Hi Charles, List,
It is good to hear that you are doing better. I hope you don't mind if
I post a reply to your inquiry to the larger Peirce list.
One relevant post on topology and the existential graphs is copied
below (beneath your email inquiry). I'll search for the particular
post you are referring to--I believe it builds on the ideas I'm
exploring below. Towards the end of the discussion, I introduce some
ideas from projective geometry and make a few illusions about the
function of the concept of the self. I should have put the point in
simpler terms: how should the different representations of "self" be
portrayed in the gamma graphs when there is a dialogue taking place
between the Graphist and the Interpreter? In this kind of case, we
have two selves (or perspectives)--and it is especially important to
note the differences between these two perspectives when we
characterizes things in terms of different subjective possibilities.
While it isn't obvious from the post to the list, I was thinking
explicitly about the remarks Kant makes in the lectures on logic about
adopting a higher perspective--such as when one is guided by the ideal
of seeking the truth. If we think like a logician about our
representations from this higher point of view, then we should picture
the conceptions we are using to reason about things as so many Euler
graphs--where one is contained with another or where two overlap with
one another--as forming a great system (like the tree of Porphyry).
I'll send some material on the diagrams I am using to understand the
way Peirce is using the phenomenological categories to inform his
inquiries in semiotics separately.
--Jeff
Jeffrey Downard
Associate Professor
Department of Philosophy
Northern Arizona University
(o) 928 523-8354
Hi Jeff-
Hope you are well.
I have two questions which I've neglected to ask for so long that
they're way past any reasonable expiration date.
First: I've been trying to piece together some exchanges from Peirce L
from mid November 2015 and cannot locate some diagrams you refer to,
mention having attached, in a November 16 2015 post. Any chance you
could help with this?
Second: In the Stjernfeldt slow read you graciously responded to a
post from me to Peirce-L about P's theory of the self and offered some
assistance. I had questions about your use of an example from
projective geometry and you responded by suggesting a look at more
basic issues in plane geometry. I faltered in responding being at
that time pressed by health problems. I'm better now and would be
interested in picking up on your comments. I think it'd be best to
start with the exchange I am thinking of but cannot locate it in my
notes or Peirce-l archives. Any chance you could help with this even
greater obscurity?
Best,
Charles
________________________________________
From: Jeffrey Brian Downard
Sent: Sunday, January 25, 2015 5:02 PM
To: [email protected]; Peirce List
Subject: RE: [biosemiotics:7983] Re: NP 8.3 and the
Hello Ben, Gary F., and Mary,
First of all, a quick response to Mary's last post. I think there is
no need to bow out of the conversation. The purpose of the Peirce
list, as I understand it, is for people with different backgrounds to
work together to sort through the texts with the aim of improving our
shared understanding of the ideas and arguments. We can, of course,
go beyond the texts and talk about our own ideas or the ideas of
people other than Peirce. It will be helpful, of course, if we tell
the others on the list when we are trying to interpret Peirce and when
we are really trying to do something very different. My assumption is
that, whatever particular ends might be guiding each of us as we make
an entry or give a reply, we are all trying to find the truth about
the kinds of questions that Peirce was also trying to answer. As
such, the different particular ends we might have—including that of
making our own points or of interpreting Peirce’s texts—are all
subservient to this larger goal of engaging in honest inquiry in the
search for truth.
If protocol demands that we wait in silence for the experts on the
graphs to chime in, then I'll need to bow out as well. Despite the
fact that I've been working on these texts for a while, the Gamma
Graphs are still quite a puzzle to me. These graphical systems of
logic may have been Peirce's chef d'oevre, but I'm afraid that I lack
the cooking skills needed to set foot in his kitchen.
The underlying topological ideas that Peirce is drawing on are also a
source of puzzlement, so it is often hard for me to see what the
implications are of introducing a many of the topological conceptions
into the graphical system of logic. In my own studies, I've found
that introductory texts on topology could only take me so far. There
were just too many unstated assumptions that were being left out of
the explanations, and too many of the texts moved too quickly to
algebraic explanations--so it was difficult for me to improve my
intuitive grasp of the basic ideas. Peirce's own writings in topology
are helpful (see the New Elements of Mathematics, vol. 4), but often
it is difficult to see what he is doing when he imports the
topological conceptions into his work on the logical graphs.
So, if you want to study some topology for the purposes of
understanding how Peirce is using these mathematical ideas in the
development of the existential graphs, here are two recommendations of
resources that have been helpful for me.
1. Jeffrey Weeks, The Shape of Space
2. Norman Wildberger, Online Course on Algebraic Topology,
https://www.youtube.com/playlist?list=EC41FDABC6AA085E78
Let me add a few suggestions for thinking about Peirce's work in the
graphs--and the relevance of questions about such things as Mobius
bands and Klein bottles. It is possible that what I have to say is
only so much review for you and many others. Regardless, I'll provide
a short summary of some key ideas to see if we are on the same
page--and to give others who have a better understanding of geometry a
opportunity to spell out where I might be confused or mistaken.
Peirce firmly believes that one of the great advances that Cayley and
then Klein made in geometry was to show how projective geometry
supplies us with a basis for understanding the relations between
metrical geometries and how topology provides us with a basis for
understanding the relationships between and ordering of the other two
main branches of geometry. In effect, topology studies the properties
of connectivity and dis-connectivity of spaces, but there are no
postulates that enable us to work with straight lines or with lengths
of lines and degrees of angle. Given the difficulties of clarifying
what is involved in relations of continuous connectivity in space, the
mathematicians have made an executive decision to treat these matters
separately under the heading of topology. For a number of reasons,
Peirce thinks that this executive decision on the part of the
mathematicians was wise because it enables us to see how all of
hypotheses that lie of the foundations of geometry are related one to
another. Projective geometry takes up those properties of
connectivity and studies the properties of proportion that hold when
straightness is introduced, and metrical geometry adds the study
properties such as length and degrees of degree of angle. One of the
great advances in the 19th century was to see with greater clarity
that the kind of metrical geometry one is working with depends on the
system of homoloids that one takes to be dominant in setting up the
postulates for a metrical system.
Historically, mathematicians came at these issues by thinking about
different ways of stating what might obtain when one tries to draw
lines through a point that are parallel to a given line. In a system
of metrical geometry, there are three possibilities: (1) there might
be only one line that can be drawn, or (2) there might be no lines
that can be drawn that do no intersect with the given line, or (3)
there might be more than one line that can be drawn that do not
intersect. Mathematicians like Cayley and Klein thought about these
matters by working with the complex plane that has both real and
imaginary dimensions, and Klein then used group theory to spell out
the relations between the geometries in more general terms. The idea
of the imaginary dimensions of the complex plane adds a further
complication that Peirce shows us how to set to the side when he
provides an "introductory" explanation of these different kinds of
metrical spaces: see “The Non-Euclidean Geometry Made Easy” in vol. 8
of the Chronological Writings.
What does a Mobius strip or a Klein bottle have to do with any of this
discussion about the relationship between parabolic, hyperbolic and
elliptical systems of metrical geometries? The short answer is that
the questions are not directly related. Each of these kinds of
geometries can be orientable, and each can be non-orientable. The
orientable character of a space is determined by whether or not it has
a cross-cap. We can figure out where or not a given surface has such
a twist in it by decomposing the space and seeing if the parts match
up in the way that a Mobius band is connected.
The “curvature” (or non-Euclidean character) of a surface is something
we can figure out by putting a polyhedron over the space and counting
up the number of vertices, sides and edges. The Euler equation
enables us to see whether the space has a value for the Euler
characteristic that is negative, 0, or positive. In Sung’s response
to you, he seems to conflate the orientability of a space with the
Euclidean or non-Euclidean character of the space. For the sake of
simplicity, we can think of the orientability and the “curvature” of
surfaces as separate matters. Look at the tables that Weeks and
Wildberger each give, where they categorize surfaces by the number of
cross caps and/or holes that are found in them. Here is an
abbreviated version of their tables:
Euler # Orientable Nonorientable
2 Sphere
1 Disc Projective plane
0 Torus Klein bottle
-1 Double cross cap
-2 Double Torus Triple cross cap
(and so on, with higher negative values)
We can see that a torus and the Klein bottle have same Euler
characteristic of 0; as such, they are both homeomorphic with the
Euclidean plane. This is counterintuitive. After all, how can a
Euclidean plane be topologically homeomorphic with a torus? One is
clearly flat, and the other is clearly curved. The answer comes when
we remember that straightness and curvedness are something that we are
setting aside in topology. In topology, we have not settled on a
dominant system of homoloids. We’re only looking at how one part of a
surface is connected or disconnected with another part of the surface.
Topologically speaking, the parts of a one-holed torus are connected
to the other parts of the surface in the same ways that the parts of
the Euclidean plane are connected to each other—and the same holds for
a space that is shaped like a Klein bottle.
Having said this much about the Euclidean or non-Euclidean character
of two dimensional surfaces, what more can we say about the
orientability and nonorientability of such two dimensional spaces?
Once again, to simplify matters, orientability and two-sidedness are
related but separate matters. If you run around a space like the
Mobius strip—but one that has two twists and not just one—then you end
up with the same orientation that you started with once you return
back to your starting point. That is another way of saying that a
strip with two twists is really two-sided, but a strip with one twist
is really one-sided.
So far, our discussion of these ideas has been confined to
two-dimensional surfaces, and all of the points are about the
intrinsic properties of those surfaces. We could ask questions about
how those parts of the surface are related to parts of a three or
higher dimensional space of which the surface is just a part, but then
we would be looking at the extrinsic properties of the surface. The
division between intrinsic and extrinsic properties is something that
applies to any space we might study. We could, for instance, study
the intrinsic properties of three dimensional spaces, and then later
look at the extrinsic properties involved in thinking about the three
dimensional space as part of a larger four dimensional space.
With this bit of mathematical prelude, we could draw on some of these
ideas from topology—including the ideas of orientability and
two-sidedness of a surface--to try to sort out what Peirce might be
doing by introducing the idea of distinguishing between the recto and
verso sides of the sheet of assertion. I add the qualification
“might” to emphasize the fact that I take it to be an open question,
at least as far as my understanding goes, and my aim is to try to
figure it out with the help of others on the list.
When it comes to interpreting the suggestion that he has made a “new
discovery”, I think a great place to start is with the particularities
of his remarks in the essay on “The Improvement on Gamma Graphs.” Let
me start with the first thing he infers from the new discovery of how
we might work with a two-sided sheet, where the recto side represents
actual facts, and the verso side represents one or another of various
kinds of possibility. We should note that the first kind of
possibility he considers is subjective possibility. This is the kind
of possibility that involves ignorance on the part of the inquirer.
It is being used to deal with the kind of assertion we have when a
person asserts “The stove is not hot” and then adds, “but I’m not a
cook, so it is possible that I am in error.” Starting with this very
familiar sort of possibility, where we admit that there are a number
of qualifications we need to add to our assertions in order honestly
to admit it is possible we might be in error due to our ignorance of
certain relevant matters, what is involved in the very first inference
that he draws from the new discovery?
He says: “First, the cut may be imagined to extend down to one or
another depth into the paper, so that the overturning of the piece cut
out may expose one stratum or another, these being distinguished by
their tints; the different tints representing different kinds of
possibility.” (CP, 4.578)
What is implied in this sentence? We should note that there are a
number of ideas in play.
First, the paper has a depth such that a cut through it will expose
multiple stratum. Don Roberts interprets the passage in the following
way: “The cut would retain its function of negation, but the effect
of scribing a graph on the verso would be to exclude a possibility
(not simply an actuality) from the universe.” (88) Having made this
point, he adds that “Peirce arrived at this analysis while considering
the ‘anolomy’ by which in EG the relation ‘other than’ is expressed
differently from any other relation. (LN 265r). It is the only
relation requiring that a graph be partly in one area and partly in
another; it requires that a line of identity cross a cut.” Ben has
tried to explain some of the implications of seeing the anomoloy of
having to treat ‘other than’ in this unique way.
Second, the cut goes through several sheets, and it exposes one or
another of the stratum as it penetrates into the depths of the book.
As such, each sheet in the book has two sides, and the strata of the
layers are exposed by cutting through sheets having different tints of
color. Each of these tints represents different kinds of
possibility—such as the practical possibility that I am in error in
thinking that I can carry out a resolution, as well as the deeper
metaphysical possibility that my assumptions about the nature of the
real are in error.
Let me interpret the significance of this in light of what Peirce says
in the Prolegomena about the multiple sheets in the book. In this
essay, the different sheets in the book represent stages in a process
of inquiry, where the Graphist and the Interpreter are engaged in a
dialogue animated by the purpose of articulating the truth. As a
starting point, the Graphist and the Interpreter are taken to have a
number of shared assumptions. As Roberts points out, “the Phemic
sheet, before anything is scribed on it, represents whatever is taken
for granted at the outset by the Graphist and Interpreter.”
The Sheet of Assertion is devoted to the expression of propositions
held to be true, the Sheet of Interrogation is devoted to questions
one might raise about those propositions, while the Sheet of
Destination is devoted to resolutions. How might we understand
Roberts’s suggestion that the “tinctures were designed with more than
formal logic in mind; they were meant to provide a structure in terms
of which Peirce could apply his categories to propositions and
inferences, to hypotheses, questions, and commands.” (Roberts, 100)
That is, the graphs were designed to represent “all that ever could
ever be present to the mind in any way or any sense.” (Ms, 499(s)).
Consider a simple example, such as the case of a child who is learning
about a stove. Suppose the child already has associated words such as
‘stove’ with objects of this kind, and the word ‘hot’ with the
experience of heat. What is involved in the child moving beyond such
psychological associations and gaining logical self-control over the
conduct of his thought? From early in his career in “Questions
Concerning Certain Faculties” Peirce is trying to explain what is
involved in learning the logical meaning of falsity, and thereby
learning the meaning of error and the logical relationships involved
in negation. Let us suppose that the child has already formed a habit
of belief to the effect that all mid-sized objects that capture his
attention are suitable for being touched. What is involved in
learning that this assertion is false? In the case of the child, the
parents will try to stop him from touching the stove and will say “The
stove is hot, don’t touch!” Logically speaking, what is involved in
the child forming hypotheses to the effect that the representations
about the stove being suitable for being touched are false, and that
there is a real difference between what he has represented to be the
case and what his parents have represented to be the case? In
learning this, he gains a conception of his own mind as being distinct
from the minds of others.
In his discussion of the logic of relatives of second intention,
Peirce says the following:
“The general method of graphical representation of propositions has
now been given in all its essential elements, except, of course, that
we have not, as yet, studied any truths concerning special relatives;
for to do so would seem, at first, to be ‘extralogical.’ Logic in this
stage of its development may be called paradisiacal logic, because it
represents the state of Man's cognition before the Fall. For although,
with this apparatus, it is easy to write propositions necessarily
true, it is absolutely impossible to write any which is necessarily
false, or, in any way which that stage of logic affords, to find out
that anything is false. The mind has not as yet eaten of the fruit of
the Tree of Knowledge of Truth and Falsity. Probably it will not be
doubted that every child in its mental development necessarily passes
through a stage in which he has some ideas, but yet has never
recognised that an idea may be erroneous; and a stage that every child
necessarily passes through must have been formerly passed through by
the race in its adult development. It may be doubted whether many of
the lower animals have any clear and steady conception of falsehood;
for their instincts work so unerringly that there is little to force
it upon their attention. Yet plainly without a knowledge of falsehood
no development of discursive reason can take place.” (CP, 3.488)
Let’s see if we can draw on some ideas from topology to try and
clarify what might be going on in his claims about the method of
graphical representation that he is exploring in the Prolegomena and
the essay on the improvement in the Gamma Graphs. For the sake of our
discussion, let us suppose that the child’s way of experiencing such
things as the blackness and hotness of the stove are similar to our
own, but that they have not yet been sorted out in the way we’ve
sorted things out. It is as if the child has wandered into a cave
filled with experiences of various odors, warmth and cold, and
textures of smoothness and roughness, and has not yet figured out what
is up and what is down with respect to the feelings of such things as
hot and cold. Putting the matter in less metaphorical terms, let us
suppose the child does not yet have a conception of where to draw the
line between hotter and colder so that he can understand the real
difference between saying “the stove is suitable for being touched
because it is not hot,” and “the stove is too hot, don’t touch it!”
In the child’s way of seeing the world, the relationship between hot
and cold might be a matter of one feeling shading into the other, but
he has not yet established a suitable set of breaks in this
relationship. His sense of what is possible in the way of such
feelings is like a Mobius band because he has not yet established a
break in the continuity between one and the other kind of experience.
The key to explaining the break in the continuity between these
experiences is that he discovers, contrary to what he had supposed,
that he is unable to hold his hand to the stove despite his resolution
to do so. As such, the general supposition made in the child’s
initial hypothesis now runs up against a hard fact in the child’s
experience. The discovery that he is unable to hold his hand to the
stove connects the facticity of the balking of his will with the
possibilities that were represented under the hypothesis about stoves
generally being well suited for being touched. How might we represent
what is being learned here in a graphical system of logic? Peirce is
pointing out that a system like Euler’s graphs is unable adequately to
represent the connection that is being made between the possibilities
covered under the general concepts represented in the child’s
hypothesis and the actual fact that is learned through the experience
of being unable to carry out the practical resolution. The same is
true, he seems to be saying, about the Alpha and Beta graphs. They do
not contain the kinds of logical relations needed adequately to
represent what the child is learning about the relationship between
the general conceptions represented in the hypothesis and the
particularities of his experience of being unable to carry out his
resolution.
How is this represented in the Gamma Graphs? That is, how do the
Gamma Graphs represent the logical relations in a way that will
clarify the connections between asking questions, carrying out
resolutions, and revising our assertions in the light of experience?
We start by taking all that is shared in the child’s and parents’
understanding to be represented on the blank Phemic sheet. After
that, the child scribes his hypothesis on the Sheet of Assertion, and
then the parent’s challenge what is ascribed on that sheet by
interpreting things differently. At this point, there is a dispute
between the Graphist and the Interpreter, and we need to understand
how this dispute might be resolved through inquiry. Despite the
parent’s warnings, the child persists in trying to touch the stove,
and “I will touch the stove” is scribed on the Sheet of Resolution.
These are all separate sheets in a book. When the practical
resolution proves to be something the child cannot do, this creates a
cut through the sheets in the book exposing the strata of the layers.
At this point, it is difficult to sort out what, logically speaking,
might going on when the child has conducted the test of his hypothesis
and has discovered that it is false. In this Gamma system of
graphical logic--where we are trying out different ways of trying to
picture the relations as iconically as possible for the sake of
developing better philosophical explanations of these fundamental
logical conceptions--we are supposing that the child’s ability to use
his actual experiences of hot and cold as signs that refer to the
possible experiences he might have in the future are only relations of
references.
That is, these experiences stand in a dyadic relation of reference to
one another—but this is merely a matter of similarity. As such, the
child can understand that the feeling of the hotness of the cookie
when he eats it is a particular experience that is contained in the
more general experience of hotness, and the same holds for the
experience of touching the hot stove. Both experiences are similar
because they are both contained in the same general representation.
But the relation is only one of particular feeling that is contained
in the more general representation that serves as container. What is
needed to establish a referential relation, and then to establish even
richer relations that are materially and formally ordered?
In order to sort out the moves that Peirce is making in the
development of the Gamma Graphs, we’d need to try to picture how he is
thinking about the kinds of dyadic relations that are needed to get
transitivity in this ordering of hotter and colder tested against
actual experiences, and we would need to see how these dyadic
relations are being conceived of as part of larger triadic
relationship under general rules. So, we’ve got a lot to sort out.
Having said that, I’m hoping that we can see that the introduction of
the idea that we can distinguish between the recto and verso sides of
the sheets is something that is being applied to all of the sheets in
the book. So, we picture the child beginning to understand the
differences between what is on the recto side of the Sheet of
Resolution and the Verso side of that same sheet. And, what the child
sees is that the fact that he is unable to touch the stove establishes
that this resolution was not possible for him to carry out in this
case, and he sees that this has a number of logical implications for
what he has scribed on the sheet of assertion.
Let me make something of a leap at this point and offer the following
suggestion in the hopes of holding a number of threads together in my
hand—even if they are only being held together quite loosely. In
learning that his hypothesis about the stove being suitable for being
touched is false, the child is learning how to order his possible
experiences of hotter and colder and connect them up to what he can
and can’t do. On this line of thought, the balking of the will is the
ground for setting up an order so that he might, for instance, treat
colder as a positive values and hotter as negative values, and the
line between the two as a kind of zero point. (note: what matters is
the break between positive and negative, and not that one is positive
and the other is negative)
It should be clear that the assignment of the quantitative differences
of positive and negative values to the qualities of his experience
must be something of an achievement for the child. Lacking something
like a thermometer, how does he sort out the feelings of hotter and
colder? We can picture the connections that are made in terms of one
two-dimensional surface of assertion intersecting with another
two-dimensional surface of resolution. The possibility of having
these kinds of intersections may require that we conceive of each of
the sheets as both two-sided and orientable, and not as non-orientable
single-sided sheets. And this may be what is taking place when the
cut that is established on the sheet of resolution is connected to the
sheet of assertion, so that the child is able sort his actual and
possible experiences of hot and cold into a system that is both
materially and formally ordered under a rule.
In this case, the rule that is connecting them is a logical rule of
negation. And, we could try to picture how this sorting is grounded
by trying to develop a graphical system of logic where the recto and
verso on the sheet of resolution is connected to the recto and verso
on the sheet of assertion by a cut, and where the lines that connect
one experience of a stove to another later experience of the stove
establish both the dyadic and triadic relations necessary for the
child to understand the implications of saying to his parents “My
assertion was false, and I was in error.”
Ok, I admit that I am groping my way around in something like a dark
cave trying to get a handle on a knotted set of problems. How can one
make any progress if each of the problems seems to be a thread that is
firmly knotted together with the others? How can we begin to sort
things out so that we might grasp one clear question with a firmer
grip and then follow this thread through the knotted maze? Peirce
tries to explain the question I’m struggling with in the following way
in the Critic of Arguments:
Diagrams and diagrammatoidal figures are intended to be applied to the
better understanding of states of things, whether experienced, or read
of, or imagined. Such a figure cannot, however, show what it is to
which it is intended to be applied; nor can any other diagram avail
for that purpose. The where and the when of the particular experience,
or the occasion or other identifying circumstance of the particular
fiction to which the diagram is to be applied, are things not capable
of being diagrammatically exhibited. Describe and describe and
describe, and you never can describe a date, a position, or any
homaloidal quantity. You may object that a map is a diagram showing
localities; undoubtedly, but not until the law of the projection is
understood, nor even then unless at least two points on the map are
somehow previously identified with points in nature. Now, how is any
diagram ever to perform that identification? If a diagram cannot do
it, algebra cannot: for algebra is but a sort of diagram; and if
algebra cannot do it, language cannot: for language is but a kind of
algebra. It would, certainly, in one sense be extravagant to say that
we can never tell what we are talking about; yet, in another sense, it
is quite true. … It is requisite then, in order to show what we are
talking or writing about, to put the hearer's or reader's mind into
real, active connection with the concatenation of experience or of
fiction with which we are dealing, and, further, to draw his attention
to, and identify, a certain number of particular points in such
concatenation. CP, 3.419)
So, to end with a question: how should we conceive of the cut of
negation cutting through the Sheet of Resolution and the Sheet of
Assertion--thereby exposing a strata of layers?
My suggestion for interpreting the Gamma Graphs in light of the goals
established at 3.419 for developing graphical systems of logic is that
the cut forms a kind of boundary that exposes the relations between
the strata in the different sheets, and thereby establishes lines of
intersection between those sheets. The intersection is something that
causes us to re-interpret the connections between a given resolution
(e.g., the child says: “I can touch the stove”) with the discovery of
what he practically is unable to do (e.g., represented in his mind as
“I couldn’t do it”). This cut through the sheet of Resolution is
based on a point where the child has discovered a practical
impossibility. Many such discoveries can be connected together by a
rule—such as a rule of induction--thereby forming connected lines. In
this way, each of the points of practical impossibility are connecting
to the sheet of assertion—thereby establishing something like a
homoloidal relationship between what is asserted actually to be the
case and what might possibly follow from such an assertion.
Thanks for your patience with this overly long post, but I wanted to
try my hand at connecting some dots,
Jeff
Jeff Downard
Associate Professor
Department of Philosophy
---------------------------------------------------------
From: Libertin, Mary [[email protected]]
Sent: Sunday, January 25, 2015 6:42 AM
To: [email protected]
Subject: [biosemiotics:7983] Re: NP 8.3 and the
Dear Ben, Jeff, Gary F.,
Your comments make much sense. Upon rereading Peirce I realized one of
my
mistakes. I had been reading the verso as being on the underside of the
recto. It is rather on the same side. I did notice Peirce briefly
referring to Klein in his discussion of existential graphs. Klein
bottles
work in the fourth dimension and operate can be seen as consisting of
two
mobius strips in some fashion. I do not wish to pursue this except to
mention that my reference to the mobius strip in the context of
boundaries
and continuity was not totally off the wall, but it was a miss.
I will bow out of the conversation and listen to the experts. Thanks
for
your explanations and tactfulness with regards to my mistake.
Best,
Mary Libertin
On 1/22/15 3:39 PM, "Jeffrey Brian Downard" <[email protected]>
wrote:
Lists,
Ben has made a quick remark offlist, and I wanted to respond to the
Lists. He says, "A surprising thing to me is that Peirce in the Gamma
graphs treats possibility, necessity, etc. without mentionng that he
is
not starting like in probability theory from a set of given data
parameters like in probability theory, but instead (somewhat like
contemporary modal logic) supposing, for instance, unspecified
conditions, or an unspecified state of information, in virtue of which
which g is possible. Of course if one does it like probability theory,
then the possibilities and necessities are merely logical
possibilities
and necessities and don't belong to a separate province within logic.
The
approach of leaving unspecified the data parameters, the states of
information, etc., that one might like to specifically know, suggests
to
me the idea of devising deductive formalisms with special utility for
inductive inquiries. But that's just an initial impression."
Here is my response: interesting remarks, Ben, especially the "idea
of
devising deductive formalisms with special utility for inductive
inquiries." One of the moves Peirce makes as he transitions from the
Beta to the Gamma graphs is to think of the lines of identity as being
really composed of branching relations--at least potentially. In the
essay on the improvement of the Gamma Graphs, he says:
"The truth is that concepts are nothing but indefinite problematic
judgments. The concept of man necessarily involves the thought of the
possible being of a man; and thus it is precisely the judgment, "There
may be a man." Since no perfectly determinate proposition is possible,
there is one more reform that needs to be made in the system of
existential graphs. Namely, the line of identity must be totally
abolished, or rather must be understood quite differently. We must
hereafter understand it to be potentially the graph of teridentity by
which means there always will virtually be at least one loose end in
every graph. In fact, it will not be truly a graph of teridentity but
a
graph of indefinitely multiple identity." (CP, 4.583)
This shouldn't be too surprising, I think, because the lines of
identity
in the Beta system are thought of extensionally as existing objects
that
are joined by actually having or not having specific qualities. As
such,
the lines are an iconic representation of the dyadic relation of an
actual matter of fact. As Peirce says in his discussion of the
nomenclature and division of dyadic relations: "The author's writings
on
the logic of relations were substantially restricted to existential
relations; and the same restriction will be continued in the body of
what
here follows." (CP, 3.574)
Once we move from the Beta to the Gamma system, we are connecting
things
with different modal characteristics, and we are connecting things
across
different universes of discourse. As such, the character of the
connection between qualities that are present in an existing thing are
being connected to the possible qualities that possible things might
have
--including the possible changes that might occur to this object if
certain conditions were to obtain. Peirce sees that the specification
of
such possibilities is governed by some rule (i.e., either one in our
understanding or one that is in the world). Connecting qualities,
individuals and objects under rules requires some way of dealing with
the
generality of the rule itself and the way that it holds across
different
possible states of affairs.
So, here is a suggestion for Gary F., as he thinks about the character
of
the sheet of assertion in the Beta and Gamma systems. In effect, the
movement from the Beta to the Gamma graphs forces us to reinterpret
the
meaning of the empty spaces found on the recto side of the sheet of
assertion, and the relationships between those empty spaces and those
that are occupied on the verso side of that sheet. This gives new
meaning to the boundaries between spaces and the connections between
those spaces. Instead of thinking of the relationship between recto
and
verso extensionally as, "it is actually the case that this object x
has
this property F," and "it is not the case that this object x does not
have this property F," we are thinking differently about how the two
sides of the sheet are related one to the other. As I mentioned in an
earlier email, Peirce is classifying the referential relation as a
species of dyadic relation proper (i.e., one that is genuine and not
degenerat
e--as a reference happens to be). In the movement to the Gamma
Graphs,
Peirce is trying to find a way to represent--as iconically as
possible--the introduction of a triadic relation between reference to
ground, reference to object and reference to interpretant. All
legisigns
bring these three functions together and binding them together--under
a
rule--as it were.
One reason I find your last remark especially interesting, Ben, is
that
the leading principles of induction and abduction are rules of a
special
sort, and Peirce is trying to understand how we might clarify the
relationship between the rules of synthetic inference and the leading
rule that governs deductive inference. As a side remark, it is really
interesting to see him explore the limits of what could and couldn't
be
done with Euler graphs in his entry on that subject. A comparison
between Peirce's remarks on the limitations of the Euler system of
diagrams and what is introduced--piece by piece--in the development of
the Alpha, Beta and Gamma systems, is really quite instructive for
thinking about these big questions about the leading principles of
synthetic inference--and the grounds of the validity of these
principles.
Like you, I think that Peirce was very much motivated by these kinds
of
philosophical questions--and that they are helping him clarify many of
the goals that are guiding him in the development of the existential
graphs generally, and especially in the development of gamma graphs.
Peirce is focusing on the question of how to understand the nature of
different kinds of conditionals (and not just those that are
conditional
propositions de in esse) because we want to gain greater insight into
what is involved in the illative transformation when the reasoning is
synthetic and not just when the transformation is deductive.
Consider,
for instance, Peirce's remarks in the "Apology for Pragmatism" when he
explains why he chose the scroll as an iconic representation that
enables
us to see what is going on when we draw inferences from a conditional
proposition de inesse. (CP 4,564) Very quickly, he clarifies the
permissions (one might label them postulates, if you are thinking like
a geometer) called "the rule of deletion and insertion,"the rule of
iteration and reiteration,"and " the rule of the double cut," etc.
Let me close by saying that I place great weight on Peirce's
conclusion
that, ultimately, there are only three such permissions in the
existential graphs that are needed to understand the nature of the
illative transformation. Those are colligation, iteration and
erasure.
(CP, 5.579) My assumption is that he is making a point about any kind
of
illative transformation when he says this, and not just the
transformation involved in a deductive inference. After all, his main
point in this passage is that these three permissions are precisely
what
is needed in order to gain a deeper understanding of the self
correcting
character of any kind of inference--including inferences by induction
and
abduction.
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Benjamin Udell [[email protected]]
Sent: Thursday, January 22, 2015 11:19 AM
To: Gary Fuhrman; Jeffrey Brian Downard
Subject: Re: OFF-LIST Re: Contradictories, contraries, etc. WAS Re:
[PEIRCE-L] Natural Propositions : Chapter 8 - On the philosophical
nature of semiosis?
________________________________________
From: Benjamin Udell [[email protected]]
Sent: Tuesday, January 20, 2015 10:52 AM
To: [email protected]; Peirce List
Subject: Re: [PEIRCE-L] RE: NP 8.3 and the Improvement on the Gamma
Graphs
Mary, Gary F.,
Gary F., thanks for changing the subject title. I had renamed it
'Contradictories, Contraries [etc]' and then it unexpectedly veered
back
toward the original subject, I should have changed the title when that
happened.
Mary, you did indeed write the starting post in this subthread. (You
sent
it only to the biosemiotics list, but Gary Richmond forwarded it
peirce-l
(I provide these links so everybody can peruse)
(gmane) http://thread.gmane.org/gmane.science.philosophy.peirce/15394
(IUPUI)
https://list.iupui.edu/sympa/arc/peirce-l/2015-01/msg00102.html
and replied to it:
(gmane) http://thread.gmane.org/gmane.science.philosophy.peirce/15404
(IUPUI)
https://list.iupui.edu/sympa/arc/peirce-l/2015-01/msg00112.html
and your text (originally in reply to Jeffrey Brian Downard) follows
Gary R.'s reply.)
I wasn't active in the subthread till a bit later but I did read your
original post. As regards the questions that you posed there:
1. From your original post:
For example I, like many readers, relate the dicisign overall as
Stjernfelt has presented it to his far-reaching cpt. 8: "Operational
and
Optimal Iconicity in Peirce¹s Diagrammatology.² How do the two kinds
of
iconicity (chapter 8) Optimal and Operational Icons), make sense when
I
relate them to or place them in dialogue with the dynamic and
immediate
objects of the index? I wonder, does a dicisign posit or ³say² that
there exists (may exist, hypothetically exists) a written or spoken
proposition SRO (Subject Relation Object)? Š that the whole
proposition
(seen completed after the fact or seen hypothetically completed
before
the fact of writing or utterance or action) is made up of two parts?
To
distinguish the object as optimal and operational in relation to the
dicisign, I consider the index as it operates in an icon and the
index
as it operates as an index. (The node between the two, the index and
icon, as they reach out and for that moment exist. Is Stjernfelt
saying,
in other words, that there (1) exists an object, undistributed in
relation to the subject and that there (2) exists an object of this
specific subject under discussion that is distributed (that are under
discussion,that are being thought, that are coming into a realer or
more
iconic existence)? What and who have or will have placed these in
discussion may be the Grapheus and the Graphist, the realist and the
doubter, but the Universe.
[End quote]
I confess that I didn't understand it! I admit that I was feeling kind
of
obtuse. I was hoping that others' subsequent discussion would clarify
it.
2. From your original post:
I find some loose ends in my thinking about Peirce, amplified
somewhat
by NP. Is the recto/verso Sheet of Discourse, the ³leaf² pointed to
by
Stjernfelt, boundless, and in what dimension? I always imagine it as
a
mobius strip when the sign is in process, but the boundaries of the
Universe of Discourse that are discussed by linguists and others are
raised. Just now I continue with the leaf (sheet of assertion)
analogy
and consider the node of life at the stem as it grows. I will
continue
to think through these icons.
[End qote]
I thought about the Mobius strip idea but I stopped because I was
uncertain about whether existential graphs have chirality, but I think
that they don't, and anyway it doesn't matter (I was wondering about a
graph that locally seems on the verso, what happens to it when one
moves
it around the Mobius strip to what locally seems the recto). Shaping a
sheet into a Mobius strip makes it all recto and no verso, as Gary F.
said, and eliminates the ability to negate a graph. I think Peirce
somewhere talks about logic without negation. Anyway it can have only
particular affirmatives and conjunctive compounds of particular
affirmatives, a one-sided logic so the Mobius strip is actually
perfect
for it. If you want it to be unbounded, the surface of a Klein bottle
would do that
https://people.math.osu.edu/fiedorowicz.1/math655/Klein2.html . Anyway
I
guessed that you were trying to think of a way for there to be a
referential relation between a recto graph and a verso 'possibility'
grap
h. I remember once trying to think of some topological trick for that.
Best, Ben
On 1/20/2015 10:36 AM, Gary Fuhrman wrote:
Mary,
The subject line got truncated in your post so I made up a new and
shorter one to continue the thread.
I can only speak for myself — I read your post carefully more than
once,
but left it to others to reply to it (which Gary R had already done,
actually) because I had no answers to the questions you raised in it.
I
couldn't make a connection between your suggestion of “would-be
hypothetical situations, such as the mobius strip” and Peirce's idea
of
using the verso of the sheet of assertion as the area inside a cut. In
fact I still don't see a connection. A mobius strip, being a bounded
surface with only one side, doesn't have a verso, and I don't see how
it
relates to Peirce's “discovery” that the verso of the sheet represents
“a
kind of possibility” and not just the negation of the graph within the
cut. I also couldn't get a handle on your question “Would boundedness
exist in a mobius strip?” or its relevance to the issue we’ve been
discussing yesterday and today.
Maybe it’s just my obtuseness, but you’ll need to explain what you
were
driving at before I can see its relevance to Jeff’s post that I did
reply
to. (I assume you want to be given credit for more than just
mentioning
the “verso” in your post, but I don’t yet see what else in it
anticipates
Jeff’s post).
gary f.
-----Original Message-----
From: Libertin, Mary
Sent: 20-Jan-15 8:36 AM
To: [email protected]<mailto:[email protected]>
Subject: [biosemiotics:7975] Re: Contradictories, contraries, etc. WAS
Jeffrey, Gary R, Lists,
I brought up significance of the verso side of the existential graphs
in
my most recent post last week, but have not been acknowledged as the
initiator of this thread. Gary R. responded to my comment on the
distributed and undistributed significance of the immediate and direct
objects of the dicisign. I wrote:
"I do think we should go on. Stjernfelt places his discussion of the
dicisign in as large a Universe of Discourse as is practical for his
audience. We need to be more tolerant of interdisciplinary analogies.
I
also think we need some instruction when we find it necessary, which
means we should ask. Here are some of the questions that came to mind
after the third time reading NP: how is the sheet of assertion, recto
and
verso sides, to be understood in various ³would¹be² hypothetical
situations, such as the mobius strip. Would boundedness exist in a
mobius
strip? The concepts of in/out, the whole or the part of the universe
of
discourse are in chapter 8, along with many other important thoughts,
juxtapositions, questions, and musings. . . ”
I have been researching this area and find it surprising that my
initial
discussion has been overlooked. If this is the first time the issue
has
been discussed I wish to be given credit or acknowledged in the
discussion.
Mary Libertin
Mary Libertin, PhD
Professor of English
Shippensburg University of PA
Shippensburg PA 17257
[email protected]<mailto:[email protected]>
On 1/19/15, 11:16 PM, "Jeffrey Brian Downard" wrote:
