Models in Science
A mathematical model is
a simplified idealization of the real world; include what is important; omit
what is not.
Computer Model
Construction:
1)
Construct Algorithm [An algorithm is a set of precisely
defined steps guaranteed to arrive at an answer to a problem or set of
problems. As this implies, a set of
steps that might never end is not an algorithm. In mathematics and computer
science, an algorithm usually means a small procedure that solves a recurrent
problem. Broadly-defined, an algorithm
is an interpretable, finite set of instructions for dealing with contingencies
and accomplishing some task. Algorithms often have steps that repeat (iterate)
or require decisions (logic and comparison) until the task is completed (such
as converting digital 1's and 0's to analog waveforms). Correctly performing an algorithm will not
solve a problem if the algorithm is flawed, or not appropriate to the problem.]
2)
Define Parameters (Inputs)
3)
Write Computer Program
4)
Run Simulations
5)
Validation of Model
6)
Sensitivity Analysis
7)
Adjust Model Design
Models in Science
Physics can be though of
as the "Art" of constructing models which approximate the Real World.
Physicist do not unearth some Absolute Truth about Nature, but rather build
mathematical structures which reflects a close facsimile of Nature. This
facsimile is like a geological map of some territory. A map has a lot of useful
information about the territory, but the map is not the territory.
* Models of Science are
useful maps which approximate Nature.
* The Laws of Physics
are mathematical Models that reflect the underlying order found in Nature.
* Models in Science are
not equivalent to, identical with, or a one-to-one match with the aspects of
Nature they describe.
* There is always some
limited range over which a Model is a useful predictor of Nature.
* The fundamental
criteria for the acceptance or rejection of a Model is determined by how close
the Model predicts the outcome of measurements and observations.
* Models of Science are
not unique. There may be two or more Models which describe the same
observations equally well.
Preference between competing Models is judged
by:
a. Size of the error. The smaller the size of the error between actual measurements and predictions, the more accurate the Model. A good Model will be able to predict the uncertainty within its predictions.
b. Range of Application. The larger the range over which a Model faithfully reflects Nature, the more universal the Model. If the range is big enough we might even call the Model a Law of Physics.
c. Simplicity. A subjective and practical property that makes a Model easier to both understand and manipulate. In Keats words, Truth is beauty, and beauty is truth."
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Example 1: Mechanics,
Relativity, and Quantum Mechanics
Newton's Laws of Motion
faithfully reflect the motion of a body as long as the speed of the body is
small compared to the speed of light. When the speed of the body approaches the
speed of light, Einstein's Theory of Relativity predicts results closer to the
actual values measured than Newton's Laws of Motion predict. The Theory of
Relativity is itself only a better approximation; it has a bigger but still
limited range over which it can be applied.
Both Relativity Theory
and Newton's Laws give inaccurate predictions when trying to explain the
behavior of matter on the atomic scale. In this range, a model call Quantum
Mechanics has proven to make more accurate predictions. None of the
well-established Models represent the Absolute Truth about Nature, but they are
very close likenesses of the Nature under certain conditions.
Example 2: Flat Earth -
Round Earth
A carpenter building a
house uses the model that the Earth is flat expect for local irregularities,
which are usually flattened out before building. A carpenter makes no practical
errors in the accuracy of his measurements in assuming that the Earth is flat.
For an airline pilot, the model that the Earth is perfectly spherical, like a
ball, is useful for navigating over long distances. The path that uses the
least fuel, the shortest path, is the arc of a great circle on a sphere, and
not a straight line on a map.
A more complicated model
of the shape of the Earth's surface is that of a triaxial ellipsoid of
rotation, since the Earth is slightly flatter (about 17 km) at the poles. At
one time astronomers did use this model in order to compare measurements of
celestial objects made at different observatories scattered across the Earth's
surface.
To within the limit of
accuracy of a carpenter's tape measure, the Flat Earth Model gives as good
predictions as that of the spherical trigonometry of a Round Ball Model or the
Triaxial Ellipsoid trigonometry. The Triaxial Ellipsoid Model is more accurate
that the Spherical Ball Model which is more accurate than the Flat Earth Model.
None of these models, constructed by scientists, represent the "TRUE"
shape of the Earth. Which Model you use depends upon the amount of accuracy
needed in your measurement.
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Science and Modeling
A. The Scientific
Process and Model Building
1.
Science
a) A rational, dialectical process of exploring the universe based
on observation, hypothesis formation and hypothesis testing.
Observation > Inductive Reasoning (pattern recogition) >
Hypothesis > Predition > Confirmation >>> "Law"?
b) Dialectical; discussion and reasoning by dialogue as a method
of intellectual investigation.
c) Instead of a method for revealing "truths," science is most accurately characterized as a process used to determine the degree of uncertainty inherent in worldly knowledge.
d) Science: An orderly process by which information is tested, assembled into a widely recognizable pattern, and upon which certain predictions can be precisely described to reveal underlying likenesses among often divergent qualities, quantities, and distributions of things in the natural world. (http://fox.rollins.edu/~jsiry/SCIENCE.HTML)
2.
What are Models?
Simplification, idealization of the real world. Include what is important; omit what is not
Scale Models
Maps as Models
Maps as Scale Models
York, ME Quadrangle 1893
(www.maptech.com)
Conceptual Models:
Systems
Model of Earth's Systems :
Atmosphere
Hydrosphere
Lithosphere
Biosphere
Mathematical Models
Computer
Model Construction:
Construct Algorithm
Define Parameters (Inputs)
Write Computer Program
Run Simulations
Validation of Model
Sensitivity Analysis
Adjust Model Design
2. Development of Sun-Centered
Model of Solar System
Eratosthenes estimates
the diameter of Earth in 247 B.C.
Claudius Ptolemy
formalizes the system of Latitude and Longitude ~150 A.D.
Ptolemy like Aristotle
and Pythagoras put Earth as the center of the solar system
Nicolas Copernicus
around 1540 challenges the Earth centered based on astronomical observations
Galileo validates
Copernicus with observations made through newly developed telescope around 1632
and brought to trial for heresy in 1633.
Newton's Laws of Motion
and Laws of Gravity 1660 - 1690.
Examples of Other
Models:
Darwin's Theory of
Biological Evolution ~1850
Alfred Wegener's model
of Continental Drift 1912 and the development of Plate Tectonics in 1960's and
70's
3. Earth Systems
Concepts
Systems
Theory Concepts:
Systems Theory
Open System
Closed System
System Feedback
Negative Feedback
Positive feedback
Examples from Enhanced Greenhouse Model
System Equilibrium
Steady-State Equilibirum
Dynamic Equlibirum
Threshold to change
Search phrase:
"electrical model"capacitor
Algorithm:
Word the problem.
My three points are by
definition, the point where the ellipse comes out of the earth's surface at
(-5ft, 0) (cartesian), the point where the observed elliptical arc peaks at (0,
10ft), and the point where the elliptical arc re-enters the earth at (5ft, 0).
The center of the earth,
which seems to be one of the foci, would be at (0, -7928 miles).
The radius at the 1st
& 3rd points is 7928 miles, assuming a perfectly spherical earth for this
problem & situation.
The radius at the 2nd
point is 7928 miles & 10 feet minus a very small fraction of an inch (a
trifle over 3.8 thousandths, if my math is right).