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:
[cid:0e8ecb79-c72a-41d3-94b7-96e15748d79c]
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
On Fri, Sep 6, 2019 at 8:09 PM Jeffrey Brian Downard
<[email protected]<mailto:[email protected]>> wrote:
Jon, List,
Generally speaking, one way to get clearer about a conception is to draw a
contrast to its opposite. As such, we stand to learn something about continuity
by closely examining discreteness. What is more, many things admit of matters
of degree. Continuity may be one of those things. As such, we will learn
something about what is most continuous by looking at the gradations of what is
less and then more connected.
When it comes to the mathematical conception of continuity, we need to remember
that Peirce holds that the differences between (a) topology and (b) symbolic
algebras is a matter of the dress they are wearing and not anything essential
to the subject matter. Anything that we might study in topology using diagrams
can also be studied algebraically using equations.
Set theory is really helpful in studying basic operations like addition,
subtraction, multiplication, and division in various algebras--including
logical algebras. That is especially true when it comes to the analysis of
various forms of order that matter when the algebra is applied to various
systems of number.
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.
Higher mathematics draws on the study of more abstract things, such groups and
categories, to look at transformations between different systems of
mathematics--such as transformations between topologies and sets. The study of
those transformations is especially enlightening for arriving at a fully
adequate conception of mathematical continuity.
The table that Francisco presents provides a comparison of different systems of
number. Examining the properties that each system does or does not have that
makes those fields more or less connected when assigned to lines, surfaces or
higher-dimensional manifolds might very well give us insight into the methods
for generating and analyzing constructions that possess all of the properties
needed to be fully continuous. That, at least, is what I take Francisco to be
doing in the paper. What is more, Francisco does examine the key passage where
Peirce articulates properties that are essential for an adequate conception of
continuity.
In response to the points you make at the end of your post, consider four
figures:
[cid:16d0966b134c14e7a1f3]
One might claim that circle A is continuous, while circles B and C is not.
After all, f and g are discrete points of intersection between B and C, while A
has no such discrete points.
In my view, A, B and C were all generated from the same general rule for the
circle. As such, each has a continuity that is different in kind from E or G.
What is more, h and i, as well as j and k are discrete endpoints. Are those
endpoints discrete in the same way that point l is discrete? My view is that
they are not. Consider the questions that might be asked about the figures.
Figure A is a 1-manifold. Is B also a 1-manifold? How many paths will take a
particle from f to g in the second figure, and how many will take a particle
from h to i in the third?
Consider the following examples of sets:
{A}
{B,C}
{D,E}
{f,g}
{h,i,j,k}
{l}
Are these sets all discrete in the same way? When it comes to the types of
things collected in each set, I think not. As such, I am more than willing to
consider how mathematicians such as Cantor have drawn on sets to clarify the
mathematical conception of continuity. I think we can lear a lot about
continuity using constructions that involve one or another type of discreteness.
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. Note that
the examples I offer here are similar to those he offers in the EM and NEM.
Yours,
Jeff
Jeffrey Downard
Associate Professor
Department of Philosophy
Northern Arizona University
(o) 928 523-8354
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