Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Philosophy of Science, Mathematical Models in

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_407-3

A Brief Historical Introduction

The subject of mature mathematical models in the form of equations has its roots in post-Newtonian developments of classical mechanics, hydrodynamics, electromagnetism, and kinetic theory of gases. It came on the scene of applied mathematics gradually, during the analytic period before 1880, thanks to the innovative efforts of great scientists, including, among many others, the Swiss mathematician Leonhard Euler (1707–1783), Italian-French mathematician Joseph Louis Lagrange (1736–1813), French astronomer–physicist Pierre Simon de Laplace (1749–1827), Scottish physicist James Clerk Maxwell (1831–1979), English physicist Lord John William Strutt Rayleigh (1842–1919), and Austrian physicist Ludwig Boltzmann (1844–1906). It was the genius of the French mathematician Henri Poincaré (1854–1912) that generated many of our current topological and differential methods of mathematical modeling in the world of dynamical systems. A structuralistset-theoretic...

Keywords

Target System Prey Population Predator Population Prey Model Biological Population 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

Bibliography

  1. Atmanspacher H, Primas H (2003) Epistemic and ontic realities. In: Castell L, Ischebeck O (eds) Time, quantum and information. Springer, New York, pp 301–321CrossRefGoogle Scholar
  2. Bailer-Jones DM (2002) Scientists’ thoughts on scientific models. Perspect Sci 10:275–301CrossRefGoogle Scholar
  3. Batitsky V, Domotor Z (2007) When good theories make bad predictions. Synthese 157:79–103CrossRefMATHGoogle Scholar
  4. Bueno O, French S, Ladyman J (2002) On representing the relationship between the mathematical and the empirical. Philos Sci 69:497–518CrossRefMathSciNetGoogle Scholar
  5. Chakravartty A (2004) Structuralism as a form of scientific realism. Int Stud Philos Sci 18:151–171CrossRefMathSciNetGoogle Scholar
  6. Da Costa NCA, French S (2003) Science and partial truth. A unitary approach to models and scientific reasoning. Oxford University Press, New YorkGoogle Scholar
  7. Domotor Z, Batitsky V (2010) An algebraic-analytic framework for measurement theory. Measurement 43:1142–1164CrossRefGoogle Scholar
  8. Dukich JM (2012) Two types of empirical adequacy: a partial structures approach. Synthese 185(1):1–20CrossRefGoogle Scholar
  9. Engel K-J, Nagel R (2000) One-parameter semigroups for linear evolution equations, vol 194, Graduate texts in mathematics. Springer, New YorkMATHGoogle Scholar
  10. Ermentrout G, Edelstein-Keshet L (1993) Cellular automata approaches to biological modeling. J Theor Biol 160(1):97–133CrossRefGoogle Scholar
  11. French S (2006) Structure as a weapon of the realist. P Aristotelian Soc 106:1–19MathSciNetGoogle Scholar
  12. Fulford G, Forrester P, Jones A (1997) Modelling with differential and difference equations. Cambridge University Press, New YorkCrossRefMATHGoogle Scholar
  13. Giere R (2004) How models are used to represent reality. Philos Sci (Proc) 71:742–752CrossRefGoogle Scholar
  14. Hellman G (2001) Three varieties of mathematical structuralism. Philos Math 9:184–211CrossRefMATHMathSciNetGoogle Scholar
  15. Ladyman J (1998) What is structural realism? Stud Hist Philos Sci 29:409–424CrossRefGoogle Scholar
  16. Lambek J, Rattray BA (1979) A general Stone-Gelfand duality. T Am Math Soc 248:1–35MATHMathSciNetGoogle Scholar
  17. Lasota A, Mackey MC (1994) Chaos, fractals and noise. Stochastic aspects of dynamics, 2nd edn. Springer, New YorkCrossRefMATHGoogle Scholar
  18. Lawvere FW (2005) Taking categories seriously. Theory Appl Categor 8:1–24Google Scholar
  19. Lotka AJ (1956) Elements of mathematical biology. Dover, New YorkMATHGoogle Scholar
  20. Mac Lane S (1996) Structure in mathematics. Philosophia Mathematica 20:1–175MathSciNetGoogle Scholar
  21. May RM (2004) Uses and abuses of mathematics in biology. Science 303:790–793ADSCrossRefGoogle Scholar
  22. Oberst U (1990) Multidimensional constant linear systems. Acta Appl Math 68:59–122MathSciNetGoogle Scholar
  23. Pincock C (2011) Mathematics and scientific representation. Oxford University Press, New YorkGoogle Scholar
  24. Polderman JW, Willems JC (1998) Introduction to mathematical systems theory: a behavioral approach, vol 26, Texts in applied mathematics. Springer, New YorkGoogle Scholar
  25. Röhrl H (1977) Algebras and differential equations. Nagoya Math J 68:59–122MATHMathSciNetGoogle Scholar
  26. Sargent RG (2003) Verification and validation of simulation models. In: Chick S, Sanchez PJ, Ferrin E, Morrice DJ (eds) Proceedings of the 2003 winter simulation conference. IEEE, Piscataway, pp 37–48Google Scholar
  27. Steiner M (1998) The applicability of mathematics as a philosophical problem. Harvard University Press, New YorkMATHGoogle Scholar
  28. Suarez M (2003) Scientific representation: against similarity and isomorphism. Int Stud Philos Sci 17:225–244CrossRefGoogle Scholar
  29. Suppes P (1988) Scientific structures and their representation. preliminary version. Stanford University, StanfordGoogle Scholar
  30. Teller P (2001) Twilight of the perfect model. Erkenntnis 55:393–415CrossRefMathSciNetGoogle Scholar
  31. Van Fraassen BC (1994) Interpretation in science; science as interpretation. In: Hilgevoord J (ed) J. Physics and our view of the world, Cambridge/New York, pp 169–187Google Scholar
  32. Walcher S (1991) Algebras and differential equations. Hadronic Press, Palm HarborGoogle Scholar
  33. Willems JC (1991) Paradigms and puzzles in the theory of dynamical systems. IEEE T Automat Contr 36:259–294CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University of PennsylvaniaPhiladelphiaUSA