Vol.:(0123456789) 1 3 Topoi https://doi.org/10.1007/s11245-017-9538-9 Helmholtz’s Vortex Motion: An Embodied View of Mathematics in the Heuristics of Fluid Mechanics Alain Ulazia 1  · Enetz Ezenarro 1 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract Some viewpoints on the foundations of mathematics and its philosophy are more connected to scientifc practice and its heu- ristics, 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 coopera- tion with other heuristic instruments like extreme thinking, thought experiment and analogy) for an embodied mathematical explanation about vortex dynamics. Keywords Heuristics · Embodied mathematics · Helmholtz · Vortex dynamics · Fluid mechanics 1 Introduction Although the main results of this paper are involved with heuristics, the historical discovery processes in fluid mechanics and, particularly, in the case of Helmholtz’s vor- texes, show other aspects and make other levels of analy- sis emerge within the interface between mathematics and reality. Firstly, the aspect about the foundations of math- ematics and its connection with human activities is treated (Mac Lane’s view). After that (second section), Magnani’s eco-cognitive view and Lakof’s comparison between men- tal schemas of cognitive linguistics and schemas of math- ematics are presented to defend an embodied view of math- ematics based on human activities. Thirdly (third section), a gradual heuristics of hypothesis generation is presented, which is developed via the cooperation between mental schemas, analogies or thought experiments with the back- ground of abductive reasoning, that is, the obtention of a suitable explanation using diferent strategies. In the last sections the examples of fuid dynamics (fow calculus) and vortex dynamics (Helmhotz’s vortexes) are studied, showing that the mathematical construction of a theory is a gradual process in which diferent heuristic instruments cooperate on the basis of mental schemas and human activities to obtain the best explanation about the target phenomenon. In this introduction, the frst point is discussed: the importance of human activities in the foundations of mathematics and their relevance in order to explain the general structure and the branches of mathematics. For centuries many philosophers and mathematicians have addressed the question of the nature of mathematics. The viewpoint that is considered mainstream nowadays largely stems from the work developed by Frege in the fnal decades of the nineteenth century (Davis and Hersh 1981). Frege is considered to have started the modern philosophy of mathematics, establishing at the outset what were to be the main concerns of the renewed discipline. This is an issue of paramount importance for understanding the subsequent development of philosophy of mathematics as a distinct area of knowledge. In fact, Frege’s interests and works largely limited the concerns of the philosophy of mathematics to the main task of searching for secure foundations for mathemat- ics. According to Davis and Hersh (1981), the mainstream * Alain Ulazia alain.ulazia@ehu.eus 1 Institute for Logic, Cognition, Language and Information (ILCLI), University of the Basque Country (UPV/ EHU), Carlos Santamaria Zentroa, Elhuyar plaza 2, 20018 Donostia, Basque Country, Spain