# Models

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A model is a representation of some aspect of reality that is created for a specific purpose, and if the representation involved mathematics, it is referred to as a mathematical model. Usually, the model is a representation of a very complex real-life situation that is highly simplified to promote understanding, and there is no single "correct" model for any situation. For example, maps and plans are models, which show a representation of reality that includes only the features relevant to the purpose of the map. By including different details, maps are produced for different purposes. Models do not have to be absolutely correct; they only have to work well enough to be truly useful. For example, linear functions are often used in modelling over short ranges where linear approximation to the true state of affairs is reasonable, because they are mathematically simpler than most other functions. One description of the main stages of the process of mathematical modelling is the following: Specify the purpose (e.g. to make predictions, improve understanding); Create a model (e.g. identify relevant variables, specify assumptions, write the mathematical function); Do the mathematics ; Interpret the results; Evaluate the model (e.g. revise assumptions and adjust the model). An abstract model (or conceptual model) is a theoretical construct that represents physical, biological or social processes, with a set of variables and a set of logical and quantitative relationships between them. Models in this sense are constructed to enable reasoning within an idealized logical framework about these processes and are an important component of scientific theories. Idealized here means that the model may make explicit assumptions that are known to be false in some detail. Such assumptions may be justified on the grounds that they simplify the model while, at the same time, allowing the production of acceptably accurate solutions, as is illustrated below.

## Examples

### Mathematical models

Main article: Mathematical model

• Model of a particle in a potential field. In this model we consider a particle as being a point of mass m that describes a trajectory modelled by a function x: RR3 given its coordinates in space as a function of time. The potential field is given by a function V:R3R and the trajectory is a solution of the differential equation
$m \frac{d^2}{dt^2} x(t) = - \operatorname{grad} V(x(t)).$
Note this model assumes that the particle is a point mass, which is certainly known to be false in many cases where we use the model, e.g. when we use it as a model of planetary motion.
• Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labelled 1,2,...,n each with a market price p1, p2,..., pn. The consumer is assumed to have a cardinal utility function U (cardinal in the sense that it assigns numerical values to utilities), depending on the amounts of commodities x1, x2,..., xn consumed. The model further assumes that the consumer has a budget M which she uses to purchase a vector x1, x2,..., xn in such a way as to maximize U(x1, x2,..., xn). The problem of rational behavior in this model then becomes one of constrained maximization, that is maximize
$U(x_1,x_2,\ldots, x_n)$
subject to
$\sum_{i=1}^n p_i x_i = M.$
This model has been used in models of general equilibrium theory, particularly to show existence and Pareto optimality of economic equilibria. However, the fact that this particular formulation assigns numerical values to levels of satisfaction is a source of criticism. But this is not an essential ingredient of the theory and again, the model is an idealization.

### Other types of models

These two models are examples of mathematical models; following are examples of models that are not mathematical (or at least not numerical).

• Myers-Briggs personality type. Myers-Briggs Type Indicator® is a technique that claims to produce a representation of a person's preferences, using four scales. These scales can be combined in various ways to produce 16 personality types. Types are typically denoted by four letters--for example, INTJ (introverted intuition with extraverted thinking)--to represent a person's preferences. This model is claimed by CPP (formerly known as Consulting Psychologists Press, Inc.) to produce a good predictor of a person's career and marriage partner preference. It should be pointed out, see [1], that there is considerable disagreement among psychologists on whether this assessment technique (and the implied idealized personality model) is of any value.
• Model of political contagion. Some versions of this model are sometimes referred to as the domino theory. In the broadest possible terms, according to this model, political movements that take hold in one country are likely to spread to geographically neighboring ones. This model is surprisingly popular, although as it stands, it is extremely impoverished conceptually, saying nothing about the type of political movement, the degree of geographical proximity, the time scale at which these events take place, etc.

## Use of models

The purpose of a model is to provide an argumentative framework for applying logic and mathematics that can be independently evaluated (for example by testing) and that can be applied for reasoning in a range of situations. Models are used throughout the natural and social sciences, psychology and the philosophy of science. Some models are predominantly statistical (for example portfolio models used in finance); others use calculus, linear algebra or convexity, see mathematical model. Of particular political significance are models used in economics, since they are used to justify decisions regarding taxation and government spending. This often leads to hotly contested debates in the academic world as well as in the political arena; see for instance supply side economics.

Abstract models are used primarily as a reusable tool for discovering new facts, for providing systematic logical arguments as explicatory or pedagogical aids, for evaluating hypotheses theoretically, and for devising experimental procedures to test them. Reasoning within models is determined by a set of logical principles, although rarely is the reasoning used completely mathematical.

In some cases, abstract models can be used to implement computer simulations that illustrate the behavior of a system over time. Simulations are used everywhere in science, especially in economics, engineering, biology, ecology etc., to discover the effects of changing a variable. The validity of different simulation methodologies is a subject of debate in the philosophy and methodology of science.

## Structure of models

Main article: Conceptual schema

A conceptual model is a representation of some phenomenon by logical and mathematical objects such as functions, relations, tables, stochastic processes, formulas, axiom systems, rules of inference etc. A conceptual model has an ontology, that is the set of expressions in the model which are intended to denote some aspect of the modeled object. Here we are deliberately vague as to how expressions are constructed in a model and particularly what the logical structure of formulas in a model actually is. In fact, we have made no assumption that models are encoded in any formal logical system at all, although we briefly address this issue below. Moreover, the definition given here is oblivious about whether two expressions really should denote the same thing. Note that this notion of ontology is different from (and weaker than) ontology as is sometimes understood in philosophy; in our sense there is no claim that the expressions actually denote anything which exists physically or spatio-temporally (to use W. Quine's formulation).

For example, a stochastic model of stock prices includes in its ontology a sample space, random variables, the mean and variance of stock prices, various regression coefficients etc. Models of quantum mechanics in which pure states are represented as unit vectors in a Hilbert space include in their ontologies observables, dynamics, measurement operators etc. It is possible that observables and states of quantum mechanics are as physically real as the electrons they model, but by adopting this purely formal notion of ontology we avoid altogether this question.

## Modeling

Modeling, especially scientific modeling refers to the process of generating a model as a conceptual representation of some phenomenon as discussed above. Typically a model will refer only to some aspects of the phenomenon in question, and two models of the same phenomenon may be essentially different, that is in which the difference is more than just a simple renaming. This may be due to differing requirements of the model's end users or to conceptual or esthetic differences by the modellers and decisions made during the modeling process. Esthetic considerations that may influence the structure of a model might be the modeller's preference for a reduced ontology, preferences regarding probabilistic models vis-a-vis deterministic ones, discrete vs continuous time etc. For this reason users of a model need to understand the model's original purpose and the assumptions of its validity. Having found a model for some desired aspect of reality, it can serve as the basis for simulation, the only way for non-invasive examination of physical reality besides real-world experiments.