ONT Re: Inquiry Into Models
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Jon Awbrey wrote (JA):
C.S. Peirce wrote (CSP):
Howard Pattee wrote (HP):
Heinrich Hertz wrote (HH):
HP: I would like to wind up a hanging tail or thread that Jon finds "too long to pursue now."
I guess I should not have put my tale out there if I did not want to be forced to chase it.
JA: Doleful experience has taught me that it is best to expand our focus a bit
to compass "sign relations", taken as wholes, over and above just isolated
signs, and taken at least at first in extension as sets of 3-tuples of the
form <o, s, i>, with o, s, i the "object", "sign", "interpretant sign" of
the "elementary sign relation" (ESR) <o, s, i>.
JA: At this point, I personally find the comparsion with group theory to be compelling.
A "group" is another sort of set of 3-tuples that is subject to a terse definition,
and yet the theory of groups encompasses a wealth of imaginative possibilities and
utilitarian potentials that can scarcely be con-&-sur-veyed in any finite lifetime.
JA: As it happens, one of my many "returns" to mathematics,
after a time in the wilds of philosophy and psychology,
was through the slits of "group representation theory",
but the tale thereby hanging is too long to pursue now.
HP: Let me pursue it briefly because it relates closely to what science is
all about. Group representation theory is a theory of homomorphisms, or
how one formal structure can be mapped into another formal structure so
that interesting or significant properties are preserved. If the mapping
(image, observation, projection, coding, measurement) is from a physical
system (i.e., matter and energy in space and time) to a formal system
then we have a physical model. Here is Hertz's statement of this
epistemic homomorphism (formally a commutation relation):
HH: | We form for ourselves images or symbols of external objects;
| and the form which we give them is such that the logically
| necessary (denknotwendigen) consequents of the images in
| thought are always the images of the necessary natural
| (naturnotwendigen) consequents of the thing pictured.
|
| For our purpose it is not necessary that they [the images] should be
| in conformity with the [external] things in any other respect whatever.
| As a matter of fact, we do not know, nor have we any means of knowing,
| whether our conception of things [our models] are in conformity with
| them [external things] in any other than this one fundamental respect.
|
| H. Hertz (1857-1894),
|'The Principles of Mechanics', Dover, NY, 1984, pp. 1-2.
| Original German Edition: 'Prinzipien Mechanik', 1894.
HP: This is a terse statement. A commutation diagram makes it clearer:
Howard, this diagram appears to have been messed up by your line wrap.
I am going to make an attempt to reconstruct your commutative diagram.
Let me know if I get it right or not. I always have to switch to
a fixed width font when I try to do this sort of ASCII-glyphics.
| EXTERNAL OBJECTS ____ WE FORM FOR OURSELVES ____ IMAGES, SYMBOLS, OR PICTURES
| | / [SIGNS, BRAIN STATES, WHATEVER]
| | / |
| | / |
| [NATURAL LAWS] . . . SUCH THAT . . . / [LOGICAL, MATHEMATICAL MODEL]
| | / |
| | / |
| | / |
| NECESSARY NATURAL /___ ARE THE SAME AS THE _____ LOGICALLY NECESSARY
| CONSEQUENTS CONSEQUENTS OF THE MODEL
HP: "WE FORM FOR OURSELVES ..." includes all forms of sensing, perception, observation,
measurement, coding, etc. which must be initiated by an agent or organism. In physics,
this is the essential cut between the world and the observer that is necessary whenever
a measurement is made. It cannot be considered as resulting from natural laws. That is
why there is a "measurement problem" in physics.
Okay, now there is a big problem here, one that I noticed quite a while back,
and one to which I have returned on a periodic basis whenever I try to think
very seriously about the puzzles of causality, on those increasingly sparse
occasions when I trick myself into supposing that causality is a sensible
or a solvable problem. The contretemps this time is not with Peirce but
with his formidable precursor Duns Scotus, at least, as he was read by
W.S. McCulloch. I have cited this passage numerous times, but the bit
that I think should give readers pause, hopefully of the reflective
variety, appears instead to pass under their gnosis without so much
as a moment of notice, or else to 'sink beneath their wisdom like
a stone'. So I think, perhaps, that it may now be time to call
e-special attention to the underlying incongruities of views.
Many people just blithely assume that the arrow of causality
and the arrow of implication just naturally must run in the
same direction, but it ain't necessarily so, nor has it
always been taken for granted by thoughtful thinkers on
the binding contract between eternality and secularity.
Here is the theme of my pandoric gloss:
| Please remember that we are not now concerned with
| the physics and chemistry, the anatomy and physiology,
| of man. They are my daily business. They do not contribute
| to the logic of our problem. Despite Ramon Lull's combinatorial
| analysis of logic and all of his followers, including Leibnitz with
| his universal characteristic and his persistent effort to build logical
| computing machines, from the death of William of Ockham logic decayed.
| There were, of course, teachers of logic. The forms of the syllogism
| and the logic of classes were taught, and we shall use some of their
| devices, but there was a general recognition of their inadequacy to
| the problems in hand. Russell says it was Jevons -- and Feibleman,
| that it was DeMorgan -- who said, "The logic of Aristotle is inadequate,
| for it does not show that if a horse is an animal then the head of the horse
| is the head of an animal." To which Russell replies, "Fortunate Aristotle,
| for if a horse were a clam or a hydra it would not be so." The difficulty
| is that they had no knowledge of the logic of relations, and almost none
| of the logic of propositions. These logics really began in the latter
| part of the last century with Charles Peirce as their great pioneer.
| As with most pioneers, many of the trails he blazed were not followed
| for a score of years. For example, he discovered the amphecks -- that
| is, "not both ... and ..." and "neither ... nor ...", which Sheffer
| rediscovered and are called by his name for them, "stroke functions".
| It was Peirce who broke the ice with his logic of relatives, from
| which springs the pitiful beginnings of our logic of relations of
| two and more than two arguments. So completely had the traditional
| Aristotelian logic been lost that Peirce remarks that when he wrote
| the 'Century Dictionary' he was so confused concerning abduction, or
| apagoge, and induction that he wrote nonsense. Thus Aristotelian logic,
| like the skeleton of Tom Paine, was lost to us from the world that it
| had engendered. Peirce had to go back to Duns Scotus to start again
| the realistic logic of science. Pragmatism took hold, despite its
| misinterpretation by William James. The world was ripe for it.
| Frege, Peano, Whitehead, Russell, Wittgenstein, followed by a
| host of lesser lights, but sparked by many a strange character
| like Schroeder, Sheffer, Gödel, and company, gave us a working
| logic of propositions. By the time I had sunk my teeth into
| these questions, the Polish school was well on its way to glory.
| In 1923 I gave up the attempt to write a logic of transitive verbs
| and began to see what I could do with the logic of propositions.
| My object, as a psychologist, was to invent a kind of least psychic
| event, or "psychon", that would have the following properties: First,
| it was to be so simple an event that it either happened or else it did
| not happen. Second, it was to happen only if its bound cause had happened --
| shades of Duns Scotus! -- that is, it was to imply its temporal antecedent.
| Third, it was to propose this to subsequent psychons. Fourth, these were
| to be compounded to produce the equivalents of more complicated propositions
| concerning their antecedents. (McCulloch, WIANTAMMKIAAMTHMKAN?, EOM, pages 7-8).
|
| Warren S. McCulloch,
|"What Is a Number that a Man May Know It,
| and a Man, that He May Know a Number",
| The Ninth Alfred Korzybski Memorial Lecture,
|'General Semantics Bulletin', Numbers 26 & 27,
| Institute of General Semantics, Lakeville, CT, 1961.
|'Embodiments of Mind', MIT Press, Cambridge, MA, 1965.
I have to break here. Maybe we can discuss the problem next time.
Jon Awbrey
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HP: Jon's (Peirce's?) definition of sign as a triad, <o, s, i>, with
o, s, i the "object", "sign", "interpretant sign" might correspond
to the first line of the above diagram, but there is no model or
homomorphism, which is why I find it epistemologically inadequate.
Peirce's definition of sign is too vague and ambiguous for me to
unravel. When he has more time, perhaps Jon could diagram it,
and say what "determined or created by" constructively entails.
And what implements the "sort of correspondence" that Peirce
has in mind? Peirce's definition:
CSP: | 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 itself stands to 'C'.
|
| CSP, NEM 4, pages 20-21, & cf. page 54, also available at:
| http://www.door.net/arisbe/menu/library/bycsp/L75/L75.htm
JA: Howard, please excuse me if I wax a bit tetchy at this point --
as nobody knows the troubles I've seen over this one scruple --
and not all of the obscurities that one finds being credited
to Peirce are really the obscurities of Peirce, nor even due
to his way of writing, which I think is admirably clear here.
JA: This is a perfectly good -- ok, a nearly perfectly good -- mathematical definition
of a particular family of combinatorial structures, that I know as "sign relations".
JA: A sign relation L is a SET of 3-tuples of the form <o, s, i>,
where o is an element of the set O, called the "object domain",
where s is an element of the set S, called the "sign domain", and
where i is an element of the set I, called the "interpretant domain".
JA: In other words, a sign relation L is a SUBSET of the cartesian product OxSxI,
a circumstance which Asciians write as "L c OxSxI".
JA: Thus CSP did not say, and JA will not say, assertively, what JA mentions here
in the form of a statement that says "a sign is a triad of the form <o, s, i>",
nor any such thing as that.
JA: Many of these further issues are tackled in the running accumulation of links
that I have already posted, but I will admit that the dialogical organization
of some of these sub-sutras is not always the best for finding quick answers,
and so I will work on extracting some more caustically focused e-lucidations.
JA: The link-o-rama that I started on the topic of "Determination" is meant to begin
addressing what Peirce meant by "determined or created" in this sign definition,
and also elsewhere, in general.
JA: To make a long story short, what Peirce means by "correspondence" in this definition
is just the whole 3-adic sign relation itself, which he occasionally describes as
a "triple correspondence". He does not mean to suggest any sort of pallid 2-adic
"imaging" or "mirrortying" as implicated in a "correspondence notion of truth".
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