Recently I saw a question on Answer.com that was not answered, so I took the opportunity to provide a response. First, not all models are mathematical. Modeling in general is to pretend that one deals with a real thing while really working with an imitation. In operations research the imitation is a computer model of the simulated reality. A flight simulator on a PC is also a computer model of some aspects of the flight: it shows on the screen the controls and what the "pilot" (the youngster who operates it) is supposed to see from the "cockpit" (his armchair).
In order to understand what it means to model a phenomena or process, we must first understand the term "model" and understand its limitation. A model is a physical, mathematical, or otherwise logical representation of a system, entity, phenomenon, or process . A model can also be thought of as an abstraction of the real world, or an approximation of it. If you think about the problem of modeling a human being, or just the mind of a human, you can immediately see the limitations of modeling. We can use the term "system" to encompass systems, entities, phenomenon, or process. Since a modeling is a representation, abstraction, or approximation of the "system" being modeled, we must understand that it is not an "exact" representation, i.e., we can't model every aspect of the system. First, we don't know everything we need to know in order to model the system. We may not be able to define a process of the system with mathematical precision, or with heuristic algorithms, and many of the processes may not appear logical. Second, even if we were to know everything about the system, we may not have enough computing power to model every process, at least for complex systems, e.g., a human being, the earth's ecosystem, etc.
Yet with their limitations, models are a good way to gain understanding of how a system operates. George E.P. Box said, "All models are wrong; some models are useful." They are wrong in that there is not a one-to-one mapping form the real system to the model; they are useful in that we can use them to understand the system, or at least certain aspects of the system. Even though we often use models to predict behavior, this is a dangerous process, and we must do it with caution.
There are many types of models that one can use when trying to represent a system. These can be divided into several classes, and for simplicity we will consider two classes: physical models and symbolic models . Physical models include mock-up models (e.g., a vehicle mock-up), scale models (e.g., a 1:48 scale aircraft model), Iconic model, natural model, or a fashion model (which represent what we all want to look like).
Symbolic models include narrative models, graphical models, tabular models, software models, and mathematical models. These models are not necessarily mutually exclusive. For example a tabular model might contain data derived from a mathematical model, or a mathematical model may be embedded in software code.
So, a mathematical model is a symbolic model, or a symbolic representation or imitation of a real system or phenomenon. An example of a model is the equation for a line (a liner model) y = mx + b. A more complicate mathematical model is a model model is the waiting time for the drive through window at McDonalds at lunchtime (I will not confuse you with the mathematical formula). We do have models that cover all times during the day (at the drive through), but the solutions to these models may not be derived, "mathematically." We use numeric approximations, heuristic models (or algorithms) and simulations to solve these.
Selected mathematical modeling references:
1. http://en.wikipedia.org/wiki/Mathematical_model
2. http://www.math.montana.edu/frankw/ccp/modeling/topic.htm
3. http://www.meaningfulmath.org/modeling
4. http://www.learn.motion.com/products/modeling/index.html
5. http://www.stt.msu.edu/~mcubed/modeling.html
6. http://www.blurb.com/bookstore/detail/1461238
7. http://www.mat.univie.ac.at/~neum/model.html
8. http://www.math.colostate.edu/~pauld/M331/
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