# Is the physicist a mathematician that takes care of reality and mathematician a physicist that cares for reals? The case of the falling and bouncing ball.

Thomas Lecorre, Imène Ghedamsi

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Thomas Lecorre, Imène Ghedamsi. Is the physicist a mathematician that takes care of reality and mathematician a physicist that cares for reals? The case of the falling and bouncing ball.. The learning and teaching of calculus across Disciplines., The Centre for Research, Innovation and Coordination of Mathematics Teaching at the University of Agder (Norway) and hosted at the University of Bergen (Norway), Jun 2023, Bergen, Norway. [https://hal.science/hal-04281301](https://hal.science/hal-04281301) [hal-04281301v1](https://hal.science/hal-04281301v1)

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# Is the physicist a mathematician who takes care of reality and the mathematician a physicist who cares for reals? The case of the falling and bouncing ball

Thomas Lecorre¹ and Imène Ghedamsi¹

¹CY Cergy Paris University, France; thomas.lecorre@cyu.fr; ighedamsi@yahoo.fr

## Mathematicians and physicists: a context of ontological divorce?

Mathematicians and physicists have always worked together, and sometimes have both statutes. Mathematics is not only a tool for physics since it sometimes generates physical entities long time before it had been observed. Physics too is not only a field of application for mathematics, since it is an infinite space of inspiration and experimentation for mathematics. For instance, when mathematical theory was not yet developed but mathematical procedures were already proposed, experimentations in physics played a crucial role in determining the effectiveness of those procedures which then would be theorized (Visser, 2018). This was, for example, the case of calculus and infinitesimal procedures when both physicists and mathematicians agreed on why such calculations may work (Garber, 1999). At the end of the nineteenth century, real numbers were submitted to logical constraints far away from physical concerns and their formalization signalled an ontological gap between the two disciplines. The foundational work of Bourbaki's group in the mid-1930s represents probably the apogee of that ontological divorce which led objects, such as real numbers, that were shared by both disciplines not to have the same nature anymore (Plotnitsky, 2020). At the end of the seventies this separation started to reverse since theoretical physicists (re)discovered the power of abstract mathematics (Urquhart, 2008). Yet, in spite of this renewed interaction between the two communities, many studies have highlighted differences in how they use and interpret mathematical objects including in teaching. For instance, Redish et al. (2015) stress the need for more investigation of the meaning ascribed to these objects in both communities. This paper is a preliminary report on an ongoing research project that aims to study the practices of mathematicians and physicists involved in real number calculations. From the community of practices (CoP) perspective (Biza, 2014; Biza et al., 2014), we see mathematicians (respectively physicists) as a community that share meanings of mathematics they do in both research and university mathematics (respectively physics) teaching. This paper initiates an exploration of the practices involved in real numbers through the bouncing ball phenomenon by addressing the following questions: How can we clarify the meaning given by physicists and mathematicians to real numbers in the bouncing ball phenomenon? Are physicists and mathematicians aware of the differences in these meanings, if any? If yes, how do they deal with in their teaching and scholarly practices?

## Reals and physics in the bouncing ball phenomenon

When one studies the throw of a ball, one has to know that the dynamic of the ball is a deterministic phenomenon shaped by its initial conditions (height and speed). In order to mathematize the problem, physicists usually have to ask this question: What are the forces applying on the ball *just after the throw*? The underlined expression permits to use Newton's law of motion when weight is the only force that determines the dynamic of the ball, whereas in $t=0$ (the time of the throw), the force of the
hand also holds. For a physicist, this expression is meaningful because it has a physical reality. For a mathematician, if $t=0$ stands for the time of throw, "just after the throw" corresponds to no time. More precisely, if $t$ is the time "just after the throw" then $t/2$ is also just after the throw, and this expression (or expressions such as "immediately before the collision with the ground" in De Luca, 2021) is no more meaningful in the model of real numbers because of their density. Using Newton's law of motion (the acceleration equals the sum of the forces at stake), the answer to the physic problem is purely mathematical: $x''=mg$ where $x$ is the position of the ball, $m$ its mass, $g$ the constant of gravity and $x''$ the acceleration. Then $x'=mgt+k$, and $x=1/2mgt^2+kt+h$, and initial conditions ($t=0, x'=V_0$) are used to determine constants $k$ and $h$.

In the case of a bouncing ball, physicists use the bounce coefficient to determine the ratio between the successive heights when the ball hits the floor. For instance, if we suppose that the ball is bouncing exactly to the half of its height, any other coefficient will give the same answer to the central question: does it stop of bouncing or not? From the equations above, we can calculate the time of each bounce and then we can get the total time of bouncing which is a convergent geometric series: the ball is bouncing an infinite number of times in a finite time (De Luca, 2021). There are no more Zeno's of Elea paradoxes for mathematicians since they have been solved a long time ago when real numbers and limit were formalized. For physicists, the mathematical calculation remains problematic because they have to determine its consistency regarding the observed reality. This reality does not behave as reals do: if the time of bouncing is consistent with experimental measures, the number of bounces seems to be finite and heights become too small to be measured. Physicists are also not so keen on tolerating infinity. While they agree with the finitude of bounces by means of the limit process, they argue for a finite number of bounces by cutting the tail of the process using several arguments. One common argument is related to the precision of the height under which they decide that there is no more bounce. However, even reality is not so obvious to get since many factors are involved in the outcome of the measurements in the bouncing ball phenomenon. For example, when one has to find the bounce coefficient, studies show that, to measure it using the sound method, particles have to be very slightly deviated from the mathematical sphere (Heckel et al. 2016). Reality of the phenomenon differs of the theory because of tiny differences in experimental conditions and in method of measurements. According to physicists, whatever the nature of numbers that shape reality, we can only capture decimals from it. Thus, differences between reality and limit model are definitively buried in experimental precisions (De Lucas, 2020). The bouncing ball phenomenon constitutes a shared problem where both disciplines are at stake, and specificities in dealing with (real)numbers appear as obvious.

## An overview of the method

As stated by the literature, both mathematicians and physicists seem to be aware of the bouncing ball phenomenon and both use limit process to determine the time of bouncing. Yet, while mathematicians do not see any more paradox, physicists feel always the need to give arguments to eliminate it. The experiment draws on data obtained from two open questions given to three French mathematicians and three French physicists. We seek to understand how these scholars deal with the bouncing ball phenomenon in their teaching and scholarly practices. For that, we made the choice to clearly underline key literature results about this phenomenon before giving the questions. Our aim is to
provoke the need to justify one's choices as well as their interpretations. The introduction by the researcher was structured as follows: the context of the bouncing ball, the way it is usually solved with the laws of Newton and the paradox that emerges are explained. It is also explained that mathematicians are using the properties of reals to evaluate the time of bouncing using an infinite number of bounces and physicists usually use the concept of precision (the height of bouncing is lower bonded) to limit the number of bounces. Two open questions are then asked about the explained situation: 1) To what extent does this situation cause discomfort in your research practices? 2) How do you manage this situation in your teaching practices?

## Preliminary results

Except one physicist who declare not being ready to answer questions about this phenomenon which is neither his domain of research nor his domain of teaching, all answers from physicists and mathematicians show sufficient familiarity with the phenomenon. The case of this one physicist to may indicate that: 1) not all physicists are aware of the bouncing ball phenomenon even though it is a paradigmatic case of physicist practices with infinitesimal calculations; 2) not all physicists seem to have faced the complexity of this case in their own university specialized studies.

While the three mathematicians agreed with the researcher's introduction about mathematical calculations for both disciplines, the two other physicists reacted unexpectedly to it. Despite their agreement with the finitude of the number of bounces according to the "reality", the physicists argued differently to express their disagreement with the researcher. Specifically, one physicist believed that an infinite sum of times gives necessary an infinite time and that the time of bounces is only obtained by a limit since it is too hard to calculate all of the terms that fit a certain given precision. His explanations were mathematically contradictory since he accepts the use of the limit for pragmatic reasons but refused the possibility of the finitude of the result of an infinite process. The other physicist rejected the possibility of an infinity of bounces since a certain force of Van der Waals is at stake and stuck the ball to the ground after a finite number of bounces (Falcon et al., 1998). According to this force, there is in fact a finite number of bounces and it is not a matter of precision or measurement. So, this participant did not feel there is any difference between mathematics and physics: when using the same models, they give the same results. This explanation is also mathematically contradictory since, on one hand, he refused absolutely the possibility of the infinite number of bounces ("we know that it is finite, one just has to try") and, on the other hand, he accepted the use of models implying infinity in the same way as mathematicians do.

A significant part of mathematicians and physicists' answers did not tackle the two questions and remained at the level of discussion of the researcher's introduction. However, the results show stability in mathematicians' explanation of the phenomenon and disparities among physicists' answers. While both physicists agreed that the mathematical model is not reality, their interpretation of this model was quite different. Particularly, they did not seem to share the same epistemology regarding the mathematization of the bouncing ball phenomenon. However, almost all physicists and mathematicians noticed the differences within real numbers practices between the two disciplines and expressed the necessity to guide students towards a more unifying object. While these mathematicians' answers show an *alignment* (Biza et al., 2014) with classical mathematics practices,
these physicists do not seem to belong to the same *CoP*. This result needs further investigation among the physicists' *CoPs*. However, both mathematicians and physicists recognize students' dilemma and agree to adapt them for more coherence across the two disciplines. This result firstly highlights the necessity to distinguish the researchers *CoP* from the teachers *CoP* for each discipline (Biza, 2014). It also echoes the idea of *critical alignment* to traditional teachers *CoPs* for both mathematicians and physicists (Biza et al., 2014). Following Biza et al.'s (2014) conceptualization of this idea as a type of *inquiry*, this means that teachers' *CoPs* of both disciplines may evolve together in a form of *community of inquiry* because of their common interest in learning issues. Further studies should examine the effectiveness of the *community of practice/inquiry lens* in investigating physicists and mathematicians' practices with reals and reality in order to identify didactical, motivational, ontological and epistemological aspects, to trigger common understandings in university teaching.

## References

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Biza, I., Jaworski, B., &, Hemmi, K. (2014). Communities in university mathematics. *Research in Mathematics Education*, 161-176. https://doi.org/10.1080/14794802.2014.918351

De Luca, R., Di Mauro, M., & Naddeo, N. (2021). Bounces of a marble and Zeno's paradox. *Journal of Physics: Conference Series* 1929. https://doi.org/10.1088/1742-6596/1929/1/012071

De Luca, R., Di Mauro, M., Naddeo, N., Onorato, P., &Rosi, T. (2020). Achilles overtakes the turtle: experiments and theory addressing students' difficulties with infinite processes. *Physics Education*, 55 (3). https://doi.org/10.1088/1361-6552/ab6eb3

Falcon, E., Laroche, C., &Fauve, S. (1998). Behavior of one inelastic ball bouncing repeatedly off the ground. *The European Physical Journal B*, 3, 45–57. https://doi.org/10.1007/s100510050283

Garber, E. (1999). *The Language of Physics: The Calculus and the development of Theoretical Physics in Europe*. Springer Science & Business Media.

Heckel, M., Glielmo, A., Gunkelmann, N., &Pösche, T. (2016). Can we obtain the coefficient of restitution from the sound of a bouncing ball? *Physics Review*, 93(3). https://doi.org/10.1103/PhysRevE.93.032901

Plotnitsky, A. (2020). The Ghost and the Spirit of Pythagoras. In Sriraman, B. (ed), *Handbook of the History and Philosophy of Mathematical Practice*. Springer, Cham.

Redish, E.F., & Kuo, E. (2015). Language of Physics, Language of Math: Disciplinary Culture and Dynamic Epistemology. *Sci & Educ*, 24, 561–590. https://doi.org/10.1007/s11191-015-9749-7

Urquhart, A. (2008). The boundary between Mathematics and physics. In Mancosu P. (ed.), *The Philosophy of Mathematical Practice* (online edn, Oxford Academic).

Visser, M. (2018) The utterly prosaic connection between Physics and Mathematics. *Philosophies*, 3(4), 25. https://doi.org/10.3390/philosophies3040025