Abstract
This work is about computer-based mathematical modeling tools. But before we can implement these tools, we have to understand modeling and, above all, mathematical modeling. The purpose of this chapter is to give a precise definition of the term model. The overview will begin with general, unspecified notions, and then proceed to more formal concepts. Finally, a short historical digression will be presented to suggest further arguments for the importance of mathematical modeling.
“Der Satz ist ein Modell der Wirklichkeit, so wie wir sie uns denken.”
— Wittgenstein, Tractatus logico-philosophicus, 4.01.
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References
Popper criticizes this manner of defining simplicity. He thinks that such considerations are entirely arbitrarily [Popper 1976, p. 99]. He also refuses the aesthetic-pragmatic concept of simplicity. Popper identifies these concepts with the concept of degree of falsification [p. 101].
Aesthetics, being an emotional reaction to simplicity, have an important adaptive function which is in no way the unique privilege of human beings. Charles Darwin stated that some female birds have an aesthetic preference for bright markings on males. For a biological foundation of aesthetics see: Rentschler I., Herzberger B., Epstein D., [ 1988 ], Beauty and Brain, Biological Aspects of Aesthetics, Birkhäuser, Basel.
The American Eliseo Vivas’ theory of disinterested perception,which asserts that the key concept in aesthetics is disinterestedness does not necessarily contradict the above theory of aesthetics as a motivating force. We have all had the experience that when we concentrate consciously on a problem by forcing an issue, it does not work. Sometimes, we have to relax, to step back from the problem in order to make some progress.
See, for instance, Thompson M., The Process of Applied Mathematics, p. 10, in: [Brams/Lucas/Straffin 1983]; see also [Feyerabend Paul, 1986, Wider den Methodenzwang, Suhrkamp, p. 113 and p. 133].
It is said that the outstanding American physicist J.W. Gibbs (1839–1903), during a discussion on the question of whether more attention should be given to ancient languages or to mathematics, pronounced in irritation: “But mathematics is also a language.”
Of course, if “deriving” means “implication” instead of “equivalence” then the consequence is weaker than the antecedence. Knowledge is lost in this case.
This definition only expresses the declarative part of a problem. We shall extend the definition of model in Chapter 7 to englobe also the algorithmic part.
Throughout this text, we use the term parameter for the known data in models, and variable for the unknown elements. Since these two terms clashes with well known concepts in programming languages, we use the convention to call parameters in functions and procedures headings always formal parameters. Using these parameters in a function or procedure call will be called actual parameters. The term variable used in programming languages, which is a name for a memory location, will be called memory variable (see also the Glossary).
A good modern introduction to model theory is: CHANG C.C., KEISLER H.J., [1990], Model Theory. 3rd ed., North-Holland, Amsterdam.
For instance, Karl Popper claims that there is no universal law in evolutionary theory, that this theory is merely “historical” (Popper K, [1979], Objective Knowledge: An evolutionary approach, Clarendon, Oxford, pp. 267–270). (Popper revoked this judgement later in [Popper K., Letter to new Scientist, 21 Aug. 1980 p. 611].)
For an introduction and further references (Sneed, Suppes, etc.) on the structuralist view of theories see: Stegmüller W. [1979], The Structuralist View of Theories, A Possible Analogue of the Bourbaki Programme in Physical Science, Springer, Berlin. A consequent approach of the structuralist view — the author calls it the “semantic view” — in evolutionary theory is worked out in: Lloyd E.A., [1988], The Structure and Confirmation of Evolutionary Theory, Princeton University Press, Princeton. An up-to-date and comprehensive discussion of the subject can also be found in [Herfel al., 1995 ]. ( I am grateful to Daniel Raemi who drew my attention again to the structuralist view of theories. )
The misleading view, that the distinction between model and theory is only a gradual one, is widespread in many scientific communities outside of the philosophy of the science community. According to this view, it is certainly not the complexity nor the size (considering models in operations research which contain many thousands of variables and constraints), nor is it the language, nor the purposes that makes the difference. It is rather the stability over time that distinguishes theories from models. Models are more volatile, subject to frequent changes and modifications [Zimmermann 1987, p. 4].
David Hilbert (1862–1943) once noted that the content of geometry does not change if we replace the words point, line,and plane by, for example, chair, table,and bar [cited in: Yaglom p. 7]. Wigner [1960] in his famous article stated that “mathematics is the science of skilful operations with concepts and rules invented just for this purpose.”
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© 1999 Springer Science+Business Media Dordrecht
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Hürlimann, T. (1999). What is Modeling?. In: Mathematical Modeling and Optimization. Applied Optimization, vol 31. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5793-4_2
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DOI: https://doi.org/10.1007/978-1-4757-5793-4_2
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