Modeling The World Highlights
Modeling the World Highlights
Review by Craig McClain
“In 1894 Heinrich Hertz published his Principles of Mechanics which attempted […] to purge mechanics of metaphysical, mystical, undefined, unmeasured entities such as force and to base the theory explicitly on measurable quantities. Hertz wanted to be as clear, rigorous, and concise as possible, so that implicit, and perhaps unnecessary, concepts could be eliminated from physical theories, [which he thought should be based solely on measurable quantities].”
A model is any complete and consistent set of verbal arguments, mathematical equations or computational rules which are thought to correspond to some other entity, its prototype. The prototype can be a physical, biological, social psychological or other conceptual entity. In the natural sciences, models are used as tools for dealing with reality. They are caricatures of the real system specifically build to answer questions about it.
Modeling serves several purposes: Data analysis, interpretation, control, prediction and understanding are central. Data analysis and interpretation allow hypothesis testing, parameter testing and making statements about the object phrased in a language that only refers to the model. Prediction, as Hertz emphasized, is obviously the crucial objective of modeling. Using models, Science attempts to forecast and control consequences at many levels from, for example, drug effects to atmospheric increase in carbon dioxide. Control can be viewed as a special kind of prediction answering how to modify or perturb a system to achieve a desired end.
The concept of understanding is illusive, but implies a simplification in modeling to the degree that the model is immediately and intuitively understandable. This does not mean that the model used can be understood directly, but that key aspects should be. It could well be possible to have admirable models that are opaque: A complete model of a cell, in all its details, capable of perfectly predicting its behavior would be a major scientific and computational achievement from a prediction standpoint, but it would fail to explain the cell, as it would be in some sense “just” a formal copy of a cell. For a model to increase our understanding of the World, it needs to strip reality from unnecessary details, and capture the fundamental processes that cause the phenomenon we wish to understand with the model. Indeed, presented with the fully detailed model of the cell, a student might rightly ask “How does it work?” Understanding implies some simplification and approximation relative to reality – and its perfect facsimiles.
Models (or theories) and experimental observations go hand in hand. Models cannot exist without experiments. The experiment is the basis of the scientific method, while the theory is only relevant once the experimental observations have been made. The role of theory is to reduce complexity.
In summary, the modeling process can be conveniently divided into the following three
Step: i) The formulation of the scientific problem in mathematical or computational terms.
Step ii) the solution of the mathematical or computational problem thus created.
Step iii) the interpretation of the solution and its empirical verification in scientific terms.
Step
1
Understand the problem. We must read it carefully to discover what it asks us to find. This is the unknown. We must discover what facts are given to us. These are the data. Explain the question to other people. Draw a figure. Introduce suitable notation.
Step
2
Formulate a plan (a model!). Find the connection between the data and the unknown. You may need to consider auxiliary problems. We develop a suitable plan for modeling our problem; using the data we have been given. We ask ourselves such questions as is this a familiar problem? Have you seen it before? Do you know a related or analogous problem? Do I have sufficient information to answer it?
Step
3
Carry out our plan. Calculate the model using all data and conditions. Do all the calculations, and check them as they go along. Ask: “Can I see it is right?” and then,
“Can I prove it is right?”
Step
4
Looking back. Examine the solution obtained. Can you check the result? Can you derive the solution differently? Were the modeling predictions correct?
posted by Camerashy at 6:52 PM
1 comments
