Jeff, List: I *think *I get it now--infinitely many parallel lines would all converge at a single point on the horizon; and there are infinitely many *potential *collections of such lines, each of which would converge at a *different *point on the horizon; so the horizon itself is a *continuum *of those potential points. Right?
Thanks, Jon On Wed, Mar 29, 2017 at 10:41 AM, Jeffrey Brian Downard < [email protected]> wrote: > Hi Jon, List, > > In response to your remarks about points and lines on the horizon. > > Let's focus on the case of perspective geometry and leave projective > geometry to the side for the time being. I recommend thinking of the points > on the horizon I have drawn in several figures that use the idea of > railroad tracks running off to distant points as examples of what all > of the possible parallel lines would look like that run through every part > of the space. All of those possible parallel lines form a line on the > horizon. > > That is what I am trying to illustrate with the diagrams having larger set > of parallel lines going to the same two points. Every possible point on the > horizon has an unlimited number of possible parallels line that could run > through it from every possible part of the larger two dimensional space. > That is what one is able to see quite readily when one alters the diagrams > by processes of continuous transformation. > > --Jeff > Jeffrey Downard > Associate Professor > Department of Philosophy > Northern Arizona University > (o) 928 523-8354 <(928)%20523-8354> > > ------------------------------ > *From:* Jon Alan Schmidt <[email protected]> > *Sent:* Tuesday, March 28, 2017 6:43 PM > *To:* Jeffrey Brian Downard > *Cc:* [email protected] > *Subject:* Re: Diagramming Inquiry (was Super-Order and the Logic of > Continuity) > > Jeff, List: > > JBD: As in projective geometry, the points on the horizon form lines. > > > I am not very familiar with projective geometry, and that is probably why > I am having a hard time understanding how points on the horizon can form > lines, hyperbolic or otherwise. Is it a change in perspective that > facilitates this transformation? Is it a limitation of your > two-dimensional diagram that you have to show multiple points along each > line? > > JBD: Inquiry starts with surprising phenomena that are a source of doubt. > As such, the points on the far left where the lines originate are meant to > pick out the sources of those doubts, It is likely that they will need to > be represented in a way that better captures the vagueness that is part and > parcel of such doubts (as a kind of grey area on the line, and not a > definite point). > > > In that case, should each group of parallel lines of inquiry perhaps > originate from the same vague region of the line on the left, rather than > discrete points? Maybe the lines of inquiry themselves should also be > fuzzy at first. Or is the idea that distinct lines of inquiry that > ultimately converge are initially prompted by *different *sources of > doubt? > > JBD: In a way, the collection of meeting points at the infinitely distant > horizon that form the hyperbolic curves are a kind of origin from which the > continuity of the larger space gets its ultimate shape. > > > This is nitpicking, but obviously Peirce would not accept a collection of > points as constituting a continuum. Again, maybe I am just not > understanding how projective geometry works. Normally we think of only *one > *horizon, but your diagram has *two *curves that are separate. I > understand that this represents how the end is different from the origin, > but it is not very intuitive for most people (myself included). > > JBD: In effect, our understanding of the aims and principles of > reasonable inquiry grows as our beliefs and theories grow. > > > That makes sense, but it still seems contrary to the idea that the lines > of inquiry *converge*. My thought was that the number and variety of > beliefs and theories being considered *shrinks *over time, until the > final opinion consists of "the truth, the whole truth, and *nothing but* > the truth." > > JBD: As such, the converging lines of inquiry are supposed to represent > the manner in which, over the course of time, inquiry makes incremental > improvements in the proportions ... > > > Right, so my thought is that maybe the circles should represent the > measurement of total *falsity *in the collection of all believed > propositions, which would cause it to get smaller rather than larger, until > it disappears at the end of inquiry. > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Professional Engineer, Amateur Philosopher, Lutheran Layman > www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt > > On Tue, Mar 28, 2017 at 11:15 AM, Jeffrey Brian Downard < > [email protected]> wrote: > >> Jon S, List, >> >> Thanks for your questions about the sketch of a diagram I've offered as >> part of an interpretation of Peirce's remarks about conceiving of the >> starting and ending points of inquiry in terms of a conception of the >> absolute drawn from projective geometry. >> >> Here are some initial responses to the first four questions you've asked. >> I will need to think a bit more about the 5th question before responding in >> greater detail. >> >> Having said that, let me offer these initial responses to your thought >> provoking questions. >> 1. With respect to the sketch of a diagram, I refer to the starting and >> ending *points *of inquiry, but then represent them as hyperbolic >> *curves*. As in projective geometry, the points on the horizon form >> lines. In this diagram, the lines are hyperbolas, and they are meant to >> represent in iconic form what Peirce says about a hyperbolic philosophy in >> the passage quoted. Is the idea that each point on the left curve is the >> start (in the indefinite past) of a distinct line of inquiry, some of which >> eventually converge (in the indefinite future) at a single point on the >> right curve? Inquiry starts with surprising phenomena that are a source of >> doubt. As such, the points on the far left where the lines originate are >> meant to pick out the sources of those doubts, It is likely that they will >> need to be represented in a way that better captures the vagueness that is >> part and parcel of such doubts (as a kind of grey area on the line, and not >> a definite point). >> 2. Is there a connection between these curves and points with the >> concepts of continuity and discontinuity, respectively, within Peirce's >> synechism? Yes, in a quite a keep way. The absolute is, within projective >> geometry, a very special part of the space, especially when it is >> represented by a hyperbolic curve. In a parabolic system (e.g., Euclidean >> geometry) parallel lines never meet. In a hyperbolic system, they do meet >> and in a real place within the space. In a way, the collection of meeting >> points at the infinitely distant horizon that form the hyperbolic curves >> are a kind of origin from which the continuity of the larger space gets its >> ultimate shape. That is, every possible point in the space is characterized >> in its relation to every other point by the way they and their >> relations ultimately relate to the infinitely distant horizon--and the rays >> extending from one horizon to the other makes this something that we can >> observe in the diagram. >> 3. Does semeiosis fit into the picture somehow--in particular, the >> progression of immediate, dynamic, and final interpretants? Yes, looking >> back in the past, we can see that subsequent sheets of assertion >> would contain the expression of observations that have caused us surprise, >> and then the expression of our doubt, and then the formulation of question, >> hypotheses, deductions from hypotheses, tests, inductive conclusions drawn >> on the basis of data gathered as part of the tests--and so on through the >> cycle of inquiry. All of these would be expressed as books on each sheet of >> assertion, where the various pages in the gamma book represent what is >> found in the past sheets of assertion and what we expect to find in the >> future. >> 4. What exactly does it mean to say that "inquiry seeks to enlarge the >> circumference of the circle," given that multiple lines meeting at one >> point is the ultimate goal? Over time, as our understanding expands, there >> are more positive assertions that have, up to that point, met the test of >> experience. As such, we have observed more, and there is more that we can >> explain. In this respect, I am drawing on Emerson's essay "Circles" to >> explain some points Peirce is making about the growth of our understanding. >> I suspect that Peirce is drawing on this essay quite self consciously, >> largely because Peirce is making the same sort of points about Kant's >> discussion of the ever-expanding horizon of our experience that Emerson is >> making. Part of that ever expanding horizon is the growth of our beliefs >> and theories--but it also includes the growth of our ability to feel, >> to observe, to infer and to guide ourselves. In effect, our understanding >> of the aims and principles of reasonable inquiry grows as our beliefs >> and theories grow. >> 5. Let me offer some initial remarks about your last question. Note that >> I am using multiple circles (titled a bit) to represent past, present and >> future stages of our understanding. As such, we might image one stage >> containing a statement of some surprising phenomena, and then subsequent >> circles containing questions, hypotheses, consequences that are drawn from >> the hypotheses, tests, and the results of tests. The inductive inferences >> that we draw as part of conducting the tests and evaluating the data >> connect our assertions through so many observations--both actual and future >> possible. As such, the converging lines of inquiry are supposed to >> represent the manner in which, over the course of time, inquiry makes >> incremental improvements in the proportions that are represent in our >> general explanations based on the observations we've made thus far the >> inductive inferences one can permissible draw on the basis of such data. >> Hope that helps, >> Jeff >> Jeffrey Downard >> Associate Professor >> Department of Philosophy >> Northern Arizona University >> (o) 928 523-8354 <(928)%20523-8354> >> >
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