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\begin{document}
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%Slide 1
\title{{\LARGE\bf Making Sense}\newline \newline \newline}
\author{Tom Carter
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%Santa Fe Institute\newline
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\date{\today}
\maketitle
%Slide 2
%% \sectionhead{Making Sense}
%% %%tth:\begin{itemize}
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\sectionhead{Introduction / theme / structure}
%% %%tth:\section{Introduction / theme / structure}
\begin{itemize}
\item This is a brief introduction to some thoughts about making sense
of the world.
We all make sense of the world in many ways all the
time, but we don't always do it consciously, nor do we necessarily
have rigorous / structured approaches to the project \ldots
My goal here is to talk some about how I go about building theories,
models and simulations that I can use to make the world make more sense
to me -- and also that I can use as parts of explanations
to help others see the world in potentially more useful ways.
As a teacher/researcher, I'm always looking for more illumination, and
better ways to reveal that illumination.
\pagedone
\item One important point for me is that in general in these sorts of projects,
I am much more interested in epistemology than in ontology.
Just briefly, what are ontology and epistemology? I tend to think about
them this way:
Ontology: ``What is there?" (i.e., questions of ``being'')
Epistemology: ``How do we know?" (i.e., questions of ``knowledge'')
In many respects, I see these as two of the main (or the main two) branches
of philosophy, and especially of philosophy of science. They tend to have
convergences and divergences, but they drive much philosphizing (and much
argumentation over angels and heads of pins . . . :-)
\pagedone
\item Ontological questions tend to litter the fields of science: What are the
fundamental elements? Are there just four of them? What is the Fifth
Element? Is it the quintessence, or just a so-so sci-fi movie? Does
caloric exist? Does phlogiston exist? Are there
atoms, or not? Do electrons exist? Does the force of gravity exist?
What is a gene? What is a species? What is life? Does Truth exist?
For the most part, I think these questions are largely irrelevant.
In many respects, I think they are very often the wrong kinds of questions
for scientists to ask \ldots
\item Epistemological questions also abound: What can we know? How can we
best go a trying to learn (gain knowledge) about the world? What role
does evidence play in understanding systems? How meaningful is deduction
within axiomatic contexts?
\pagedone
\end{itemize}
\pagedone
\sectionhead{Language and meaning}
%% %%tth:\section{Language and meaning}
Hmmm \ldots This is a placeholder for some things I want to write about,
but haven't yet.
\begin{itemize}
\item Reference and Platonism
\item Use and Wittgenstein
\item What is meaning, and how does it happen?
\end{itemize}
\pagedone
\exercises{Language and meaning}
%% %%tth:\subsection{Language and meaning -~exercises}
\begin{enumerate}
\item Explain why "meaning is use" is meaningful.
\end{enumerate}
\pagedone
% \sectionhead{Mathematics}
% %% %%tth:\section{Mathematics}
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% \begin{itemize}
% \item Axiomatics and applicability
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% \item Probability and information
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% \item Why is mathematics so unreasonably effective?
%
% \end{enumerate}
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% \pagedone
\sectionhead{Theories, models and simulation}
%% %%tth:\section{Theories, models and simulation}
\begin{itemize}
\item What is a theory? This turns out (at least in a socio-cultural
sense) to be a rather difficult question.
One fairly traditional notion is to use ideas from mathematics. In
this form, a {\em theory} is a collection of axioms, definitions,
rules of inference, symbols, ``objects,'' relations, etc. (e.g., the
theory of Euclidean Geometry).
One can (in theory :-) derive results within the theory, and
engage in a general hypothetico-deductive cycle. One ``makes
a hypothesis'' within the framework of the theory, and then
checks to see if the hypothesis is derivable within the theory.
A fundamental question one can ask at this stage is whether the
theory is consistent -- i.e., whether the collection axioms, etc., is
logically consistent.
\pagedone
The next step, then, would be to try to construct an {\em interpretation}
of the theory (what might also be called a {\em model}). In general,
this would be a mapping from ``object'' and ``relation'' (etc.) symbols
in the theory to specific ``objects,'' ``relations,'' etc., external
to the theory. This process can be remarkably problematic \ldots
We can now enter into another level of hypothetico-deductive cycle.
We make observations in the realm ``external'' to the ``theory,''
then ``turn the crank'' to get ``predictions,'' make more observations,
and see if the results ``match'' (e.g., we ``do experiments'').
At this point, we are likely to have to make some sense of what we have
seen, and decide what to do next.
\pagedone
\item Perhaps it would be worth going through a specific example, to see
some of the issues.
Imagine for a moment that it is around 1300 C.E., and you work for
the Grand Vizier. He (and the King) believe that the planets affect
one's life, and they demand that you ``cast the King's horoscope."
In other words, you are to describe (in some detail) what the
sky would look like at a particular time and place (perhaps some
35 years prior).
Your first ``observation'' is that most of the points of light (stars) in
the night sky are ``fixed.'' Of course, even this much requires a
significant degree of abstraction -- if you hold your gaze fixed
with respect to the ground you stand on, the stars will ``move'' --
they will ``rotate'' during the night. The stars are ``fixed'' with
respect to each other. Notice that there is also a (covert) assumption
that the stars I see tonight are the same stars I saw last night.
I can have my graduate students (apprentices :-) draw maps of the
(relative) positions of the stars on successive nights, noting the
strong similarities between the maps, I can then
make the (simplifying) assumption that they are the same stars.
Part of what I am pointing out here is that in building a ``theory''
there are innumerable background (often unspoken) assumptions
necessarily underlying the ``theory.'' It is probably worth noting
that if the ``theory'' ``doesn't work,'' it may be (is?) a nontrivial
exercise to figure out which of the explicit and/or implicit
assumptions might be changed to get the ``theory'' to ``work'' \ldots
We can now ``observe'' (with various caveats \ldots) that the planets
``move'' with respect to the fixed stars. Making various assumptions
about regularity, continuity, and simplicity in general, we want to
build a ``model'' (an orrery?) of the (relative) motions of the planets.
What are the constraints within which we will build our ``model?''
A first major constraint is that the planets, being celestial objects,
will move in ``perfect'' ways, and since the circle is the most perfect
of shapes, they will move along circular paths. Thus we start building
our theory.
Our theory: The fixed stars are on an encompassing immense sphere. The
planets move on circular paths within the sphere. The planets move
continuously, smoothly, and at a constant rate along their paths.
Each (circular) path has a fixed center and a fixed radius.
We now build a specific model. For each planet, we choose (determine)
a specific center, radius, and rate of travel. Without much thought, the circles
are all coplanar.
\pagedone
Here is a first picture:
\vspace{1cm}
% \centerline{\includegraphics[width = 4in]{geo1}}
\centerline{\includegraphics[width = 5in]{ptolemy}}
\pagedone
Or, perhaps a better way to think about it:
\vspace{1cm}
\centerline{\includegraphics[width = 5.5in]{ptolemy2}}
Here the observer (us) looks at the world through the theory/model (from
within a perhaps unacknowledged paradigm). The theory/model becomes a
lens through which the world is viewed. This lens serves to select/emphasize
certain aspects of the world.
Notice that the ``prediction'' of the model does not exactly match the
world \ldots
%% \pagedone
Now what do we do? Given that the ``prediction'' of our model does not
exactly match the world, we have several choices. First, we could
declare the match ``good enough,'' take the money from the Grand Vizier,
and go on our way. Second, we could modify our model, by changing various
parameters (radii, centers, rate of motion, etc.). We could then
check each of these revised models to see if one was ``good enough.''
If none of the revised models was ``good enough'' (e.g., if, say, we noticed
the ``retrograde motion'' of Mars), we might replace our theory with a
different one. In this case (remaining in the same ``paradigm'') we might
``allow'' our planets to move in ``circles upon circles'' (i.e., epicycles),
so that sometimes the planet would move ``backward.''
\pagedone
Here's the next:
%% \vspace{1cm}
\centerline{\includegraphics[width = 5in]{geo3}}
We now have lots of parameters we can adjust. We could also
add epicycles upon epicycles upon \ldots (and, by
Fourier, make things match pretty much as well as we
want \ldots).
\pagedone
\item On the other hand, we could even go so far as to step to a new paradigm,
and allow our theory to include the earth as one of the moving bodies
(no longer distinguishing between celestial and terrestrial), and even
allow the paths to be other conic sections, such as ellipses. We might
make an orrery like this, with an actual physical crank
we can turn:
%% \vspace{0.2cm}
\centerline{\includegraphics[width = 5in]{orrery}}
\pagedone
\item Hmmm. Are we done now? What do we want a theory/model to do for
us? It seems to me in various contexts we would like it to
\begin{enumerate}
\item Give us a good description of the system
\item Allow us to predict behavior of the system
\item Give us an explanation of the system
\item Allow us to control the system
\end{enumerate}
A nice heliocentric theory/model (Keplerian, say) can do a very good job
of describing the solar system, and can allow us to make good predictions
(or retrodictions) of the behavior, but it doesn't give us much in the
way of an explanation of the system. In the next step, things get very
interesting.
Enter Isaac Newton. In the new paradigm, we allow new entities to
exist in our theory/model -- forces. In particular, gravitational force.
The picture looks very similar to what we had before:
\vspace{0.5cm}
\centerline{\includegraphics[width = 5.5in]{newton7}}
Note, though, that the new entity ``gravitational force,'' shown as an arrow in
the model, does not correspond with any observed entity in the world \ldots
\pagedone
\item One more, to clarify the last observation. In general
relativity, an object follows a geodesic with a curved
spacetime, with curvature determined by local masses.
\centerline{\includegraphics[width=5.5in]{general-rel1}}
Note that there is no ``force of gravity'' in this model \ldots
\pagedone
%
% \begin{enumerate}
%
% \item History and context, purposes / uses
%
% \item Theories, models, instances, paradigms
%
% \item Lenses
%
% \item Objects
%
% \end{enumerate}
\item On building models:
My general policy when building a model is to start with the absolute minimum
to get the model off the ground. In order to do this, I have to observe /
think about the system long and hard, thinking about what might possibly
be irrelevant to the aspects of the system's behavior in which I am
interested. (See, for example, my ``economics'' models \ldots)
There must be more to come, but I'll stop writing here for now, and
talk instead :-)
\end{itemize}
\pagedone
\exercises{Theories, models and simulation}
%% %%tth:\subsection{Theories, models and simulation -~exercises}
\begin{enumerate}
\item What sorts of relations can there be between models and reality?
\end{enumerate}
\pagedone
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\footnotesize
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\bibitem{fox-keller-1}
Fox Keller, Evelyn,
{\em Making Sense of Life},
Harvard University Press, Cambridge, 2002.
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