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Applied catastrophe theory in the social and biological sciences

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Abstract

Catastrophe theory is a mathematical theory which, allied with a new and controversial methodology, has claimed wide application, particularly in the biological and the social sciences. These claims have recently been heatedly opposed. This article describes the debate and assesses the merits of the different arguments advanced.

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Literature

  • Amson, J. C. 1975. “Catastrophe Theory: A Contribution to the Study of Urban Systems?”Envir. Planning,B2, 177–221.

    Google Scholar 

  • Arnol’d, V. I. 1972. “Integrals of Rapidly Oscillating Functions and Singularities of Projections of Lagrangian Submanifolds.” (English translation)Funct. Analyt. Applic.,6, 222–224.

    Article  MATH  Google Scholar 

  • Benham, C. J. and J. J. Kozak. 1976. “Denaturation: An Example of a Catastrophe. II. Two-state transitions.”J. theor. Biol.,63, 125–149.

    Article  Google Scholar 

  • — and J. J. Kozak. 1977. “Denaturation: An Example of a Catastrophe. III. Phase Diagrams for Multistate Transitions.”J. theor. Biol.,66, 679–693.

    Article  Google Scholar 

  • — and J. J. Kozak. 1978. “Catastrophes, in Statistical Biophysics.”Behavl Sci.,23, 355–359.

    Google Scholar 

  • Berlinski, D. 1975. “Mathematical Models of the World.”Synthèse,31, 211–227.

    Article  MathSciNet  Google Scholar 

  • Berry, M. V. 1976. “Waves and Thom’s Theorem.”Adv. Phys.,25, 1–25.

    Article  MathSciNet  Google Scholar 

  • Bröcker, Th. 1975. “Differentiable Germs and Catastrophes” (Trans. L. Lander).London Mathl Soc. Lecture Notes, Vol. 17. Cambridge University Press.

  • Chillingworth, D. R. J. 1976. “Structural Stability of Mathematical Models: The Role of the Catastrophe Method.” InMathematical Modelling, Ed. J. G. Andrews and R. R. McLone. London: Butterworths.

    Google Scholar 

  • Cobb, L. 1978. “Stochastic Catastrophe Models and Multimodal Distributions.”Behavl Sci.,23, 360–374.

    MathSciNet  Google Scholar 

  • Cobb, L. in press a. “Parameter Estimation for Ensembles of Nonlinear Stochastics Systems.”

  • Cobb, L. in press b. “Parameter Estimation for Cusp Catastrophe Models.”

  • Cooke, J. and E. C. Zeeman. 1976. “A Clock and Wavefront Model for Control of the Number of Repeated Structures During Animal Morphogenesis.”J. theor. Biol.,58, 455–476.

    Google Scholar 

  • Croll, J. 1976. “Is Catastrophe Theory Dangerous?”New Scient.,70, 630–632.

    Google Scholar 

  • Deakin, M. A. B. 1977. “Catastrophe Theory and Its Applications.”Mathl Scient.,2, 73–94.

    MATH  Google Scholar 

  • Deakin, M. A. B. to appear. “The Impact of Catastrophe Theory on the Philosophy of Science.” Preprint. Monash University Mathematics Department.

  • Dodgson, M. M. 1975. “Quantum Evolution and the Fold Catastrophe.”Evolut. Theory.,1, 107–118.

    Google Scholar 

  • — 1976. “Darwin’s Law of Natural Selection and Thom’s Theory of Catastrophes.”Mathl Biosci. 28, 243–274.

    Article  Google Scholar 

  • —. 1977. “Catastrophe Theory.”Nature,270, 658.

    Article  Google Scholar 

  • — and A. Hallam. 1977. “Allopatric Speciation and the Fold Catastrophe.”Am. Nat.,111, 415–433.

    Article  Google Scholar 

  • Duistermaat, J. J. 1974. “Oscillatory Integrals, Lagrange Immersions, and Unfoldings of Singularities.”Communs pure appl. Math.,27, 207–281.

    MATH  MathSciNet  Google Scholar 

  • Elsdale, T. R., M. J. Pearson and M. Whitehead. 1976. “Abnormalities in Somite Segregation Induced by Heat Shocks toXenopus embryo.”J. Embryol. exp. Morph.,35, 625–635.

    Google Scholar 

  • Fowler, D. H. 1972. “The Riemann-Hugoniot Catastrophe and Van der Waals’ Equation.” InTowards a Theoretical Biology, Vol. 4: Essays, Ed. C. H. Waddington. Edinburgh University Press.

  • Fox, V. 1971. “Why prisoners riot.”Fed. Prob.,35, 9–14.

    Google Scholar 

  • Golubitsky, M. 1978. “An Introduction to Catastrophe Theory and its Applications”SIAM Rev.,20, 352–387.

    Article  MATH  MathSciNet  Google Scholar 

  • Guckenheimer, J. 1978. “The Catastrophe Controversy.”Math. Int.,1, 15–20.

    Article  MATH  MathSciNet  Google Scholar 

  • Hardin, G. (1960). “The Competitive Exclusion Principle.”Science,131, 1292–1297.

    Google Scholar 

  • Hodgkin, A. L. and A. F. Huxley. 1952. “A Quantitative Description of Membrane Current and its Applications to Conduction and Excitation in Nerve.”J. Physiol.,117, 500–544.

    Google Scholar 

  • Isnard, C. A. and E. C. Zeeman. 1976. “Some Models from Catastrophe Theory in the Social Sciences.” InThe Use of Models in the Social Sciences, Ed. L. Collins. London: Tavistock.

    Google Scholar 

  • Klahr, D. and J. G. Wallace. 1976.Cognitive Development—An Information-Processing View. New York: Wiley. (Note, in particular, pp. 201–208).

    Google Scholar 

  • Kolata, G. B. 1977. “Catastrophe Theory: The Emperor has no Clothes.”Science,196, 287, 350–351.

    MathSciNet  Google Scholar 

  • Kozak, J. J. and C. J. Benham. 1974. “Denaturation: An Example of a Catastrophe.”Proc. natn. Acad. Sci. U.S.A.,71, 1977–1981.

    Article  MATH  Google Scholar 

  • Lavis, D. A. and G. M. Bell. 1977. “Thermodynamic Phase Changes and Catastrophe Theory.”Bull. Inst. math. Applic.,13, 34–42.

    MathSciNet  Google Scholar 

  • Lewis, M. 1977. “Catastrophe Theory.”Science,196, 1271.

    Google Scholar 

  • Lorenz, K. 1966.On Agression. London: Methuen.

    Google Scholar 

  • Lu, Y.-C. 1976.Singularity Theory and an Introduction to Catastrophe Theory. Berlin: Springer.

    MATH  Google Scholar 

  • Mees, A. I. 1975. “The Revival of Cities in Medieval Europe: An Application of Catastrophe Theory.”Reg. Sci. Urb. Econ.,5, 403–425.

    Article  Google Scholar 

  • Onsager, L. 1944. “Crystal Statistics. I. A Two-Dimensional Model with Order-Disorder Transition.”Phys. Rev.,65, 117–149.

    Article  MATH  MathSciNet  Google Scholar 

  • Panati, C. 1976. “Catastrophe Theory.”Newsweek, 19/1/76, 46-47.

  • Popper, K. 1959.The Logic of Scientific Discovery (English Translation). New York: Basic Books.

    MATH  Google Scholar 

  • Poston, T. 1978. “The Elements of Catastrophe Theory or the Honing of Occam’s Razor.” InTransformations: Mathematical Approaches to Culture Change, Ed. K. Cooke and C. Renfrew. New York: Academic Press.

    Google Scholar 

  • Poston, T. in press. “On Deducing the Presence of Catastrophes.”Math. Sci. Hum.

  • Poston, T. and I. Stewart. 1976.Taylor Expansions and Catastrophes. Research Notes in Mathematics Vol. 7. London: Pitman.

    Google Scholar 

  • — 1978.Catastrophe Theory and Its Applications. Surveys and Reference Works in Mathematics Vol. 2. London: Pitman.

    Google Scholar 

  • — and A. G. Wilson 1977. “Facility Size vs Distance Travelled: Urban Services and the Fold Catastrophe.”Envir. Planning,A9, 681–686.

    Google Scholar 

  • Renfrew, A. C. and T. Poston. 1978. “Discontinuities in the Endogenous Change of Settlement Pattern.” InTransformations: Mathematical Approaches to Culture Change, Ed. K. Cooke and C. Renfrew. New York: Academic Press.

    Google Scholar 

  • Rosenhead, J. 1976. “Prison ‘Catastrophe’.”New Scient.,71, 140.

    Google Scholar 

  • Rybak, B. and J. J. Béchet. 1961. “Recherches sur l’Électromécanique Cardiaque.”Path. Biol.,9, 1861-1871, 2035–2054.

    Google Scholar 

  • Senechal, M., M. Lewis, R. Rosen and M. A. B. Deakin (separately). 1977. “Catastrophe Theory.” (Letters to the editor.)Science,196, 1271–1272.

    Google Scholar 

  • Sewell, M. J. 1975. “Kitchen Catastrophe.”Mathl. Gaz.,59, 246–249.

    Google Scholar 

  • Stewart, I. N. (1975). “The Seven Elementary Catastrophes.”New Scient. 68, 447–454.

    Google Scholar 

  • Stewart, I. N. and A. E. R. Woodcock. In press. “On Zeeman’s Equation for the Propagation of the Nerve Impulse.”

  • Sussman, H. J. 1975. “Catastrophe Theory.”Synthèse,31, 229–270.

    Article  Google Scholar 

  • — 1976. “Catastrophe Theory—A Preliminary Critical Study.” InPSA 1976: Proceedings of the 1976 Biennial Meeting of the Philosophy of Science Association, Ed. F. Suppe and P. Asquith. East Lansing: Phil. Sci. Assn.

    Google Scholar 

  • — 1977. “Catastrophe Theory: A Skeptic.” (Letter to the Editor.)Science,197, 820–821.

    Google Scholar 

  • — 1979. Review ofCatastrophe Theory: Selected Papers 1972-1977 (by E. C. Zeeman).SIAM Review,21, 268–276.

    Article  Google Scholar 

  • — and Zahler, R. S. 1977. “Catastrophe Theory: Mathematics Misused.”The Sciences,17 (6), 20–23.

    Google Scholar 

  • — and-— 1978. “Catastrophe Theory as Applied to the Social and Biological Sciences: A Critique.”Synthèse 37, 117–216. Reprinted in modified form as “A Critique of Applied Catastrophe Theory in the Social Sciences.”Behav. Sci.,23, 383–389, 1978.

    Article  Google Scholar 

  • Thom, R. 1972a.Stabilité Structurelle et Morphogénèse. Reading, MA: Benjamin.

    Google Scholar 

  • Thom, R. 1972b. “Structuralism and Biology.” InTowards a Theoretical Biology, Vol. 4: Essays, Ed. C. H. Waddington. Edinburgh University Press.

  • — 1975. “Answer to Christopher Zeeman’s Reply.” InDynamical Systems—Warwick 1974 Ed. A. Manning.Lecture Notes in Mathematics Vol. 468. Berlin: Springer.

    Google Scholar 

  • — 1976a. “The Two-fold way of Catastrophe Theory.” InStructural Stability, the Theory of Catastrophes, and Applications in the Sciences Ed. P. Hilton.Lecture Notes in Mathematics Vol. 525. Berlin: Springer.

    Google Scholar 

  • Thom, R. 1976a. “Applications of Catastrophe Theory.” Compiled from notes taken by A. W.-C. Lun, Ed. M. A. B. Deakin, Preprint, Monash University Mathematics Department.

  • Thom, R. 1977. “Structural Stability, Catastrophe Theory, and Applied Mathematics.” (The John von Neumann Lecture, 1976.)SIAM Rev.,19, 189–201.

    Article  MATH  MathSciNet  Google Scholar 

  • — and M. Dodgson (separately). 1977. “Catastrophe Theory.” (Letters to the Editor.)Nature,270, 658.

    Article  Google Scholar 

  • Thompson, M. 1976. “Class, Caste, the Curriculum Cycle and the Cusp Catastrophe.”Stud. Higher Ed.,1, 31–46.

    Article  Google Scholar 

  • Thompson, M. to appear. “The Geometry of Confidence.”

  • Woodcock, A. E. R. and M. Davis. 1978.Catastrophe Theory. New York: Dutton.

    MATH  Google Scholar 

  • Zahler, R. S. 1978. “Catastrophe Theory Reply.” (Letter to the Editor.)Nature,271, 401.

    Article  Google Scholar 

  • — and H. J. Sussmann. 1977. “Claims and Accomplishments of Applied Catastrophe Theory.”Nature,269, 759–763.

    Article  Google Scholar 

  • Zeeman, E. C. 1971. “The Geometry of Catastrophe.”Times Literary Supplement 10/12/71, pp. 1556-1557.

  • Zeeman, E. C. 1972a. “Different Equations for the Heartbeat and Nerve Impulse.” InTowards a Theoretical Biology, Vol. 4: Essays, Ed. C. H. Waddington. Edinburgh University Press.

  • Zeeman, E. C. 1972b. “A Catastrophe Machine.” InTowards a Theoretical Biology, Vol. 4: Essays Ed. C. H. Waddington. Edinburgh University Press.

  • — 1974a. “Primary and Secondary Waves in Developmental Biology.” InSome Mathematical Questions in Biology VI, Ed. S. A. Levin.Lectures on Mathematics in the Life Sciences, Vol. 7. Providence, R.I.: Am. Math. Soc.

    Google Scholar 

  • — 1974b. “On the Unstable Behaviour of Stock Exchanges.”J. Mathl Econ.,1, 39–49.

    Article  MATH  MathSciNet  Google Scholar 

  • — 1976a. “Catastrophe Theory.”Scient. Am.,234 (4), 65–83.

    Article  Google Scholar 

  • — 1976b. “Euler Buckling.” InStructural Stability, the Theory of Catastrophes, and Applications in the Sciences Ed. P. Hilton.Lecture Notes in Mathematics Vol. 525. Berlin: Springer.

    Google Scholar 

  • — 1976c. “Gastrulation and Formation of Somites and Amphibia and Birds.” InStructural Stability, the Theory of Catastrophes, and Applications in the Sciences, Ed. P. Hilton).Lecture Notes in Mathematics Vol. 525. Berlin: Springer.

    Google Scholar 

  • — 1976d. “A Mathematical Model for Conflicting Judgements Caused by Stress, Applied to Possible Misestimations of Speed Caused by Alcohol.”Br. J. math. statist Psychol.,29, 19–31.

    MATH  Google Scholar 

  • — 1976e. “Brain Modelling.”In Structural Stability, the Theory of Catastrophes, and Applications in the Sciences, Ed. P. Hilton.Lecture Notes in Mathematics Vol. 525. Berlin: Springer.

    Google Scholar 

  • — 1976f. “Duffing’s Equation in Brain Modelling.”Bull. Inst. math. Applic.,12, 207–214

    MathSciNet  Google Scholar 

  • — 1977a. “Catastrophe Theory: Draft for a Scientific American Article.” InCatastrophe Theory: Selected Papers 1972–1977 (by E. C. Zeeman). London: Addison-Wesley.

    Google Scholar 

  • — 1977b.Catastrophe Theory: Selected Papers 1972–1977. London: Addison-Wesley. [This reprints the eleven preceding papers and the next one in this reference list.]

    MATH  Google Scholar 

  • —, C. Hall, P. J. Harrison, H. Marriage and P. Shapland. 1976. “A Model for Institutional Disturbances.”Br. J. math. statist. Psychol.,29, 66–80.

    MATH  Google Scholar 

  • —, R. Bellairset al., I. Stewart, M. Berry, J. Guckenheimer, and A. E. P. Woodcock (separately). 1977. “In Support of Catastrophe Theory.” (Letters to the Editor.)Nature,270, 381–384.

    Article  Google Scholar 

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Deakin, M.A.B. Applied catastrophe theory in the social and biological sciences. Bltn Mathcal Biology 42, 647–679 (1980). https://doi.org/10.1007/BF02460985

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