Jeff: Parallelograms of forces about interacting electrical charges require a spherical mode of description. Correspondingly, an explanation of spherical forces requires categorial illations.
Can the diagram be extended to spheres? Cheers Jerry > On Mar 17, 2017, at 10:19 AM, Jeffrey Brian Downard <jeffrey.down...@nau.edu> > wrote: > > Jon A, John S, Jon S, Clark, List, > > I'd like to refer back to an earlier discussion of the last lecture in RLT in > order to take up the question of how we try to supply a diagram for better > understanding the relationship between truth and the starting and ending > points of inquiry. The diagram below is a sketch of how I would like to > develop the philosophical implications of what Peirce says about the > mathematics of projective relations. Let me know if you have suggestions for > making things clearer. > > Peirce argues that, in regard to the principle of movement from these two > points, only three types of philosophical positions are possible. Here are > the passages I'm trying to interpret: > > 1. Elliptic philosophy. Starting-point and stopping-point are not even > ideal. Movement of nature recedes from no point, advances towards no point, > has no definite tendency, but only flits from position to position. > 2. Parabolic philosophy. Reason or nature develops itself according to one > universal formula; but the point toward which that development tends is the > very same nothingness from which it advances. > 3. Hyperbolic philosophy. Reason marches from premisses to conclusion; nature > has ideal end different from its origin. (CP, 6.581-2) > > Pierce argues that the conception of the absolute in philosophy “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 favour of the first.” [W > 8.22; 1890], [CP 4.145; 1893]) Drawing on the series of mathematical > examples copied below, let's construct a diagram to illustrate Peirce's claim > that“[j]ust as geometry has its descriptive and its metrical portions, the > former considering whether points coincide or not, the latter measuring how > far distant from one another they are... so logic has first to decide whether > a proposition or reasoning to be true or false, and secondly in the latter > case, to measure the amount of its falsity” [W 4.241; 1881], [W 5.166; 1885]) > > <Absolute for Inquiry diagram.png> > > While I'm not able to fit it into this diagram, one idea I'd like to capture > is that of the relations of proportion in quantitative inductive inferences > between what has been observed and what might observed in the future. In > order to understand the significance of picturing the horizon as hyperbolic > lines, see the discussion below. > > --Jeff > > > Jeffrey Downard > Associate Professor > Department of Philosophy > Northern Arizona University > (o) 928 523-8354 > > > From: Jeffrey Brian Downard <jeffrey.down...@nau.edu > <mailto:jeffrey.down...@nau.edu>> > Sent: Wednesday, November 16, 2016 10:04 AM > Cc: peirce-l@list.iupui.edu <mailto:peirce-l@list.iupui.edu> > Subject: Re: [PEIRCE-L] Re: Super-Order and the Logic of Continuity (was > Metaphysics and Nothing (was Peirce's Cosmology)) > > List, > > Following up on Peirce's line of argument about the differences between > elliptical, parabolic and hyperbolic philosophies in Logic and Spiritualism > (from 1905, CP 6.581-585), I'd like to offer some relatively simple examples. > The purpose of the examples is to illustrate how ideas from perspective and > projective geometries can be used to clarify the points Peirce is making > about the formal differences between these different ways of orienting > ourselves with respect to the beginning and ending points of inquiry when we > go about the business of framing hypotheses. > > For starters, let us draw out the analogy between the ending point of inquiry > and a point on the distant horizon. Like travelers who head off to the > horizon when they board different trains, different inquirers may try to > answer a question using different methods. What do we suppose might happen > when these different inquirers follow out these parallel lines of inquiry? > Should we suppose that such lines of inquiry will tend to converge on some > point of agreement--call it the truth--in the long run? > > I am interested in taking a closer look at what Peirce says about the analogy > between converging parallel lines in a perspective or a projective geometry > and converging lines of inquiry. My hunch is the Peirce is offering more than > just a mere analogy. What he is doing is exploring the formal features in the > mathematical conceptions used in projective geometry--including the > conceptions of invariance under transformation, proportion, the infinite, the > absolute, etc.--in order to see how those formal features and conceptions > might be employed in both formal logic and in semiotics in order to analyze > the role of these kinds of formal conditions and conceptions in reasoning > generally. > > These conceptions come into the semiotic theory in a number of ways. Like > Aristotle and Kant, Peirce is trying to provide a more adequate answer to the > questions of how we are able form general conceptions and how we are able to > apply them to particular cases in a more measured manner. Just as we face > the difficulty of determining what kinds of classifications really represent > natural classes, we face an analogous problem in determining how to establish > and apply standards of measurement that are not arbitrary. Some forms of > measurement are, for a given phenomenon, more natural and others are less so. > > So, we are studying the way that mathematicians have drawn on projective > geometry in order to clarify the formal conditions for applying any system of > space measurement (i.e., any sort of extensive measurement). Consider two > sets of train tracks that are heading off into the distance. Each set of > tracks will converge on the horizon. As the tracks converge in the image > plane, the distance between the ties on each track gets progressively closer > together. In fact, it appears to the eye that there is a proportionality > between the rate at which the tracks converge on the horizon and the rate at > which the railroad ties get closer and closer together. In a more faithful > representation, the railroad ties get so close together as the tracks > approach the horizon that it is no longer possible to distinguish one from > another. That is, they lose their individuality in the perspective > representation of these relations. > > <Perspective two sets railroad tracks horizon.png> > Figure 1. > > While these two sets of railroad tracks might appear to be running in > parallel directions, we can see that such an appearance is an illusion. After > all, if they were really parallel to each other, the two sets of tracks would > converge on the same point on the horizon. That is, in a perspective space, > all parallel lines converge on the same point at the infinitely distant > horizon. If the perspective from which we view these relations were to change > radically, the proportions between the railroad ties and lengths of the track > between ties would appear to be preserved. > > <railroad tracks perspective 2.png> > Figure 2. > > We can generalize on the relationships in the perspective images by noting > that what is true of railroad tracks running in parallel on the ground is > also true of contrails in the sky. That is, lines running above the horizon > converge in the same manner that lines do that are below the horizon. In > fact, if we look at the image plane, the lines below the horizon are > continuous with parallel lines running above the horizon. What is more, we > can consider what happens with there are larger numbers of lines all running > parallel to each other. > > <many parallel lines to horizon.png> > Figure 3. > > If we change the perspective taken on the figure above, we get a clearer > sense of how those lines intersect through the perspective space to form a > large number of quadrilateral figures. What is more, we see that the red > railroad track, which is on the right of the blue track below the horizon, is > on the left of the blue track above the horizon. > > <lines filling perspective space.png> > Figure 4. > > Now, for the part that is a bit harder to make intuitive. Remember that, in a > perspective space, we can imagine turning around a given point of > perspectivity and gazing at the horizon as it sweeps by. If the image plane > goes all of the way around the point of perspectivity when we turn around in > a circle, then the horizon, too, will circle around us on a cylindrically > shaped image plane. > > As we mentioned before, an intuitive way of thinking about a two dimensional > projective space is to try to picture it as a generalization of a slice > through the the image plane, the object plane and the point of perspectivity. > In a projective space, we can see that the horizon is a part of the fabric of > that space. In this way, the horizon line that consists of the points where > all parallel lines converge is a conception of infinity that is different in > kind than the conception of infinity that we get in Euclidean geometry. That > is, instead of conceiving of what is infinitely distant as what happens when > parallel lines are extended further and further and never diverge or > converge, we get a conception of infinity as the collection of connected > points where all parallel lines would (i.e., have the potential to) converge. > > Let us now see what follows when we consider the proof of Desargues's 6-point > theorem from within such a projective space. Let us construct the diagram for > the proof--and let's do it in such a way that we place the two points that > are the origins of the rays in the construction (i.e., the points of > perspectivity at P and Q) on an ellipse that represents the horizon. The > central feature of the diagram is the quadrilateral PQSR formed by the > intersecting rays, along with two other lines that connect the opposite > vertices QR and PS of the quadrilateral figure. The 6 points on the line from > A to C' represent the pairs of points formed by extending the opposites sides > of the quadrilateral QS and PR and also the lines running through the pairs > of vertices QR and PS. As such, we have the pairs of points AA', BB' and CC'. > The proportionality between those pairs of lines forming the opposite edges > and lines through the quadrilateral are preserved under transformations--and > those proportional relationships are mapped to the 6 points on the line AC'. > > <Desargues 6-point horizon.png> > Figure 5. > > Now, let us generalize on the diagram by noting that, in the figure 4, the > quadrilaterals that are formed by the intersections between the parallel > lines are akin to the quadrilateral PQRS. That is, for any quadrilateral > formed by rays projecting from "opposite" sides of the line that forms the > horizon, we have the same relations of proportionality. Imagine all of the > parallel lines that can be drawn from the points P and R and the > quadrilaterals that are formed by the intersection of those lines. Now, > generalize and imagine all of the parallel lines that can be extended from > all of the points on the horizon. This is a rich form of super-order, and it > forms a tout ensemble for this particular kind of space as a whole having > that particular sort of horizon. > > Now, let us notice that the relations of proportionality between all of the > possible intersecting rays from all of the possible points on the horizon are > preserved under different transformations of the horizon. The horizon can be > moved and shaped into a parabola or an hyperbola. In this way, we get a yet > more general kind of super-order of super-orders, and a yet larger tout > ensemble for all of these 2-dimensional spaces. > > The general conclusions we have illustration by drawing on Desargues 6-point > theorem can also be illustrated by drawing on other important theorems in > projective geometry, such as the fundamental theorem. Let us construct this > theorem in a space in which the horizon is parabolic in order to illustrate > the idea. > > <Horizon fundamental theorem hyperbola.png> > > Peirce is correct when he suggests that these types of examples drawn from > mathematics need to be explored by the reader with pencil and paper in order > to sharpen the powers of imagination and reasoning--and to really see in a > more intuitive way what follows in a more general way about the space as a > whole from the proofs of the different theorems. Fortunately, we have > software such as CarMetal and the Geometer's Sketchpad. I recommend > constructing these diagrams, running through the proofs and seeing what > happens when the diagrams are transformed in a continuous way. The same is > true when it comes to the other examples in RLT that are drawn from topology. > The same holds for the diagram consisting of lines on the blackboard. The > movement from individual lines to the curve can be understood from within the > framework of projective geometry as having a number of important implications > about the way in which that curve is being generated from the lines. > > Thus far, in our exploration of these ideas from projective geometry, we made > use of a number of the conceptions that are italicized in RLT. Here is the > incomplete list that I included in the earlier email: (e.g, completed > aggregate, multitude brought to an end, potential, topical singularity, > furcation, perissid, artiad, fornix, line, filament, surface, film, space, > tripon, chorisis, cyclosis, immensity, ensemble, etc.). A number of the > conceptions have been used explicitly, and most of the rest have been used > implicitly as we have drawn out the implications from the diagrams. > > Now, let us ask: What is involved in the hypotheses that lie at the bases of > projective geometry? Consider what Peirce has to say here: > > In the pre-Lobachevskian days, elementary geometry was universally regarded > as the very exemplar of conclusive reasoning carried to great lengths. It had > been the ideal of speculative thinkers in all ages. Metaphysics, indeed, as > an historical fact, has been nothing but an attempt to copy, in thinking > about substances, the geometer's reasoning about shapes. This is shown by the > declarations of Plato and others, by the spatial origin of many metaphysical > conceptions and of the terms appropriated to them, such as abstract, form, > analogy, etc., and by the love of donning the outer clothing of geometry, > even when no fit for philosophy. For instance, one of the remarkable features > of geometry is the small number of premises from which galaxies of theorems > result; and accordingly it has been an effort of almost all metaphysicians to > reduce their first principles to the fewest possible, even if they had to > crowd disparate thoughts into one formula. It did not seem to occur to them > that since a list of first principles is a work of analysis, it would not be > a small number of elementary propositions so much as a large number that > would bespeak its thoroughness. > > As we continue the exploration of the topological diagrams that he offers > next as examples in RLT, we'll need to consider--in a more direct way--the > philosophical implications that he is drawing from within both phenomenology > and semiotics. As such, we will need to ask: how might we more carefully > analyze the role of the different formal elements in what we are observing > when we construct the diagrams and when we move things around to observe what > happens under continuous transformations. What is the role, for example, of > iconic representations and relations in our thinking about such > transformations. Furthermore, what is the role, of the logical interpretants > in drawing out the various implications? > > --Jeff > > > Jeffrey Downard > Associate Professor > Department of Philosophy > Northern Arizona University > (o) 928 523-8354 > > > From: Jeffrey Brian Downard <jeffrey.down...@nau.edu > <mailto:jeffrey.down...@nau.edu>> > Sent: Monday, November 14, 2016 5:06 PM > Cc: peirce-l@list.iupui.edu <mailto:peirce-l@list.iupui.edu> > Subject: Re: [PEIRCE-L] Re: Super-Order and the Logic of Continuity (was > Metaphysics and Nothing (was Peirce's Cosmology)) > > Jon S, List, > > Before I try to provide an outline of where I'm heading, let me anchor what > I'm trying to do in a specific text. The philosophical points that Peirce is > trying to draw from the mathematical examples in RLT are offered, at least in > part, in support of the following kind of argument: > > In regard to the principle of movement, three philosophies are possible. > 1. Elliptic philosophy. Starting-point and stopping-point are not even ideal. > Movement of nature recedes from no point, advances towards no point, has no > definite tendency, but only flits from position to position. (CP, 6.582) > 2. Parabolic philosophy. Reason or nature develops itself according to one > universal formula; but the point toward which that development tends is the > very same nothingness from which it advances. (CP, 6.582) > 3. Hyperbolic philosophy. Reason marches from premisses to conclusion; nature > has ideal end different from its origin. (CP, 6.582) > > The choice of elliptic philosophy, which refuses to acknowledge the ideal, > supposes more interest in nature than in reason. The philosophy which sees > nothing in nature but the washing of waves on a beach cannot consistently > regard mind as primordial, must rather take mind to be a specialization of > matter. Bent on outward studies, it will find the statement that nerve-matter > feels, just as carmine is red, a convenient disposition of a troublesome > question. Elliptic philosophy is irreconcilable with Spiritualism. (CP, 6.583) > > He who feels himself and his neighbors under the constraints of overwhelming > power, from which they long to take refuge in annihilation – situation little > life as rounded with a sleep, readily accepts the idea that the world, too, > sprang out of the womb of nothingness to evolve its destiny, and into > nothingness back to return. Such life as this philosophy recognizes -- a > fatal struggle, a mere death-throe --it should extend throughout nature. Soul > should be a mere aspect of the body, not tied to it, therefore, but identical > with it. Nothing can be more hostile to Spiritualism than this Parabolic > philosophy. (CP, 6.584) > > Hyperbolic philosophy has to assume for starting-point something free, as > neither requiring explanation nor admitting derivation. The free is living; > the immediately living is feeling. Feeling, then, is assumed as > starting-point; but feeling uncoördinated, having its manifoldness implicit. > For principle of progress or growth, something must be taken not in the > starting-point, but which from infinitesimal beginning will strengthen itself > continually. This can only be a principle of growth of principles, a tendency > to generalization. Assume, then, that feeling tends to be associated with and > assimilated to feeling, action under general formula or habit less common in > this country and age than in other places and times -- viewing this tending > to replace the living freedom and inward intensity of feeling. This tendency > to take habits will itself increase by habit. Habit tends to coordinate > feelings, which are thus brought into the order of Time, into the order of > Space. Feelings coordinated in a certain way, to a certain degree, constitute > a person; on their being dissociated (as habits do sometimes get broken up), > the personality disappears. Feelings over whose relations to their neighbors > habit has acquired such an empire that we detect no trace of spontaneity in > their actions, are known as dead matter. The hypothesis here sketched, whose > consequences, traceable with precision to considerable detail in various > directions, appear to accord with observation, to an extent of which I can > here give no idea, affords a rational account of the connection of body and > soul. This theory, so far as I have been able as yet to trace its > consequences, gives little or no countenance to Spiritualism. Still, it is > evidently less unfavorable than any other reasonable philosophy. (CP 6.585) > > He makes these sorts of claims in a number of places (e.g., CP, 8.317). The > three types of "philosophies" that Peirce characterizes here are emblematic > of three approaches to framing hypotheses about beginning and ending points > in a developmental sequence, including those that are more spiritual (i.e., > inward) in character as well as those that are more natural (i.e, outward) in > character. For the purposes of developing a logical account of the growth of > understanding, he is arguing for (3) as a plausible hypothesis and rejecting > (1) and (2) because they fail to explain a number of surprising phenomena > that are associated with the growth of understanding. What is more, both (1) > and (2) have the effect of closing the door to further inquiry in a > particularly problematic manner. A parallel argument is made about the growth > of order in nature. > > The mathematical examples in RLT that draw on projective geometry and the > relation to metrical geometries as well as the examples that draw on the more > fundamental ideas in topology are offered by Peirce for the purposes of > getting clearer about a number of key mathematical conceptions that he is > putting to use in the development and refinement of the conception of > continuity within the logical theory. Many of these conceptions are > identified in italics in the text (e.g, completed aggregate, multitude > brought to an end, potential, topical singularity, furcation, perissid, > artiad, fornix, line, filament, surface, film, space, tripon, chorisis, > cyclosis, immensity, ensemble, etc.), and he is applying these mathematical > conceptions for the purpose of addressing the problem of how to clarify a > logical conception of continuity that will be adequate for a normative theory > of semiotics. > > Peirce focuses on the evolution of more determinate dimensions from a vague > potentiality in the account of the logical conception of continuity at 253-4 > in RLT because he is trying to articulate an explanation of how the various > dimensions of our thought might evolve. Those logical dimensions can be > divided, in quite a fundamental way, according to three dimensions of yet > further dimensions of possibles, existents and necessitants of signs, > objects, interpretants and their relations within a growing system. > > --Jeff > > > Jeffrey Downard > Associate Professor > Department of Philosophy > Northern Arizona University > (o) 928 523-8354 > > > From: Jon Alan Schmidt <jonalanschm...@gmail.com > <mailto:jonalanschm...@gmail.com>> > Sent: Monday, November 14, 2016 3:17 PM > To: Jeffrey Brian Downard > Cc: peirce-l@list.iupui.edu <mailto:peirce-l@list.iupui.edu> > Subject: Re: [PEIRCE-L] Re: Super-Order and the Logic of Continuity (was > Metaphysics and Nothing (was Peirce's Cosmology)) > > Jeff, List: > > I am definitely interested, but it would be helpful to me if you could first > outline where you see this ultimately going, and then proceed in smaller > steps. As you could probably tell, I had trouble making the connection > between Desargues' theorem and Peirce's conception of continuity, not to > mention the subsequent blackboard diagram; and my own intuition (or perhaps > lack thereof) is such that discussing "the projective absolute" and "metrical > relations in elliptical, parabolic or hyperbolic geometries" is not (at least > so far) helping me understand your/Peirce's point "about the kind of > hypothesis that is needed to make sense of ... the growth of order in the > cosmos." Also, I still believe that Peirce's "table of contents" in "A > Neglected Argument" was for a future book that he had not yet written and > never did manage to write, rather than anything specific in his previous > material such as RLT. > > 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 Mon, Nov 14, 2016 at 3:45 PM, Jeffrey Brian Downard > <jeffrey.down...@nau.edu <mailto:jeffrey.down...@nau.edu>> 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 <tel:928%20523-8354> > > <message-footer.txt> > ----------------------------- > 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 > <mailto:peirce-L@list.iupui.edu> . To UNSUBSCRIBE, send a message not to > PEIRCE-L but to l...@list.iupui.edu <mailto:l...@list.iupui.edu> with the > line "UNSubscribe PEIRCE-L" in the BODY of the message. 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