List, Jeffrey, John S. First, I would support further discussion along these lines.
Of course, I would think that a degree of balance should be introduced into the nature of premises. There are deep mathematical issues involved in this rather casual conversations. With this regard, John S’s slides of 1 August 2015 provide a counter point. (see slides 4, 5, and 6.) Cheers Jerry > On Nov 14, 2016, at 3:45 PM, Jeffrey Brian Downard <[email protected]> > wrote: > > Jon S, Gary R, Edwina, John S, List, > > If others are interested, I'd like to continue the discussion of the last > lecture on continuity in RLT. The goal, I took it, was to draw on it for the > sake of filling in some the details in the "table of contents" for a larger > set of inquiries that he sketched in "A Neglected Argument." > > My proposal is to march through more of the mathematical examples he offers > in the hopes of getting more clarity about the logical conception of > continuity that he articulates. Then, the aim is to work up to the example of > the lines on the blackboard and the way that he uses that example to frame > some hypothesis in cosmological metaphysics. > > Given the fact that my post on Desargues 6-point theorem did not generate > much in the way of comments or questions, I am concerned that I overdid it > and managed to smother some of the interest in the questions--both > interpretative and philosophical--that we were considering. As such, I'm > asking for feedback to make see if continued discussion of the mathematical > examples is welcome. > > Late last week, I thought of a way to illustrate Peirce's larger point about > how the 6 point theorem is connected to the larger idea that Cayley and Klein > make about the character of the projective absolute and how it provides the > basis of any system of metrical relations in elliptical, parabolic or > hyperbolic geometries. The illustration helps to see, in a more intuitive > way, the point Peirce seems to be making about the kind of hypothesis that is > needed to make sense of the possibility of progress with respect to the > growth of our understanding or, more generally, with the growth of order in > the cosmos. > > So, let me ask if there are any takers for continuing the discussion of RLT > along these lines? > > --Jeff > > Jeffrey Downard > Associate Professor > Department of Philosophy > Northern Arizona University > (o) 928 523-8354 > > > From: Jon Alan Schmidt <[email protected]> > Sent: Tuesday, November 8, 2016 10:55 AM > To: Jeffrey Brian Downard > Cc: [email protected] > Subject: Re: [PEIRCE-L] Re: Super-Order and the Logic of Continuity (was > Metaphysics and Nothing (was Peirce's Cosmology)) > > Jeff, List: > > Did Peirce retain the notion of "the absolute" as he developed his conception > of a continuum in the last RLT lecture? > > CSP: The other rule that if A is r to B and B is r to C then A is r to C > leads to no contradiction, but it does lead to this, that there are two > possible exceptional individuals[,] one that is r to everything else and > another to which everything else is r. This is like a limited line, where > every point is r that is, is to the right of every other or else that other > is to the right of it. The generality of the case is destroyed by those two > points of discontinuity,--the extremities. Thus, we see that no perfect > continuum can be defined by a dyadic relation. But if we take instead the > triadic relation, and say A is r to B for C, say to fix our idea that > proceeding from A in a particular way, say to the right, you reach B before > C, it is quite evident, that a continuum will result like a self-returning > line with no discontinuity whatever. (RLT 250) > > Yesterday I came across an interesting paper by Nicholas Guardiano, "The > Categorial Logic of Peirce's Metaphysical Cosmogony" (The Pluralist 10:3, > Fall 2015, 313-334; early version at > http://www.american-philosophy.org/saap2014/openconf/modules/request.php?module=oc_program&action=view.php&id=28 > > <http://www.american-philosophy.org/saap2014/openconf/modules/request.php?module=oc_program&action=view.php&id=28>). > He describes Peirce's theory about the origin and development of the > universe from the standpoint of each Category, always in terms of three > stages. > Secondness - chaos (1ns), reaction (2ns), regularity (3ns). > Thirdness - spontaneity/chance/freedom (1ns), evolutionary process (3ns), > fixed end (2ns). > Firstness - continuum (3ns), Platonic world of qualities (1ns), brute > existence (2ns). > Guardiano thus presents the three most common interpretations, and does so in > this particular order. I offer three observations about this, which may or > may not be significant. > The Category that provides the point of view for the analysis always > corresponds to the second stage. > The stage associated with Firstness always precedes the one associated with > Secondness. > The stage associated with Thirdness moves from third to second to first in > the sequence. > Gary R. and I have been advocating the Firstness perspective, while my > understanding is that Edwina primarily adopts the Secondness perspective. > The Thirdness perspective involves the hyperbolic absolute with two distinct > points, as Jeff described below. > > Regards, > > Jon > > On Mon, Nov 7, 2016 at 2:04 PM, Jeffrey Brian Downard > <[email protected] <mailto:[email protected]>> wrote: > Jon, Edwina, Gary R, List, > I've been studying projective geometry for a while in the hopes of getting a > better handle on these kinds of questions. Unfortunately, I am slow on the > uptake and my grasp of these mathematical ideas is still in its formative > stages. > Having said that, we don't need to have mastery of this area of mathematics > in order to explore some examples. So, let's consider the construction of > the 6 point theorem. Some background might be helpful for those who haven't > studied projective geometry. The general idea that drove the development of > this area of mathematics is taken from perspective drawing in art and > perspective geometry in mathematics. > Here is a woodcut by Albrecht Dürer that nicely illustrates the key ideas. > From a point of perspectivity (the nail on the wail to which the string is > attached), a ray (the string) is extended through an image plane to a point > on some object, such as a lute. The image of the lute is constructed on the > image plane by taking a number of points where the rays intersect the image > plane. One could imagine a number of different perspective drawings being > constructed by altering the position of the lute in relation to the image > plane and in relation to the point of perspectivity. > <pastedImage.png> > In order to get an idea of what the absolute is in projective geometry, it > helps to think of it as a generalization of the horizon within perspective > geometry. If we compare what happens to parallel lines in Euclidean geometry > and perspective geometry, we see that parallel lines never meeting in > Euclidean geometry no matter how far they are extended, whereas all parallel > lines meet at the horizon in perspective geometry. > <pastedImage.png> > Projective geometry is a generalization on perspective geometry. Let us take > the horizontal line in the construction of the 6-point theorem to be > analogous to a slice through the image plane in perspective geometry. Taking > a point Q as an origin that is analogous to a point of perspectivity, three > rays are extended through the horizontal line and the points of intersection > are marked A, B and C. > > <Desargues 1.png> > > Taking a point on C as another origin (akin a picking another point of > perspectivity when drawing the lute), two more rays are extended through the > horizontal line and through the other rays with points A' and B' marked on > the horizontal line and points > > <Desargues 2.png> > > Then, construct a line through the points P and S where the rays intersect, > and extend it through the horizontal line to a point C'. > > <Desargues 3.png> > > How are the points on the horizontal line related under different > transformations? Well, move the origins around and see what you get (if you > want to try this, then download CarMetal, which is free, and make the > diagram). Here is one such transformation where the two origins have been > moved. > <Desargues 4.png> > > The diagram we have constructed is an example of a ensemble of the first > genus. There are a finite number of points and lines in the figure. Having > said that, we can see what is involved in an ensemble of the second genus if > we consider the continuous transformations of this diagram that can be made > by moving around the points at the origin. In this way, we now have an > ensemble involving an infinite number of lines and points. In order to > conceive of what a tout ensemble involves, we need to consider the space as a > whole in which the diagram has been constructed. Just as the drawing in the > perspective space has a horizon where the parallel lines meet, there is > effectively a generalization of the "infinitely distant horizon" that is > built into the very nature of the projective space. What is really neat about > projective geometry is that this notion of the infinite is part of the space > itself. It is unlike Euclidean geometry, where the infinite is an exception > to the rule that all lines intersect. In projective geometry, we get greater > symmetry and harmony in the construction of the space because all lines > intersect--including those that are parallel. > > What more follows when we think about the character of the different kinds of > order that obtain between the points on the horizontal line? The key idea, I > take it, is that the line segments between these points A, B, C, A', B' C' > maintain relations of proportionality under all transformations. One reason > that the proof of this theorem is so important in projective geometry is > that, in effect, the rays from the two origins do not map onto some object > (such as a lute). Rathter, they "map" onto each other. It would take some > time to explain what follows from this feature of the construction. So, for > those who would like a summary of the conclusions, here is a link: > http://www2.washjeff.edu/users/mwoltermann/Dorrie/63.pdf > <http://www2.washjeff.edu/users/mwoltermann/Dorrie/63.pdf> > > The reason I have spent some time running through this example is that it > provides the resources needed to understand what Peirce is saying about the > mathematical conception of the absolute. For the sake of providing an > intuitive approximation of the conception of the absolute, note that the > horizon in perspective geometry is something that can be extended all of the > way around the point of perspectivity. Imagine standing at that point, > turning all the way around, and looking at the line that would be constructed > on the image plane if that plane were to circle around you. In effect, there > would be a circle draw on the image plane, where the image plane itself would > form a circular band (like a cylinder) around the point of perspectivity. > > In order to generalize that idea and apply it to projective geometry, we need > to remember that the projective plane is a generalization of a space (i.e., > one unified and homogeneous plane) in which any number of points of > perspectivity (origins) can be located, as can any number of image planes > (what we are treating as the horizontal line in our example). So, if we > generalize, we effectively get an ellipse, parabola or hyperbola that is in > the projective space and that functions as a generalization of the infinitely > distant horizon in perspective geometry. The ellipse, parabola or hyperbola > are related to one another as so many different conic sections. Just as we > can transform the diagram by moving around the origins, so too can the > character of the infinite horizon be transformed in that space. Those > transformation were seen by Cayley and Klein to be the key to understanding > how all metrical geometries are related to each other. > > Peirce is arguing that we should picture the starting and ending points in > the evolution of the cosmos to be analogous to the starting and ending points > of inquiry. In both cases, if we map the transformations in our understanding > or in the universe onto a surface, they are related to each other in a > hyperbolic fashion and not in a parabolic or elliptical fashion. Only in the > hyperbolic case are we able to treat the loci on those curves that serve as > the bases of the metrical systems as both real and different. > > Peirce is drawing on these points when he offers the examples in order to > work his way up to the logical conclusion that he draws about the conception > of continuity on page 253-4. Before turning to questions of metaphysics and > the example of the diagram of the lines on the chalkboard, it would probably > be worth asking if there are any questions about how Peirce is drawing on > these mathematical examples in order to clarify the logical conception? Much > of what he is doing is backing up from these sorts of points about the > absolute in projective geometry in order to ask yet more fundamental > questions about how the parts of such spaces (e.g., the projective space) are > connected when we study them in the context of topology. > > --Jeff > > Jeffrey Downard > Associate Professor > Department of Philosophy > Northern Arizona University > (o) 928 523-8354 <tel:928%20523-8354> > From: Jon Alan Schmidt <[email protected] > <mailto:[email protected]>> > Sent: Monday, November 7, 2016 11:41 AM > To: Jeffrey Brian Downard > Cc: [email protected] <mailto:[email protected]> > Subject: Re: [PEIRCE-L] Re: Super-Order and the Logic of Continuity (was > Metaphysics and Nothing (was Peirce's Cosmology)) > > Jeff, List: > > I need to offer a correction (in bold) to one of your Pierce quotes. > > CSP: The Absolute in metaphysics fulfills the same function as the absolute > in geometry. According as we suppose the infinitely distant beginning and > end of the universe are distinct, identical, or nonexistent, we have three > kinds of philosophy. What should determine our choice of these? Observed > facts. These are all in favor of the first. (R 928:7, W 8:22; 1890) > > As for your concluding question, I hope that you will share with us your own > answer to it. > > 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> > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] > . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] > with the line "UNSubscribe PEIRCE-L" in the BODY of the message. 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