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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