So, you want to learn about modeling ...

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So, you want to learn about modeling ...
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http//plato.stanford.edu/entries/models-science/
Models in Science Models are of central
importance in many scientific contexts. The
centrality of models such as the billiard ball
model of a gas, the Bohr model of the atom, the
MIT bag model of the nucleon, the Gaussian-chain
model of a polymer, the Lorenz model of the
atmosphere, the Lotka-Volterra model of
predator-prey interaction, the double helix model
of DNA, agent-based and evolutionary models in
the social sciences, or general equilibrium
models of markets in their respective domains
are cases in point. Scientists spend a great
deal of time building, testing, comparing and
revising models, and much journal space is
dedicated to introducing, applying and
interpreting these valuable tools. In short,
models are one of the principle instruments of
modern science.
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Vehicles for Understanding and Doing Science
Models
See http//plato.stanford.edu/entries/models-sci
ence/
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I have deeply regretted that I did not proceed
far enough at least to understand something
of the great principles of mathematics for
men thus endowed seem to have an extra sense.
- The Autobiography of Charles Darwin
(pg. 8)
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Chapter 1 Introduction There is a story that
during his stay at the court of Catherine II of
Russia, the great Swiss mathematician Euler got
into an argument about the existence of God. To
defeat the Voltairians in the battle of wits,
the great mathematician called for a blackboard,
on which he wrote (x y)2 x2 2xy
y2 Therefore God exists. Unable to
dispute the relevance of the equation (which
they did not understand), and unwilling to
confess their ignorance, the literati were
forced to accept his argument! Much of the vigour
of behavioural ecology stems from the use of
mathematical models. But, as this little story
illustrates, there is a certain mystique
surrounding mathematics and mathematical
modelling. ...
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Paul Thagard Philosophy Department
University of Waterloo
The widespread use of analogies in cognition,
including scientific reasoning, has been
well documented ... people use analogies to solve
problems.
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Metaphors and Analogies in Scientific Thinking
Holyoak and Thagard (1995) identify three
constraints that must be satisfied by a good
analogy 1. Similarity The source and the
target must share some common properties.
Both minds and computers process and store
information ... solve problems. 2. Structure
... there should be an overall correspondence in
structure. Here the mind/computer analogy
becomes more slippery. 3. Purpose The creation
of analogies is guided by the problem-solver's
goals. Analogies are not fixed forever -
they can be modified.
An analogy is comparing one thing that we
understand to something that we do not
understand to gain insight into the unknown.
Therefore, analogies are used to explain or
clarify.
A good analogy seems to accurately or
completely represent critical relationships and
concepts while less important attributes are
considered irrelevant. 
Sir William Thomson (later Lord Kelvin) "I
never satisfy myself until I can make a
mechanical model of a thing. If I can make
a mechanical model, I can understand it.
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It was six men of IndostanTo learning much
inclined,Who went to see the Elephant(Though
all of them were blind),That each by
observationMight satisfy his mind.The First
approached the Elephant,And happening to
fallAgainst his broad and sturdy side,At once
began to bawl"God bless me! but the ElephantIs
very like a wall!"... And so these men of
IndostanDisputed loud and long,Each in his own
opinionExceeding stiff and strong,Though each
was partly in the right,And all were in the
wrong!
The challenge is to make useful analogies that do
more good than harm, not to get too hung up on
which one is the correct or true one.
The Blind Men and the Elephant John Godfrey
Saxe's ( 1816-1887) version of the famous Indian
legend http//www.noogenesis.com/pineapple/blind_m
en_elephant.html
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If the premise If turns out to not be valid
in particular cases, the theoretical hyp
of external validity is rejected, but the model
If then isnt really wrong it just isnt
very useful because it doesnt specify the else
part.
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If God exists then he will save me NOT
saved implies NOT he exists.
Is the logic is internally valid? Does the
conclusion follows properly from the premise? Is
the premise externally valid?
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Any system can be represented by a block diagram
consisting of an input, an output, and a law
(or function f () that determines (maps) input
to output out f(in).
in law out
x(t) f(x(t)) y(t)
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Empirical vs Mechanistic Models?
In hierarchical systems like the biosphere, the
distinction between empirical phenomena and
causal mechanism is frame sensitive
In reductionist approach, every mechanism can be
treated as a phenomenon needing an
explanation one level down. Your unmeasured,
statistical parameters become my dependent
variables.
In practice, there is some mechanism implicit in
all empirical models there are empirical
parameters in all mechanistic models.
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Levins, R. 1966. The strategy of model building
in population biology. American Scientist 54
421-431.
Mathematical models in population biology can
be evaluated in terms of 3 general
characteristics (Levins 1966) precision,
generality, and realism. Any model will
represent a compromise in these characteristics.
It is of course desirable to work with manageable
models which maximize generality, realism, and
precision toward the overlapping but not
identical goals of understanding, predicting, and
modifying nature. But this cannot be done.
Therefore, several alternative strategies have
evolved 1. Sacrifice generality to realism and
precision. This is the approach of Bolling,
(e.g., 1959), of many fishery biologists, arm of
Watt (1956). These workers can reduce the
parameters to the" relevant to the shortterm
behavior of their organism, make fairly accurate
measurements, solve numerically on the computer,
and end with precise testable predictions
applicable to these particular situations. 2.
Sacrifice realism to generality and precision.
Kerner (1957), Leigh (1965), and most physicists
who enter population biology work in this
tradition which involves setting up quite general
equations work which precise results may be
obtained. Their equations am clearly unrealistic.
For instance, they use the Volterra predator-prey
system which omit time lags, physiological
states, and the effect of a species population
density on its own rate of increase, But them
workers hope that their model is analogous to
assumptions of frictionless systems or perfect
gases They expect that many of the unrealistic
assumptions will cancel each other, that small
deviations from realism result in small
deviations in the conclusions, and that, in any
case the way in which nature departs from theory
will suggest where further complications will be
useful. Starting with precision they hope to
increase realism. 3. Sacrifice precision to
realism and generality. This approach is favored
by MacArthur (1965) and myself. Since we are
really concerned in the long run with qualitative
rather than quantitative results (which are only
important in testing hypotheses) we can resort to
very flexible models often graphical, which
generally assume that functions are increasing or
decreasing, convex or concave, greater or less
than some value, instead of specifying the
mathematical form of an equation. This means that
the predictions we can make am also expressed as
inequalities
Orzack, S.H. Sober, E. 1993. A critical
assessment of Levins's The Strategy of Model
Building in Population Biology (1966). The
Quarterly Review of Biology 68533-546. Levins,
R. 1993. A response to Orzack and Sober formal
analysis and the fluidity of science. The
Quarterly Review of Biology 68547-555.
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