Abstract
Some viewpoints on the foundations of mathematics and its philosophy are more connected to scientific practice and its heuristics, mainly with the construction of physical theories and the search for the best explanations of physical phenomena by means of abduction or the solution of problems by the analytical method. Some researchers have introduced the importance of human cultural activities into the cognitive aspects of the mental processes of scientists, proposing an embodied approach in the bridge between mathematics and reality. Fluid mechanics is an interesting area in this sense due to its position if the network of mathematics. By means of an historical example on vortex motion by Helmholtz, we show that the intuitive idea of eddy (or vortex) contains cognitive properties of a mental schema and that it gives many heuristic options (via cooperation with other heuristic instruments like extreme thinking, thought experiment and analogy) for an embodied mathematical explanation about vortex dynamics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barrow I (1660) Letiones geometricae (J. M. Child Trans., 1916): the geometrical lectures of Isaac Barrow. The Open Court Publishing Company, London
Cellucci C (2008) The nature of mathematical explanation. Stud History Philos Sci A 39(2):202–210
Cellucci C (2013) Top-down and bottom-up philosophy of mathematics. Found Sci 18:1–14
Cellucci C, Gillies D (2005) Mathematical reasoning and heuristics. King’s College Publications, London
Clement J (2008). Creative model construction in scientists and students: the role of imagery, analogy, and mental simulation. Springer, New York
Darrigol O (2005) Worlds of flow: a history of hydrodynamics from the Bernoullis to Prandtl. Oxford University Press, Oxford
Davis PJ, Hersh R (1981) The mathematical experience. Birkhäusser, Basel
Gillies D (2015) Why do scientific revolutions begin? In: Ippoliti E (ed) Heuristic reasoning. Springer, New York, pp 89–112
Hanson NR (1958). The logic of discovery. J Philos 55:1073–1089
Helmholtz HV (1867) LXIII. On Integrals of the hydrodynamical equations, which express vortex-motion. Lond Edinb Dublin Philos Mag J Sci 33(226):485–512
Hintikka J (1985) True and false logic of scientific discovery. Commun Cognition 18(1/2):3–14
Horsten L (2016) Philosophy of mathematics. In: Zalta EN (ed) The stanford encyclopedia of philosophy (Winter 2016 Edition). Stanford University, Stanford. https://plato.stanford.edu/archives/win2016/entries/philosophy-mathematics/>
Ippoliti E (2013) Generation of hypotheses by ampliation of data. In: Magnani L (ed) Model-based reasoning in science and technology: theoretical and cognitive issues. Springer, New York
Ippoliti E (2014) Dynamic generation of hypotheses: mandelbrot, soros and far-from-equilibrium. In: Ippoliti E (ed) Heuristic reasoning. Springer, New York
Ippoliti E (2016) Mathematical models of time as a heuristic tool. In: Magnani L, Casadio C (eds) Model-based reasoning in science and technology. Springer, New York
Jaccard J, Jacoby J (2010). Theory construction and model-building skills: a practical guide for social scientists. Guilford Press, New York
Lakoff G, Nuñez RE (2000) Where mathematics come from. Basic Books, New York
Leroux J (2001) ‘Picture theories’ as forerunners of the semantic approach to scientific theories. Int Stud Philos Sci 15(2):189–197
Mac Lane S (1986) Mathematics: form and function. Springer, New York
Magnani L (2001) Abduction, reason, and science: processes of discovery and explanation. Kluwer Academic/Plenum Publishers, New York
Magnani L (2007) Abduction and chance discovery in science. Int J Knowl Based Intell Eng 11:273–279
Magnani L (2014) Are heuristics knowledge-enhancing? Abduction, models, and fictions in science. In: Ippoliti E (ed) Heuristic reasoning. Springer, New York, pp 29–56
Paavola S (2004) Abduction as a logic and methodology of discovery: the importance of strategies. Found Sci 9(3):267–283
Psillos S (1996) Ampliative reasoning: induction or abduction. ECAI96 workshop on abductive and inductive reasoning. Springer, New York
Rouse H, Ince S (1957). History of hydraulics. Edwards Brothers, Ann Arbor
Sintonen M (1996) Structuralism and the interrogative model of inquiry. In: Balzer W, Moulines CU (eds) Structuralist theory of science: focal issues, new results. Walter de Gruyter, Berlin/New York, pp 45–75
Ulazia A (2016a) The cognitive nexus between Bohr’s analogy for the atom and Pauli’s exclusion schema. Endeavour 40(1):56–64
Ulazia A (2016b) Multiple roles for analogies in the genesis of fluid mechanics: how analogies can cooperate with other Heuristic strategies. Found Sci 21(4):543–565
University of Iowa Gallery (2017). Retrieved http://user.engineering.uiowa.edu/~cfd/gallery/vortex.html
Venturi GB (1797) Recherches experimentales sur le principe de communication laterale dans les jluides, Paris.
Venturi GB (1826) Experimental inquiries concerning the principle of the lateral communication of motion in fluids. In: Tredgold T (ed) Tracts on hydraulics, London. J. Taylor (translation by W. Nicholson, 2nd edn)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ulazia, A., Ezenarro, E. Helmholtz’s Vortex Motion: An Embodied View of Mathematics in the Heuristics of Fluid Mechanics. Topoi 39, 949–961 (2020). https://doi.org/10.1007/s11245-017-9538-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11245-017-9538-9