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 - twitter.com/JonAlanSchmidt On Wed, Aug 28, 2019 at 3:48 PM Jeffrey Brian Downard < [email protected]> 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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