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# Model (abstract)

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 a 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, but by their simplification of the model allow the production of acceptably accurate solutions, as is illustrated below.

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## 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 which describes a trajectory which is modeled 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 trajetory is a solution of the differential equation
$m \frac{d^2}{dt^2} x(t) = - \operatorname{grad} V(x(t)).$
Note this model assumes the particle is a point mass, which is certainly known to be false in many cases we use this model, for example, 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 an cardinal utility function U (cardinal in the sense that it assign 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 equilbria. However, the fact that this particular formulation assigns numerical values to levels of satisfaction is the source of criticism (and even ridicule). However, it is not an essential ingredient of the theory and again this is an idealization.

### Other types of models

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

• Myers-Briggs personality type. Myers-Briggs Type Indicator® is a technique which 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 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 neighborhing 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, providing systematic logical arguments, as explicatory or pedagogical aids and for evaluating hypotheses theoretically or 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 over time of a system. Simulations are used everywhere in science, especially in economics, biology and ecology etc., to discover the effects of changing a variable.

## Structure of models

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.

## Abstract models vs models in mathematics

The notion of conceptual model used in physical or social sciences is different from the notion of model as used in mathematics and mathematical logic. In mathematics a model requires two components: a formal deductive component and a semantic one. The formal component is summarized in the concept of mathematical theory. A model of a mathematical theory is an interpretation of syntatic objects by objects defined in terms of the semantic domain, such that formulas that are provably true in the mathematical theory are interpreted by those that are true in the semantic domain. An abstract model, on the other hand, resembles more closely a formal (that is, uninterpreted) mathematical theory itself, including the machinery of inference to discover new facts.

The key difference therefore, between the two notions of model is this: Whereas in mathematical models, interpretation (and validity) occurs entirely within mathematics, for an abstract model interpretation and validity requires the following two activities. First it requires an expression (or perhaps more realistically, an explanation) of the model's ontology (that is what the model deals with) in terms of ordinary language. Secondly, it requires a test of the validity of the model's assumptions. These tests of validity are set up by experimental procedures such as laboratory tests, statistical data gathering or polling. In some cases, both these activities can be expressed by operational definitions.

However, it is possible to interpret one abstract model in terms of another abstract model in such a way that validity of assertions is preserved. This relation between models can actually be used to justify the validity of a model based on the validity of another more refined model.

Finally if we suppose that an abstract model has a formulation as a deductive mathematical theory (not necessarily involving quantitative relationships), then it also should admit a semantic interpretation as a mathematical theory. In fact, one can argue that a model without such a mathematical formulation is ill-posed.

## Modeling: the process of generating a model

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.

## References

• I. Briggs Myers with P. Myers, Gifts Differing. Understanding Personality Type, CPP Books, 1993.
• K. Lancaster, Mathematical Economics, Dover Publications, 1968.
• W. Quine, From a Logical Point of View, Harper Torchbooks, 1961.