OUTLINE:
DON MIKULECKY 28-JAN-1996 19:14:13.28
Outline Chapter 2 of Kampis:
DON MIKULECKY 29-JAN-1996 16:31:34.18
here are some notes for Wednesday afternoon!
DON MIKULECKY 1-FEB-1996 15:13:56.74
for next week's discussion!
Jeff Prideaux 8-FEB-1996 09:00:59.48
complexity discussion group
DON MIKULECKY 8-FEB-1996 09:23:54.12
here's a try at a running summary!
Cindy SHILLADY 8-FEB-1996 09:50:34.12
chapters on membership perspective
DON MIKULECKY 8-FEB-1996 11:33:08.17
some further comments to seed discussion
DON MIKULECKY 13-FEB-1996 14:11:48.29
some further notes and an "agenda" change!
DON MIKULECKY 13-FEB-1996 14:38:16.57
a post script to the previous message
DON MIKULECKY 13-FEB-1996 20:04:25.16
quantum measurement problems, etc.
DON MIKULECKY 14-FEB-1996 10:48:56.35
entailment, causality, and arrows
DON MIKULECKY 14-FEB-1996 14:09:12.72
classification, models, and individuality
DON MIKULECKY 14-FEB-1996 15:20:55.67
another mapping
Seth Roberts 14-FEB-1996 15:24:37.14
RE: another mapping
------------------------------------------------------------------------
DON MIKULECKY 28-JAN-1996 19:14:13.28
Subj: Outline Chapter 2 of Kampis:
A constructive approach to models
His index:
2.1 FOUNDATIONS
2.1.1 The notion of "model"
2.1.2 The role of Mathematics
2.1.3 Prediction and dynamic Prediction
2.1.4 theories of modeling
2.2 MODELS AND THE IDEA OF MATHEMATICAL CONSTRUCTIVISM
2.3CAUSALITY AND DETERMINISM: Why's and Hows
2.3.1 Phenomenology, Functionalism, and realism
2.4 ENTITIES
2.4.1 Levels of representation
2.4.2 Natural, abstract, and formal entities
2.4.3 Essentialism
2.4.3.1 Procedural and declarative knowledge
2.4.3.2 Observational and functional units
2.5 TIME AND INFORMATION
2.5.1 The concept of time
2.5.2 The limited accessibility of Information
2.6 OBSERVABLES
2.6.1 Dynamic and static observables
2.7 THE CONCEPT OF INFORMATION SET
2.7.1 Discrete event systems
2.7.2 Definition of information sets
2.7.3 Examples
2.7.4 Information sets and Turing computers
2.7.5 Complexity Theory and Modeling
2.8 ENCODINGS OF OBSERVABLES INTO VARIABLES
2.9 THE MODELLING RELATION
2.9.1 Natural systems
2.9.2 Abstraction
2.9.3 Description frame
2.9.4 Questions and systems
2.9.5 The modeling relation
2.9.6 The notion of interpretation
2.9.7 Summary: Set theoretic and constructive modeling
2.10 IMPLICATIONS
2.10.1 The relevance of models
2.10.2 Reduction and equivalence of models
2.10.2.1 Weak and strong equivalence
2.10.2.2 conditions for reducibility
As you can see...we will need many sessions to cover this chapter. I'll have
some notes by Wednesday.
------------------------------------------------------------------------
From: GEMS::MIKULECKY "DON MIKULECKY" 29-JAN-1996 16:31:34.18
Subj: here are some notes for Wednesday afternoon!
Some notes on:
Chapter 2 of Kampis:
A constructive approach to models
2.1 FOUNDATIONS
2.1.1 The notion of "model"
He starts by reminding us that our consciousness deals with the external world
by using models. Therefore, models are both a tool and an important goal of
inquiry as well.
"In the Western tradition of thought, there ia always a separation, a logical
distinction drawn between the parts of reality to be studied and the ones that
perform the study........This implies the object-subject separation."
"...they are not studied immediately as they are. They are filtered through
us and are studied as viewed by us, BUT ARE TREATED AS IF THEY WERE ORIGINALLY
INDEPENDENT OR SEPARATE. [my emphasis] ....OBSERVATIONS ARE CO-DETERMINED BY
US AND BY NATURE and we can not tell who contributes what...."
Here he suggests a third "observer" would lead to a fourth....ad infinitum.
we seek a correspondence between subject and object....models do this for us.
He cites Feyerabend (1975) "Against Method" to say that ANY DESCRIPTION IS A
MODEL. (lots more in Feyerabend...we need to look at!)
He then offers the positivist school as a counter view.
He suggests that positivism is doomed from the start because of the limitations
of which "facts" are available to posit! Here is where COMPLEXITY enters!
(see comparison with constructivism below)
2.1.2 The role of Mathematics
All models are not logically equivalent.
Models are a substitute for reality.
Models are useful if they allow us to make inferences.....especially
predictions.
Mathematics is especially useful in this process. Can help eliminate hidden
assumptions and logical failures. Language can not be made independent of
its content as it is being evaluated, mathematics can. (but there's the rub!)
Thus, science has adopted mathematical models and the discussion will focus
on these.
These models are deductive. Strictly speaking, they are therefore ALWAYS
hypothetical. They will always need empirical confirmation.
2.1.3 Prediction and dynamic Prediction
Prediction is an operation which allows anticipation! This has become the
main criteria for evaluating models.
What predictions can be made?
1) those based on empirical regularity
accumulated experience
example: Ohm's law
[note that nothing IN Ohm's law helps determine whether or not it will work.
Its verification is totally outside its content!]
2) deductive theoretical hypothesis supplies a prediction
theories...corroboration or falsification necessary
example: relativity theory
3) dynamical prediction (see 1.1.2)
the Newtonian paradigm: a VALID dynamical model removes prediction
from the realm of the hypothetical.
The "discovery" of complexity makes it seem a wonder that we can have these
models at all!
2.1.4 theories of modeling
two types of modeling strategy:
1) paradigmatic: based on some belief structure rather than
constructed to match observations
2) operative: models constructed on the basis of observed phenomena
now we must deal with what constitutes an observation. This also suggests
that we use phenomenology before we use mechanistic hypothesis.
(Non-equilibrium thermo for example)
He is leading us to the MODELLING RELATION which we have met before (Rosen)
2.2 MODELS AND THE IDEA OF MATHEMATICAL CONSTRUCTIVISM
Here he discusses the history which Rosen also cites...The hope of formalizing
mathematics, Goedel, and the failure of the formalists. (Russel, Hilbert,
etc.)
Constructivism was an "underground" development which saw the formalists
concept as an "asylum".
Here we can once again appreciate Rosen's use of number theory as an example.
Rosen asked us to see number theory as an object of study as if it were a
natural system. He then proceeded to instruct us about the failure of
formalization in order to lead us to see the problems inherent in the
positivist/reductionist/mechanist approach. It is worth refreshing our
memories about this.
Formalization collapsed because there were always significant aspects of
numbers which we could recognize to be true which failed to be described
within any given formalism. Expanding the formalism to include the excluded
facets of our knowledge of numbers resulted in still more excluded facets
being revealed in spite of the greater extent of the expanded formalism.
Goedel showed that it WAS NOT merely a matter of further extension of the
formalism. The regress is infinite!
[Here we can probably benefit from some historical perspective: let me try to
supply the seeds of an idea!]
The following is in large part due to contributions Norma Geddes is making in
spite of scheduling problems: The source is some course material from Dr.
Mary K. Rodwell in the school of social work.
Let me try my idea before I get to that material:
For the longest time, certainly most of my career, the social sciences were
considered "soft" science, if science at all. Texts had introductory chapters
selling the field as a "real" science. Reductionism seemed to be conquering
even the soft. Humans were atoms in a Newtonian social world. The sterility
of this parody on physics has now evidently broken through and new
methodologies are springing up. One of these is "constructivism". I do not
yet know the history of this approach, but Kampis seems to be strongly
impressed by it for he declares his methodology will adopt it.
Let me suggest the following modest proposal. To the extent that the
imposition of "hard" science on the social sciences seems to have had limited
success if any, maybe the opening up of hard science to ideas used in the soft
sciences is a better strategy! WE have been searching for an approach to
complexity as if we must create it and then pass it to our "less scientific"
colleagues in the "soft" sciences. Patronizing as usual! A close look at the
material I got from Norma may fit in well at this point. I will have copies
for anyone interested.
I'll send this now in order that those interested in preparing for wednesday
afternoon will have something to chew on (few seem to want to buy the
book...it is so expensive!) More will follow as time permits!
------------------------------------------------------------------------
From: GEMS::MIKULECKY "DON MIKULECKY" 1-FEB-1996 15:13:56.74
Subj: for next week's discussion!
Some notes from Rosen's book: "Fundamentals of measurement and
representation of natural systems".
the issues we are interested in involve:
system: some part of the real world that comprises your object of study
observable: a characteristic of the system that can be measured directly
state: a specification of what our system is like at a specific instant in time
[note that these intuitive definitions are accompanied by rigorous mathematical
definitions which I will not go into here]
he then puts forth two key propositions:
Proposition 1: The only meaningful physical events which occur in the world
are those represented by observables on states.
Proposition 2: every observable can be regarded as a mapping from states to
real numbers.
He then discusses the implications of these propositions in the context of
quantum mechanics...
in short.....
"one of the aims of our formalism is to point up the essential equivalence
of the measurement problem in physics with all types of recognition or
classification mechanisms based on observable properties of the objects being
recognized or classified."
Meters and Observables:
The crucial part of evaluating an observable on a state is a particular
physical system called a measuring instrument, or a meter. These must satisfy
the following properties:
1) Many identical properties of a meter can be prepared.
2) A meter can always be brought to a particular reference state.
3) Some of the states of the meter are labeled or indexed with real numbers.
4) The meter, when in its reference state, can interact with states of
other systems.
5) the interaction results in the passing from the reference state to
another one which is labeled with a real number.
This in principle, is enough to define what we mean by an observable.
HOWEVER, it is the intuitive version. The rigorous mathematical version is
the real contribution of Rosen here.
Linkage between observables:
Again we defer from the rigorous math to a more intuitive discussion:
As best as I can summarize in words, linkage exists if there is a
correspondence in the set of states of one to the set of states of the other
of a particular kind. This correspondence can be either total or partial.
given maps from states to observables, when there is linkage, the map may go
DIRECTLY from state to observable or via a second observable linked to the first.
This has a rigorous definition in terms of refinement os sets by equivalence
relations and the notion of COMMUTIVITY of set theoretical diagrams.
To summarize a lot of what Rosen then goes on to deal with rigorously, the
linkage issue is important in two respects:
1) There must be significant linkage between the observable sought and the
observable in the meter which is to be used to determine the sought observable's
value.
In particular, there may be observables for which a suitable meter does not exist!
2) Linkage between the sought observable and others (besides the meter) can be a problem!
One strong outcome of these considerations is the restricted character of
those families of observables we call STATE VARIABLES.
One requirement is that the aforementioned mappings be one-to-one and onto.
There can be no BIFURCATIONS!
There are also strong restrictions about the NUMBER of observables to be
associated with each state and the desire to have only those having the MINIMAL
number of members.
There are also questions about the EQUIVALENCE of alternative descriptions.
Next comes the issue of reducibility of representations. Here is where Rosen
first rigorously demonstrated the fallacies of REDUCTIONISM!
Next, it is necessary to introduce a class of formal mathematical objects, whose
properties are in conformance with the ideas developed up to this point.
Definition:
A formal system shall consist of a pair (S,F) where S is a set and F is a set
of real-valued mappings on the set. The ELEMENTS of F will be called the
OBSERVABLES of the formal system.
Given such a system, we are in a position to begin to construct models.
Next we move to meters and dynamics.
A key idea is:
"dynamics is the agency through which linkage relations among observables are
generated, modified, or broken."
How do linkage relations arise?
Dynamics imposes an interaction...this means it forces the system to occupy
states with linkages of a definite character, effectively excluding all others.
These linkages are between the system generating the dynamics and the system
upon which they are imposed.
Likewise, linkages may be broken by dynamics.
Reductionist modes of analysis of natural systems do this producing unlinked
"fractions".
Not all abstract dynamics are realizable in natural systems. Thus, even though
in principle, certain linkages may be broken, in fact, they may indeed be irreducible.
A further consequence of these considerations is that dynamic interactions
require a much bigger state set than that required for non-interacting systems.
EMERGENCE AND EMERGENT NOVELTY
evolution is replete with new modes of organization and behavior. In a rough
heuristic way, emergence is the "failure" or inadequacy of a particular mode
of description.
Rosen argues that this is the result of THE TACIT ASSUMPTION THAT IT IS
APPROPRIATE TO DESCRIBE A NATURAL SYSTEM BY A SINGLE SET OF STATES and that
all the dynamics has trajectories in that set.
Thus if linkages pull systems out of the set of states in which they lie, this
provides a natural way to describe emergence!
The characteristic feature of emergent novelty is the need to pass to a new
mode of system description after the emergent novelty is realized. New
observables are needed.
Examples:
1) The Benard phenomenon
2) The discussion of hidden variables in quantum mechanics.
This is a first installment. Send back questions if they arise.
More to follow.
------------------------------------------------------------------------
From: GEMS::JPRIDEAUX "Jeff Prideaux" 8-FEB-1996 09:00:59.48
Subj: complexity discussion group
Dear Eric,
My name is Jeff Pridaux and I am a PhD student under Donald Mikulecky.
He said that you are interested in participating over e-mail in our
on-going complexity discussion group. I just wanted to say a few
words about what complexity means to me. (Of course I am still
formulating my opinions).
Complexity is associated with the saying "The whole is more than
the sum of the parts". This implies that there must be some
context-dependence. Note that in main-stream science it is
assumed that any function attributed to a part is independent
of the context of which it is found. (For example a resistor
has the same transfer function whether it is in a radio,
television, or computer...). The idea of context dependence
is that (for complex systems) there are attributes to the system
(although not parts in the traditional sense) whose functionality
depends on the context in which the attribute is embedded. And
this context will be continually changing.
The following is brain-storming....
My understanding is that a mechanistic model requires a static
(or unchanging) context. If the context is indeed unchanging
in a system, it is always possible to come up with a mechanistic
model of the system. The values of the state variables can change
in a mechanistic model, but the existence of the state variables
themselves are static. The context is related to the existence
or non-existence of state variables. In a complex system,
the state variables will be dynamic (meaning that they can come
into and out of existence over time). This moving context
prohibits one from being able to form a single largest
mechanistic model that is valid over time. Also, the change
in context is not related (in a mechanistic way) to the values
of the existing state variables.
A mechanism (or formalism) cannot know a meta statement about
itself. A mechanism cannot know anything about the context of
which it is embedded. Or, a mechanism doesn't have to know
anything about the context of which it is embedded because this
context doesn't change. It can be given from outside once and
for all (for a mechanism). In the general case, though, with
a moving or changing context, we are forced to model this
property (changing context) non-mechanistically. We must have
models that are "closed to efficient cause" (a la Roesn).
Dynamic contexts are the general case. Static contexts are
a special case. Mechanistic (static context) science is a
special case of a more general dynamic context science.
Jeff Prideaux
------------------------------------------------------------------------
From: GEMS::MIKULECKY "DON MIKULECKY" 8-FEB-1996 09:23:54.12
Subj: here's a try at a running summary!
Complexity Seminar: Summary to date ( February 8, 1996)
So far we have discussed the first chapter and a half of Kampis" book and also
a brief summary of ideas from Rosen's "Fundamentals of Measurement and
Representation of Biological Systems"
Let me try to summarize what I think are some of the hogh points up to now and
then suggest some direction for next week.
The Newtonian Paradigm gave us the dynamic systems approach. This, simply stated,
is the reduction of all events to some set of dynamics, most often in a set
of ordinary differential equations. This, at this point in time, seems a very
inadequate way to do science. Using ideas from Rosen's latest manuscript,
which we passed out yesterday, it not only focuses on the science of the
material, but in a very special way! Rosen has said that physics is the
specialized science relative to biology which is far more general. Here is
the crux!
A dynamic system is a set if equations of the form:
dxi(t)/dt = f(x1,x2,...,xn).
These can be highly non-linear leading to chaos, etc. If we are lucky, we get
a solution:
xi(t)=F(x1,x2,...,xn)
Here's what we have:
These variables are generally HUGE simplifications! (In Newton's case, the
planets became points
There is generally more than one system that can lead to the same set of equations.
Thus any given dynamic system represents, as a model, a class of natural systems.
This is like thermodynamics which is so general that all realizable mechanisms
MUST fit it so that it is unable to distinguish between mechanisms!
In our present state of technological development, computers are IT!
Hence, the ability to reduce real systems to syntactical entities like
dynamic systems is the apparent "right way" to go. Hence we focus totally on
the computable aspects of the problem.
In recent years, this approach has dominated all of human intellectual
endeavor. One especially insidious manifestation of this is the "hard"
vs "soft" science snobbery. The social sciences, for example, were shamed
into trying to reduce humans to "atoms" of interaction, virtually to "point
particles" in the social "atmosphere". This failed miserably and the
alternative is an approach which has come to be known as constructivism.
Constructivism rejects the positivist's narrow, unrealistic view of the world
and tries to construct a world view which accepts the highly interactive and
"linkage" rich nature of reality. We therefore, have to turn the tables, and
humbly look to the social sciences for guidance.
As an example of the kind of interactive, self-referential problem we seek to
be able to deal with, we looked at Rosen's example of the active site of an
enzyme. We noticed the following:
The protein subunits are the "parts" we can reduce the system to. However the
active site is an entity which is very different in character. Some of the
important differences are:
1) The actual molecular interaction with the substrate has no "location"
on these subunits.
2) The final existence of the active site may strongly depend on the
substrate being near it. (Dr. Kier has written about this and will bring some
material and hopefully tell us more about it next time!)
3) The material aspects of the active site are accompanied by a notion of
"function" which is ONLY manifest if the reaction actually occurs!
4) Even if it were possible to treat this problem as a "dynamic system"
it would be intractable. Furthermore, even if it were tractable, nothing in
the solution would be recognizable as an "active site". It takes a meta-system
which involves us to make that semantic interpretation!
We said some things about time. We will have much more to say about this.
For example, I suggested that cycles are a way of breaking away from any "absolute"
time frame and bring a lot of intrinsic times into a system.
We speculated on the idea that the cell cycle is an irreducible
synchronization of the material dynamics of the parts of the cell DOMINATED by
a dynamic imposed on them by the functional components, which in turn are
responding to the fact that the cell is a cyclic entity from the start!
This idea needs some work, obviously, but can go somewhere, I think.
Next, we need to compare Kampis description of the modeling relation with Rosen's.
That, along with some enlightenment from Dr. Kier, should keep us very busy!
------------------------------------------------------------------------
From: GEMS::LSHILLADY 8-FEB-1996 09:50:34.12
Subj: chapters on membership perspective
Hi Dr. Mikulecky!
I have just been reading over those chapters in Dr. Falck's book
that I mentioned yesterday. I just came across a nice paragraph...
"Despite this danger [of oversimplification -- from previous
paragraph] and its potential deficits, it is possible to present a
perspective that accounts, in principle, for the functions of the body,
social interaction, the assignment of meaning of the human experience,
and the internalization of relationships in order to build and maintain
the ego. Through it, one repeatedly discovers a certain unity that
many aspects of life portray with surprising consistency. In other
words, the membership perspective involves certain predictable
constants. Everything within it is COMPONENTIAL; NOTHING IS MERELY A
PART. THE WHOLE STAYS INTACT AT EVERY LEVEL OF DISCOURSE, FROM THE
HIGHEST TO THE LOWEST LEVEL OF ABSTRACTION. The concepts that portray
this unity are "member" and its derivatives, "membership" and
"membership behavior". THE TERM MEMBER IMPLIES OTHER MEMBERS, AND IT
IS THIS CORE CONDITION THAT MAKES THE MEMBERSHIP PERSPECTIVE DIFFERENT
FROM ALL OTHER SOCIAL WORK PERSPECTIVES.
The concept of member RULES OUT INDIVIDUALISM WITH THE SAME ENERGY
WITH WHICH IS REJECTS COLLECTIVISM. IT NEITHER LOCATES LIFE'S CORE IN
THE PHYSICAL NOR IN THE PSYCHOLOGICAL OR THE SOCIAL ALONE. Instead
it holds that the member is a functioning human being, A COMPONENT OF A
WORLD FULL OF PEOPLE. MEMBERSHIP IS TREASURED AND PROTECTED AS
INDISPENSABLE TO THE LIFE OF THE HUMAN COMMUNITY." (from SOCIAL WORK:
THE MEMBERSHIP PERSPECTIVE. by Hans S. Falck. New York: Springer
Publishing Company 1988).
I have added the emphasis where the words seem strongest to me,
but both of the chapters I have read so far focus on these same ideas.
I think I will try to find the book in the library and read more.
Where is your office again, and I will bring you a copy of what I have
at the moment. I also have a book I bought and never read for another
honors module with Dr. Kuhn in the English department on Noam Chomsky
and his ideas in linguistics. It is a short book ("Syntactic
Structures"), but I am guessing that it will not be easy to get
through. I make try it though, as I have been wanting to read it since
I took that class and just never seem to make the time. I would love
to approach this idea of complexity from a more sociological angle
because although my undergraduate degree was in chemistry, I really
find myself drawn to the social sciences and don't always understand
the "hard" science examples we talk about in our meetings.
Cindy
------------------------------------------------------------------------
From: GEMS::MIKULECKY "DON MIKULECKY" 8-FEB-1996 11:33:08.17
Subj: some further comments to seed discussion
some comments to seed discussion...next time or after
The discussion of Phenomenology, Functionalism, and Realism closely parallels
Rosen's concern about simulation. It also is central in our strongly negative
evaluation of "Artificial Life". Note that the Turing test makes no attempt
to establish or even ask questions about the causal relations and whether they
are matched in the object being tested and what it is supposed to be "as good as"
In the same way, a successful simulation is merely an external resemblance
which ignores causal realities.
Take network thermodynamics for example. In my book I devote but a small part
to the SPICE simulations. They are using the Turing algorithmic method to mimic
system behavior. The Models being simulated, on the other hand, are a more
faithful representation of the actual causal relations which result in the
phenomenon being studied.
An aside, but related:
I spent a long time trying to simulate the benard cells on SPICE using
network Thermodynamics. It is possible, and trivial...ONLY if the model is
cooked to incorporate the final result as a foregone conclusion.
There is no way to set up a naive model of the watery system which will
spontaneously self-organize! The semantics of that event are outside the realm
of network thermodynamics as we know it! Like the discussion of the active
site, the issue is not resolvable to classical dynamics! There is much more
involved of a distinctly different kind!
I suspect the same to be true of CA as well, but now it is more difficult to
sort out technical difficulties from things impossible in principle!
A note on Jeff's post to Eric about context dependence: This is Jeff's
paraphrase of Rosen's discussion ...and appears in still a slightly different
form in Kampis in 2.4.3 Essentialism ( in the form of a discussion of atomism).
Procedural and declarative knowledge: This is the idea that a computer
program which asks for data as input can always be rewritten as a program
which requires no input if the data is already included. This will be very
relevant in trying to understand autopoiesis.
Kampis discusses "units" in a very important way:
(2.4.3.2)
"A natural entity is selected on the basis of our interactions that mark it
as a compact unit in terms of the information manifested and conveyed by
these interactions. If some thing is not a unit in this sense, we never define
it as an object and we never select it as a target of study."
This is crucial!!!!
Atomism and the mechanistic world view propped up by positivism demand that we
therefore focus only on entities which are produced by reducing or fragmenting
the systems. We are en route to break free from this and see NEW units which
are defined by function, self-reference, and strong interaction!
OK I will pause here for a bit.
FOR YOU WHO DO NOT HAVE THE BOOK: One purpose for doing this is to give you a
glimpse of the text to see if you are following in some way or need more
specifics. PLEASE ASK FOR MORE DETAIL, CLARIFICATION, ETC, IF YOU NEED IT!
------------------------------------------------------------------------
From: GEMS::MIKULECKY "DON MIKULECKY" 13-FEB-1996 14:11:48.29
Subj: some further notes and an "agenda" change!
On category theory, functors, etc. (and a preview of things to come)
Category theory is a mathematician's attempt to try to step outside of
mathematics in order to look at it while still using some of the clout
that mathematics itself can offer.
I think it is safe to say that in the second chapter, as he discusses
models and measurements, information sets, etc., Kampis is trying to
use a "user friendly" version of categorical thinking.
Also, as I will expand upon in another communication, his discussion of
dependence of information on timing as in the "prisoner's dilemma" he is
giving us a simplified version of "Meixner's Paradox" which is so important
in thermodynamics. Briefly stated, Meixner showed that it is IMPOSSIBLE
to define entropy in real, (we would now say "complex" ) systems.
I also want to devote another communication to the relationship between
Rosen's idea of "linkage" so crucial in any consideration of measurement,
and the concept of "coupling" in non-equilibrium thermodynamics. As
we will see, these are the ideas we need to see complexity, especially
emergence and self-organization, as both compatible with thermodynamic
reasoning and simultaneously opening it up to new ways of thinking.
Now back to category theory:
Here is a quick and dirty introduction, based on this reference:
"ARROWS, STRUCTURES, AND FUNCTORS: The Categorical Imperative"
by Michael A. Arbib and Ernest G. Manes
Academic Press, NY (1975)
Quote from their preface:
"What, then, is the categorical imperative - the set of core concepts of
category theory which should be shared by this diverse audience
[talented undergraduate students, graduate students in the sciences other than pure
mathematics, and to professionals in control theory, automata theory,
theoretical linguistics, philosophy, mathematical biology.....{Now! if you
are feeling left out realize that the contents of their book is NOT the topic
of our discussion, but that I have staked out a broad goal.....to see if we
find clues as how to proceed in a new approach to complexity using every tool
that may have some promise. We already have begun to explore constructivism
and are looking to the social sciences for help in that area. Earlier,
a core group of us set out on the long path towards understanding the approach
of Bob Rosen, which is where we encountered category theory. Now we hope to
achieve either a contrast or a synthesis.}]
before they pursue more specialized avenues tailored to their own area of
interest? Our answer is threefold. First is the ability to think with
ARROWS: to express key concepts in terms of mappings ....rather than in
terms of sets of elements. Second is the realization that collections of
mathematical STRUCTURES find convenient characterization in terms of arrows.
Third is the use of FUNCTORS as the appropriate tools with which to compare
different domains of mathematical discourse."
Ok, at the risk of having turned EVERYONE off, let me explain what I am up to.
We seek to answer a fundamental question:
Now that we have awakened and realized that the positivist dominated,
reductionist dominated world of science may not be what it was sold to us
to be, how do we:
1)avoid dumping the baby with the bath water?
2)do something positive rather than revel in how good it feels to have come
this far?
3)bring together as much as we can from the efforts of those who were
smart enough, or at least fortunate enough, to have seen this all before we
did?
So...we are investigating the core of our thinking process, the making of
models. We are now at the stage where foundations have been shaken in order to think
about unconscious activity in a conscious, systematic way.
We examined the world of dynamics...."hard science's" bread and butter.
We see a host of problems because it is clear that the CORRESPONDENCE
between dynamic models and "reality" may not be as clean as we wish.
This is a categorical finding! We discovered something about an arrow.
Or, more precisely, a set of arrows. The "mapping" from a given natural
system to a dynamic may be OK, but there may be OTHER natural systems that
map to the same dynamic! Now we have a problem. The problem gets
compounded when we realize that within a given, non-linear dynamic there may
be "bifurcations". These again are properties of arrows. In this case the
map from time to system description (states) has ambiguity.
Another example to hold up as a goal:
Rosen has given us a definition of what it means to be an "organism".
It took him many books to get us there. And what do we get?
A little diagram with letters connected by funny arrows! Either he
is pulling our leg, or they may be some real rewards to learning to think
with arrows.
Tomorrow, we have a slight change of plans. Dr. Kier has a conflict, so his
discussion of the "active site" example is deferred to next week. I will,
however, pass out some reprints of his papers in the new journal "Complexity".
We will continue our discussions around the second chapter of Kampis' book,
which, since most of you don't have the book, means these and the other notes
from Jeff which we passed out. I hope to have more copies on hand tomorrow.
Then following week, Dr Kier will rejoin us and talk about his papers.
I hope this entices you to join us tomorrow. The discussion will center around
our "measurement" and "modeling" of the natural world....hopefully with a
constructivist perspective.
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From: GEMS::MIKULECKY "DON MIKULECKY" 13-FEB-1996 14:38:16.57
Subj: a post script to the previous message
A post script to the previous transmission:
The idea of thinking with arrows is important for the following reason:
A mapping between sets in traditional math focused on the members of the sets.
members of set A were identified with members of a set B. Can you
sense how it is a focus on set membership?
Now we are studying a subtle seeming but really profound way of looking at
things.
Symbolically, a mapping, f between the two sets might be represented as
f(a) = b where a is a member of set A and b a member of set B.
Alternatively, the same idea can be represented by an arrow:
A --------------> B
So what is the big deal?
the arrow, if we let it, can focus our attention on the PROCESS which the
mapping really is rather than focusing us on the starting and ending points
of that process, which the set elements are. Notice how quickly we can go
from mathematical abstraction to real world events with that notion!
From: GEMS::MIKULECKY "DON MIKULECKY" 13-FEB-1996 20:04:25.16
Subj: quantum measurement problems, etc.
For those of you who are not into quantum measurement problems, fear not!
We are using them for a number of purposes. Not the least of which is to
become aware of the fact that there is trouble in paradise! but even
more important is Rosen's parallel between that manifestation of the measurement problem and the fact that it IS only one manifestation! We can find a
manifestation in all forms of classical measurement! Hence our need for a new approach (constructivism?)
------------------------------------------------------------------------
From: GEMS::MIKULECKY "DON MIKULECKY" 14-FEB-1996 10:48:56.35
Subj: entailment, causality, and arrows
Arrows and entailment:
One of the breaks we are making with the positivist approach to science
involves the notion of causality. What this does is shift the question about
things from a "how?" to a "why?" Something that brings us back to
Aristotle. Aristotle identified four aspects to the cause for anything.
For example, the causes for a house, are the answer to having asked why
is the house there?
material cause- "entails" the house by being the material basis for its
existence. This is the collection of bricks, mortar, glass,
wood, etc. which make up the material content of the house.
In the Newtonian paradigm this becomes the initial conditions
from which the next state is to be computed.
efficient cause- entails the house by actually providing a means for its
realization. In other words....there is a house because there
were builders. In the Newtonian paradigm, this function is
performed by the dynamics in the form of "equations of motion".
formal cause- entails the house by giving a form or template to which it
ultimately conforms. In other words, there is a house because
there was a plan or blueprint. in the Newtonian paradigm,
this is the "trajectory" or the function of time resulting
from the integration of the equations of motion.
FINAL CAUSE- entails the house by recognizing a PURPOSE for the house! The
fact that someone needed a dwelling place. THIS IS TOTALLY
ABSENT FRON THE NEWTONIAN PARADIGM AND IS AN ANETHMA!
Rosen takes great pains to distinguish between final cause and teleology. He
also identifies final cause with ANTICIPATION. In anticipation lies the idea
that complex systems contain a "model" of their environment which is used to
allow an "idea" about a future event to influence the present dynamics of the
system. This is in total distinction from Newtonian dynamics which can ONLY
use present and past information in its calculation! This is precisely why we
are struggling to come up with another way of formulating dynamics. Remember
that it is dynamics through which we either make or destroy linkages between
observables.
Now, back to arrows. Clearly if we are to represent processes such as the
kind of metabolic events that go on inside living cell, or some social
phenomenon, we might want to see these events in terms of their causal roots.
Here is a common application of mappings between sets. These events or
processes are then capable of being dealt with categorically, but in this
context, the purely mathematical version of category theory becomes
inadequate. This is why, in "Fundamentals of Measurement", Rosen develops his
own version of category theory which is only finally completed in "Life
Itself" when the definition of an organism is offered for the first time. The
concept Rosen focuses on there is that of BEING CLOSED UNDER EFFICIENT CAUSE.
We will spend a great deal of time learning to understand the implications of
that concept for they seem to be revolutionary and in the spirit of the
constructivist ideas we are exploring.
The only competing notion that I know of comes via Maturana and Varella's
notion of autopoiesis. Later in Kampis' book he discusses the value and the
limitations of this idea. When I summarized his argument in my review and
suggested that Rosen's concept of closure under efficient cause was moth more
useful and less limiting, I received complete agreement from John Stewart,
Varella's coworker in France! John seems to have "graduated" to Rosen's view
of complexity as a result of the talks I gave in France this summer. I say
this only to remind us that we have quickly moved to the edge. Where we go
next is mostly up to us. Hopefully, as we delve into Kampis' and the other
ideas we have been discussing, we will
see a direction.
A final word on arrows. The biggest consequence of focusing on causality is
that category theory must be seen to include different sorts of arrows. Each
cause may either require a separate one (certainly true in the Newtonian
paradigm) or they can become more complicated, in the sense that single arrows
represent mixed causalities as in complex systems. This is a point is
implicit in Kampis' presentation and it seems appropriate to call your
attention to it.
------------------------------------------------------------------------
From: GEMS::MIKULECKY "DON MIKULECKY" 14-FEB-1996 14:09:12.72
Subj: classification, models, and individuality
Some further comments from Rosen's "Fundamentals of Measurement"
The measurement process has been defined as a procedure involving the
assigning of numbers to states via an interaction with a meter. It also
can be generalized to include the more general process of recognition, or
discrimination or classification.
Some examples:
1) A mapping which associates the lengths of a set of curves in a
Euclidean space with real numbers.
2) A mapping which defines a measure on a field of subsets of a set.
3) A mapping which assigns a topological invariant to a class of
topological spaces.
4) A mapping which defines the cardinality of a class of sets.
5) A mapping which assigns coefficients or roots to a set of polynomials
over the reals.
6) A mapping which assigns an eigenvalue to a set of linear transformations.
7) A mapping which assigns a "fitness" to a class of organisms in a fixed environment.
8) A mapping which assigns a market value to a class of economic commodities.
9) A mapping which assigns a number of steps or algorithmic operations to
a class of computer programs.
10) A mapping which assigns a call number to a class of books in the library.
You should now be able to supply some examples more familiar to you. Please
share them with us by e-mailing them to me.
Some observations and comments:
Notice the word "class" or "set" is used in each example. This has implicit
in it a notion of equivalence. Let us explore that further. Two lists (sets)
have a kind of equivalence if the following properties obtain among their members:
Let us speak of comparing the lists members in pairs, one member of the pair
being from one list and he other from a different list (or in some cases, even
the same list). We will call this comparison a RELATION between the members
of one list and the members of the other. We can then make a discrimination
among possible relations. A very special relation we might discover would be
the EQUIVALENCE relation (so named because it has a strong resemblance to the
equality of numbers). This relation holds if our comparison of the two lists
turns out to yield the following:
Reflexivity: whatever the comparison is, it is met by the same list member
being compared with itself.
Symmetry: If a member from the first list is comparable to a member of the
second, the second is also comparable to the first. (notice that this will NOT
work with comparisons like "greater than" or "less than" for any quality).
Transitivity: If a member of list one compares favorably with a member of
list two, and the member of list two compares favorably with a member of list
three, then the member of list one will ALSO compare favorably with the member
of list 3. (Once again the notion of "more" or "less" works well here.)
Why get involved in such notions? They are the basis for all classification
we do. Hence, if members of any list are EQUIVALENT under such scrutiny
the list is divisible into DISJOINT categories! No overlap or ambiguity.
That is what is implied in the above examples. This has a profound
meaning in our measurement and modeling enterprises. We usually BELIEVE we
are dealing with distinct entities as we measure and clasify, yet our
METHODOLOGY intrinsically has only the limited capability of reducing things
to members of an equivalence class, not individuals! This has profound
implications for reductionism and positivism as we will see!
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From: GEMS::MIKULECKY "DON MIKULECKY" 14-FEB-1996 15:20:55.67
Subj: another mapping
from Seth Roberts: a mapping between colors of the rainbow and
various wavelengths of light
------------------------------------------------------------------------
From: GEMS::SBROBERTS 14-FEB-1996 15:24:37.14
Subj: RE: another mapping
another example, perhaps more evocative of the common usage of the
word mapping:
a mapping between points on a street map of Richmond and actual
locations in this city.
(just testing my understanding of this concept)
seth