1. Introduction
The role of convention in scientific theories has been widely examined in the literature, and the consequences of conventional theories have also been explored. Classical examples are Willard van Orman Quine (Reference Quine1936), Rudolf Carnap (Reference Carnap1937), or David Lewis (Reference Lewis1969). Convention is also linked to the work of some scientists that, by the end of nineteenth century, employed the term to characterize the fundamental principles of a theory. Prominent among these are Duhem and Poincaré. Conventionalism, as a general philosophical position, has been linked to Poincaré’s philosophy of science, due to his use of the term convention to characterize the axioms of geometry and the fundamental principles of mechanics and physics. Classical works which consider and explain geometrical conventionalism in depth are Torretti (Reference Torretti1978), Jerzy Giedymin (Reference Giedymin1982), Michael Friedman (Reference Friedman1999), and Elie Zahar (Reference Zahar2001). Among these, only Giedymin (Reference Giedymin1982) and later Giedymin (Reference Giedymin1991) distinguished between the use of convention in geometry and in mechanics. More recently, Stathis Psillos (Reference Psillos2014), Milena Ivanova (Reference Ivanova2015), and David Stump (Reference Stump2015) have analyzed the role of convention in geometry and also in the principles of mechanics. It is also clear that Poincaré used the concept of convention first for geometry (the first use is in 1891a) and only later for mechanics (the first use dates from 1901). However, it has been proved that he was not the single author using this concept and, since 1994, Helmut Pulte has published several works in which he shows the use of this concept by the German mathematician Carl Gustav Jacobi to characterize the principles of mechanics before Poincaré’s use.
The work of Pulte paved the way for a historical survey on the genesis of the concept of convention in mechanics, and we aim to contribute to that survey here. However, neither Pulte nor other approaches mentioned have inquired or speculated on the origin of the concept: Where did it come from? and What is the relation between that origin and its application to natural science? Pulte (Reference Pulte2000) stated the following: “Jacobi, well educated in classical philology and very conscious of linguistic subtleties, consequently called first principles of mechanics ‘conventions,’ exactly 50 years before Poincaré did.” So, it looks as if the introduction of this concept that works as a new epistemological category (at least for both Jacobi and PoincaréFootnote 1) appears only due to a “linguistic subtlety.” We will suggest that it is more than that.
In fact, to our knowledge no one has tried to examine conventions as concepts originated in another field of knowledge and to expose them as concepts transferred from that field to science (whether it is to geometry or to mechanics). In this paper, I suggest that the origin of the use of “convention” as an epistemological category in nineteenth-century natural science was connected with its use as a concept in the field of law and jurisprudence, and I show how some of its distinctive features were retained in its use in mechanics. Although further historical research is required to fully clarify the introduction of this notion in the natural sciences, I will show that this connection is elucidating to understand the use of convention in mechanics by analyzing some implications of this transfer.
2. The Notion of Convention in Jurisprudence
When considering convention, it is common to appeal to its social origin. Rescorla (Reference Rescorla and Zalta2015) points out that the two typical places for convention are language and money, which however does not mean that the origin of the concept lies in linguistics or economics.Footnote 2 Almost in every dictionary (independently of the language consulted), convention is explained as a synonym of agreement. And agreements are regulated by law. This opens the possibility of speculating about the origin of the term in the field of jurisprudence, together with the fact that we find it always associated in the first place (when looking at a dictionary) to the field of law and to the idea of a legal regulation.
That is precisely what happened in the French Encyclopédie, where the entry on convention dates back to its first edition of 1751 and was written by D'Alembert (the first four lines), Boucher D'Argis (the major part of the article), and Mallet (the final lines).Footnote 3 In this entry, the authors traced its use back to Roman jurisprudence and asserted that “the word convention is a general term which includes every kind of covenant, agreement, stipulation, pledge and obligation” (D'Argis et al. Reference D'Argis, Boucher, Diderot and D'Alembert1751, 161). So, conventions cover every kind of agreement, written and non-written, with explicit clauses or not. They all have in common that they are self-imposed by the contracting parties and these parties force themselves to conform to them. This refers, of course, to the idea of a contract. Also in the Encyclopédie, in the entry “contract” it is said: “what forms the contract is the mutual and reciprocal consent of the contracting parties; where it follows that those who cannot give a free consent cannot make contracts,” and as an example, it is mentioned that prisoners are not capable of making contracts (D'Argis and Boucher Reference D'Argis, Boucher, Diderot and D'Alembert1751, 122). This could also be retained for conventions in the sense that a convention establishes a regulation on something and to be established it requires the free decision of the parties to establish it: they freely agree to accept it and to conform to it.
The existence of conventions is seen as a natural consequence of a civil society as becomes clear when we think about the social contract tradition. Conventions are agreed to satisfy certain needs for living in society: they regulate certain commitments and some of them are agreed upon in order to perform certain actions. For example: a sale contract is a convention that stipulates the conditions by which the action of selling something is regulated. Another feature of conventions is that they are able to be modified: the parties can add, or suppress, or change clauses in a convention. So they are not rigid and they can always be re-examined.
The entry also asserts that “in order to be valid, every convention must have a legitimate cause” (D'Argis et al. Reference D'Argis, Boucher, Diderot and D'Alembert1751, 162). And it gives the following example to illustrate what is meant: “in the case of borrowing, the money borrowed is the cause of the convention: in the same way a donation must have a cause, such as to reward the beneficiary's merits, or because of the friendship that the donor feels for him.” This means that conventions have always a raison d’être. A cause is just the reason to establish the convention. This idea of having a cause, trivial as it seems, may also suggest something more relevant: that conventions are not arbitrary. Opposed, for instance, to the case of linguistic conventions, as referred to in the classical example of Plato's Cratylus where Hermogenes says: “No name belongs to a particular thing by nature” (Plato Reference Cooper1997, 384c), meaning that words are associated with objects for no particular reason, thus making linguistic conventions arbitrary.
The last idea that we would like to stress regarding the role of convention in jurisprudence is this: “the effect of valid conventions is to conform not only to what is expressed in them, but also to everything which is a natural consequence” (D'Argis et al. Reference D'Argis, Boucher, Diderot and D'Alembert1751, 163; emphasis in original). This indicates that the establishment of a convention imposes conditions that might be implicit but that are a natural consequence of it, such as the assumptions on which the application of the convention depends, or the frame and the context in which it is applicable. So, once the convention is established, its “natural consequences” have also to be fulfilled or have to be considered as defining its range of applicability.
As one of the aims of this paper is to deal with the emergence and meaning of the category “convention” in science and mathematics in the later nineteenth century, it might be helpful to consider some reference works from the end of the eighteenth century (and some of the nineteenth), in order to underpin the main meanings of “convention” in law and jurisprudence detected a century earlier. We have consulted the Oxford English Dictionary as the main reference for the English Language,Footnote 4 a total of seven French dictionaries,Footnote 5 and a total of six German dictionaries.Footnote 6 It is curious that frequently in English as well as in German the origin of the word is stated as coming from French or Latin, stating first the French etymology and then this from Latin. This shows the influence of French language (and probably also of French culture) on the other languages. It is also interesting to state that the word “Convention” (the modern spelling is “Konvention”) in German dictionaries only appears in Adelung's (1791) and in Zedler's (1731-1754). So it was probably used only in the High German dialect and the definition is that of agreement (and contract) and the conclusions derived from those agreements. When consulting the DWDS (Digitales Wörterbuch des deutschen Sprache), it refers to the usage from the eighteenth century, precisely in jurisprudence, in the use of “Konventionalstrafe” or conventional penalty, as the punishment for non-compliance of a treaty. This is why we have decided to enlarge our search by consulting five specific German lexicons for law and jurisprudence and we have found the term in four of them.Footnote 7 In those, the word appears to refer to a contract and to an agreement, just like in French dictionaries and in the Encyclopédie, and also to conventional penalties, thus showing the technical use of the term in the field. The fact that the term does not appear in general dictionaries of German language but it does in technical lexicons for law and jurisprudence reinforces the idea that this notion might come from this field of knowledge.
Both in English and French dictionaries a new meaning appears more or less from 1792 on, which is the use of “Convention” (usually with capital ‘C’) as a political assembly (parliament, a political party meeting, etc.), probably because of the French National Assembly designed as the “Convention National.” But before that the common use is for legal conventions and, particularly in France, for those regulating marriage and pecuniary interests of the spouses, meaning that the convention is not only the agreement (the act of marriage) but also the consequences derived from that agreement.
To summarize, the features that we have stressed as distinctive in jurisprudence's conventions are: the free decision, that is, the parties are free to adopt the convention; they are regulative in the sense that they regulate certain actions, they establish rules to act, they are guides for action, as in the case of selling; the non-rigidity, i.e. conventions are able to be changed and are flexible because they could be adjusted, or some parts could be suppressed; they are not arbitrary, meaning that they are grounded in some reason; and finally, they define their own range of use and their adoption implies the adoption of what is a consequence of them.
In the next section, we will show how these features are retained in the notion of convention used by three nineteenth-century authors to characterize the principles of mechanics.
3. Convention in Mechanics: Jacobi, Reech, and Poincaré
One might question why we have chosen to analyze the transfer of convention to mechanics instead of geometry since, as we said above, Poincaré, who is the most famous “conventionalist,” used it first for geometry (Reference Poincaré1891a) and only later for mechanics (Reference Poincaré1901). However, as Pulte (Reference Pulte2000) has shown, there are reasons to consider the independence, both historical and philosophical, of physical conventionalism (mechanical, in this case). Thus, we will consider its use, first by two authors who employed the term only for mechanics, and then its use by Poincaré, but only regarding the principles of mechanics.Footnote 8
So, we will analyze the use of convention in mechanics by Jacobi, Reech, and Poincaré. We have chosen Jacobi, since, to our knowledge, he is the first to apply the notion of convention to the principles of mechanics. There are previous uses of this concept for mathematics, such as Diderot's critical remark that “the Thing of the mathematician has no more existence in nature than that of the gambler. On both sides, it is a matter of convention” Footnote 9 and passages in the work of John Radford Young (Reference Young1841) regarding the use of negative and positive signs,Footnote 10 but these appear to refer more to a kind of linguistic convention, different from the more substantial uses of convention that we would like to consider. In the second place, we will consider the work of Fréderic Ferdinand Reech, because of three main reasons: first, its chronological proximity to the work of Jacobi; second, the striking similarities of his position to that of the German mathematician; and third, because he is French, and as a contemporary of Jacobi, it will be interesting to compare the views regarding mathematics and mechanics of these authors, taking into account Pulte's considerations about the demolition of axiomatic Euclidianism in mechanics being “primarily a German enterprise” (Lützen Reference Lützen2009). Finally, we will consider Poincaré’s use. Despite being the most famous of the three and despite his views having been widely discussed, it would not be useless to provide an analysis framed by the terms defined for the use of convention in jurisprudence.
3.1. Carl Gustav Jacob Jacobi
The work of Jacobi that we are considering here are his 1847–1848 Berlin Lectures on Analytical Mechanics (Vorlesungen über analytische Mechanik). As Pulte has remarked on several occasions, in these lectures, Jacobi criticized the dogmatic character of Lagrange's axiomatic-deductive approach to mechanics (Pulte Reference Pulte1998; Pulte Reference Pulte2000). Jacobi's criticism of Lagrange is particularly significant because it is not based on practical problems regarding the applicability of Lagrangian mechanics, but on theoretical considerations in a sense that affects the foundations of mechanics. Jacobi was trained as a “pure” mathematician and in some way this gave him the ability to acknowledge that the systems of pure mathematics, being axiomatic-deductive, do not follow the same epistemological standards as the system of analytical mechanics, despite their formal similarity (Pulte Reference Pulte1998, 179).
Although the content of pure mathematics changed during the nineteenth century,Footnote 11 for Jacobi it referred to the idea of an a priori knowledge independent of every empirical element, based only on the laws of thought (Pulte Reference Pulte2016, 114). As a priori knowledge, it possesses absolute certainty and evidence. Given this conception of pure mathematics, mechanics has to be excluded from that field, because it must refer to physical (empirical) bodies, transcending, thus, “the limits of pure mathematics” (Pulte Reference Pulte2000, 61). Jacobi separates the object of pure mathematics from that of mechanics: this last has to deal with bodies in nature (Jacobi Reference Jacobi1996, 193).
This implies a conception of mechanics different from the previous Lagrangian perspective, in which mechanics was conceived as an axiomatic-deductive science based on self-evident principles, a pure mathematical science.Footnote 12 But Jacobi did not propose an inductivist account of the principles of mechanics, assuming them to be empirical generalizations. For him, the principles of mechanics were not self-evident axioms nor were they based on mathematical proofs.Footnote 13 So the choice was no longer between an axiomatic-deductive and an empirical-inductivist account.Footnote 14 Jacobi proposed a new epistemological category which did not fit into any of these schemes: “From the point of view of pure mathematics these laws cannot be demonstrated; [they are] mere conventions, yet they are assumed to correspond to nature – they are not proved a priori, but the nature of their correspondence must be shown through experiment” (Jacobi Reference Jacobi1996, 3; my emphasis).
It is clear that empirical knowledge is not deductive, and thus, mechanics as an empirical science cannot be based on an axiomatic-deductive scheme. However, although related to experience, conventions to Jacobi are not mere inductions. They are empirically relevant, since experiments “show” their correspondence to nature. However, as they are mathematical principles, they are not taken form nature in the sense of being induced. So, it could be said that Jacobi's concern was about applying mathematical principles to nature, and about the epistemic status of these principles. It is in this context that he introduced “convention” as a new epistemic category to characterize the principles of mechanics.
He chose conventions to set a different category that opened the door to what has been called a “third-way-epistemology” (Pulte Reference Pulte2000). This is something philosophically very relevant, but it does not explain why this precise word fits so well with what Jacobi thought that the principles of mechanics should be. Our purpose is to show the relationship between Jacobi's choice and the features of conventions described above as a concept originated in jurisprudence – and thus to understand it as a concept transferred.
Resuming the features stressed from conventions in jurisprudence, we start with the idea of free decision. It happens exactly the same with the principles of mechanics: when Jacobi adopts the inertia principle and as such uniform and rectilinear movement as a privileged movement he introduces the role of decision in the constitution of science. The scientist is free to adopt that movement or any other one:
From the point of view of pure mathematics, it is a circular argument to say that rectilinear motion is the proper one, [and that] consequently all others require external action: because you could define as justly any other movement as the law of inertia of a body, if you only add that external action is responsible if it does not move accordingly. (Jacobi Reference Jacobi1996, 3–4)
Conventions in mechanics are also regulative in the sense that they serve as a guide for certain actions (de Paz Reference de Paz2015), as regimented stipulations. For example, once the principle of inertia is established, it works as a guiding rule in the sense that any deviation of that movement guides the scientist in the direction of searching for a cause that perturbs the movement. In this sense, they work exactly as conventions in jurisprudence. If we recall the example of a sale contract, it regulates the conditions for the action of selling something. In the example of inertia, that principle regulates the conditions in which a movement should be considered as being caused by a force.
Then we come to non-rigidity or flexibility. This is one of the fundamental changes that the idea of convention brings into mechanics. When considered as an axiomatic-deductive science, such as in the early Lagrangian conception, the principles are rigid, indubitable, and they cannot be changed. By introducing the idea that the principles are in no way imposed on us either by the laws of thought nor by the empirical features of the external world; rather it is by our decision, it brings together the view that our decision could be changed. Let us recall again the example of the law of inertia: Jacobi explicitly states that “you could define as justly any other movement [different from the rectilinear and uniform one] as law of inertia of a body” (Jacobi Reference Jacobi1996, 3–4). This implies that there could be a range of optional movements from which we could indeed have chosen and set one as “the” inertial movement, establishing it as the privileged one, as the first law of motion. This agrees with the non-rigidity discussed regarding conventions in jurisprudence: the parties can adjust, change, or suppress parts (or clauses) in a convention.
The setting of uniform and rectilinear motion is thus not necessary or mandatory, but a free and flexible decision that guides the investigation of any interacting system of mechanics starting from a defined motion. However, arbitrary as the choice of rectilinear and uniform motion might appear because of its free and flexible character, it also obeys a raison d’être. By studying mechanics and understanding the notion of force as the reason why a body is deviated from its original path, the need of establishing a privileged motion becomes obvious. What is not obvious is the fact that that motion is uniform and rectilinear. As conventions are introduced as an epistemic category different from empirical and a priori assertions, the ground to choose inertial motion as rectilinear and uniform cannot be found in the pure laws of our thought or in experience. In fact, Jacobi states what we might call pragmatic criteria in order to justify the adoption of certain principles instead of others as the fundamental principles of mechanics: “One can demand that the form of this principle is as simple and plausible as possible.” So, the reasons to adopt a certain principle are its simplicity and plausibility. A principle is simple, according to Jacobi, when it is “most clearly represented analytically” and it is plausible when “connected with natural laws”. (Jacobi Reference Jacobi1996, 5)
We regard simplicity and plausibility as pragmatic criteria because they could be understood with respect to their functional character. For example, the simplicity of inertial motion when defined as rectilinear and uniform means that it is easier to handle a principle postulating a motion traversing equal paths in equal times (uniform) than a principle postulating an irregular motion. We could consider that it also happens the same way when postulating rectilinear motion, but it is clear from the history of science that rectilinear motion has not always been considered as the “most simple one.” However, considering Jacobi's conception of simplicity, it is patent that the analytical representation of a straight line is clearer than any other, and thus, we could consider as justified the introduction of rectilinear and uniform motion according to Jacobi's criteria without it being an arbitrary convention.
Next we have to analyze plausibility. It is given by the connection with natural laws. We have already said that conventions are not inductions, so they cannot be directly “extracted” from experience, but we have also pointed out that they are empirically relevant, meaning that there has to be a way to prove the principles by experiments (their “correspondence to nature”). Taking again the case of inertia, Jacobi asserts: “If we can physically show (aufweisen) external action in any case where a body deviates, we are entitled to call the law of inertia, which is now at the basis [of our argument] a law of nature” (Jacobi Reference Jacobi1996, 4). We only have to be able to show that in any experiment in which a body deviates from a rectilinear and uniform motion, the deviation is due to an external cause. Since, according to Newtonian mechanics, gravitational force is always acting, every motion is constantly deviated from its rectilinear and uniform path, thus there is a connection with experience, an “assumed” (angenommene) connection, but there is where the possibility of showing the empirical relevance of these principles lies.
As a result, both simplicity and plausibility are criteria to introduce some principles better than others and to make their choice not arbitrary, that is, different from linguistic conventions and similar to conventions in jurisprudence. Also, the need to establish a privileged movement when we are studying the science of motion (mechanics) is the reason to establish the law of inertia, the very first convention that we introduce to make a rigorous study of motions.
It remains to analyze the last feature in order to finish our approximation between the features of conventions as defined in jurisprudence and conventions as mechanical principles in Jacobi's conception: that is the idea that the adoption of a certain convention defines the range of use of that convention and implies the adoption of what is a natural consequence of it. In order to understand this point, we can resort to the conceptual analysis of the principle of inertia to determine the underlying conditions on which the use of the principle depends.Footnote 15 We can think that the introduction of the inertia principle as uniform and rectilinear obeys, as we have shown, its criteria of simplicity and plausibility; however, a deeper analysis of its correct application in the practice reveals a commitment to a certain conception of time as uniform and a certain conception of space as Euclidean. The principle defines the field of application of inertia, a Euclidean space in which time flows uniformly, because it presupposes that field in its very formulation. It is, thus, a natural consequence of the adoption of the principle of inertia as rectilinear and uniform, the idea that time flows uniformly and that space has a Euclidean configuration. This analysis discloses also the semantic relevance of the principle, which, according to Pulte (Reference Pulte2000, 64), fixes the meaning of the terms, in the sense that the principle is the definition of “movement free of external forces.”
As a result, all the distinctive features of conventions in jurisprudence are fulfilled by Jacobi's conception of mechanical principles as conventions, from the free decision to the definition of its range of application. The notion of convention according to its use in the field of jurisprudence fits with its use in mechanics. In this sense, we do not think that it could be attributed just to a “linguistic subtlety,” but to a choice based on its meaning, in order to be applied to a field where conventions were not commonly considered at the time (they did not usually appear in lectures and treatises, in fact Jacobi was the first one to introduce them). The fact that the word “convention” did not appear in common dictionaries and only in technical works for jurisprudence suggests a possible contact with a certain juridical tradition. As a cultivated man, Jacobi could have been aware of or even well acquainted with the works of the Encyclopédie. Apparently, after Hegel's death in 1831, the German interest in the Encylopédie became bigger than it had been in the previous century (Schalk Reference Schalk1951, 88).Footnote 16 Also, Rotteck and Welker's Staats-Lexikon (one of the technical works we have consulted in which the word appears) was very well known among liberals and it had influence during the period between 1815 and the March Revolution of 1848 in which Jacobi participated on the liberal side. Taking into account all these reasons, linking the concept of convention in Jacobi's Lectures on Analytical Mechanics (Reference Jacobi1996) with the field of law and jurisprudence where it was part of the established terminology of the time, brings new light to the work started by Pulte by contributing to the contextualization of this concept.
In the next section, we will try to show a similar use of the same word for the same principles in a slightly different context, that is, in the work of Fréderic Ferdinand Reech.
3.2. Fréderic Ferdinand Reech
Reech is what would usually be considered a “minor figure” in the history of science. Compared to the other two authors whose work we analyze in this paper, Reech's work is relatively unknown and had a relatively minor impact on the scientific community.Footnote 17 However, we will try to show the value of studying his work, not only as a contemporary of Jacobi,Footnote 18 but also as the first French author, to our knowledge, to introduce the notion of convention to characterize the principles of mechanics. Just as in the case of Jacobi, the work we are considering here are also lectures: Reech's Lectures on Mechanics according to the flexible and elastic nature of bodies (Cours de mécanique d'après la nature généralement flexible et élastique des corps). Unlike Jacobi's lectures, which remained unpublished until Pulte's edition of Reference Jacobi1996, Reech's course was published in 1852.
Reech's works were well known to his contemporaries. He was the director of the École du Génie Maritime (a famous military naval engineering school)Footnote 19 during almost forty years, and his works on thermodynamics and the construction of ship models were frequently quoted at the time (cf. Bertin Reference Bertin1867).Footnote 20 His course on mechanics is also mentioned, but it is most famously known thanks to the work of Andrade (Reference Andrade1898), who considers him as the founder of a new school in mechanics (1898, III). These lectures are motivated by the reform in the teaching programs of the École Polytechnique, in which, according to Reech, they eliminated the teaching of statics from the syllabus of mechanics. As a result, just like Jacobi's, these lectures prompt a critique to properly ground (and teach) the science of mechanics. Thus, he declared as his purpose: “To separate the tares from the good grain in the science of mechanics, by classifying on the one hand all that is but geometry or kinematics, with the incomplete notions of the words force and mass, and on the other hand all that belongs to true mechanics” (Reech Reference Reech1852, VII).
The main target of his criticism was the notion of force. He censured Lagrange's and Ampère's approach because, to him, they tended to eliminate the idea of force, and he considered that as a step backward in mechanics (ibid.). It is interesting to recall here that Jacobi himself admitted that analytical mechanics in Lagrange's treatment did not “study the effects of the forces any longer,” meaning that as such, “mechanics lacks any justification” (Jacobi Reference Jacobi1996, 193). However, unlike Jacobi,Footnote 21 Reech considered Lagrange's demonstration of the principle of virtual velocities as “brilliant” (Reech Reference Reech1852, 69). This probably has to do with Reech's general conception of how the science of mechanics and mathematics in general should be, which shows a striking contrast with Jacobi.
Like Jacobi, Reech considered mathematics to be an exact science whose concepts should be precisely defined and free from ambiguity (Reech Reference Reech1852, Post-scriptum). But contrary to Jacobi's idea that it only includes pure mathematics, and that mechanics does not correspond to that field because it should be applicable to the empirical realm (cf. Pulte Reference Pulte2012), Reech thought that mechanics should be grounded as a mathematical science (Reech Reference Reech1852, 33). In fact, to him, statics should be established prior to dynamics, as a science of equilibrium, whose assertions grounded on a true basis would have “the same status of certainty as the truths of geometry” (ibid., 41). This idea of using geometry as a model for science means that Reech must be embedded in what Pulte calls the “traditional euclidianism” of the mechanical tradition,Footnote 22 in line with authors such as Lagrange. According to Pulte, Jacobi, by introducing conventions as a new epistemic category different from the empirical and the a priori, has to be considered as a critic of this tradition and his 1847–1848 lectures on mechanics as a ground-breaking work in this sense.
Reech also asserted that mechanics should be developed in a way that it becomes a mathematical science but a real one as well (Reech Reference Reech1852, I), and in this sense, he criticized the abuse of idealizations such as the hypothesis of the rigidity of bodies (ibid., VII). He wanted to found mechanics “according to the flexible nature of bodies,” as indicated in the title of his lectures, but he did not intend to lay the foundation for an empirical science. In a certain sense, Jacobi and Reech appear to agree on what mathematics should be (i.e. a pure science, free from empirical elements), but they seem to disagree as to the content of this science, and this places mechanics in a problematic situation.
Pulte explains Jacobi's demarcation of pure mathematics from anything outside its field (including mechanics) by appealing to the influence of Neohumanism in Jacobi's views, thus developing a new conception of mathematics that would be particularly influential in the development of German mathematics along the nineteenth century.Footnote 23 Precisely the introduction of the notion of convention as an epistemic category is one of the signs of this new perspective. “From the point of view of pure mathematics,” as Jacobi stresses, “the principles of mechanics are conventions” (Jacobi Reference Jacobi1996, 3). This vision of mathematics was different from the French tradition of the “physique-mathématique” to which Lagrange belonged. The only similarity with French positions (as noticed by Pulte Reference Pulte, Heidelberger and Schiemann2009, 83) is when Jacobi regretted that Lagrange's Mécanique analytique lacks justification because it does not describe the behavior of physical bodies, and this could be understood in the line of French authors calling for a “mécanique physique,” such as Poisson (Reference Poisson1829). Reech, nevertheless, does not seem to share this point with Jacobi, because he explicitly asserted that if we are not rigorous in the study of mechanics “we shall fall into that perpetual confusion of which we have spoken, so as to cause to arise, alongside the mechanics that we call mathematical, another mechanical science called physical and experimental, which really will be a jarring contrast to the first”Footnote 24 (Reech Reference Reech1852, Post-scriptum). So, he did not share the empiricist conception of mechanics of some of his countrymen and contemporaries and, although his perspective on mathematics (in general) shared some features with Jacobi, he did not share the view that mechanics was outside mathematics as an empirical science as well, and apparently he did not renounce the “Euclidian ideal” in mechanics, since geometry was still the model according to which the true science of mechanics should be founded. However, despite all these differences, he employed also the word convention to characterize the principles of mechanics, and by recalling the features stressed in jurisprudence, we aim to show how Reech's use of the word is similar to Jacobi's:
But then there will be a convention to be made. It will be a matter of knowing what kind of movement, rectilinear or curvilinear, uniform or varied, we shall have to admit as that of a material point entirely free in appearance. And because we shall have complete latitude in this respect, as we have already suggested in the last section of the first part, with the sole advantage or inconvenience of seeing more or less great simplifications in the mechanical relations of systems, we shall naturally be led to make use for that purpose of the state of uniform rectilinear motion, and to encounter that famous law of inertia of matter, which will no longer be a principle [i.e. a self-evident axiom] or a fact from experience, but a pure convention, the simplest of all those among which we are forced to choose. (Reech Reference Reech1852, 49)
It is clear from the above text that there are several possibilities among which we can choose one to set the motion of a free material point, which amounts to a free decision. Reech asserts that we “have complete latitude in this respect.” Secondly, exactly as we explained regarding Jacobi's setting of the principle of inertia as a convention, to Reech, the choice of inertia guides the scientist in acknowledging the geometrical effects of forces as a deviation from the inertial path of a certain mass. In this sense, the choice of a certain principle, of a certain convention, indeed, regulates as a commitment the conditions under which any deviation of that convention is informative. Thirdly, regarding non-rigidity, as a convention it has not an absolute truth-value, since its truth does not depend on experience or on the laws of thought. Also, from the latitude we have in the choice, we could derive the flexibility. Although Reech does not assert it explicitly, apparently if we could find a simpler convention, the one chosen could be substituted, so they are provisional. When coming to arbitrariness, we find in Reech similar criteria as we found in Jacobi: i.e. simplicity. He states that the convention adopted would be the simplest “regarding any system of rectangular axes” (Reech Reference Reech1852, VII). Similarly to Jacobi it refers to simplicity in representation, thus not assuming as evident that uniform and rectilinear motion is the simplest one.
Finally, we have to consider if the adoption of conventions, to Reech, defines their range of use and the adoption also of what is a natural consequence of them. It is interesting to notice that to set the foundations of mechanics in order to establish it as a mathematical science, Reech considers that there has to be a hierarchy in the understanding and teaching of mechanics (ibid., III, 3 and Post-scriptum). First of all, motion has to be understood as a change of distance in time, which requires a conception of space and also a conception of time. So, conceptions of space and time are prior to any possible establishment of the science of mechanics. Then, the knowledge of the purely geometrical relations of the motion of bodies is required. After that, comes dynamometry, that is, the experimental measure of forces, for which it is important to acquire an intuitive meaning for the word force. With all this and without including the notion of mass or velocity, we can establish statics as the science of the equilibrium of forces. The extension of the laws of statics to all the cases of the motion of bodies lays the ground for dynamics. These are the bases to a universal science of mechanics that can study any case from “what is on the surface of the Earth to the most distant celestial bodies” (ibid., Post-scriptum).
According to this hierarchy, any convention established after the understanding of motion as a change of distance implies the adoption of the preconditions that make possible the setting of the convention, i.e. the notion of distance and, subsequently a conception of space and time. That is precisely the case of the adoption of the principle of inertia, whatever the meaning we give to it, as uniform and rectilinear or any other. It is also important to notice that the inertial convention defines its range of use, since it defines the meaning of motion of a free material point (Reech Reference Reech1852, 49).
As a result, not only the features of conventions in jurisprudence apply to Reech's conception, but also the use he makes of them is in agreement with Jacobi's, despite the fact that he is French, and that there is no proof of the possible influence of the Neohumanist conception of mathematics in his views, he is still in the Euclidian Lagrangian tradition of mechanics and he does not consider the object of mechanics as separated from that of pure mathematics.Footnote 25
Reech's use of the word convention is restricted to the principle of inertia, whereas Jacobi employs it more generally referring to the laws of mechanics, but the most detailed reference is also to the principle of inertia when he considers it as a circular definition (Jacobi Reference Jacobi1996, 3). Despite its restricted use in Reech, we can also find all over his work the use of what has been called “conventionalist ideas and terminology” (cf. Giedymin Reference Giedymin1977, 286), that is, words like “choice,” “adoption,” “decide,” “simplicity” appear repeatedly, as happens also in Jacobi's work.
However, in spite of the similarities and the historical proximity, we cannot prove that Reech knew anything about Jacobi's last lectures. Indeed, it is quite hard to imagine how they could possibly have come into his hands. In the light of this we are left to speculate that the resemblances had a common source, pointing at the field of jurisprudence as a possibility. Reech, first as a polytechnicien (he followed Ampère's courses) and later as a military man who reached the rank of admiral, was a cultivated man, who could have known one of the numerous editions of the Encylopèdie. Further historical work is required, but the proximity of both conceptions makes the suggestion plausible.
Maybe the reasons to introduce a new label in the sense of a new epistemic category in the work of Reech cannot be attributed to a new conception of mathematics, as Pulte insists when interpreting Jacobi's work, but certainly its introduction is relevant in the process of the loss of certainty that mechanics undergoes along the nineteenth century and that we will refer to in more detail in the fourth section of the paper. Now, it is time to turn to the last conventional mechanics: that of Henri Poincaré.
3.3. Jules Henri Poincaré
As we stated at the beginning and we have repeatedly stressed along the paper, we are dealing only with the use of convention in mechanics. Regarding geometrical conventionalism, there are classical works that characterize and discuss it: (Torretti Reference Torretti1978; Giedymin Reference Giedymin1982; Stump Reference Stump1989; Friedman Reference Friedman1999; Zahar Reference Zahar2001, etc.). All of these agree more or less that Poincaré’s position regarding geometry is based in his understanding that the axioms of geometry are not empirical or a priori propositions and that there are no spatial features that can determine the applicability of a geometry better than another. Hence, the axioms of geometry have a conventional status and the applicability of geometry to space is also conventional. Although there still are some controversies regarding this position, it has been widely discussed in the literature. On the contrary, mechanical conventions have not attracted so much interest and taking into account that the two authors previously considered did not apply the notion of convention to geometry, we will restrict our discussion to mechanical conventions also in the work of Poincaré.
In order to consider the concept of convention in mechanics we will focus mainly on Poincaré’s paper “On the principles of mechanics” (“Sur les principes de la mécanique”) that he presented at the International Congress of Philosophy in Paris in 1900 and published a year later in the Proceedings. In this paper, Poincaré made an examination of what he considered to be the fundamental principles of mechanics and he interpreted them as conventions.Footnote 26 This paper is interesting not only because it provides an analysis of the fundamental principles of mechanics, but also because in it, Poincaré developed his conception of mechanics as a science and stresses the way it had to be taught: “The English teach mechanics as an experimental science; on the Continent it is taught always more or less as a deductive and a priori science. The English are right, no doubt” (Poincaré Reference Poincaré1901, 457).
It is clear from this statement that Poincaré conceives mechanics as an empirical science, much closer to Jacobi than to Reech. We have no evidence that Poincaré knew Jacobi's lectures, despite the fact that a draft of them was in circulation. But he knew Reech's work, at least through Andrade's which is referred to in this text (Poincaré Reference Poincaré1901, 469). In the conclusions of the text, he made explicit the conception that mechanics is not a mathematical science, since he drew a line separating geometry, a science that belongs to the mathematical realm, from mechanics, an experimental (non mathematical) science (ibid., 492).Footnote 27
In order to understand Poincaré’s conception of mechanics and to properly compare it to Reech's and Jacobi's, it is useful to understand the difference with respect to geometry. This could be explained in terms of the object and structure of both sciences. Regarding the object, we have to consider the classical distinction between an experimental science, in the sense that it says something about the world and some of its claims could be tested by experiments (that is mechanics) and a science that is not empirical but starts from the axioms, and can be constituted deductively as an exact science. There is no experiment able to prove the correspondence between nature and a certain geometry. Geometry is the study of the groups of the movements of idealized solids (Poincaré Reference Poincaré1902, 93). On the contrary, the object of mechanics is the natural bodies we find in nature or bodies analogous to those (Poincaré Reference Poincaré1901, 493). As a result, it is not a deductive discipline, its status is changeable and lacking the rigor of geometry, it is that which corresponds to an empirical science. Regarding structure, that of geometry is an axiomatic-deductive system (or several, since there are several possible and consistent geometries). There can also be several mechanical systems (as Poincaré shows in 1897), but the structure of the science has to be different because its components are subject to change in the light of new empirical evidence and are not established as axioms.
As a result, mechanics is not a mathematical science and does not share the same status as geometry, i.e., that of an exact science. This is similar to Jacobi's perspective. However, Poincaré not only drew a line between mathematics and anything outside its field. It is more complex than that, since arithmetic is also a mathematical science and neither does it share the status of geometry. And it is also more complex because there is a deep interrelation between mathematics and the natural sciences, since mathematics “gives the physicist the only language he can speak” (Poincaré [Reference Poincaré1905] 1970, 105). However, the explanation of the whole structure of science in Poincaré is beyond the scope of this paper and once his conception of mechanics is established, we should focus on his understanding of the fundamental principles of this science as conventions.
Principles of mechanics are defined as propositions whose origin is empirical but are not empirically verifiable (Poincaré [Reference Poincaré1902] 1968, 177), responding thus to the idea of a different epistemic category: they are conventions or definitions in disguise (ibid., 165). As such, they are different from a priori and empirical principles and are neither true nor false (ibid., 76); in fact, truth is not applicable to them. As any other convention, they are not imposed upon us, it is the scientist who freely decides to introduce them, thus complying with the first feature of conventions in the field of law. In several places Poincaré states the free character of our decision when introducing conventions and also states the idea that they are not arbitrary: “They [the principles of mechanics] reduce in final analysis to a simple convention that we have a right to make . . . . This convention, however, is not absolutely arbitrary; it is not the child of our caprice” (Poincaré Reference Poincaré1901, 491).
This refers to the fourth feature that we have indicated for conventions in jurisprudence, and before entering into the raison d’être we would like to show its regulative character. The principles of mechanics are heuristic rules that guide us in the systematization of our theories. As we have seen in the cases of Jacobi and Reech, the decision to include a certain principle works also as a guide that regulates the way in which we shall understand further notions or facts in the theory. For example, it is according to the classical theory of astronomic perturbations and by using the classical principles of mechanics, such as the law of force or the principle of gravitation that Le Verrier came to discover Neptune in 1846.Footnote 28 The principles have guided the coordination between mathematical astronomy and observational astronomy so as to discover a new celestial body, proving the efficiency of the Newtonian system (cf. Roseveare Reference Roseveare1982, 16).
Despite this experimental success, and despite the fact that Poincaré asserts that the principles cannot be directly contradicted by experience (Poincaré Reference Poincaré1902, 177), this does not mean that the conventions are fixed implying that they are rigid, as is clear from the following quotation: “If a principle ceases to be fecund, experiment without contradicting it directly will nevertheless have condemned it” (Poincaré [Reference Poincaré1905] 1970, 110).
Conventions have frequently been claimed to have a fixed character or to be rigid (e.g. Friedman Reference Friedman1999, 74; Ivanova Reference Ivanova2015, 120). However, it never occurred to Poincaré to consider the science of mechanics as a definitively established one and the character of conventions in mechanics is always stated as provisional, since in the light of new empirical evidence the system could be discarded. What is not possible is a direct empirical refutation of a certain convention, but just as it happened when he considered the relativity principle for Lorentz's electrodynamics,Footnote 29 conventions could be changed, they are “responsive to change,” to recall Folina's expression (Reference Folina, de Paz and DiSalle2014, 40). This means that conventions could be adjusted to new situations or could even be demoted, stating thus their flexibility.
Considering now the reason to establish a convention, and going back where Poincaré states that they are not arbitrary, he adds the following: “This convention, however, is not absolutely arbitrary; it is not the child of our caprice. We admit it because certain experiments have shown us that it will be convenient, and thus is explained how experiment has built up the principles of mechanics, and why, moreover, it cannot reverse them” (Poincaré Reference Poincaré1901, 491–492).
Conventions are chosen to provide a convenient systematic description of phenomena, but they are not arbitrary because experience guides us in the choice of the simpler and more convenient ones. There are good reasons to assume a principle as a convention – it would be a guide for further research, it is inspired by experience, and it simplifies the theory and the further steps in it. Simplicity to Poincaré can mean, for example, the intervention of the least possible number of elements in the explanation of something (cf. Poincaré Reference Poincaré1908a, 19). Simplicity is a way to explain, a criterion that we impose on nature because we cannot assert the simplicity of nature itself (cf. Poincaré [Reference Poincaré1902] 1968, 161). Just as in Jacobi and Reech, it is a criterion that helps us in the representation of phenomena. The other criterion, that is, convenience, could be understood in a double dimension: an empirical (cf. During Reference During2001, 83) and a theoretical one (cf. Zahar Reference Zahar2001, 16). The empirical understanding of convenience makes reference to the ability to explain observed phenomena and the predictive ability regarding not-yet observed phenomena. The theoretical understanding is close to the idea of unity in a theoretical explanation: that explanation which reveals a multiplicity of connections between different theoretical domains is more convenient. In this sense, conventions are not chosen arbitrarily or “come what may,” but, according to Poincaré’s views, there have to be good empirical and theoretical reasons to adopt them.
Finally, we will consider whether the adoption of conventions, to Poincaré, implies also the definition of their context of application which means understanding conventions as context-providers, and also whether their adoption implies the adoption of what is a natural consequence of them. Taking into account that on several occasions Poincaré states that conventions are “definitions in disguise” or “disguised definitions” (cf. Poincaré [Reference Poincaré1902] 1968, 24; Poincaré [Reference Poincaré1905] 1970, 160), it is pretty clear that they are semantically relevant and they provide the definitions of relevant concepts of a theory. In fact, according to DiSalle, a definition in disguise is “a principle that has the form of a factual assertion, but says nothing directly about the world; instead, it fixes the meanings of the concepts that occur in it” (DiSalle Reference DiSalle, de Paz and DiSalle2014b, 171).
In Poincaré’s Reference Poincaré1901 paper, before examining the principles of mechanics, he asserts that there are a few conditions usually admitted when working in mechanics, that is, when applying its fundamental principles. These are absolute space and time, simultaneity, and Euclidean geometry. Poincaré considers that absolute space is not a necessary condition for the application of mechanics, probably none of them are necessary, but he admits provisionally absolute time (and simultaneity) and Euclidean geometry (Poincaré Reference Poincaré1901, 458). When he defines the principle of inertia not only as a disguised definition of a free-moving particle, showing thus its semantic relevance, but also as rectilinear and uniform motion, he insists on the idea that it is not a priori, nor empirical and that other principles could have been adopted, but it is the simplest one, because he has previously admitted Euclidean geometry. In fact, there has to be a geometrical configuration previously established in order to consider rectilinear motion as the simplest one and that happens in the Euclidean plane. The acceptance of the principle in the terms provided by Poincaré (as uniform and rectilinear) presupposes the acceptance of the Euclidean configuration of space, and the uniformity of time.
The epistemic implications of Poincaré’s understanding of the principles of mechanics as conventions are quite relevant because, as DiSalle remarks, he “uncovers a subtle philosophical error,” that is, “mistaking a definition for an empirical principle, through a failure to see the assumptions on which the content and the application of the principle really depend” (DiSalle Reference DiSalle2006, 85). The fundamental principles of mechanics are not self-evident axioms or empirical propositions, rather they are “disguised” (some would say implicit) definitions of the central concepts occurring in the theory.
Several reasons have been pointed out to explain Poincaré’s use of the notion of convention. Pulte approaches Poincaré’s and Jacobi's positions also on the basis of their background as pure mathematicians, having both in mind the possibility of demarcating pure mathematics from anything outside its field (Pulte Reference Pulte1998 and Reference Pulte2000). Although this understanding is justified, we have shown that Poincaré’s position is a bit more complex, and it is true that he separates pure or a priori mathematics (arithmetic) from mechanics, but also from geometry – and he does not consider geometry as “applied” mathematics (in the sense of “applied to nature,” i.e. as empirical).
It is also difficult to justify in Poincaré’s work a direct influence of the Neohumanist understanding of mathematics, although it is interesting to stress that his assertion “science for its own sake” (Poincaré [Reference Poincaré1905] 1970, 142) is close to the Neohumanist Weltanschauung according to which “science and scientific education had ends in themselves” (Pulte Reference Pulte2000, 57). In fact, by the time Poincaré wrote his philosophical views, he was already familiar with Riemann's work, and as Pulte has stressed, Riemann attended Jacobi's 1847–1848 lectures in Berlin, pointing thus at a possible influence of Jacobi's views in Riemann's works (Pulte Reference Pulte, Heidelberger and Schiemann2009). Common to Jacobi's perspective is also the idea that mechanics is not a mathematical and axiomatic-deductive science, that is, the dismissal as well of the Euclidian perspective.
It is difficult to trace the genesis of the concept of convention from Poincaré back to Jacobi, since its use by the end of the century starts to be relatively common (contrary to what happens in Jacobi's time),Footnote 30 and the sources could be varied. What is clear is the similar meaning that these three authors give to the word convention in spite of the differences in their conception of mechanics, mathematics, and science in general. And we would stress that the use of this word is not random in any of them, given the precise meaning and its implications that they give to it, as it has been shown. The concept “convention” continued to be a technical term in jurisprudence by the end of the century, as is proved by the 1878 edition of the Dictionnaire de l'Académie française, which keeps the essential meaning as regulated agreement. However, according to the famous Dictionnaire de la Langue Française (1873) by Émile Littré, its use became popular, since it appears in several fields such as poetry and theatre, but at the very beginning it is asserted that is a term from law.
After having analyzed these conceptions and having reflected on them in terms of a concept whose use is similar to that of jurisprudence, we would like to consider briefly in the next section the impact of the introduction of convention in mechanics.
4. The Impact of Convention in Mechanics
In order to understand the transformation that mechanics undergoes during the nineteenth century, it is interesting to contrast Lagrange's views at the end of the eighteenth century with those of some authors at the end of the nineteenth. In his Thèorie des Fonctions Analytiques Lagrange stresses the idea that “mechanics can be understood as a geometry with four dimensions” and that we have to understand the “analysis of mechanics as an extension of geometrical analysis” (Lagrange [Reference Lagrange1797] 1813, 337). This supports the idea of mechanics as a branch of mathematics (of geometry or even of analysis) and also the idea of an axiomatic-deductive science following the model of Euclidean geometry. In 1883 Mach says: “The view that makes mechanics the basis of the remaining branches of physics and explains all physical phenomena by mechanical ideas is in our judgment a prejudice,” stressing his criticism and even questioning the desirability of considering mechanics as a model for the rest of the sciences (Mach [Reference Mach1883] 1912, 596). A few years later, Hertz showed the possibility of having different mechanical systems, undermining the unique character of mechanics as a science:
In this sense the fundamental ideas of mechanics, together with the principles connecting them, represent the simplest image which physics can produce of things in the sensible world and the processes which occur in it. By varying the choice of the propositions which we take as fundamental, we can give various representations of the principles of mechanics. Hence we can thus obtain various images of things; and these images we can test and compare with each other in respect of permissibility, correctness, and appropriateness. (Hertz Reference Hertz1894, 4)
These views, together with Poincaré’s suggestion that mechanics is an experimental science, witness the loss of certainty (cf. Pulte Reference Pulte2005) experienced by mechanics during the nineteenth century, from the evidence and certainty of Lagrange's work to the criticism, dubious character and revisability of the end of the century.
The introduction of conventions as a new epistemic category is one of the indicators of this change, going hand in hand with the changes that the concept “science” underwent during the nineteenth century leading to the Wahrheitsgewissheitsverlust characterized by Schiemann (Reference Schiemann1997). But more than an index of the loss of certainty, or just a consequence of the process, it could be also understood as a cause in itself, if we recognize its introduction as a meditated decision by the scientists that use it, more than as a tacit assumption. In fact, conventions as a distinct epistemic category reduce the truth pretensions of empirical science and have also the ability of showing that most of the elements in it are not true in light of our intellectual capabilities. As a result came the necessity of introducing conventions to express the different status of the principles. Our idea is to see the introduction of the new epistemic category not as a consequence of something (as Pulte Reference Pulte, Heidelberger and Schiemann2009 argues), but as a cause in itself, a contributing cause indeed (together with other factors) to the changing of a discipline.
Indeed the discussion regarding non-Euclidean geometry that took place in the last third of the nineteenth century might support this thesis. At that period, the work of several scientists showed a concern for epistemological questions (e.g. Riemann, Helmholtz, Poincaré). And when the debate about geometry became public, those figures were particularly language-sensitive regarding the categories that should be applied to characterize first principles – axioms, hypotheses, conventions, etc. This introduction of new epistemic categories in geometry contributed to changing the status of the science itself. It might well be considered that something similar happened when the category convention was introduced into mechanics, helping to change the status of this discipline and excluding it from the mathematical realm, to the more physical (empirical) one.
The concept of convention in mechanics, if understood as transferred from jurisprudence, rather than from language or arbitrary stipulations, regulates certain commitments such as the use of concepts, and points in the direction of determinate research, leading to a provisional agreement with empirical consequences. It also fits into the picture of a concept proper to the natural sciences in which empirical content is made relevant because conventions have to be able to explain and predict observed phenomena (as in Poincaré’s empirical dimension of convenience). Thus, even if it is interpreted as a concept transferred, as it happens in conceptual transference, it does not imply an absolute identification with the concept in its original field of meaning, because conventions in mechanics are not the derivation of a social agreement. Despite the impact it had on science, the signature of the metric system convention (Convention du Mètre) in 1875 (cf. Galison Reference Galison2003) is not comparable to the understanding of the principles of mechanics as conventions. First, because this happened prior to that historical event. And also because in the metric convention there is arbitrariness that is not present in jurisprudence or mechanics. But the point is not to identify conventions in jurisprudence and in mechanics, but to suggest a transfer connecting the use of both notions, given their similarities.
The notion of convention as a cause of the change in the development of mechanics, situating its principles in a revisable (not absolutely certain) status, entails considering the hypotheticality of the science, as a fact that scientific assertions are not simply true or false and that certain assumptions are required for the subsequent development of the theory. The gradual introduction of this notion from the half to the end of the century allows interpreting this process as a gradual one, rather than considering it a sharp rupture as if nothing happened from Lagrange's perspective to Mach's.
The hypotheticality goes hand in hand with the recognition of conventions as outlining their own context of application, i.e. as interpretive principles that define the fundamental concepts of the theory (cf. DiSalle Reference DiSalle, Psillos and Curd2014a, 8). As disguised definitions, they set the interpretation of the fundamental concepts, but as conventional, not being true or false, they can only work “as if” they were true, that is, as mere hypotheses that can be perfected or even replaced. It is not a casual nor a trivial consequence that these authors explicitly assert the conditions of application of the principles, whether for a simpler representation of them (e.g. Jacobi and Reech) or by admitting a hierarchy in our knowledge (e.g. Reech) or the necessity of a previous spatial-temporal configuration (e.g. Poincaré). By making these conditions explicit and by acknowledging their non-empirical or a priori character they are deliberately undermining the undoubted character of mechanics and making of it a hypothetical science. That is the sense in which we want to understand the introduction of convention as a cause in the changing development of mechanics.
The impact of convention in mechanics is also noticed by looking at the relation between mathematics and natural sciences and the respective conceptions of both. The principles of the scientific theory about nature no longer possess the rigor proper to a mathematical discipline. Even in the work of Reech, whose purpose is to free mechanics from any ambiguity, the loss of certainty appears in the recognition of the principle of inertia as a mere convention. And this happens in spite of the different conception of mathematics and mechanics that these authors uphold. Beyond any doubt, the impact that the use of this epistemic label has in science is dramatic: mechanics will no longer recover its certain status and the explanation of phenomena in terms of matter and motion will come to an end with the end of the century (cf. Klein Reference Klein1973, 58).
5. Conclusion
The introduction of new epistemic categories to characterize the principles of mechanics was a common point throughout the nineteenth century, in the sense that several authors introduced new categories,Footnote 31 but Jacobi, Reech and Poincaré are particularly interesting regarding this topic since they employ exactly the same epistemological label, “convention,” and since this label permits a connection with other fields of knowledge opening the space for a possible transfer of knowledge from jurisprudence to mechanics.
Although there is no direct textual or historical evidence, as stated above, to prove that the concept of convention in mechanics was directly taken by Jacobi, Reech, and Poincaré from jurisprudence,Footnote 32 it has been illuminating to think of this concept as a transferred one in order to clarify some of the ambiguities that frequently surround the notion of convention, and given its technical character as the search in several dictionaries has shown, it makes our suggestion at least plausible. We have shown their free but non-arbitrary character, their ability to work as rules, and their flexibility as non-definitively established rules, and also their function as definitions of their context of application (context-providers). All of this reinforces the idea that the choice of this concept by these authors is not random nor a mere subtlety in the use of language, but probably a choice that aims to show that the status of the principles of mechanics is different from other scientific assertions depending only on the pure laws of thought or on empirical considerations. Conventions as they are employed in the field of jurisprudence are free from the ambiguities involved in taking the conventions of mechanics to be similar to linguistic or monetary ones.
By stating these differences with other kinds of convention we challenge the common idea that “there exist alternative conventions that are in some sense equally good” (Rescorla Reference Rescorla and Zalta2015). Although the use of conventions in Jacobi, Reech, and Poincaré implies of course the existence of alternative conventions, just as in jurisprudence, this does not imply at all that those conventions are in some sense “equally good,” because they are not arbitrarily chosen. Some criteria have been introduced to justify the choice and to provide a raison d’être, that makes the choice of them not an “indifferent” one, but – to these authors – the best choice among the different possibilities. Just as Varzi states: “it is up to us to make the choice. We may adopt a convention precisely because it is the best choice” (Varzi Reference Varzi, Campbell, O'Rourke and Slater2011, 148) – we would add, because there are good reasons (theoretical, pragmatic) to consider it as the best choice. When conventions are better than others, they are not equally good or equivalent alternatives.
This opposes also to a rapprochement between conventionalism and any kind of social “decisionism,” because as DiSalle states: “conventionalism would be an absurd doctrine if it asserted, that (for example) straight lines are defined by light rays in accord with some explicit decision by a social group. For Poincaré, at least, the connection between the geometrical straight line and the physical propagation of light arises from a long, successful, and largely unexamined history of empirical practice” (DiSalle Reference DiSalle, Psillos and Curd2014a, 7). This means that the decision to introduce certain conventions is, of course, informed by experience, but also by the historical knowledge of the discipline itself, proving that they are dependent on the use that they have previously had, or on the previous scientific practice as DiSalle remarks. This is also clear if we recall Jacobi's and Reech's criterion of simplicity, because it involves the practice of graphically representing the simpler trajectories in coordinate axes or in analytical functions. Thus, although conventions imply decisions, they are not the product of a mere social contract, but of an informed decision implying a deep understanding of how the principles of a discipline work, of the best practice to provide a satisfactory systematic description, i.e. the better theory, and of the assumptions on which the use of the convention depends.
An intriguing question is whether the historical context may have had a significant influence in this transference of notions from the field of law to the sciences. The temporal coincidence of Jacobi and Reech in their proposals is quite noteworthy, and one cannot simply ignore the fact that they were almost simultaneous with the great bourgeois Revolutions of 1848. In fact, Jacobi took action and was politically involved in Berlin during the revolution, and even presented his candidature for the Parliament in Frankfurt; his engagement actually caused him trouble as the royal pension on which he was living was cut off. Considering things more generally, the whole period between 1789 and 1848 was one in which debates about social reform and political conventions were widespread, so it is pertinent to ask if a metaphorical or analogical transfer of notions from the fields of jurisprudence, law, and social organization to science might have been possible.
This particular kind of transfer implied a transformation, or at least modulation. The transfer is a consequence of scientific practice: the use of a new epistemic category brings to light the loss of certainty that the principles are undergoing. By using it and by exploring its consequences, scientists realized that some principles could no longer be considered as entirely empirical or entirely a priori, but as something else, something whose correspondence to nature is assumed, decided, conventioned, although never forced:
“In conclusion, it is our mind that furnishes a category for nature. But this category is not a bed of Procrustes into which we violently force nature, mutilating her as our needs require. We offer to nature a choice of beds among which we choose the couch best suited to her stature” (Poincaré Reference Poincaré1898, 31).
Acknowledgments
I thank the Junta de Andalucía Excellence Research Project, “The Genesis of Mathematical Knowledge: Cognition, History and Practices” (P12-HUM-1216), for funding support; and Department I of the Max Planck Institute for the History of Science (MPIWG) for the Visiting Fellowship during which I finished this work. I also want to thank José Ferreirós for insightful comments on a previous version of this paper. Finally, I want to thank two anonymous referees and the editors of this journal for their comments.
