reconsidering his conception of continuity and the continuum [always used in his sense quite distinctly from the common usage of those terms by mathematicians, though he often used them as starting points for contrast] in very fundamental ways, reflected in conceptions like the imperfect continuum and incipient cohesion [the former obviously reflecting the latter in specific instantiations, thus, again, illustrating the difference in usage but integral connection in concept]. I would say that what you are addressing in late CSP, and what CSP was late addressing, is grades of continuity, again reflected in corresponding grades of continua. But the key point which does carry over from the prior discussion is this pairing: interleaving more and less perfect/complete examples/degrees of continuity necessarily involves a parallel refinement in the continua instantiated. That merely illustrates that the conceptions are integrally I stress integrally and necessarily related. The purported separation of these conceptions, other than in usage and diction, is simply not to be found in CSP anyway. [Im not sure I even know what such a separation would mean: as with differentiation and continuity, I assume one would have to illustrate this purported separation with an example of a grade of continuity that reflects a connectivity different from that reflected in what is an independently correlated continuum which seems to me utter nonsense.] And your exposition below does not indicate, or purport to indicate, otherwise. So, yes, I indeed think there is little I would emphatically say NO difference between the adjectival use of continuous, and the conception of continuity that we arrive at by hypostatic abstraction other than their diction. (4) So to answer your simple-minded question [--] is an imperfect continuum continuous, or is it not continuous? I say, with intentionally boring obviousness, An imperfect continuum [necessarily] exhibits imperfect continuity, just as a true continuum exhibits genuine continuity; and refinement into finer grades of continuity will be reflected in corresponding grades of continua and in each such case, this is necessarily so. Its all just usage -- the conceptions of continuous, continuity, and continuum at each grade of however many grades necessarily go hand in hand. I cannot imagine Peirce taking exception to these, indeed, simple-minded statements, all masking the genuinely substantive point that his late reconsideration of continuity and related continua reflect a new uncertainty about the nature of continuity and continua themselves, even in the perfect sense he had propounded in prior years. Im not sure I see quite where CSP imagined he was heading with these hypothetic refinements, which, to me, seem only to classify and label stages of crowding, degrees of connectedness, en route to his genuine continuum. So, yes, an imperfect continuum is imperfectly continuous; and no, an imperfect continuum is not perfectly continuous; and we all knew this all the time. And, to return to your first comment below, these elementary, even simple-minded, considerations contribute only a first, but critically important, step in the full understanding and implications of synechism. -----Original Message----- From: Jeffrey Brian Downard [mailto:[email protected]] Sent: Tuesday, November 18, 2014 2:29 AM To: Peirce-L Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. Gary R., List, My aim in raising a different sort of question was to move the recent discussion about the conceptions of continuity and the continuum in a different direction. The passage Michael copied into the email is one that I've spent some time trying to make out--and I find that it is not a simple point that Peirce is making. As such, I was trying to return to the starting point in our recent discussion of continuity--which began with a question about the relationship between generality and continuity and the motivations behind Peirce's synechism. Gary R., would you clarify your last remark, where you say: "I don't see that it offers any particular difficulties in answering (Peirce is quite explicit), and Michael in passing did offer a remark clearly articulating this distinction (I didn't quickly find it)." My initial hunch reading this line was that you are referring in both clauses to Michael's response to Jerry. Is this hunch on the right track? Or, are you responding to me by saying that Peirce is being quite explicit about what he means in CP 4.642--and that it shouldn't take much work on our part to clarify what is being said? If you are saying that latter, then I disagree. At the risk of defeating my own aim in trying to move the discussion in a different--and hopefully more productive--direction, let me consider the question: "How are the conceptions of continuity, continuous and continuum connected to each other?" Let me say that I am working on the assumption that the root conception we use in talking about things that appear to be connected in the way that our experience of something like time is connected is to say, "time is continuous." What is the function of the monadic predicate: "-is continuous"? I tend to think that it often functions like an adjective. That is, we say about our common experience of moving from what we were thinking a few seconds ago to what we are thinking now is "of a continuous series of changes." The term "continuity" is the result of taking this adjective and turning it into an abstract noun. As such, it is the result of hypostatic abstraction. How is the conception of the continuum connected to the conception of continuity? One place where we use the conception of "the continuum" is in mathematics. If we move from one area of mathematical inquiry to another, we come across a number of different examples of things that seem--at least to some mathematicians--to be cases of continua. For example: there is the line in geometry; and the field of real numbers in analysis; and the power set of the real numbers in set theory. After considering a number of such examples, the mathematicians ask: do these different cases that appear to involve continuity all have something in common? That is, can we arrive at a conception of "The continuum" that is at the root of the plurality of examples of things that appear to be continuous? That, I take it, is the kind of thing Gödel's was talking about when he posed the question: What is Cantors continuum problem? (1947) Peirce also uses the term 'continuum' in philosophy to talk about things that appear to be continuous. He is following the lead of other philosophers who, in the mid-1700's, appear to have coined the term by taking the Latin "continuus" and the English "continuous" and then used the new term to refer the thing that has the property of being continuous. In this way, Peirce asks questions about the nature of the continuum of feelings, and the continuum of space, etc. We might, if we were thinking like mathematicians, ask if these different examples all have something in common. Is there a philosophical conception of "The continuum" that is at the root of these different continua? It might appear that there is little difference between the adjectival use of continuous, and the conception of continuity that we arrive at by hypostatic abstraction, and the thing we refer to by calling it a continuum. But, we should keep in mind that Peirce labored over the differences between the conceptions of relative, relation and relationship. I think we could probably draw on the points he makes about the conceptions of relative, relation and relationship in order to get clearer about the conceptions of continuous, continuity and continuum. But that is just my hunch. Ill leave that task to another time. With this much having been said, let me ask a simple-minded question about what Peirce seems to be saying in CP 4.642. In this passage, he makes a distinction between a perfect continuum and an imperfect continuum. Elsewhere, he offers the following definition about the construction of Cantor's set: but now I define a pseudo-continuum as that which modern writers on the theory of functions call a continuum. But this is fully represented by, and according to G. Cantor stands in one-to-one correspondence with, the totality of real values, rational and irrational; and these are iconized, in their turn, according to these writers [by the] entire body of decimal expressions carried out to the right to all finite powers of 1/10 without going on to Cantor's {ö}th place of decimals. (CP 6.176) Peirce uses this definition in an argument that the pseudo-continuum is not really continuous. So, here is my simple-minded question: is an imperfect continuum continuous, or is it not continuous? --Jeff Jeff Downard Associate Professor Department of Philosophy NAU (o) 523-8354 ________________________________________ From: Gary Richmond [[email protected]] Sent: Monday, November 17, 2014 10:39 PM To: Peirce-L Subject: Re: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, etc. Michael, Jeff D., Jerry, list, Michael wrote: I part with this very late gem [ca. 1908], where I think the continuity-continuum seamlessness is quite evident continuous, if you will. I quite agree, and if one could say of a quotation offered to prove a point that it did, one might say of this one: Q.E.D. Jeff asked: 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? As Michael noted earlier, this is a very different question than the one he was arguing with Jerry. I don't see that it offers any particular difficulties in answering (Peirce is quite explicit), and Michael in passing did offer a remark clearly articulating this distinction (I didn't quickly find it). Best, Gary R [Gary Richmond] Gary Richmond Philosophy and Critical Thinking Communication Studies LaGuardia College of the City University of New York C 745 718 482-5690< <tel:718%20482-5690> tel:718%20482-5690> On Mon, Nov 17, 2014 at 11:53 PM, Michael DeLaurentis < <mailto:[email protected]%3cmailto:[email protected]> [email protected]<mailto:[email protected]>> wrote: 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> www.avg.com<http://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. PEIRCE-L posts should go to <mailto:[email protected]%3cmailto:[email protected]> [email protected]<mailto:[email protected]> . To UNSUBSCRIBE, send a message not to PEIRCE-L but to <mailto:[email protected]%3cmailto:[email protected]> [email protected]<mailto:[email protected]> with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at <http://www.cspeirce.com/peirce-l/peirce-l.htm> http://www.cspeirce.com/peirce-l/peirce-l.htm . No virus found in this message. Checked by AVG - <http://www.avg.com%3chttp:/www.avg.com> www.avg.com<http://www.avg.com> Version: 2015.0.5577 / Virus Database: 4213/8588 - Release Date: 11/17/14 ----------------------------- PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to <mailto:[email protected]%3cmailto:[email protected]> [email protected]<mailto:[email protected]> . To UNSUBSCRIBE, send a message not to PEIRCE-L but to <mailto:[email protected]%3cmailto:[email protected]> [email protected]<mailto:[email protected]> with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at <http://www.cspeirce.com/peirce-l/peirce-l.htm> http://www.cspeirce.com/peirce-l/peirce-l.htm .
----------------------------- PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
