Abstract
We can distinguish between two different ways in which mathematics is applied in science: when mathematics is introduced and developed in the context of a particular scientific application; when mathematics is used in the context of a particular scientific application but it has been developed independently from that application. Nevertheless, there might also exist intermediate cases in which mathematics is developed independently from an application but it is nonetheless introduced in the context of that particular application. In this paper I present a case study, that of the Lagrange multipliers, which concerns such type of intermediate application. I offer a reconstruction of how Lagrange developed the method of multipliers and I argue that the philosophical significance of this case-study analysis is twofold. In the context of the applicability debate, my historically-driven considerations pull towards the reasonable effectiveness of mathematics in science. Secondly, I maintain that the practice of applying the same mathematical result in different scientific settings can be regarded as a form of crosschecking that contributes to the objectivity of a mathematical result.
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Notes
An overview of the connections that the applicability problem has with several ongoing debates in philosophy of science and philosophy of mathematics is offered in Panza and Molinini (forthcoming).
An exception is Steiner (2005), to which I shall return below.
A similar distinction has been made by Mark Steiner (2005). In the next section I will consider Steiner’s proposal and I will spell out two respects in which my proposal is different from Steiner’s.
I thank José Ferreirós for having drawn my attention to the case of the Dirichlet’s principle.
Obviously, some work was done to find the best way of modeling the biological scenario in terms of a system of differential equations. But this does not affect the point I want to stress here, namely that the mathematical resources employed by Volterra and Lotke were already available and they were not introduced ex novo in the modeling process.
More formally, the delta Dirac is not a function in the ordinary mathematical sense because if a function is zero everywhere except at one point, and the integral of this function over its entire domain of definition exists, then the value of this integral is necessarily equal to zero (this is not the case for the delta Dirac, which has integral equal to 1).
The delta function was interpreted as a distribution by Laurent Schwartz, who observed: “I believe I heard of the Dirac function for the first time in my second year at the ENS. I remember taking a course, together with my friend Marrot, which absolutely disgusted us, but it is true that those formulas were so crazy from the mathematical point of view that there was simply no question of accepting them. It didn’t even seem possible to conceive of a justification. These reflections date back to 1935, and in 1944, 9 years later, I discovered distributions. The original reflections remained with me, and became part of the accumulated material I was talking about earlier, which remained in a corner of my mind, only to explode suddenly the night I discovered distributions, in November 1944. This at least can be deduced from the whole story: it’s a good thing that theoretical physicists do not wait for mathematical justification before going ahead with their theories!” (Schwartz 2001, 218).
It is important to note that Steiner’s purported cases of canonical and non-canonical applications are not accepted unanimously by historians and philosophers of science. For instance, many scholars would not agree on the claim that differential calculus was introduced by Newton in order to describe accelerated motion. Although interesting, I will not address those criticisms here and I limit my treatment to a presentation of Steiner’s proposal.
In some passages of his memoir on the libration of the Moon (e.g., Lagrange 1780, 16–17, 20–21, 25–26) Lagrange puts forward the idea of a method for the elimination of dependent variables that can be identified with what he will explicitly introduce, in Mécanique analytique, as the multiplier’s method.
The letter is reported by Varignon in his Nouvelle mécanique ou statique (Varignon 1725, tome 2, 174–176) and it is dated 1717.
The term “moment” will be adopted for a long time. It began to be replaced by “virtual work” only after Coriolis introduced this terminology in his treatise Du calcul de l'effet des machines (1829). In what follows I shall use “virtual work” instead of “moment”, since this makes the treatment clearer from a modern point of view.
For a more detailed illustration of the principle of the pulleys and Lagrange’s treatment of it see Capecchi 2012, 260, and Dugas 1988, 334–335. According to Capecchi, Lagrange considers the principle of the pulleys as “self-evident” (Capecchi translates “évident” with “self-evident” and therefore, for him, Lagrange is giving a particular epistemic status to the principle of the pulleys). Differently from Capecchi, I translate “évident” with “clear”, since Lagrange does not seem to use “évident” with the epistemic force that we attach to “self-evident” (I am grateful to one referee for drawing my attention to this point).
Helmut Pulte has defined “mechanical Euclideanism” as the ideal of reducing mechanics to an axiomatic-deductive system (Pulte 1998). Lagrange’s program of reducing mechanics to analysis provides an example of mechanical Euclideanism. As Pulte shows, the project of reducing mechanics to mathematics belongs to a tradition that was already present when the first edition of Lagrange’s Mécanique analytique was published.
A more rigorous justification came soon after the publication of the 2nd edition of Mécanique analytique and can be found in lesson 11 of Cauchy's lessons on infinitesimal calculus (Cauchy 1823). I am indebted to two anonymous referees for having helped me to clarify the content of this section and to give a better rendering of the way in which the multiplier’s method is introduced and justified on purely mathematical grounds.
Note that the multiplier’s case is not even a case of intended mathematics, as the multipliers are not developed by keeping the content of the application at the forefront, as it happens in cases such as that of the delta Dirac.
I have already quoted the passage in Sect. 3. For the sake of easy reading, I offer it again here: “Je remarque maintenant que les termes λδL, μδM, etc. de l’équation générale de l’équilibre, peuvent être aussi regardés comme représentant les momens de différentes forces appliquées au même système” (Lagrange 1811, 76–77).
Comments provided by one referee have been particularly helpful to clarify this point. Let me note, however, that the status (empirical or not) of the principle of the pulleys in Lagrange is not uncontroversial among historians of mechanics.
It seems important to me to clarify that I am not claiming that such a position could not be defended and should be ruled out. Rather, I am pointing to some difficulties of it, and because of these difficulties I am opting for a more moderate philosophical standpoint.
It may be objected that the forces that Lagrange sees as corresponding to the multipliers are virtual, and that therefore these forces do not correspond to any actual force that is operating on the system. It should be noted, however, that as far as the system is considered at equilibrium there will be no difference, from a physical point of view, in treating virtual forces as real forces.
My feeling is that this idea can also be accounted for in terms of the inferential conception of the applicability of mathematics advocated by Otávio Bueno and Mark Colyvan (Bueno and Colyvan 2011). However interesting, I shall not pursue this issue here for reasons of space and I leave it for future work.
Obviously, I am not claiming that all cases of unintended mathematics involve constrained optimization and equilibrium considerations. My intuition is that the investigation of the method of Lagrange multipliers provides insights on a specific type of application and other successful applications require different analysis.
I suspect that applications of the multipliers in other empirical domains, as for instance in economics, could receive a similar analysis. Nevertheless, a presentation of these cases would require a more elaborated discussion that I am not addressing here for reasons of space.
This attitude is reflected in the role that applied mathematics plays, according to the platonist, in the enhanced (or explanatory) indispensability argument (cf. Baker 2009).
The opinion that some parts of mathematics, as for instance elementary geometry, are grounded in basic cognitive skills is shared by some philosophers of mathematics (cf. Giaquinto 2007). Giuseppe Longo has named “cognitive foundation of mathematics” the project of accounting for the intersubjective and conceptually-stable character of mathematics in terms of early cognitive processes (Longo 2003).
Here I am oversimplifying for the sake of simplicity. Indeed, it would be more appropriate to see the principle of virtual work as the result of a sequence of practices.
Michèle Friend has recently offered an account of the objectivity of mathematical knowledge in terms of the notion of crosschecking (Friend 2014). Although she does not explicitly connect her analysis to the problem of applicability, Friend’s idea of crosschecking well fits within the picture I am sketching here.
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Acknowledgements
I would like to thank two anonymous reviewers for their helpful and constructive comments. Their suggestions greatly contributed to improving the final version of the paper. I also wish to thank the organizers of the workshop “Mathematics and Mechanics in the Newtonian Age” and the members of the audience for useful discussion of an earlier version of this paper. This work was supported by the Portuguese Foundation for Science and Technology through the FCT Investigator Programme (Grant Nr. IF/01354/2015) and the project PTDC/FER-HFC/30665/2017.
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Molinini, D. Intended and Unintended Mathematics: The Case of the Lagrange Multipliers. J Gen Philos Sci 51, 93–113 (2020). https://doi.org/10.1007/s10838-019-09483-5
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DOI: https://doi.org/10.1007/s10838-019-09483-5