Skip to main content
SpringerLink
Log in

Philosophy of Science, Mathematical Models in

  • Reference work entry
Mathematics of Complexity and Dynamical Systems

  • 538 Accesses

Article Outline

Glossary

Definition of the Subject

Introduction

Mathematical Models: What Are They?

Philosophical and Mathematical Structuralism

Three Approaches to Applying Mathematical Models

Validating Mathematical Models

Future Directions

Bibliography

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

Philosophy of science:

Broadly understood, philosophy of science is a branch of philosophy that studies and reflects on the presuppositions, concepts, theories, 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 mathematical models representations of the structure and behavior 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.

Mathematical model:

Stated loosely, models are simplified, idealized and approximate representations of the structure, mechanism and behavior of real-world systems. From the standpoint of 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 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, 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 difficult to come up with a workable concept of a mathematical model that is adequate in most fields of applied mathematics and anticipates future extensions.

Target system:

There are many definitions of the concept of ‘system’. Here by a target system we mean an effectively isolated (physical, 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 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.

Bibliography

Primary Literature

  1. Arnold L (1998) Random dynamical systems. Springer, New York

    MATH  Google Scholar 

  2. Atmanspacher H, Primas H (2003) Epistemic and ontic realities. In: Castell L and Ischebeck O (eds) Time, Quantum and Information. Springer, New York, pp 301–321

    Google Scholar 

  3. Bailer‐Jones DM (2002) Scientists' thoughts on scientific models. Perspectives Sci 10:275–301

    Google Scholar 

  4. Balzer W, Moulines CU, Sneed J (1987) An Architectonic for science: The structuralist program. Reidel, Dordrecht

    Book  MATH  Google Scholar 

  5. Balzer W, Moulines CU (1996) Structuralist theory of science. Focal issues, new results. de Gruyter, New York

    Book  Google Scholar 

  6. Batitsky V, Domotor Z (2007) When good theories make bad predictions. Synthese 157:79–103

    Article  MATH  Google Scholar 

  7. Bourbaki N (1950) The architecture of mathematics. Am Math Mon 57:221–232

    Article  MathSciNet  MATH  Google Scholar 

  8. Chakravartty A (2004) Structuralism as a form of scientific realism. Int Stud Philos Sci 18:151–171

    Article  MathSciNet  Google Scholar 

  9. Colonius F, Kliemann W (1999) The dynamics of control. Birkhäuser, Boston

    Google Scholar 

  10. Da Costa NCA, French S (2003) Science and partial truth. A unitary approach to models and scientific reasoning. Oxford University Press, New York

    Google Scholar 

  11. Erkal C (2000) The simple pendulum: a relativistic revisit. Eur J Phys 21:377–384

    Article  Google Scholar 

  12. Fulford G, Forrester P and Jones A (1997) Modelling with differential and difference equations. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  13. Giere R (2004) How models are used to represent reality. Philos Sci (Proc) 71:742–752

    Article  Google Scholar 

  14. Giry M (1981) A categorical approach to probability theory. In: Banaschewski B (ed) Categorical Aspects of Topology and Analysis, Lecture Notes in Mathematics, vol 915. Springer‐Verlag, New York, pp 68–85

    Chapter  Google Scholar 

  15. Krantz DH, Luce RD, Suppes P, Tversky A (1971) Foundations of measurement. Academic Press, New York

    MATH  Google Scholar 

  16. Ladyman J (1998) What is structural realism? Stud Hist Philos Sci 29:409–424

    Google Scholar 

  17. Lambek J, Rattray BA (1979) A general Stone‐Gelfand duality. Trans Am Math Soc 248:1–35

    MathSciNet  MATH  Google Scholar 

  18. Landry E (1999) Category theory: The language of mathematics. In: Howard DA (ed) Proceeedings of the 1998 Biennial Meeting of the Philosophy of Science Association. The University of Chicago Press, Chicago, pp S14–S26

    Google Scholar 

  19. Lasota A, Mackey MC (1994) Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, 2nd edn. Springer‐Verlag, New York

    MATH  Google Scholar 

  20. Lawvere FW (2005) Taking categories seriously. Theor Appl Categ 8:1–24

    Google Scholar 

  21. Le Cam L (1986) Asymptotic Methods in Statistical Decision Theory. Springer‐Verlag, New York

    Book  MATH  Google Scholar 

  22. Mac Lane S (1971) Categories for the working mathematician. Springer‐Verlag, New York

    Book  MATH  Google Scholar 

  23. Mac Lane S (1996) Structure in Mathematics. Philosophia Mathematica 20:1–175

    MathSciNet  Google Scholar 

  24. Moore C (1991) Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4:199–230

    Article  MathSciNet  MATH  Google Scholar 

  25. Morgan M (2005) Experiments versus models: New phenomena, inference and surprise. J Econ Methodol 12:317–329

    Article  Google Scholar 

  26. Oberst U (1990) Multidimensional constant linear systems. Acta Appl Math 68:59–122

    MathSciNet  Google Scholar 

  27. Polderman JW, Willems JC (1998) Introduction to mathematical systems theory: A behavioral approach. Texts in Applied Mathematics 26. Springer‐Verlag, New York

    Google Scholar 

  28. Röhrl H (1977) Algebras and differential equations. Nagoya Math J 68:59–122

    Google Scholar 

  29. Sargent RG (2003) Verification and validation of simulation models. In: Chick S, Sanchez PJ, Ferrin E, and Morrice DJ (eds) Proceedings of the 2003 Winter Simulation Conference. IEEE, Piscataway, pp 37–48

    Google Scholar 

  30. Schlomiuk DI (1970) An elementary theory of the category of topological spaces. Trans Am Math Soc 149:259–278

    Article  MathSciNet  MATH  Google Scholar 

  31. Shapiro S (1997) Philosophy of Mathematics. Structure and Ontology. Oxford University Press, Oxford

    MATH  Google Scholar 

  32. Steiner M (1998) The applicability of mathematics as a philosophical problem. Harvard University Press, New York

    MATH  Google Scholar 

  33. Suarez M (2003) Scientific representation: Against similarity and isomorphism. Int Stud Philos Sci 17:225–244

    Article  Google Scholar 

  34. Walcher S (1991) Algebras and differential equations. Hadronic Press, Palm Harbor

    Google Scholar 

  35. Willems JC (1991) Paradigms and puzzles in the theory of dynamical systems. IEEE Trans Automatic Control 36:259–294

    Article  MathSciNet  MATH  Google Scholar 

  36. Worrall J (1996) Structural realism: the best of both worlds? In: Papineau D (ed) The Philosophy of Science, Oxford University Press, New York, pp 139–165

    Google Scholar 

Books and Reviews

  1. Brin M, Stuck G (2002) Introduction to Dynamical Systems. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  2. Hellman G (2001) Three varieties of mathematical structuralism. Philosophia Mathematica 9:184–211

    Article  MathSciNet  MATH  Google Scholar 

  3. Margani L, Nersessian N, Thagard P (eds) (1999) Model-based Reasoning in Scientific Discovery. Kluwer, Dordrecht

    Google Scholar 

  4. Margani L, Nersessian N (eds) (2002) Model-based Reasoning: Science, Technology, Values. Kluwer, Dordrecht

    Google Scholar 

  5. Morgan M, Morrison M (1999) Models as Mediators. Perspectives on Natural and Social Science. Cambridge University Press, Cambridge

    Book  Google Scholar 

  6. Psillos S (1999) Scientific Realism: How Science Tracks Truth. Routledge, London

    Google Scholar 

  7. Suppes P (2002) Representation and Invariance of Scientific Structures. CSLI Publications, Stanford University, Stanford

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Pennsylvania, Philadelphia, USA

    Zoltan Domotor

Authors
  1. Zoltan Domotor
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Domotor, Z. (2012). Philosophy of Science, Mathematical Models in. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_89

Download citation

Publish with us

Policies and ethics

Search

Navigation