Equation or Algorithm: Differences and Choosing Between Them
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Equation or Algorithm: Differences and Choosing Between Them C. Gaucherel • S. Be´rard • F. Munoz
Received: 29 July 2009 / Accepted: 22 August 2010 / Published online: 5 September 2010 Springer Science+Business Media B.V. 2010
Abstract The issue of whether formal reasoning or a computing-intensive approach is the most efficient manner to address scientific questions is the subject of some considerable debate and pertains not only to the nature of the phenomena and processes investigated by scientists, but also the nature of the equation and algorithm objects they use. Although algorithms and equations both rely on a common background of mathematical language and logic, they nevertheless possess some critical differences. They do not refer to the same level of symbolization, as equations are based on integrated concepts in a denotational manner, while algorithms specifically break down a complex problem into more elementary operations, in an operational manner. They may therefore be considered as suited to the representation of different phenomena. Specifically, algorithms are by nature sufficient to represent weak emergent phenomena, but not strong emergent patterns, while equations can do both. Finally, the choice between equations and algorithms are by nature sufficient to represent weak emergent phenomena, but not strong emergent patterns, while equations behave conversely. We propose a simplified classification of scientific issues for which both equation- and/or algorithm-based approaches can be envisaged, and discuss their respective pros and cons. We further discuss the complementary and sometimes conflicting uses of equations and algorithms in a context of ecological theory of metapopulation dynamics. We finally propose both conceptual and practical guidelines for choosing between the alternative approaches.
C. Gaucherel (&) INRA—EFPA, UMR AMAP, TA-A.51/PS2, 34398, Montpellier, Cedex 5, France e-mail: [email protected] S. Be´rard F. Munoz Universite´ Montpellier 2, UMR AMAP, TA-A.51/PS2, 34000 Montpellier, France
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Keywords Emergence Ecology Metapopulation Modeling Semantics Spatial dynamics Theory
1 Introduction Powerful computers have changed the way we address mathematical issues, and to some extent tool-centred approaches have come to compete with formal mathematics. Wolfram (2002) has claimed that an approach based on algorithms and intensive computing is more efficient for scientific demonstration, while other mathematicians emphasize the ‘‘unreasonable effectiveness’’ of a formal mathematical representation (e.g. Wigner 1982). Our basic purpose here is to investigate whether, in practice, equations and algorithms are redundant or complementary tools, and in what situations one approach might outperform the other. Whether one or the other is able to characterize a ‘‘true’’ reality or instead a derived, subjective perception, is outside the scope of the present study, and for the purposes of the present we assume that equations and algorithms do not diff
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