Hi, Can it be that continuum is the complementary union of continuity and discontinuity, in the same sense that light can be viewed as the complementary union of wave and particle ? Are there any Peircean texts that would either support or deny this possibility ?
With all the best. Sung __________________________________________________ Sungchul Ji, Ph.D. Associate Professor of Pharmacology and Toxicology Department of Pharmacology and Toxicology Ernest Mario School of Pharmacy Rutgers University Piscataway, N.J. 08855 732-445-4701 www.conformon.net > For the deeply skeptical, and finicky, who see only continuous and > continuum but not continuity, one need only back up a few lines in the > same text to find that third integrally related term thrown into the mix, > where CSP contrasts the Dedekind/Cantor notion of continuity [of the line] > with his own, and asks whether I am right in contending, solus, that > though > such a series of points [omega-1] no doubt has what is called continuity > in > the calculus and theory of functions, it has not the continuity of the > line > [which for Peirce is the true continuum]. > > > > A [truly/imperfect] continuous object is by definition a [true/imperfect] > continuum, which by definition exhibits [true/imperfect] continuity. > > > > This can be beaten up quite a bit more, Im certain, but I havent yet > seen > a step away from this obvious truism. > > > > From: Michael DeLaurentis [mailto:[email protected]] > Sent: Tuesday, November 18, 2014 10:14 AM > To: 'Jeffrey Brian Downard'; 'Peirce-L' > Subject: RE: [PEIRCE-L] Continuity, Generality, Infinity, Law, Synechism, > etc. > > > > 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]> . 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