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Philosophy of Science, Mathematical Models in

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Abbreviations

Philosophy of Science:

Philosophy of science is a branch of philosophy that studies and reflects on the presuppositions, concepts, theories, models, data, arguments, methods, and aims of science. Philosophers of science are concerned with general questions which include the following: What is a scientific theory and when can it be said to be confirmed by its predictions? What are mathematical models and how are they validated? In virtue of what are models representations of the structure and possible behaviors of their target systems? In sum, a major task of philosophy of science is to analyze and make explicit common patterns that are implicit in scientific practice.

Target System:

A target system is an effectively isolated (physical, chemical, biological, or other empirical) part of the universe, made to function or run by some internal or external causes, whose interactions with the universe are strictly delineated by a fixed (input and output) interface, and whose structure, mechanism, or behavior are the principal objects of mathematical modeling. Changes produced in the target system are presumed to be externally detectable via measurements of the system’s characterizing quantitative properties.

Mathematical Model:

A mathematical object of a given type is said to be a mathematical model of a target system just in case it provides substantive information about the system’s modes of being, mechanisms, and behavior in all possible situations in which the system might find itself. The amount of information carried by the model is contingent upon the scope and depth of encoding of the target’s features in the representing mathematical object’s structure. Why model at all? Answer: because investigators who think with mathematical models consistently outperform those who do not. Models help generate testable predictions and explanations and thereby support a better understanding of their targets.

From the standpoint of classical set-theoretic model theory, a mathematical model of a target system is specified by a nonempty set, called the model’s domain, endowed with some algebraic operations and relations, delineated by suitable axioms and intended empirical interpretation. No doubt, this is the simplest definition of a model that, unfortunately, plays a limited role in scientific applications of mathematics. Because applications exhibit a need for a large variety of vastly different mathematical structures – some topological or smooth; some algebraic, computational, stochastic, order-theoretic, or combinatorial; some measure-theoretic or analytic; and so forth, no useful overarching definition of a mathematical model is known even in the edifice of modern category theory. It is invariably difficult to come up with a workable concept of a mathematical model that is applicable and adequate in most fields of applied mathematics and anticipates future extensions.

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Authors and Affiliations

  1. University of Pennsylvania, Philadelphia, PA, USA

    Zoltan Domotor

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  1. Zoltan Domotor
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Correspondence to Zoltan Domotor .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Domotor, Z. (2014). Philosophy of Science, Mathematical Models in. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_407-3

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