SUO: Re: Brouillon Projet, Les Yeux Des Argues, La Laine Des Cartes
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| Book 1.
| Definitions.
|
| 1. A 'point' is that which has no part.
|
| 2. A 'line' is breadthless length.
|
| 3. The extremities of a line are points.
|
| 4. A 'straight line' is a line which
| lies evenly with the points on itself.
|
| 5. A 'surface' is that which has length and breadth only.
|
| 6. The extremities of a surface are lines.
|
| 7. A 'plane surface' is a surface which
| lies evenly with the straight lines on itself.
|
| [It Continues ...]
|
| "Euclid",
| 'The Thirteen Books of Euclid's "Elements"', Second Edition,
| Translated from the Text of Heiberg, With an Introduction and
| Commentary by Sir Thomas L. Heath, Dover, New York, NY, 1956.
| Volume 1, page 153.
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Jean-Marc,
I am going to recoup one of my earlier essays on this subject --
there were so many clever things that I blurted out within it,
as the initial incitements of the topic struck me on my first
impression, that my elastic, all too elastic stores of memory
are already beginning to blur into obliviscence, that form of
resilience in impressionability that I suspect you share, too.
Besides which concern I am for the moment earnestly of a mind
and a mettle to keep on broadening out this malleable subject
to take in some aspects of what we mean by definition, in the
first place, if that is indeed the only place for definitions
to make a place for themselves, which I occasionally question.
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Note 1
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JM: A mon avis dans l'extrait que vous citez (1.551) le terme "ground"
est pris dans un sens beaucoup plus large que implement le ground
d'un signe, puisque Peirce écrit (New List ... ):
| Moreover, the conception of a pure abstraction is indispensable,
| because we cannot comprehend an agreement of two things, except
| as an agreement in some respect, and this respect is such a pure
| abstraction as blackness. Such a pure abstraction, reference to
| which constitutes a quality or general attribute, may be termed
| a ground.
|
| The conception of second differs from that of other,
| in implying the possibility of a third. In the same way,
| the conception of self implies the possibility of an other.
| The Ground is the self abstracted from the concreteness which
| implies the possibility of an other.
JM: Since no one of the categories can be prescinded from those
above it, the list of supposable objects which they afford is,
What is:
Quale -- that which refers to a ground,
Relate -- that which refers to ground
and correlate,
Representamen -- that which refers to ground,
correlate, and interpretant.
JM: C'est à dire qu'on peut penser le ground sans le signe
mais pas l'inverse. Donc il ne s'agit pas seulement du
ground du signe, mais du ground de manière beaucoup plus
générale, puisque "le ground est abstrait d'un être concrêt
(après coup identifié comme le representamen/signe) et implique
la possibilité d'un autre être (après coup identifié comme l'objet
du signe)". Donc définir le ground à partir de la notion est signe,
interprétant ... c'est mettre la charrue avant les boeufs.
JA: Arisbeans, SemioCompères, ...
JA: This is my first essay at making some remarks,
all of which have been accumulating in my mind
for quite some time, about the uses that people
frequently make of Peirce's Categories, but that
I think, in my arrogance, go against the grain of
his thought overall. This is a difficult subject
to get a handle on, and so I am likely to fail on
the first few tries, at best, if not perpetually.
JA: What I want to say, first and foremost, is that
Peirce was a relational thinker, one of the first,
one of the best, and, I am beginning to fear, one
of the last thoroughly relational thinkers that we
will ever see throughout our intellectual history.
I have had my own struggles in trying to transform
my thinking in this way, and, after a long time,
I can still see many absolutist and essentialist
habits that were ingrained in me by the standard
experiences and impressions of my rote education.
But that is another story. What is pertinent here
is the observation that Peirce's unique daimon as
a relational spirit means that we cannot interpret
his ostensible Categories in the same absolutist
and essentialist ways that we have been accustomed
to regard Aristotle's, Kant's, Hegel's, and so on.
Another time I will argue whether it was right to
interpret even Aristotle in so extreme a manner,
but another time. In particular, I think that it
would be a mistake for us to seek out in Peirce's
work, or to foist upon it, a new fundamentalism
that seeks to base itself on the idea of "ground".
JA: And so, just to 'cut to the chase', and to tell you the way
that I have personally worked out to negotiate a compromise
between this ordinarily so unrelational a term as "Category"
and what is evidently a thoroughly relational way of thinking,
let me suggest this interpretation of 1-ness, 2-ness, 3-ness,
insofar as they apply to the subject matter of sign relations.
JA: 1-ness has to do with the 1-dim projections of sign relations.
2-ness has to do with the 2-dim projections of sign relations.
3-ness has to do with the 3-dim projections of sign relations.
JA: In the 1st category we find the relations of O to O, S to S, I to I.
In the 2nd category we find the relations of O to S, O to I, S to I.
In the 3rd category we find the relations of O, S, I, in 3-foldness.
JA: Similar studies can be outlined for any other type of k-adic relation.
But we simply must begin to lift our eyes above the level of one tuple
at a time if we wish to understand what 3-adic or k-adic relations are.
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Note 2
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JM: Isn't the ground of the nature of a "form"
or a relational structure? What else could
it be like?
JA: I am tempted to agree, and I probably would if I could use the
words "form" and "relational structure" in the ways that I am
already used to, but I cannot be sure yet of the way that you
may intend them, so I must hesitate until I know your meaning.
JM: [Quotes JA:]
| And so, just to 'cut to the chase', ...
|
| In the 1st category we find the relations of O to O, S to S, I to I.
| In the 2nd category we find the relations of O to S, O to I, S to I.
| In the 3rd category we find the relations of O, S, I, in 3-foldness.
JM: There you have a circular definition.
JA: I pretend no definition.
JA: I am presenting the relations among primitive notions,
undefined in themselves and yet aphorized in relation
to one another. This is in practice a very common way,
at least among non-fundamentalists, for setting out the
underpinnings of a conceptual framework, as if to raise
the geodesic domes of our thought by gradually allowing
the 'tensegrity' of the whole structure to raise itself
in the very process of hanging together. It goes back to
Euclid, of course, where points and lines remain undefined,
but bear their mutually supportive relationship to each other.
JM: If the definition of the 1st category
is derived from the idea of S, O, and I,
as elements of a genuine triad ("1-ness
has to do with the 1-dim projections of
sign relations"), then the first category
presupposes the 3rd category. (???)
JM: Idem with the 2nd category
JA: Let me express the general principle in the words of Noam Chomsky:
| In linguistic theory, we face the problem of constructing
| this system of levels in an abstract manner, in such a way
| that a simple grammar will result when this complex of abstract
| structures is given an interpretation in actual linguistic material.
|
| Since higher levels are not literally constructed out of lower ones,
| in this view, we are quite free to construct levels of a high degree
| interdependence, i.e., with heavy conditions of compatibility between
| them, without the fear of circularity that has been so widely stressed
| in recent theoretical work in lingustics. (Chomsky, LSOLT, page 100).
|
| Noam Chomsky, 'The Logical Structure of Linguistic Theory',
| Based on a widely circulated manuscript dated 1955.
| University of Chicago Press, Chicago, IL, 1975.
JA: And, of course, everyone has heard of the "hermeneutic circle".
JA: Without understanding the power of these potentials,
I fear that semiotics will never get off the ground.
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To the present:
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JM: Isn't the ground of the nature of a "form"
or a relational structure? What else could
it be like?
JA: I am tempted to agree, and I probably would if I could use the
words "form" and "relational structure" in the ways that I am
already used to, but I cannot be sure yet of the way that you
may intend them, so I must hesitate until I know your meaning.
JM: my meaning would be, a collection of points and relations
between these points so that no point is left alone.
| "... the phaneron is made up entirely of qualities of
| feeling as truly as Space is entirely made up of points. ...
| no collection of points ... without the idea of the objects
| being brought together can in itself constitute space."
JM: What is yours?
JA: Form. From Latin "forma" = "beauty".
There's more to say, of course, but
that is all you really need to know.
JA: Relational Structure. Any relation
viewed with an eye to its form, q.v.
JA: Relation. Here I see two cases:
1. Relation in Extension = a set of tuples.
Tuple = finite sequence of elements from
a predesignated set or collection of sets.
If the tuples all have the same cardinality k,
then they are called k-tuples and the relation
is said to have "arity", "adicity", "valence" k.
2. Relation in Intension = a property ("intension")
that is common to all of the elements in a set.
Nota bene: Saying that a property is shared by
all of the elements in a set is different from
saying that the property is a property of a set.
The elements of a relation in intension are known
as "elementary relations". These are the analogues,
in intension, of the tuples in extension.
JA: For the past many years, all against my first inclinations,
I have been working to develop the extensional side of the
theory of sign relations, simply because this area is less
crowded, because far less work has been done on this face
of the mountain, and because this is the side of things
that makes a connection with empirical efforts, say,
in databases, ethology, and qualitative research.
JA: In the 1st category we find the relations of O to O, S to S, I to I.
In the 2nd category we find the relations of O to S, O to I, S to I.
In the 3rd category we find the relations of O, S, I, in 3-foldness.
JM: These would be the degenerate categories of thirdness.
I believe that it is better to build the categories so
that they are hierachized but still be independent of
each other. How do you express the fact that genuine
secondness is independent of genuine thirdness?
JA: I have the feeling that "independent" may be another one of
those words that we use in different ways from one another.
JM: Genuine thirdness requires an independent secondness
and an independent firstness, i.e. a genuine secondness
that exists independently of genuine thirdness, but all
genuine secondness is not necessarily independent of all
thirdness (ex: degenerate thirdness in the first degree).
| Thirdness it is true involves Secondness and Firstness, in a sense.
| That is to say, if you have the idea of Thirdness you must have had
| the ideas of Secondness and Firstness to build upon. But what is
| required for the idea of a genuine Thirdness is an independent
| solid Secondness and not a Secondness that is a mere corollary
| of an unfounded and inconceivable Thirdness. (CSP, EP2, p.177).
You have given me examples of citations, in your own words
and in those of Peirce, where the word "independent" is
employed in context, and this is helpful up to a point,
but does it really tell us what anybody means by it?
JA: But I may need to repeat that I am not trying to define
the Categories of 1-ness, 2-ness, 3-ness, as I consider
them to be primeval, primitive, undefined terms, and so,
in a peculiar sense, already independent "in terms of"
each other. Here, I am merely seeking to illustrate
how I understand their application to sign-theoretic
subject matter. It may help if I quote Chomsky again:
JA: [Quotes Chomsky, LSOLT, p. 100, again.]
JM: Chomsky says "Since higher levels are not literally constructed out of
lower ones, we don't need to fear circularity". But would you say that
thirdness (seen as a "level" ) is not constructed out of lower levels
(secondness, firstness)? The categories are hierachized, aren't they?
Points, lines, planes -- they are customarily regarded as falling
into a hierarchy, are they not? But consider the "definitions"
of the eponymous Euclid. Were these ever actually regarded as
strict definitions, or merely intended as assists, helpful to
an extent, if taken with a grain of sapience, distracting in
the extreme, if read with eyes too near their gradgrindstone,
almost being completely dispensable, except for gratuitously
having in joint kilter much news of points, and all the rest?
I cannot say. But I know how these elements, points, lines,
planes, and so on up the scale, if up it be, are generally
regarded today, as undefined primitives held in relation
to each other by the whole panoply of tales that can be,
up to the limits of logical consistency, told of them.
JA: It did not occur to me that anyone would take what I said
as a strict definition of anything, since it was intended
more as a way of building relations among constructs that
are either primitive or else already sufficiently defined.
First of all, we already have a good enough definition of
the sign relation -- I personally consider the one in L75
to be the most clear, detailed, explicit, and formalized
of them all -- and this defines all of the roles O, S, I
simultaneously in relation to each other. Moreover, the
definition of the cartesian product, that comes into the
game as soon as we start to develop the theory of signs
along extensional lines, and which is almost inevitable
if we want to use sign relations as models of empirical
activities and natural forms of conduct, already brings
us the utilities of the various dimensional projections.
So my purpose here was more to elucidate or rationalize
the Categories as aspects or facets of 3-adic relations
than it was to define them on any particular foundation.
JM: [Quotes L75:]
| [I define a sign as] something, A, which brings something, B,
| its interpretant sign determined or created by it, into the
| same sort of correspondence with something, C, its object,
| as that in which itself stands to C. [Peirce, NEM 4, L75].
JA: More fully:
| On the Definition of Logic [Version 1]
|
| Logic will here be defined as 'formal semiotic'.
| A definition of a sign will be given which no more
| refers to human thought than does the definition
| of a line as the place which a particle occupies,
| part by part, during a lapse of time. Namely,
| a sign is something, 'A', which brings something,
| 'B', its 'interpretant' sign determined or created
| by it, into the same sort of correspondence with
| something, 'C', its 'object', as that in which it
| itself stands to 'C'. It is from this definition,
| together with a definition of "formal", that I
| deduce mathematically the principles of logic.
| I also make a historical review of all the
| definitions and conceptions of logic, and show,
| not merely that my definition is no novelty, but
| that my non-psychological conception of logic has
| 'virtually' been quite generally held, though not
| generally recognized. (CSP, NEM 4, 20-21).
|
| On the Definition of Logic [Version 2]
|
| Logic is 'formal semiotic'. A sign is something,
| 'A', which brings something, 'B', its 'interpretant'
| sign, determined or created by it, into the same
| sort of correspondence (or a lower implied sort)
| with something, 'C', its 'object', as that in
| which itself stands to 'C'. This definition no
| more involves any reference to human thought than
| does the definition of a line as the place within
| which a particle lies during a lapse of time.
| It is from this definition that I deduce the
| principles of logic by mathematical reasoning,
| and by mathematical reasoning that, I aver, will
| support criticism of Weierstrassian severity, and
| that is perfectly evident. The word "formal" in
| the definition is also defined. (CSP, NEM 4, 54).
|
| Charles Sanders Peirce,
|'The New Elements of Mathematics', Volume 4,
| Edited by Carolyn Eisele, Mouton, The Hague, 1976.
|
| Available at the Arisbe website:
|
| http://www.door.net/arisbe/menu/library/bycsp/L75/L75.htm
JM: The problem is that the definition only says
that the sign determines the interpretant.
It says nothing about the relation between
the object and the sign, i.e., that
the object determines the sign.
JA: Are you under the impression that objects determine signs?
I will have to think about that. As you know, the proper
reading of the definition, if ever we arrive at it, will
depend on using the author's meanings for "correspondence"
and for "determination", which CSP gives in full, and at
length, needless to say, in many other prominent places.
But I still read this definition as defining a relation
among three roles of players or domains of components,
and so defining all of them in relation to each other.
JM: Definition L.75 says:
A (sign) determines B (interpretant).
A (sign) puts B (interpretant) in correspondence
with C (object) so that the correspondence between
C and B is of the same sort of as that between C and A.
JM: i.e. the sign determines the interpretant,
which as a sign determines other interpretants ...
JM: But to say that the correspondence between C and B
is of the same sort as that between C and A doesn't
imply that there should be a determination at all.
If there is a determination of the sign by its object,
there will be a determination of the Interpretant
by the object, which is consistent with Peirce's
later definitions where the object clearly
determines the sign:
JM: You write: "Are you under the impression that objects determine signs?"
JM: Jon, this is not just an impression ...
| http://www.door.net/arisbe/menu/LIBRARY/rsources/76defs/76defs.htm
|
| 32 - v. 1905 - MS 283. p.125, 129, 131. "The Basis of Pragmaticism":
|
| A Sign, on the other hand, just in so far as it fulfills
| the function of a sign, and none other, perfectly conforms
| to the definition of a medium of communication. It is
| determined by the object, but in no other respect than
| goes to enable it to act upon the interpreting quasi mind;
| and the more perfectly it fulfill its function as a sign,
| the less effect it has upon that quasi-mind other than that
| of determining it as if the object itself had acted upon it.
|
| 33 - 1906 - S.S. 196 - Letter to Lady Welby (Draft) dated "1906 March 9":
|
| I use the word "Sign" in the widest sense for any medium
| for the communication or extension of a Form (or feature).
| Being medium, it is determined by something, called its Object,
| and determines something, called its Interpretant or Interpretand.
|
| 34 - 1906 - C.P. 4-531 - "Apology for Pragmaticism":
|
| First, an analysis of the essence of a sign, (stretching that word
| to its widest limits, as anything witch, being determined by an object,
| determines an interpretation to determination, through it, by the same
| object), leads to a proof that every sign is determined by its object, ...
|
| 35 - v, 1906 - C.P. 5-473 - "Pragmatism":
|
| [...] That thing which causes a sign as such is called the object
| (according to the usage of speech, the "real", but more accurately,
| the existent object) represented by the sign: the sign is determined
| to some species of correspondence with that object. [...]
|
| 36 - v. 1906 - MS 292. "Prolegomena to an Apology for Pragmaticism":
|
| A sign may be defined as something (not necessarily existent)
| which is so determined by a second something called its Object
| that it will tend in its turn to determine a third something
| called its Interpretant ...
So it's true, you are of the impression that a sign is determined by its object?
JM: If you find a sign not determined by its object,
it will be a sign only according to Peirce's earlier definitions of a sign,
and it will not be a sign according to Peirce's later definitions.
JM: So carefully choose your definitions.
Moi? Peircenally speaking, I am learning to go with my first impressions.
JM: Now you say that the sign relation is a cartesian product <O,S,I>?
JA: No, I say that a sign relation L
is a subset of a cartesian product OxSxI.
At least, that is what I say on extensional days,
which is most days of late.
JM: OK, that is what I meant, then,
by asking what is S, O, and I.
So the question is: how do you choose them,
since you are taking an extensional approach?
Do you list all possible signs? And once
you have selected either O, S, or I, how do
you express the idea that there are three
determinations (O -> S, S -> I, O -> I)?
I have some stuff that I wrote back in the first eleven or twelve drafts
of my dissertation proposal that may fit in about here. I will find it.
JM: Take for example a photograph with your picture on.
The picture on the photograph represents you, but
you do not represent the picture on the photograph.
How do you express that mathematically?
JM: so you have three sets: O, S, and I and the cartesian product
is O x S x I = {(o, s, i) | o is in O, s is in S, i is in I}, i.e.
all possible combinations of elements from each set, corresponding
to "points in space" with coordinates (o,s,i) or ordered triplets,
which you project on lines, planes --?
JA: Yes, that is a good description of the full product space OxSxI.
A sign relation L, then, is a subset L c OxSxI.
JM: But I believe that it is only begging the question:
what are S, O, and I? what are they sets of?
and why should it matter at all?
JA: I do not understand. It is a form of description, no more.
It is not meant to tell you why you should care about this
or that sign relation. That is a matter for you to choose.
JM: see above
JA: Where?
JM: Why not simply say as Peirce that when you have a triplet
you have three pairs, and when you have a pair you have
two units, no matter what the triplet is made of?
Why does the relation have to be a sign relation?
and how do you translate into the cartesian product
that idea that O determines S, S determines I,
and O determines I?
JA: Again, this is just a form of description. As it happens,
and this is a very common tactic in mathematical practice,
it is very useful to begin by weakening it, and simply to
incorporate all subsets of such a space under a "nominal"
title of sign relations, only coming back at the second
or third pass to note that some of them qualify only in
a "trivial" way. The properties that they have are the
properties that they have. It is our job but to notice,
to describe, and to articulate them, species by species,
genus by genus, an so on. It is all very straightforward,
well, in principle, at least.
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