Jon S, Gary R, List,
On my reading of the last lecture of RLT, I think it is an error to suggest that he is making measurement intrinsic to the definition of those dimensions of either time, space or quality. Rather, the thrust of the argument is to start with mathematical conceptions and then use them for the sake of developing hypotheses in metaphysical cosmology. In doing so, he moves from the consideration of metrical geometries, to projective geometry to topology. In doing so, he is setting metrical considerations to the side and focusing primarily on topological matters. It is clear that the topological points--including those about the possible dimensions of a such a malleable space in which such things straigntness, length and degree of angle are not preserved across transformations--are being used to clarify a mathematical conception of continuity. He is then putting that refined notion of continuity to use as he engages with questions about the origins and evolution of the universe. The questions he is trying to answer include the following. How many dimensions of time, space and quality where there early in the history of the universe? How many dimensions of each are there now. How did the number of dimensions change over time? Jon quoted two passages in that last lecture: 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) 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) Consider the following sentence from the second passage: "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." For the sake of engaging in inquiry in cosmological metaphysics, I would make a distinction between the dimensions of real space at some point in the evolution of the cosmos, and the dimensions of the qualities of the objects in space. As far as I can tell, he is offering a hypothesis about the number of dimensions of (1) time, (2) space and (3) of the various qualities that were present early in the history of the cosmos. It looks to me like he is arguing that each started with a dimensions that were vague in character and not distinctly separated--one from another. Over time, as the cosmos evolved, those dimensions became (a) more determinate and (b) fewer in number. If we go back far enough, we arrive at a vague potentiality as a kind of hypothetical "beginning of all things." This vague conception of potentiality functions as a kind of starting point in the explanations being offered. My assumption is that, in this vague potentiality, there might have been--for instance--potential energy, but there was no kinetic energy. That potential energy might have taken different qualities, such as a particular charge or a particular spin, but there was no actual object having any determinate charge or spin. There might have been a potential for space and time having dimensions, but there were no actual things moving around in space and time. How many dimensions did this potential have? An indefinite vague multitude. The dimensions were continuous. There were uncountable, to say the least. In saying that the dimensions of space, time and quality were potential and not actual, I do not take him to be saying that the dimensions were not real. Possibles may, on Peirce's account, be real things. I take this starting point, which is explicated in terms of a conception of vague potentiality, as a kind of limiting idea. One thing he is trying to accomplish in clarifying such a limiting idea is to arrive at something that doesn't call out for further explanation. If someone asks, why does the original vague potentiality have the characteristics it does? His answer is: that doesn't need a further explanation. It can be illustrated using diagrams. He is offering analogy to the effect that the vague potentiality is like an empty chalkboard before any chalk streaks have been drawn on its surface. Some philosophers might claim that Peirce is wrong to think the original vague potentiality doesn't need a further explanation, but I take that to be the view he is exploring in this last lecture. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Gary Richmond <gary.richm...@gmail.com> Sent: Thursday, August 29, 2019 1:24 PM To: Peirce-L Subject: Re: Re: Re: [PEIRCE-L] Re: Peirce and the Big Bang 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 On Wed, Aug 28, 2019 at 10:12 PM Jon Alan Schmidt <jonalanschm...@gmail.com<mailto:jonalanschm...@gmail.com>> wrote: Jeff, List: JD: Peirce provides definitions for dimension, dimensional and dimensionality in the Century Dictionary. Thanks for pointing this out; it did not occur to me to look there (http://triggs.djvu.org/century-dictionary.com/djvu2jpgframes.php?volno=02&page=741). Here is his first definition of "dimension." CSP: Magnitude measured along a diameter; the measure through a body or closed figure along one of its principal axes; length, breadth, or thickness. Thus, a line has one dimension, length; a plane surface two, length and breadth; and a solid three, length, breadth, and thickness. The number of dimensions being equal to the number of principal axes, and that to the number of independent directions of extension, 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. The word generally occurs in the plural, referring to length, breadth, and thickness. (CD 1621) Here is his second definition of "dimension." CSP: A mode of linear magnitude involved (generally along with others) in the quantity to which it belongs. (a) In alg., a variable factor, the number of dimensions of an expression being the number of variable factors in that term for which this number is the largest. (b) In phys., a linear measure of length, time, mass, or any kind of quantity regraded as a fundamental factor of the quantity of which it is a dimension. (ibid) Here is his first definition of "dimensional." CSP: Pertaining to extension in space; having a dimension or dimensions; measurable in one or more directions: used in composition: as, a line is a one-dimensional, a surface a two-dimensional, and a solid a three-dimensional object. (ibid) Finally, here is his only definition of "dimensionality." CSP: The number of dimensions of a quantity. (ibid) He provides two other definitions for "dimension," and a second one for "dimensional," but they do not strike me as relevant to this discussion. JD: Nothing jumps out at me in the definitions offered, but it is worth noting that he does make a distinction between the dimensions of a mathematical space and that of a physical space. Where exactly do you see Peirce making that specific distinction? The word "space" appears only once, in a way that seems applicable to both the mathematical and physical senses. 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." This is consistent with my suggestion that discrete dimensions are arbitrary and artificial creations of thought for that purpose, rather than real characters of space-time in itself. Moreover, the second definition hints at why we typically count dimensions with whole numbers--we begin with "linear magnitude," and then build up additional discrete dimensions from there. The linked video 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. 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 vague generality of the most abstract potentiality. (NEM 3:111, RLT 253-254; 1898) The first three statements reflect his mistaken "supermultitudinous" conception of continuity, but his later "topological" (or "topical") theory would similarly require the dimensions (parts) of a perfect continuum to be indefinite unless and until they are "marked off." Nevertheless, the development of "point-set topology" indicates that the lure of discreteness remains strong in contemporary mathematics, even within the branch that Peirce described as "the full account of all forms of Continuity" (NEM 2:626; 1905). The fourth statement brings us back to the subject of this thread, obviously anticipating what "the clean blackboard" represents later in the same lecture--primordial 3ns, or what Gary R. has called "the ur-continuity." 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) Rather than "a vague infinity of dimensions," there are no distinct dimensions--no definite dimensions--no discrete dimensions at all in the original continuum that is fundamental to the constitution of being. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt<http://www.LinkedIn.com/in/JonAlanSchmidt> - twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt> On Wed, Aug 28, 2019 at 3:48 PM Jeffrey Brian Downard <jeffrey.down...@nau.edu<mailto:jeffrey.down...@nau.edu>> wrote: Jon S, Gary F, List, Peirce provides definitions for dimension, dimensional and dimensionality in the Century Dictionary. Nothing jumps out at me in the definitions offered, but it is worth noting that he does make a distinction between the dimensions of a mathematical space and that of a physical space. For the sake of understanding the points made in the last lecture of RLT, the discussion of topological dimensions in the EM and the NEM are particularly helpful resources. There, Peirce describes the various ways that a particle can be moved from a point, a filament from a line, etc. This is consonant with the contemporary way of talking about the dimensions of a topological space in terms of the degrees of freedom that something can be moved. Those mathematical ideas can be applied to physical space by asking questions about how something like an atom or a sub-atomic particle might be able to move. My sense is that Peirce is thinking in terms of continuous spatial fields as being more fundamental than discrete particles. Cosmologically speaking, the permanence of something like a Hydrogen atom is explained in terms of the parts (e.g., the proton) of that whole evolving from something more basic. So, if we consider something like an extremely high-temperature plasma in which the particles (e.g., the quarks, leptons and bosons) are moving relatively freely in relation to one another, then it is helpful to think of those "particles" as spread areas of charge in a field. If we think of the laws of physics as evolving in the early stages of the development of the universe, how might we envision gravity, and the strong and weak forces operating in a relatively dense plasma? More to the point, how might we envision the laws of time and space evolving where the universe is comprised of a dense plasma of charged areas in a multi-dimensional field? In order to conceive of the evolution of time and space as involving a trend having a decrease in number from a vague infinity of dimensions to a more determinate number (e.g., from more than 100, to 12, to 10 to 4), we need some kind of tools to picture how this might work. Two of the resources that Peirce worked with in his various studies of topology, projective geometry and metrical geometries are Riemannian manifolds and Klein groups. Those probably give us what we need for thinking, at least in broad terms, about the character of the dimensions of a space that are (1) vague and (2) infinite. Setting aside metrical considerations (which will naturally make things more vague), the question becomes a matter of explaining how a topological space (which may be folded, knotted and twisted in many ways) might evolve into a space that has projective characteristics (where there is "straightness" or homoloidal properties, but no preservation of angles or lengths under transformations). If you will, let me think out loud using very rough terms about how some of the characteristics of sub-atomic particles in a plasma might change as those particles move through a space of high dimensions. What follows is conjectural in character. In the case of a real physical space that is highly folded, knotted and twisted, where the "particles" are charged areas that move through the space, how should we conceive of the dimensionality of such a space in the initial phases where the laws of time and space themselves are evolving as the number of dimensions of that space decrease? It helps, I think to distinguish between the global character of such a space and its local character. Locally speaking, I imagine that the charged areas might "break up" into smaller areas as they move through different "branches" (i.e., handles, like a hole in a torus) that may twist (i.e., cross caps, as with a Mobius band) and that are knotted together and then recombine with other moving charged areas. We tend to think of subatomic particles (e.g., quarks) as having relatively fixed masses (voltages). Neutrinos, on the other hand, have mass values that are simply less than a particular voltage value. This seems to imply that they have an amount of energy that may vary, perhaps up to some limit. Furthermore, I suspect the value of the charge and perhaps the value of the spin (the angular momentum) of the charged areas moving through a field may change as the charged area moves through a twist in the space. How might we study something as complex as a highly folded, twisted and knotted space? As with any kind of relatively complex topological space, it helps to decompose that space into its component parts. As such, we can focus on one simpler two-dimensional surface at a time, and then think about the various ways such surfaces might be connected. What is more, we can think of the possible paths that things might travel on that surface as edges in a graph. Those, I suspect, are kinds of the techniques we might profitably employ to study the question of how the dimensions of space and time might have evolved in the early history of the cosmos. Thanks for your patience as I've tried to talk out loud. In order to make any progress in cosmological metaphysics, we will need to make a transition from these sorts of conjectural musings on matters of cosmological physics to something that is easier to get one's mind around. As such, in a future post, I'd like to take up some graph-theoretical explorations of how we might think about the dimensions of space and time. In doing so, the aim will be to create some kind of diagram that helps to picture how time and space might be evolving from a vague infinity of dimensions to a more determinate and smaller number of dimensions. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354
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