Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T14:33:05.643Z Has data issue: false hasContentIssue false
  • English
  • Français

The Emergence of Biomathematics and the Case of Population Dynamics A Revival of Mechanical Reductionism and Darwinism

Published online by Cambridge University Press:  26 September 2008

Giorgio Israel
Giorgio Israel
Affiliation:
Department of MathematicsUniversita di Roma “La Sapienza”
Get access Rights & Permissions [Opens in a new window]

Abstract

The development of modern mathematical biology took place in the 1920s in three main directions: population dynamics, population genetics, and mathematical theory of epidemics. This paper focuses on the first trend which is considered the most significant. Modern mathematical theory of population dynamics is characterized by three aspects (the first two being in a somewhat critical relationship): the emergence of the mathematical modeling approach, the attempt at establishing it in a reductionist-mechanist conceptual framework, and the revival of Darwinism. The first section is devoted to the analysis of the concept of mathematical model and the second one presents an example of a mathematical model (Van der Pol's model of heartbeat) which is a good prototype of that concept. In section 3 the main trends of mathematization of biology and the cultural and scientific contexts in which they found their development are discussed. Sections 4 and 5 are devoted to the contributions of V. Volterra and A. J. Lotka, to the analysis of the differences of their scientific conceptions, and to a discussion of a case study: the priority dispute concerning the discovery of the Volterra-Lotka equations. The historical analysis developed in this paper is also intended to detect the roots of some recent trends of mathematization of biology.

Type
Article
Information
Science in Context , Volume 6 , Issue 2 , Autumn 1993 , pp. 469 - 509
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bailey, N. T. J. 1957. The Mathematical Theory of Epidemics. New York.Google Scholar
Battimelli, G. 1984. “The Mathematician and the Engineer: Statistical Theories of Turbulence in the 20's”, Rivisia di Storia della Scienza 1 (1): 73–94.Google Scholar
Bernoulli, D. 1760. “De la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir.” Mémoires de mat hématique etphysique. Paris.Google Scholar
Bernoulli, D., and d'Alembert, J. 1971. Smallpox Inoculation: An Eighteenth-Century Controversy. Translated and with a critical commentary by Bradley, L.. Nottingham: Nottingham University Press.Google Scholar
Bloch, M. 1964. Apologie pour l'histoire ou métier d'historien. Paris: A. Cohn.Google Scholar
Bourbaki, N. 1986. “L'architecture des mathématiques.” In Les grands courants de lapensée mathématique. Presented by F. Le Lionnais, Cahiers du Sud, 35–47. Reprint. Paris: Rivages.Google Scholar
Burckhardt, H. 1979. “Learning to Use Mathematics.” Bulletin of the IMA, October.Google Scholar
Burghes, D. N., and Borries, M. S. 1981. Modelling with Differntial Equations. New York: Wiley.Google Scholar
Burghes, D. N., Huntley, I., and McDonald, J. 1982. Applying Mathematics. New York: Wiley.Google Scholar
Cannizzaro, L. 1993. “The Beginnings of the Mathematical Theory of Population Genetics.” Dipartimento di Matematica, Università di Roma “La Sapienza.” Preprint.Google Scholar
D'Ancona, U. 1954. The Struggle for Existence. Leiden: Brill.CrossRefGoogle Scholar
Darmon, P. 1986. La longue traque de la variole. Paris: Perrin.Google Scholar
Daston, L. 1988. Classical Probability in the Enlightenment. Princeton, N.J.: Princeton University Press.CrossRefGoogle Scholar
Dell'Aglio, L. 1993. “The Beginnings of the Theory of Games in Borel's and von Neumann's Works.” Rome: Università di Roma “La Sapienza.” Preprint.Google Scholar
Dessiì, P. 1989. L'ordine e il caso: Discussioni epistemologiche e logiche sulla probabilità da Laplace a Peirce. Bologna: II Mulino.Google Scholar
Diner, S. 1992. “Les voies du chaos déterministe dans l'école russe.” In Chaos et déterminisme. Edited by Dahan Dalmedico, A., Chabert, J.-L., and Chemla, K., 115–69. Paris: Editions du Seuil.Google Scholar
Duvillard de Durand, E. E. 1806. Analyse et tableaux de l'influence de lapetite vérole sur la mortalité a chaque age et de celle qu'un préservat tel que la vaccine peut avoir sur la population et la longevizé. Paris.Google Scholar
Edwards, A. 1977. Foundations of Mathematical Genetics. Cambridge: Cambridge University Press.Google Scholar
Fischer, R. A. 1922. “On the Dominance Ratio.” Proceedings of the Royal Society of Edinburgh 42:321–41.CrossRefGoogle Scholar
Fischer, R. A. 1930. The Genetical Theory of Natural Selection. Oxford: Clarendon Press.CrossRefGoogle Scholar
Galileo, G. 1932. Dialogo dei Massimi Sistemi.Google Scholar
Glass, L., and MacKey, M. C. 1988. From Clocks to Chaos: The Rhythms of Life. Princeton, N.J.: Princeton University Press.CrossRefGoogle Scholar
Haldane, J. B. S. 1924. “A Mathematical Theory of Natural and Artificial Selection.” Transactions and Proceedings of the Cambridge Philosophical Society 23:1924–32.Google Scholar
Haldane, J. B. S. 1932. The Causes cf Evolution. London: Longmans and Green.Google Scholar
lanniello, M. G., and Israel, G. n.d. “Boltzmann's Concept of ‘Nachwirkung’ and the ‘Mechanics of Heredity’.” In International Symposium on Ludwig Boltz mann, Roma, Università degli Studi “La Sapienza,” 9–11 February 1989. Vienna: Osterr. Akad. der Wissenschaften. Forthcoming.Google Scholar
Ingrao, B., and Israel, G. 1985. “General Economic Equilibrium Theory. A History of Ineffectual Paradigmatic Shifts, I & II. Fundamenta Scientiae 6 (1):1–45, (2):89–125.Google Scholar
Ingrao, B., and Israel, G. 1990. The Invisible Hand: Economic Equilibrium in the History of Science, Cambridge, Mass.: MIT Press. Originally published as La mano invisibile: L'equilibrio economico nella storia della scienza. Bari: Laterza.Google Scholar
Israel, G. 1981. “‘Rigor’ and ‘Axiomatics’ in Modern Mathematics.” Fundamenta Scientiae 2(2):205–19.Google Scholar
Israel, G. 1982a. “Volterra Archive at the Accademia Nazionale dei Lincei.” Historia Mathematica 9:229–38.Google Scholar
Israel, G. 1982b. “Le equazioni di Volterra e Lotka: una questione di priorità.” In Atti del Convegno su “La Storia delle Matematiche in Italia,” Cagliari, 29–30 September-1 October 1982. Edited by Montaldo, O. and Grugnetti, L., 495–502. Cagliari: Università di Cagliari.Google Scholar
Israel, G. 1984a. “Vito Volterra: un fisico matematico di fronte ai problemi della fisica del Novecento.” Rivista di Storia della Scienza 1(1): 39–72.Google Scholar
Israel, G. 1984b. “Le due vie della matematica italiana contemporanea.” In La ristrut turazione delle scienzefra le due guerre mondiali. Edited by Battimelli, G., De Maria, M., 1:253–87. Rome: La Goliardica.Google Scholar
Israel, G. 1985. “Sulle proposte di Vito Volterra peril conferimento del premio Nobel per la fisica a Henri Poincaré. “Atti del V° Congresso Nazionale di Storia della Fisica, Roma, 29 Ottobre–1° Novembre 1984, Rendiconti dell'Accademia Nazionale delle Scienze detta dei XL, Memorie di Scienze Fisiche e Naturali 103(5th series, Vol. 9, part II): 227–29.Google Scholar
Israel, G. 1986. Modelli matematici. Rome: Editori Riuniti.Google Scholar
Israel, G. 1987. “L'interminabile crisi del meccanicismo.” Rivista di Storia della Scienza 4 (1):73–99.Google Scholar
Israel, G. 1988. “On the Contribution of Volterra and Lotka to the Development of Modern Biomathematics.” History and Philosophy of the Life Sciences 10 (1):37–49.Google Scholar
Israel, G. 1990. “Volterra e la dinamica delle popolazioni biologiche.” In Il pensiero scientUico di Vito Volterra. Edited by Roccheggiani, A., 87–113. Ancona: La Lucerna.Google Scholar
Israel, G. 1991a. “El declive de La mathématique sociale y los inicios de La economia matematica en el contexto de Los avatares del Institut de France.” LLULL, Révista de la Sociedad Espáñola de Historia de la Ciencias y de la Técnicas 14:59–116.Google Scholar
Israel, G. 1991b. “Il determinismo e La teoria delle equazioni differenziali ordinarie.” Physis, Rivisga Internazionale di Storia della Scienza 28:305–58.Google Scholar
Israel, G. 1991c. “Volterra's Analytical Mechanics of Biological Associations,”I and II. Archives Internationales d'Histoire des Sciences 41:57–104; 307–52.Google Scholar
Israel, G. 1991d. “Approccio descrittivo e approccio ermeneutico nella modellistica matematica contemporanea.” In Immagini, Linguaggi, Concetti. Edited by Petruccioli, S., 139–66. Rome: Theoria.Google Scholar
Israel, G. 1992. “L'histoire du principe du déterminisme et ses rencontres avec les mathématiques.” In Chaos et déterminisme. Edited by Dalmedico, A. Dahan, Chabert, J. L., and Chemla, K., 249–73. Paris: Editions du Seuil.Google Scholar
Israel, G. n.d.(a). “The Two Paths of the Mathematization of the Social and Economic Sciences. The Decline of the “Mathematique Sociale” and the Beginnings of Mathematical Economics at the Turn of the Eighteenth Century.” Physis, Rivista Internazionale di Storia della Scienza. Forthcoming.Google Scholar
Israel, G. n.d.(b). La mathématisation du reel. Introduction aux themes et a l'histoire de Ia modélisation maihematique. Paris: Editions du Seuil. Forthcoming.Google Scholar
Israel, G., and Gasca, A. Millán. 1993. “La correspondencia entre Vladimir A. Kostitzin y Vito Volterra (1933–1962) y los inicios de la biomatematica.”Google Scholar
LLULL, Révista de la Sociedad Española de Historia de las Ciencias 16(30):159–224.Google Scholar
Kermack, W. O., and McKendrick, A. G. 1927. “Contributions to the Mathemati cal Theory of Epidemics.” Proceedings of the Royal Statistical Society 115:700–721.Google Scholar
Kingsland, S. E. 1982. “The Refractory Model: The Logistic Curve and the History of Population Ecology.” Quarterly Review of Biology 57:29–52.CrossRefGoogle Scholar
Kingsland, S. E. 1985. Modeling Nature: Episodes in the History of Population Ecology. Chicago: University of Chicago Press.Google Scholar
Kostitzin, V. A. 1937. Biologie mathématique. Paris: A. Cohn. English translation published by Harrap, London, 1939.Google Scholar
La, Condamine. 1773. Histoire de l'inoculation de Ia petite vérole. Amsterdam.Google Scholar
Leigh, E. 1968. “The Ecological Role of Volterra's Equations.” In Some Mathe matical Problems of Biology. Edited by Gerstenhaber, M. Providence, R.I.Google Scholar
Lotka, A. J. 1910. “Contribution to the Theory of Periodic Reactions.” Journal of Physical Chemistry 14:271–74.CrossRefGoogle Scholar
Lotka, A. J. 1920a. “Undamped Oscillations Derived from the Law of Mass Action.” Journal of the American Chemical Society 42:1595–99.CrossRefGoogle Scholar
Lotka, A. J. 1920b. “Analytical Note on Certain Rhythmic Relations in Inorganic Systems.” Proceedings of the NationalAcademy of Sciences 6:410–15.CrossRefGoogle Scholar
Lotka, A. J. [1925]1956. Elements of Physical Biology. Baltimore: Williams & Wilkins. Republished as Elements of Mathematical Biology. Dover: New York.Google Scholar
Lotka, A. J. 1934–1939. Théorie analytique des associations biologiques. Paris: Hermann.Google Scholar
Marsden, J. E., and McCracken, M. 1976. The Hopf Bifurcation and Its Applica tions. New York: Springer.CrossRefGoogle Scholar
May, R. M. 1973. Stability and Complexity in Model Ecosystems. Princeton, N.J.: Princeton University Press.Google ScholarPubMed
Gasca, A. Mill^n 1993. “Immagini matematiche della vita.” Prometeo 10 (43): 88–96.Google Scholar
Mill^n Gasca, A. n.d. The Efforts Towards the Empirical Ver of V. Volterra's Bio mat hematics through His Scienz Ic Correspondence. reprint.Google Scholar
Oster, G., and Guckenheimer, J. 1976. “Bifurcation Phenomena in Population Models.” In Marsden and McCracken 1976, 327–31.Google Scholar
Porter, T. M. 1986. The Rise of Statistical Thinking, 1820–1920. Princeton, N.J.: Princeton University Press.CrossRefGoogle Scholar
Provine, W. B. 1971. The Origins of Theoretical Population Genetics. Chicago: University of Chicago Press.Google Scholar
Scudo, F. M. 1971. “Vito Volterra and Theoretical Ecology.” Theoretical Population Biology 2:1–23.CrossRefGoogle ScholarPubMed
Scudo, F. M. 1984. “The ‘Golden Age’ of Theoretical Ecology: A Conceptual Appraisal.” Revue Européenne des Sciences Sociales 22(67):11–64.Google Scholar
Scudo, F. M. 1988. “Umberto D'Ancona e Vito Volterra: le ragioni di un'amicizia” and “Umberto D'Ancona a Vito Volterra, Lettere.” In La vita, le forme, i numeri, Biologica I, 47–78. Bologne: Transeuropa.Google Scholar
Smith, J. M. 1974. Models in Ecology. Cambridge: Cambridge University Press.Google Scholar
Thompson, W. D'Arcy 1942. On Growth and Form. Cambridge: Cambridge University Press.Google Scholar
Van der Pol, B. L. 1926. “On ‘Relaxation-Oscillations'.” The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science 2(July–December): 978–92.CrossRefGoogle Scholar
Van der Pol, B. L., and Mark, J. Van der 1928. “The Heartbeat Considered as a Relaxation Oscillation, and an Electrical Model of the Heart.” London, Edinburgh and Dublin Philosophical Magazine and Journal of Science 6(July-December):763–75.CrossRefGoogle Scholar
Volterra, V. 1901. “Sui tentativi di applicazione delle Matematiche alle scienze biologiche e sociali.” Opening lecture, Annuario della R. Università di Roma, 1901–2: 3–28. Reprinted in Giornale degli Economisti, s.II, 23, 1901:436–58.Google Scholar
Volterra, V. 1902. “Betti, Brioschi, Casorati. Trois analystes italiens et trois manières d'envisager les questions d'Analyse.” Compte-Rendu du 2e Congrès Interna tional des Mathématiciens, Paris, 1900. Paris: Gauthier-Villars.Google Scholar
Volterra, V. 1906. “Les mathématiques dans les sciences biologiques et sociales.” Trans lated by Ludovic Zoretti. La Revue du Mois 1(1):1–20.Google Scholar
Volterra, V. 1913. Lecons sur lesfonctions de ligne. Lecture notes written and edited by Pérès, J. Paris: Gauthier-Villars.Google Scholar
Volterra, V. 1926a. “Variazioni e fluttuazioni del numero d'individui in specie animali conviventi.” Memorie della R. Accademia dei Lincei 6(2):31–113.Google Scholar
Volterra, V. 1926b. “Fluctuations in the Abundance of a Species Considered Mathemati cally.” Nature 118:558–60.CrossRefGoogle Scholar
Volterra, V. 1927a. Letter to Nature. Nature 119:12.CrossRefGoogle Scholar
Volterra, V. 1927b. “Una teoria matematica sulla lotta per l'esistenza.” Scientia 41:85–102.Google Scholar
Volterra, V. 1990 Leçons sur la théorie mat hématique de la luttepourla vieu. Edited by Brélot, M. Paris: Gauthier-Villars. Reprint. Paris: Gabay.Google Scholar
Volterra, V. 1937. “Principes de biologie mathématique.” Acta Biotheoretica, Leiden, Vol. 111, Part I.Google Scholar
Volterra, V., and Ancona, U. D'. 1935. Les associations biologiques aupoint de vue mathématique. Paris: Hermann.Google Scholar
Von Bertalanify, L. 1968. General System Theory. New York: Braziller.Google Scholar
Wilson, E. O., and Bossert, W. H. 1971. A Primer of Population Biology. Stanford, Calif.Google Scholar
Wright, S. 1916. An Intensive Study of the Inheritance of Color and Other Coat Characters in Guinea-Pigs, with Especial Reference to Graded Variations. Carnegie Institution, Pubi. no. 241, Washington, D.C.Google Scholar
Wright, S. 1921. “Systems of Mating.” Genetics 6:111–78.CrossRefGoogle ScholarPubMed
Wright, S. 1930. “The Genetical Theory of Natural Selection: A Review.” Journal of Heredity 21:349–56.CrossRefGoogle Scholar