Jeff, List: Thanks for the additional clarifications. I have just one more comment for now.
JD: What is more, Peirce is not claiming that there are no discrete positions. What I take Peirce to be claiming is that discrete positions are not *real parts* of space, and discrete instants are not *real parts* of time. Instead, they are artificial creations of thought for *describing *a hypothetical instantaneous state of the dynamic process of motion, which does not *really *consist of such states because it is truly continuous. Likewise, definite propositions are not *real parts* of an argument; instead, they are artificial creations of thought for *describing *a hypothetical instantaneous state of the inferential process of semeiosis, which does not *really *consist of such propositions because it is truly continuous. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Sat, Sep 7, 2019 at 2:12 AM Jeffrey Brian Downard < [email protected]> wrote: > Jon, List, > > It appears that we are talking past each other in a number of places. Let > me try to point some of them out. > > JD: As such, I accept the claim that a set-theoretical analysis of the > real number line is not adequate for a full understanding of topological > continuity. Having said that, I reject that idea that the set-theoretical > analysis is not helpful for clarifying key notions involved in the > conception of continuity, including the character of various kinds of order > that may pertain to continuous manifolds. > > > JS: Understood. We have certainly learned a lot about about continuous > spacetime by describing it using discrete positions and instants, which are > artificial creations of thought. > > JD: I recommend separating the remarks about mathematics from those that > pertain to physics. It is clear that discrete points in math are > idealizations. The entire mathematical systems are idealizations. Peirce > *claims* that there are no actual points in physical space or time that > are like a mathematical point in having no length or breadth and I am > sympathetic to the claim. Having said that, it is not obvious that the > metaphysical assertion is correct. There are many who disagree. What is > more, Peirce is not claiming that there are no discrete positions. Physical > collisions between bodies do have some discrete characteristics, and they > do take place in positions that are relatively discrete in some respects. > Again, here is the diagram were are considering: > > JD: One might claim that circle A is continuous, while B and C are not. > After all, f and g are discrete points of intersection between B and C, > while A has no such discrete points. > > > JS: That depends on whether points f and g are considered *parts *of > circles B and C. > > JD: The construction I've provided was intended to be an intersection > between circles B and C. I am asserting that the points of intersection f > and g are parts of both circles. Later in the post, I am pointing out that > those intersections connect the circles--creating multiple paths between f > and g. As such, these discrete points--that create some kinds of > discontinuity--are also connections. > > CSP: If in an otherwise unoccupied continuum a figure of lower > dimensionality be constructed--such as an oval line on a spheroidal or > anchor-ring surface--either that figure is a part of the continuum or it is > not. If it is, it is a topical singularity, and according to my concept of > continuity, is a breach of continuity. If it is not, it constitutes no > objection to my view that all the parts of a perfect continuum have the > same dimensionality as the whole. (CP 4.642; 1908 May 26) > > JD: Peirce's point in this passage is pertinent to the distinction > between the intrinsic character of each figure A, B&C, D, and the extrinsic > character of the surface on which they have been drawn. In topology, we > often separate the two--focusing solely on the intrinsic character of a > given figure A as a 1-dimensional manifold without considering what kind of > space it might be drawn in. Circle A could, for instance, be drawn in a > 3-space. It would still have the intrinsic character of being a 1-manifold. > > JS: If so, then they are topical singularities, and therefore breaches of > continuity; at best, circles B and C are then *imperfect *continua. If > not--if we instead treat points f and g as artificial additions--then B and > C are indeed still *perfect *continua. There are no points shared by two > overlapping circles, because there are no points in a circle *at all*. > > JD: This seems to involve a number of confusions. It is possible that I > am the one who is confused, but I tend to think you are mixing things > together that need to be kept separate. I accept that B and C are imperfect > continua once they are drawn in such a way that they intersect. If we > imagine them drawn separately and then brought together by a movement of > the filaments by translation so that they come to intersect, then the > process of constructing a figure having two intersecting points does create > some discontinuities. Having said that, there are also new connections. > What is more, the figure is now a higher kind of manifold. > > CSP: In a continuum there really are no points except such as are marked; > and such interrupt the continuum. It is true that the capability of being > marked gives to the points the beginnings of *potential being*, but only > the beginnings. It should be called a *conditional being*, since it > depends upon some will's being exerted to complete it. (R 1041:13; 1906) > > > JS: Unless and until points f and g are *marked *by an exertion of some > will, they only have *potential *being. There is also a sense in which > each circle is "a figure of lower dimensionality" than the surface on which > it is drawn (or imagined). If it is considered a *part *of that plane, > then it is a topical singularity, and therefore a breach of continuity that > divides the plane into three parts--the circle itself, the area inside it, > and the area outside it. If it is *not *considered a part of the plane, > then the latter remains a perfect continuum. > > > JD: Again, I would keep the questions about the intrinsic character of > the figure separate from the questions about the character of that figure > being extrinsically considered as a part of a larger surface. I agree with > the point that any line drawn on a surface introduces a discontinuity in > the surface, just as an intersection made on a line creates a discontinuity > on the line. I did exert my will in drawing the figures, and that exertion > followed a general rule in generating the circular figures and then writing > them down as actual figures. The discontinuities that are created at the > points of intersection f and g have a lot to do with the fact that this > particular token figure involve the placing of these token circles in an > existential relation, one to another. It does not follow that the lines, > considered intrinsically, do not have certain kinds of continuity that > remain. After all, the intersection does not prevent a particle from > tracing a path all the way around B or C. > > CSP: Are these sets all discrete in the same way? When it comes to the > types of things collected in each set, I think not. > > > JS: My impression is that "the types of things collected in each set" are > supposed to be irrelevant to the *mathematical *properties of the sets > themselves. Am I wrong? > > JD: After recognizing the paradoxes that had arisen in set theory, > Russell introduced type theory into the interpretation of sets in order > to avoid the problems. The theory of types explicitly recognizes the > different sorts of things that are being collected in a given set. > Russell distinguished, for instance, between sets of individual objects, > predicates and predicates of predicates. Notice that we can have a set of > {horse, dog, cat} and another set {Secretariat, Lassie and Kitty}. The > first is a set consisting of three kinds of things. The second is a set > of three individual objects. When constructing sets, mathematicians and > logicians sometimes seem to feel that they can put anything into a given > collection (creating sets of mixed types). The problem, as Peirce is well > aware, is that there is a difference between a type and token--and that > difference is sometimes ignored thereby causing confusion later when they > try to reason about the sets. > > JD: If you want to get clearer about the nature of mathematical > continuity, I recommend considering some concrete examples and then asking > questions about them. Passages drawn from Peirce's writings will only take > us so far. > > > JS: As I have said before, although it is no doubt helpful, I do not > believe that a *mathematically rigorous* conception of continuity is a > strict prerequisite for a *philosophically adequate* conception of > continuity. The latter is what primarily interests me; in particular, > Peirce's late topical theory as it applies to his semeiotic and metaphysics. > > JD: Adequacy comes in degrees. Peirce seems to think that we can improve > the rigor and adequacy of our mathematical conception of continuity and > thereby improve the rigor and adequacy of our logical and metaphysical > conceptions of continuity. I am attracted to Peirce's suggestion that we > should do what we can to improve rigor in philosophy by drawing on richer > mathematical conceptions. It appears to be an unending task. Nonetheless, > we stand here at the beginning of the 21st century with a wealth of > resources at our disposal. I want to make use of what is available to the > best of my abilities. > > --Jeff > >>
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