Jon, List,
JAS: . . . as Pietarinen comments, "Peirce also makes the observation that the notion of dimension does not imply that the geometry of logical space is metric. If we have dimension, we already have a topological space (topical geometry, topology, topics) that is not subject to measurement" (p. 209). This means that I was wrong to dismiss Peirce's statement in the Century Dictionary definition of "dimension" that "it has become usual, in mathematics, to express the number of ways of spread of a figure by saying that it has two, three, or *n* dimensions, although the idea of measurement is quite extraneous to the fact expressed." Apparently he was there referring *specifically *to the concept as employed in topical geometry, such that measurement is *not* intrinsic to the relevant sense of the term after all. I thus stand corrected. What are the implications of this 'correction' re: dimension for Peirce's model/diagram of the earliest Universe as interpreted by you? Best, Gary R *Gary Richmond* *Philosophy and Critical Thinking* *Communication Studies* *LaGuardia College of the City University of New York* On Fri, Aug 30, 2019 at 9:27 PM Jon Alan Schmidt <[email protected]> wrote: > Gary R., List: > > This post has a twofold purpose--first, to "bump" the one below, in case > it got overlooked in the flurry of other exchanges over the last 24 hours, > including the "distraction" about real possibilities; and second, to > introduce some additional remarks by Peirce about the concept of > *dimension*, which shed further light on how he defined it. A few months > ago, we discussed a passage in "The Bed-Rock Beneath Pragmaticism" where he > suggested that the three Modalities are "different dimensions of the > logical Universe" (R 300:80[37]; 1908) and attributed this way of thinking > to his former student, O. H. Mitchell. Here is how the latter introduced > it in his chapter, "On a New Algebra of Logic," in the 1883 book that > Peirce edited, *Studies in Logic. By Members of the Johns Hopkins > University*. > > OHM: The relation implied by a proposition may be conceived as concerning > "all of" or "some of" the universe of class terms. In the first case the > proposition is called universal; in the second, particular. The relation > may be conceived as permanent or as temporary; that is, as lasting during > the whole of a given quantity of time, limited or unlimited,--the Universe > of Time,--or as lasting for only a (definite or indefinite) portion of it. > A proposition may then be said to be universal or particular in time. The > universe of relation is thus two-dimensional, so to speak; that is, a > relation exists *among *the objects in the universe of class terms *during > *the universe of time. (pp. 73-74). > > > Mitchell went on to discuss "Propositions of Two Dimensions" in some > detail (pp. 87-95), but only briefly touched on "Propositions of more than > two dimensions." For three dimensions ... > > OHM: The logic of such propositions is a "hyper" logic, somewhat > analogous to the geometry of "hyper" space. In the same way the logic of a > universe of relation of four or more dimensions could be considered. The > rules of inference would be exactly similar to those already given. (pp. > 95-96). > > > Peirce again cited Mitchell in his definition of "dimension" within his > entry on "Exact Logic" in Baldwin's *Dictionary of Philosophy and > Psychology*. > > CSP: An element or respect of extension of a logical universe of such a > nature that the same term which is individual in one such element of > extension is not so in another. Thus, we may consider different persons as > individual in one respect, while they may be divisible in respect to time, > and in respect to different admissible hypothetical states of things, etc. > This is to be widely distinguished from different universes, as, for > example, of things and of characters, where any given individual belonging > to one cannot belong to another. The conception of a multidimensional > logical universe is one of the fecund conceptions which exact logic owes to > O. H. Mitchell. (CP 3.624; 1901) > > > In his 2006 book, *Signs of Logic: Peircean Themes on the Philosophy of > Language, Games, and Communication*, Ahti-Veikko Pietarinen transcribes a > portion of Peirce's original draft entry on "Modality" for the same work, > which is considerably different from what ultimately appeared in the > published version. It mentions "Professor Mitchell's idea of a > multidimensional logical universe," elaborating as follows. > > CSP: A logical universe of two or more dimensions must not be confounded > with two or more logical universes. When we consider, in addition to the > usual limited universe of individual subjects, also a limited universe of > marks, we have two logical universes. That which is contained in the one is > not contained in the other. But if, in addition to the universe of > subjects, we conceive each of these as enduring through more or less time, > so that on the one hand, each subject exists through part or all of time, > and on the other hand, in each instant of time there exist a part or all of > the subjects, we are considering a logical universe of two dimensions, and > the same terms have their place in both. The word *dimension* is here > applied with perfect propriety; for were we to restrict it to cases in > which measurement could be applied, we should be forced to abandon its use > in topical geometry, to which no mathematician (and it is a mathematical > word) would consent. (R 1147:267-268[Modality 1-2]; c. 1901) > > > The first few sentences echo Mitchell's words, but as Pietarinen comments, > "Peirce also makes the observation that the notion of dimension does not > imply that the geometry of logical space is metric. If we have dimension, > we already have a topological space (topical geometry, topology, topics) > that is not subject to measurement" (p. 209). This means that I was wrong > to dismiss Peirce's statement in the Century Dictionary definition of > "dimension" that "it has become usual, in mathematics, to express the > number of ways of spread of a figure by saying that it has two, three, or > *n* dimensions, although the idea of measurement is quite extraneous to > the fact expressed." Apparently he was there referring *specifically *to > the concept as employed in topical geometry, such that measurement is > *not* intrinsic to the relevant sense of the term after all. I thus > stand corrected. > > Regards, > > Jon S. > > On Thu, Aug 29, 2019 at 9:54 PM Jon Alan Schmidt <[email protected]> > wrote: > >> Gary R., List: >> >> GR: Are you suggesting that it is* only* in the aboriginal (from Latin >> <https://en.wikipedia.org/wiki/Latin> *ab >> <https://en.wiktionary.org/wiki/ab#Latin> origine >> <https://en.wiktionary.org/wiki/origine#Latin> --*“from the beginning”) >> continuum that >> there are *no discrete dimensions*? >> >> >> In order to answer this question, I will begin by quoting a little-known >> passage that I have excerpted several times over the past few months, >> including just a few days ago in this thread. >> >> CSP: In the first place, then, I do not call a line, or a surface, or >> anything else, continuous unless every part of it that is homogeneous in >> dimensionality with the whole and is marked off in the simplest way is, in >> respect to the connexions of its parts, precisely like every other such >> part; although, if the whole has but a finite number of interruptions, I do >> call it "continuous in its uninterrupted portions." In the next place, I >> conceive that a Continuum has, IN ITSELF, no definite parts, although to >> endow it with definite parts of no matter what multitude, and even parts of >> lesser dimensionality down to absolute simplicity, it is only necessary >> that these should be marked off, and although even the operation of thought >> suffices to impart an approach to definiteness of parts of any multitude we >> please.* >> *This indubitably proves that the possession of parts by a continuum is >> not a real character of it. For the real is that whose being one way or >> another does not depend upon how individual persons may imagine it to be. >> (R S-30 [Copy T:5-6]; c. 1906) >> >> >> Although it has never appeared in the secondary literature--presumably >> because of the obscurity of the manuscript, which received a >> "supplementary" number from Robin--I consider it to be Peirce's clearest >> definition of his late topical conception of continuity, because I think >> that it elaborates helpfully on a subsequent one that is commonly cited. >> >> CSP: 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; 1908 May 26) >> >> >> Every material part of a perfect continuum is *indefinite *and has "the >> same dimensionality as the whole," but any number of *definite *parts of >> the same or "lesser dimensionality, down to absolute simplicity"--i.e., >> dimensionless and indivisible points--can be "marked off" as >> "interruptions" or "breaches of continuity." Therefore, since the original >> continuum has "some indefinite multitude of dimensions," all of its >> material parts must likewise have "some indefinite multitude of >> dimensions"; and any subsidiary continuum that has a *definite *number >> of *discrete *dimensions is "a topical singularity," like "an oval line >> on a spheroidal or anchor-ring surface." What is the implication of this >> for our *physical *universe? I see two alternatives. >> >> 1. Continuous space-time has no discrete dimensions in itself; it is >> a material part of the original continuum. >> 2. Continuous space-time has a definite number of discrete >> dimensions; it is "a figure of lower dimensionality" in the original >> continuum. >> >> #1 leads to my previous statement that you quoted twice--*discrete >> *dimensions >> are arbitrary and artificial creations of thought for particular purposes, >> cognitive constructions that *represent *space-time. However, Peirce >> evidently endorsed #2 instead. >> >> CSP: The whole universe of true and real possibilities forms a >> continuum, upon which this Universe of Actual Existence is, by virtue of >> the essential Secondness of Existence, a discontinuous mark--like a line >> figure drawn on the area of the blackboard. (NEM 4:345, RLT 162; 1898) >> >> >> As you already observed in your latest reply to Jeff, this would seem to >> require an external "scriber" who *chooses *to draw the line figure on >> the blackboard--i.e., to "mark off" the *discrete *dimensions of the >> physical universe as *definite *parts of the original continuum. As >> Peirce wrote elsewhere ... >> >> CSP: In a continuum there really are no points except such as are >> marked; and such interrupt the continuum. It is true that the capability of >> being marked gives to the points the beginnings of *potential being*, >> but only the beginnings. It should be called a *conditional being*, >> since it depends upon some will's being exerted to complete it. (R 1041:13; >> 1906) >> >> >> Regards, >> >> Jon Alan Schmidt - Olathe, Kansas, USA >> Professional Engineer, Amateur Philosopher, Lutheran Layman >> www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt >> >> On Thu, Aug 29, 2019 at 3:25 PM Gary Richmond <[email protected]> >> wrote: >> >>> Jon, Jeff, List, >>> >>> This message is meant to solicit clarification on what seems to be the >>> thrust of Jon's argument in support of a dimensionless ur-continuity. My >>> question is: Am I clearly grasping what you're getting at, Jon? You wrote >>> near the end of your post: >>> >>> JAS: What I notice is that *measurement *is evidently intrinsic to the >>> definition of dimension, except for the particular mathematical usage >>> mentioned in the first one, where "the idea of measurement is quite >>> extraneous." >>> >>> >>> Again, I would tend to strongly with your suggestion that: >>> >>> >>> JAS: ". . . *discrete *dimensions are arbitrary and artificial >>> creations of thought for that purpose, rather than *real *characters of >>> space-time in itself." >>> >>> >>> You continued: >>> >>> >>> JAS: [. . .] The linked video [which Jeff earlier provided] about higher >>> numbers of dimensions employs the same "bottom-up" analytic approach, using >>> the real number line--what Peirce called a "pseudo-continuum"--as the basis >>> for *defining *each individual dimension. >>> >>> >>> I would take it, then, that "pseudo-continuua," are most certainly of >>> *analytical* value as long as one remembers, as you have been positing >>> recently (and I agree) that: >>> >>> JAS: . . .*discrete *dimensions are arbitrary and artificial creations >>> of thought for that [analytical] purpose, rather than *real *characters >>> of space-time in itself. >>> >>> >>> You then asked if dimensionality would even apply in a "top-down" >>> approach and suggested that it may not, offering a Peirce quotation in >>> support of your suggestion : >>> >>> JAS: What might it look like to adopt a "top-down" synthetic approach >>> instead? Would the familiar notion of dimensions even apply? Maybe not, >>> according to Peirce. >>> >>> CSP: A continuum may have any discrete multitude of dimensions >>> whatsoever. lf the multitude of dimensions surpasses all discrete >>> multitudes there cease to be any distinct dimensions. I have not as yet >>> obtained a logically distinct conception of such a continuum. >>> Provisionally, I identify it with the *uralt * [Ger., ancient], vague >>> generality of the most abstract potentiality. (NEM 3:111, RLT 253-254; 1898) >>> >>> >>> You then quoted Peirce on the 'blackboard' as a metaphor for the >>> original, or, ur-continuum: >>> >>> CSP: Let the clean blackboard be a sort of diagram of the original >>> vague potentiality, or at any rate of some early stage of its >>> determination. This is something more than a figure of speech; for after >>> all continuity is generality. This blackboard is a continuum of two >>> dimensions, while that which it stands for is a continuum of some >>> indefinite multitude of dimensions. This blackboard is a continuum of >>> possible points; while that is a continuum of possible dimensions of >>> quality, or is a continuum of possible dimensions of a continuum of >>> possible dimensions of quality, or something of that sort. There are no >>> points on this blackboard. There are no dimensions in that continuum. (CP >>> 6.203, RLT 261; 1898) >>> >>> >>> JAS: Rather than "a vague infinity of dimensions," there are no >>> *distinct *dimensions-- no *defnite *dimensions--no *discrete *dimensions >>> at all in the original continuum that is fundamental to the constitution of >>> being. >>> >>> >>> So, finally getting back to my question: Are you suggesting that it is* >>> only* in the in the aboriginal (from Latin >>> <https://en.wikipedia.org/wiki/Latin> *ab >>> <https://en.wiktionary.org/wiki/ab#Latin> origine >>> <https://en.wiktionary.org/wiki/origine#Latin> --*“from the beginning”) >>> continuum that there are *no discrete dimensions*? That makes sense to >>> me; and, of course, it has significant implications for what you and I have >>> been arguing regarding Peirce's late view of the situation of the earliest >>> cosmos; namely, that ur-continuity is quasi-necessarily primal in the >>> constitution of reality, including, of course, existential being on "time >>> is." >>> >>> Best, >>> >>> Gary R >>> >>> *Gary Richmond* >>> *Philosophy and Critical Thinking* >>> *Communication Studies* >>> *LaGuardia College of the City University of New York* >>> >>>>
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