Re: [PEIRCE-L] [EXTERNAL] Peirce on Dimensionality (was Connected Signs Theorem)
Helmut, List: >From a topological standpoint, any line figure is one-dimensional regardless of whether it is straight or curved, and any surface is two-dimensional regardless of whether it is flat or undulating. At any arbitrarily marked point on a line figure, only a hypothetical particle moving an infinitesimal distance along a straight line tangent to the figure at that point would remain within the line figure. Likewise, at any arbitrarily marked point on a surface, only a hypothetical particle moving an infinitesimal distance along a flat plane tangent to the surface at that point would remain within the surface. Consider these definitions that Peirce provides in the draft chapter on "Topical Geometry" for his unpublished textbook, *Elements of Mathematics*. CSP: A *place *is any one of the like parts of space. ... A *point *ls an indivisible place. ... A *particle *is a body which at any one instant occupies a point. ... A *line *is a place which a particle can occupy in the course of a lapse of time. ... A *filament *is a body which at any one instant occupies a line. ... A *surface *is a place which can be generated by the motion of a filament. ... A *film *is a body which at any one in.stant occupies a surface. ... A *space* is a place which can be generated by the motion of a film. ... A *solid *is a body which at any one instant occupies a space. ... A *surface *is a place that is the boundary between two solid spaces which together form one unbroken solid space. ... A *line *is a place that is the boundary between two surfaces which have more than isolated points in common. (NEM 2:170-172, c. 1895) Notice that he first presents "bottom-up" definitions, going from a dimensionless point to a one-dimensional line to a two-dimensional surface to a three-dimensional space, but then presents "top-down" definitions, going from a three-dimensional space to a two-dimensional surface to a one-dimensional line. In fact, three-dimensional space is presupposed from the very beginning since a dimensionless point is defined as "an indivisible place," while a place is defined as "any one of the like parts of space." In other words, space is a true continuum--the whole is real, while the parts are *entia rationis*. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Sun, Oct 10, 2021 at 4:24 AM Helmut Raulien wrote: > Jon, List, > > I think, the dimensionality of a line or of a surface is only then integer > (1 or 2), if the line is straight, or the surface is even. Otherwise, the > dimensionality of the line is between 1 and 2, or of the surface it is > between 2 and 3. > > Best, Helmut > _ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
Aw: [PEIRCE-L] [EXTERNAL] Peirce on Dimensionality (was Connected Signs Theorem)
Jon, List, I think, the dimensionality of a line or of a surface is only then integer (1 or 2), if the line is straight, or the surface is even. Otherwise, the dimensionality of the line is between 1 and 2, or of the surface it is between 2 and 3. Best, Helmut 09. Oktober 2021 um 23:08 Uhr "Jon Alan Schmidt" wrote: Jack, List: I can offer a couple more thoughts related to dimensionality. First, I also suggest reading my earlier paper, "Peirce's Topical Continuum: A 'Thicker' Theory" (https://doi.org/10.2979/trancharpeirsoc.56.1.04), which quotes and comments on a previously unpublished manuscript by Peirce that includes the following definitions. CSP: [I]n order to make up a continuum, two continua must have something in common, but their common part need not be like them in complexity of its composition. By a portion, in the terminology of this memoir, is meant a part of like complexity of composition of its whole. A limit between two portions of a continuum having no common portion is the part of lower complexity of composition. The dimensionality of a continuum is the number which measures the complexity of its composition. If the limit between two portions of a continuum having no common portion is not continuous, that continuum is said to have its dimensionality equal to one, or to have one dimension. If the limit between two portions of a continuum that have no common portion is, at highest, of dimensionality, N, that continuum is said to have its dimensionality equal to N+1, or to have N+1 dimensions. (R 144:2, c. 1900) The portions of a continuous one-dimensional line are also continuous one-dimensional lines, while the limits between such portions are discrete dimensionless points. The portions of a continuous two-dimensional surface are also continuous two-dimensional surfaces, while the limits between such portions are one-dimensional lines that meet at dimensionless points. The portions of a continuous three-dimensional space are also continuous three-dimensional spaces, while the limits between such portions are two-dimensional surfaces that meet at one-dimensional lines, which meet at dimensionless points. And so on. Second, with that in mind, I suggest that we can diagram the entire universe as a semiosic continuum of three dimensions. It is a vast argument whose portions are likewise three-dimensional spaces that correspond to its constituent argument types, whose limits are two-dimensional surfaces that correspond to proposition types, whose limits in turn are one-dimensional lines that correspond to name types. The dimensionless points where different spaces, surfaces, and lines meet correspond to the discrete tokens of all three classes of signs. This reflects the "top-down" nature of a true continuum (3ns), such that its material parts are indefinite possibilities (1ns), only some of which are actualized (2ns). CSP: Experience is first forced upon us in the form of a flow of images. Thereupon thought makes certain assertions. It professes to pick the image into pieces and to detect in it certain characters. This is not literally true. The image has no parts, least of all predicates. Thus predication involves precisive abstraction. Precisive abstraction creates predicates. Subjectal [or hypostatic] abstraction creates subjects. Both predicates and subjects are creations of thought. But this is hardly more than a phrase; for creation and thought have different meanings as applied to the two. ... That the abstract subject is an ens rationis, or creation of thought does not mean that it is a fiction. (NEM 3:917-918, 1904) CSP: [A]n Argument is no more built up of Propositions than a motion is built up of positions. So to regard it is to neglect the very essence of it. ... Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, Thought (I am not talking Psychology, but Logic, or the essence of Semeiotics) cannot, from the nature of it, be at rest, or be anything but inferential process; and propositions are either roughly described states of Thought-motion, or are artificial creations intended to render the description of Thought-motion possible; and Names are creations of a second order serving to render the representation of propositions possible. (R 295:117-118[102-103], 1906) Arguments are not built up from their constituent propositions, and propositions are not built up from their constituent names, including predicates and subjects. Instead, these are all artificial creations of thought for the purpose of describing arguments, which in
Re: [PEIRCE-L] [EXTERNAL] Peirce on Dimensionality (was Connected Signs Theorem)
Jack, List: I can offer a couple more thoughts related to dimensionality. First, I also suggest reading my earlier paper, "Peirce's Topical Continuum: A 'Thicker' Theory" ( https://doi.org/10.2979/trancharpeirsoc.56.1.04), which quotes and comments on a previously unpublished manuscript by Peirce that includes the following definitions. CSP: [I]n order to make up a continuum, two continua must have something in common, but their common part need not be like them in complexity of its composition. By a *portion*, in the terminology of this memoir, is meant a part of like complexity of composition of its whole. A *limit *between two portions of a continuum having no common portion is the part of lower complexity of composition. The *dimensionality *of a continuum is the number which measures the complexity of its composition. If the limit between two portions of a continuum having no common portion is not continuous, that continuum is said to have its dimensionality equal to one, or to have *one dimension*. If the limit between two portions of a continuum that have no common portion is, at highest, of dimensionality, N, that continuum is said to have its *dimensionality *equal to N+1, or to have N+1 *dimensions*. (R 144:2, c. 1900) The portions of a continuous one-dimensional line are also continuous one-dimensional lines, while the limits between such portions are discrete dimensionless points. The portions of a continuous two-dimensional surface are also continuous two-dimensional surfaces, while the limits between such portions are one-dimensional lines that meet at dimensionless points. The portions of a continuous three-dimensional space are also continuous three-dimensional spaces, while the limits between such portions are two-dimensional surfaces that meet at one-dimensional lines, which meet at dimensionless points. And so on. Second, with that in mind, I suggest that we can *diagram *the entire universe as a semiosic continuum of three dimensions. It is a vast argument whose portions are likewise three-dimensional spaces that correspond to its constituent argument types, whose limits are two-dimensional surfaces that correspond to proposition types, whose limits in turn are one-dimensional lines that correspond to name types. The dimensionless points where different spaces, surfaces, and lines meet correspond to the *discrete *tokens of all three classes of signs. This reflects the "top-down" nature of a true continuum (3ns), such that its material parts are indefinite possibilities (1ns), only some of which are actualized (2ns). CSP: Experience is first forced upon us in the form of a flow of images. Thereupon thought makes certain assertions. It professes to pick the image into pieces and to detect in it certain characters. This is not literally true. The image has no parts, least of all predicates. Thus predication involves precisive abstraction. Precisive abstraction creates predicates. Subjectal [or hypostatic] abstraction creates subjects. Both predicates and subjects are creations of thought. But this is hardly more than a phrase; for *creation *and *thought* have different meanings as applied to the two. ... That the abstract subject is an *ens rationis*, or creation of thought does not mean that it is a fiction. (NEM 3:917-918, 1904) CSP: [A]n Argument is no more built up of Propositions than a motion is built up of positions. So to regard it is to neglect the very essence of it. ... Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or *ens rationis*), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are *entia rationis* (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, Thought (I am not talking Psychology, but Logic, or the essence of Semeiotics) cannot, from the nature of it, be at rest, or be anything but inferential process; and propositions are either roughly described states of Thought-motion, or are artificial creations intended to render the description of Thought-motion possible; and Names are creations of a second order serving to render the representation of propositions possible. (R 295:117-118[102-103], 1906) Arguments are not built up from their constituent propositions, and propositions are not built up from their constituent names, including predicates and subjects. Instead, these are all artificial creations of thought for the purpose of *describing *arguments, which in themselves are continuous inferential processes. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Fri, Oct 8, 2021 at 7:09 PM JACK ROBERT KELLY CODY < jack.cody.2...@mumail.ie> wrote: > Jon, List, > > Cheers, Jon, that's
Re: [PEIRCE-L] [EXTERNAL] Peirce on Dimensionality (was Connected Signs Theorem)
Jon, List, Cheers, Jon, that's helpful. I'm rereading your Temporal Synechism article (https://doi.org/10.1007/s10516-020-09523-6) at present which also helps clarify some of these issues. Best Jack From: peirce-l-requ...@list.iupui.edu on behalf of Jon Alan Schmidt Sent: Saturday, October 9, 2021 1:03 AM To: Peirce-L Subject: [EXTERNAL] [PEIRCE-L] Peirce on Dimensionality (was Connected Signs Theorem) *Warning* This email originated from outside of Maynooth University's Mail System. Do not reply, click links or open attachments unless you recognise the sender and know the content is safe. Jack, List: JRKC: What role does dimensionality have in Peirce's schema? I can suggest at least a couple of roles that dimensionality plays in Peirce's thought. CSP: The evolution of forms begins or, at any rate, has for an early stage of it, a vague potentiality; and that either is or is followed by a continuum of forms having a multitude of dimensions too great for the individual dimensions to be distinct. It must be by a contraction of the vagueness of that potentiality of everything in general, but of nothing in particular, that the world of forms comes about. (CP 6.196, 1898) 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 [actual] points on this blackboard. There are no [actual] dimensions in that continuum. (CP 6.203, 1898) CSP: At the same time all this, be it remembered, is not of the order of the existing universe, but is merely a Platonic world, of which we are, therefore, to conceive that there are many, both coordinated and subordinated to one another; until finally out of one of these Platonic worlds is differentiated the particular actual universe of existence in which we happen to be. (CP 6.208, 1898) In cosmology, he thus posits a continuum of potential dimensions within which our "actual universe of existence" with its four dimensions is like "a figure of lower dimensionality" (CP 4.642, 1908). CSP: But I ask you to imagine all the true propositions to have been formulated; and since facts blend into one another, it can only be in a continuum that we can conceive this to be done. This continuum must clearly have more dimensions than a surface or even than a solid ... Nevertheless, in order to represent to our minds the relation between the universe of possibilities and the universe of actual existent facts, if we are going to think of the latter as a surface, we must think of the former as three-dimensional space in which any surface would represent all the facts that might exist in one existential universe. (CP 4.512&514, 1903) CSP: [Existential graphs] are diagrams upon a surface, and indeed must be regarded as only a projection upon that surface of a sign extended in three dimensions. Three dimensions are necessary and sufficient for the expression of all assertions ... (R 654:6-7, 1910 Aug 19) In logic, he thus diagrams "the universe of actual existent facts" with two dimensions, as represented in EGs, and situates it within "the universe of possibilities," which has three dimensions. Moreover, the continuum of "all the true propositions ... must clearly have more dimensions" than three, but three "are necessary and sufficient for the expression of all assertions." Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt<http://www.LinkedIn.com/in/JonAlanSchmidt> - twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt> On Fri, Oct 8, 2021 at 5:39 PM JACK ROBERT KELLY CODY mailto:jack.cody.2...@mumail.ie>> wrote: Quick scan reveals nothing novel a dimensional framework/problematic: "Lorentz has already shown us a convenience in considering a time, if not exactly as a dimension of time-space, at least as that fourth unit that Hamilton adds to the three dimensions of space to make up a quaternion, and indeed one may say that, from the point of view of matrices, three-dimensional space appears as not altogether comprehensible without a fourth" 1913 (in EP Volume 2: 474). (Einstein worked on relativity between 1907-1915, but built on Lorentz amongst others - this Peirce quote stems from 1913, I think). Also, in Logic of Drawing History from Ancien
