10.6092/issn.2421-4124/9029
The paper deals with Pierre-Louis Moreau de Maupertuis’s (1698-1759) writings on the principle of least action (PLA). I discuss Maupertuis’s philosophical interpretation of the PLA by analysing three of his papers (1740, 1744, 1746), in order to assess the overall consistency of his argument. As I shall argue, Maupertuis presents the PLA as a universal principle of nature, this being at odds with the empiricist inspiration of his epistemology. Maupertuis’s attempt to introduce conceptual elements borrowed from the Leibnizian tradition into a Newtonian-inspired framework creates in fact a tension that is difficult to overcome.
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Marco Storni
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Maupertuis’s Principle of Least Action:  Epistemology and Metaphysics
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2019-02-02
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Marco Storni
2019/02/02
10.6092/issn.2421-4124/9029
Forschungszentrum Gotha der Universität Erfurt
2421-4124
metaphysics
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Maupertuis’s Principle of Least Action:  Epistemology and Metaphysics
https://montesquieu.unibo.it/article/download/9029/8925
11
             Maupertuis’s Principle of Least Action:
                      Epistemology and Metaphysics

                                      Marco Storni
                     (Forschungszentrum Gotha der Universität Erfurt)




The paper deals with Pierre-Louis Moreau de Maupertuis’s (1698-1759) writings on the principle of
least action (PLA). I discuss Maupertuis’s philosophical interpretation of the PLA by analysing three
of his papers (1740, 1744, 1746), in order to assess the overall consistency of his argument. As I shall
argue, Maupertuis presents the PLA as a universal principle of nature, this being at odds with the
empiricist inspiration of his epistemology. Maupertuis’s attempt to introduce conceptual elements
borrowed from the Leibnizian tradition into a Newtonian-inspired framework creates in fact a tension
that is difficult to overcome.

Keywords: Maupertuis, principle of least action, Newton, Leibniz, epistemology, metaphysics




Introduction


In 1740, Pierre-Louis Moreau de Maupertuis (1698-1759) was invited by Frederick II
of Prussia to be the chairman of the Berlin Academy of Sciences. Thenceforth,
Maupertuis would increasingly become uninterested in regular scientific practice, and
inclined more towards philosophical speculation 1. The development of Maupertuis’s
interest in philosophical matters roughly coincides with this biographical episode, and
is tightly linked to the formulation of the «principle of least action» (PLA) 2. In what
follows, I shall discuss a few epistemological questions linked to Maupertuis’s
formulation of PLA, most notably the relationship between physical laws and
metaphysical principles. This I shall do by relying on three of his papers (Maupertuis
1740, 1744, 1746), which represent three successive steps toward the formulation and
philosophical framing of the PLA. I shall start from the core epistemological

1
 See Brown 1963 and Casini 1983.
2
 I am not going here into the scientific details of the PLA. There is abundant secondary literature on
PLA: see Brunet 1929, Fee 1941, Feher 1988, Pulte 1989, Radelet De Grave 1998, Leduc 2015, and
Lyssy 2015.


                                                   1
distinction between “first-type principles” and “second-type principles,” and proceed
by assessing Maupertuis’s attempt to make natural laws – in particular, the laws of
geometrical optics – consistent with metaphysical principles of high generality. I shall
then inquire into the extent to which Maupertuis’s argument is indebted to Leibniz’s
discussion of the laws of optics, and end with a critical presentation of Maupertuis’s
own proof for the existence of God.


1. First-Type Principles versus Second-Type Principles

At the outset of the Loi du repos (Maupertuis 1740), Maupertuis distinguishes
between two different kinds of principles on which all the sciences 3 are grounded.


      If the sciences are grounded on certain simple and immediately clear principles, on which all the
      truths that are the object thereof depend, they have yet other principles, less simple indeed, and
      often difficult to discover, but which once discovered, are of very great utility. These are in
      some way the laws Nature follows in certain combinations of circumstances, and which teach us
      what it will do on similar occasions (Maupertuis 1740, 170).


    Here, Maupertuis formulates the core epistemological distinction between “first-
type principles” and “second-type principles”. Second-type principles are «less
simple» and «difficult to discover», and coincide with particular laws of nature (e.g.
the principle of the lowest centre of gravity in statics and that of the conservation of
living forces in dynamics 4), whereas first-type principles are «simple and immediately
clear». If the status of second-type principles is sufficiently clear, thanks to the
examples Maupertuis himself provides, the status of first-type principles remains on
the contrary vague 5 . Scholars have suggested various interpretations for first-type
principles: they might e.g. be general metaphysical principles 6, or physical axioms of
the mathematical sciences 7.
    Things, however, get trickier when Maupertuis specifies other features of both
kinds of principles, in particular concerning their demonstrability.


      Principles of the first type hardly require any demonstration for their evidence is clear to the
      mind as soon as it examines them. The principles of the second type, however, are not



3
  Not only mechanics, as Marco Panza suggests (Panza 1995).
4
  These are the examples Maupertuis himself mentions.
5
  See Van den Abbeel 2017, 61-63.
6
  Beeson 1992, 165.
7
  Terrall 2002, 175.


                                                  2
      susceptible of physical demonstration in the strict sense, because it is impossible to go through
      all the cases in which they occur (Maupertuis 1740, 170).


    It is impossible to provide any demonstration of first-type principles, as the
knowledge of them we have is intuitive 8. Second-type principles, on the contrary, are
demonstrable, but are not so a pedibus usque ad caput. A few lines later Maupertuis
comments further on this point:


      These principles have never been given a general demonstration in the strict sense; but no
      one who is accustomed to study the sciences, and who knows the force of induction,
      will ever doubt their truth (Maupertuis 1740, 170).


    Second-type principles are demonstrated according to the methodological lines
Newton gave in the Principia (1687) and in the Opticks (1704). As the fourth Regula
philosophandi reads,


      In experimental philosophy, propositions gathered from phenomena by induction should be
      considered either exactly or very nearly true notwithstanding any contrary hypotheses, until yet
      other phenomena make such propositions either more exact or liable to exceptions. This rule
      should be followed so that arguments based on induction may not be nullified by hypotheses
      (Newton 1999, 796).


    And again, in the General scholiu 9: «In this experimental philosophy, propositions
are deduced from the phenomena and are made general by induction» (Newton 1999,
943). Maupertuis stresses the reliability of Newton’s inductive method, although this
method provides no complete demonstration for the propositions at stak 10 . Our
understanding is too weak, and our knowledge too scanty, to enable us to follow the
whole chain of reasons and causes 11. If we could do so, our demonstrations of second-
type principles would be completely a priori:


      As of a priori demonstrations of these kinds of principles, it seems that physics is not able to
      provide them; they [a priori demonstrations] seem to belong to some superior science
      (Maupertuis 1740, 170).




8
   It is hard to see, then, how first-type principles could be identified with axioms of Newtonian
mechanics (axiomate sive leges motus), which seem to require a stronger foundation than mere
intuition.
9
  Published for the first time as an appendix to the 2nd edition of the Principia (1713).
10
    On Newton’s inductivism, see Ducheyne 2012, 107-169, and Belkind 2017.
11
    Sources for this claim are Locke and the sceptical tradition: see Tonelli 1987.


                                                  3
   Given the contrast between first-type and second-type principles which Maupertuis
had established at the outset, it seems not unreasonable to suppose that the former
might indeed be a priori. Their simplicity does not allow for demonstration though, as
Maupertuis explicitly states. The intuitive status of first-type principles might likewise
shed light on the meaning of the somewhat obscure expression «superior science». If
a science superior to physics exists, it might be one that works entirely with a priori
intuitive principles, i.e., with first-type principles.
   The epistemological discussion carried out in the Loi du repos is relevant for it
provides conceptual distinctions which will be useful in investigating Maupertuis’s
effort to integrate physics with the «higher sciences» (metaphysics, theology).
Moreover, the discussion of first-type and second-type principles, a priori and a
posteriori demonstrations, and of a «superior science» suggests that – as David
Beeson correctly stresses – «Maupertuis in this paper takes a step towards Leibnizian
architectonic reasoning» (Beeson 1992, 167).


2. Optics and Metaphysics

2.1. PLA

The 1744 paper Accord de différentes lois de la nature qui avaient jusqu’ici paru
incompatibles is well-known for it includes the first clear formulation of the PLA.
Although the argument deals with optics, this subject is for Maupertuis no more than
«a memorable example» (Maupertuis 1744, 417), useful to discuss a general
philosophical point.


      We should not expect that the different means we have to increase our knowledge should lead
      us to the same truths, but it would be shocking to see that propositions which philosophy
      presents as fundamental truths, were in fact contradicted by geometry or algebra (Maupertuis
      1744, 417).


   Maupertuis introduces here a first assumption: the truths which philosophy and the
sciences – represented here by mathematics – teach should not be at odds with each
other (the “truth-agreement” postulate). From this assumption a first problem arises,
which the formulation of the PLA should come as an answer to: although undeniably
true, the “truth-agreement” postulate seems to be contradicted by a few
counterexamples, of which optics is perhaps the most striking.



                                                4
     There is yet another assumption grounding Maupertuis’s discussion, which,
however, is made explicit only at the end of the paper. From this second postulate
another assumption follows which Maupertuis’s argument is meant to ground. God
exists, and has organised the world harmoniously. This harmony consists in the fact
that a blind mechanism – the laws of nature, viz. second-type principles – executes the
orders of a superior intelligence. The PLA will thus prove that it is good to employ
final causes in the natural sciences 12.
     I have mentioned that the discussion carried out in the Accord concerns optics, first
and foremost because it provides a “memorable” counterexample to the “truth-
agreement” postulate. Why is this so? Maupertuis believes that, whereas the first two
fundamental laws of geometrical optics – (1) light, in a uniform medium, moves in a
straight line, and (2) the angle of incidence is equal to the angle of reflection (the “law
of reflection”) – are philosophically intelligible, for they have evident mechanical
analogies (the law of inertia and elastic collision), the third law – the “Snell–
Descartes law” or “law of refraction” – is not 13 . Such a law states that, in any
refraction, the ratio between the sine of the angle of incidence and the sine of the
angle of refraction is a constant determined by the mediums involved. This law lacks
intelligibility because, as Maupertuis stresses, «when light passes from one medium to
another, the phenomena are very different than when a ball passes through different
mediums» (Maupertuis 1744, 419) 14.
     Maupertuis distinguishes three interpretative options for the laws of optics,
corresponding to three emblematic figures of seventeenth-century science: (1) A
mechanical explanation (Descartes); (2) A mechanical explanation, which resorts to
an additional non-mechanical assumption (Newton); (3) A non-mechanical
explanation (Fermat). The accounts provided by both Descartes and Newton are
immediately dismissed. If Descartes’s account is not physically accurate – «the way
in which this great philosopher tried to explain this phenomenon is imperfect»
(Maupertuis 1744, 419-420) – the same, however, does not apply to Newton’s and the
Newtonians’ account. Alexis Clairaut, e.g., in his Newtonian-inspired theory of light,

12
   As d’Alembert writes in the article «Action» of the Encyclopédie, Maupertuis’ cosmology is based
on the attempt to “ally the metaphysics of final causes to the most fundamental truths of mechanics”
(d’Alembert 1751, 119).
13
   For an account of modern discussions on optics, see Sabra 1981.
14
   In some sense, one should seek the “sufficient reason” for that law to be. As I will show, the PLA is
characterized in much the same way as Leibniz’s principle of sufficient reason. See Terrall 2002, 177-
178.


                                                   5
«deduces the phenomena of refraction with the same clarity as all the other subjects
he deals with» (Maupertuis 1744, 420). Although not explicitly stated, the problem
with the Newtonians seems to be philosophical: Maupertuis seems indeed to assume
that the Newtonian theory provides no appropriate metaphysical framework for
physics 15.
     Fermat’s method is therefore preferable, as it «seeks the explanation of these
phenomena in a wholly different and purely metaphysical principle» (Maupertuis
1744, 420). Fermat formulates, on the basis of empirical evidence, a most general
principle: «Nature, in the production of its effects, always acts in the simplest way»
(Maupertuis 1744, 421). Fermat generalizes this principle from optics to the whole of
physics, arguing that all natural movements follow the shortest-path and shortest-time
law. The metaphysical principle of economy grounds for Fermat the phenomenal
level: both the «direct and reflexed movements of light – Maupertuis stresses – […]
seem to depend on a metaphysical law» (Maupertuis 1744, 421; my emphasis).
     Maupertuis agrees with Fermat on the necessity to ground physical laws on a
“higher” principle, but disagrees on the specific contents of Fermat’s argument.
Fermat’s principle, according to Maupertuis, is incorrect, for the physical data from
which it is deduced are not accurate. Fermat’s mistake is in fact to assume that light
moves slower in denser mediums, and quicker in less dense ones. Maupertuis agrees
with Descartes (and Leibniz) that the converse is rather true. The refraction of light,
therefore, does not happen according to the shortest-path and shortest-time law. The
metaphysical principle grounding the laws of optics is another one, namely that
according to which the «quantity of action» is minimized, meaning the sum of the
distances covered, each multiplied by the velocity with which the body travels
through it. Only at this point do we find the first explicit formulation of the PLA: «It
[light] takes a path that has a real advantage: the path it takes is that whereby the
quantity of action is the least» (Maupertuis 1744, 423). «Once the true principle
discovered», Maupertuis concludes, «all laws of light can be deduced from it»
(Maupertuis 1744, 425).


2.2. Epistemology and Metaphysics

15
  When presenting Fermat’s position, Maupertuis says that Fermat rejects the Newtonian explanation,
«although it is known that this principle [attraction] was neither unknown nor distasteful to him»
(Maupertuis 1744, 420). In 1744, Maupertuis’s attitude towards Newton remains ambiguous; in 1746,
on the contrary, Maupertuis expressly criticizes Newton and the Newtonians for their physicotheology.


                                                 6
The Accord provides further elements to pursue the discussion of epistemology
started with the Loi du repos. In the above-quoted passage stating that «it would be
shocking to see that propositions which philosophy presents as fundamental truths,
were in fact contradicted by geometry or algebra» (Maupertuis 1744, 417),
Maupertuis makes a clear-cut distinction between philosophical and scientific truths.
Philosophy is about «fundamental truths», whereas the sciences are confined to
contingent truths. According to Maupertuis, mathematics and geometry also deal with
natural objects, at a higher level of abstraction than physics, and their conclusions are
therefore contingent 16. Prima facie, this opposition seems to reproduce that between a
«superior science», including most general truths, and the other sciences, constructed
with second-type principles.
   The discussion of optics and metaphysics, however, poses some challenges to this
epistemological picture. In some passages of the Accord, particular scientific truths
are said to depend on metaphysical laws – «the direct and reflexed movements of
light […] seem to depend on a metaphysical law» (Maupertuis 1744, 421) –, whereas
elsewhere it rather seems that scientific laws are the basis for the elaboration (or
modification) of metaphysical laws – «on the basis of this fact, that light moves
quicker in denser mediums, Fermat’s and Leibniz’s [metaphysical] construction is
destroyed» (Maupertuis 1744, 422). It seems therefore reasonable to raise the question
whether metaphysical principles ground physical laws, or the other way round. At the
end of the paper, Maupertuis gives indications for solving the puzzle. To find «the
causes of physical effects», we can either «[calculate] the properties of bodies», or
«[search] what it was more appropriate to make them [the bodies] do» (Maupertuis
1744, 425-426). These two methods roughly correspond to induction and deduction,
i.e., bottom-up and top-down relations. De facto, Maupertuis says, since «the first
method [induction] is within our reach, but it does not lead us very far», and «the
second [deduction] sometimes misleads us, for we do not know which is the purpose
of nature», we should «employ both methods[:] let us calculate the movements of
bodies, but also consult the design of the Intelligence making them move»
(Maupertuis 1744, 426). Our understanding is too weak and our knowledge too scanty
to enable us to follow the entire chain of causes (either bottom-up, or top-down).

16
   So Maupertuis will argue in the Examen philosophique de la preuve de l’existence de Dieu employée
dans l’Essai de cosmologie (Maupertuis 1756, 393-394).


                                                 7
There is no logical primum; the whole research relies on an imperfect empirical
procedure.
     This is how things are de facto. Such an explanation, however, does not provide
any answer to the above-raised question whether metaphysical principles ground
physical laws, or the other way round. How should things be de jure, then? The PLA
seems to be formulated on an inductive basis. The research on optics provides
evidence to formulate a general conclusion, grounded on phenomena, which is
therefore a posteriori. This, of course, would make of the PLA a second-type
principle, just like Newtonian attraction. The PLA, however, is clearly put on a
different qualitative level than the laws of optics: it is hard to see, then, how induction
can allow for such a qualitative passage. Moreover, Maupertuis argues that the PLA is
a most general principle. As he writes in a further work (Letters, 1752): «I have
showed that all laws of movement are grounded on the principle of least action: […]
attraction itself depends on the principle of least action […]: it makes the bodies move
as they should in order to obey this universal law of nature» (Maupertuis 1768b, 288-
289; my emphasis). According to the Loi du repos, induction cannot provide general
(i.e., complete) demonstrations. If the PLA is a «universal law of nature» stricto
sensu, it should therefore be a first-type principle 17 . Maupertuis’s ideas – at least
regarding the situation de jure – are unclear. All the more so, if one considers that the
PLA will become the basis of Maupertuis’s metaphysics, whose identification with a
«superior science» looks in fact plausible.


3. Leibniz and Teleology

Maupertuis’s argument is deeply indebted to Leibniz’s teleological account of the
laws of optics. Although Maupertuis never explicitly quotes Leibniz’s work on optics,
he was certainly acquainted with Leibniz’s optical theories, which he fleetingly refers
to 18. Most importantly, the argument presented in the Accord is strikingly similar to




17
   However, it is hard to see how the PLA might be «simple and immediately clear», and how it would
«hardly require any demonstration» (Maupertuis 1740, 170) for its self-evidence.
18
   Maupertuis’s knowledge of Leibniz’s optics is indirect. As he says in the version of the Accord
published in the 1756 edition of his Works: «When I read this paper [the Accord] in the Paris Academy
of Sciences, I did know Leibniz’s work on optics only by what de Mairan says in his paper on the
reflection of light, Mémoires de l’Académie de Paris, year 1723» (Maupertuis 1768c, 23).


                                                 8
Leibniz’s. A comparison between the two accounts might therefore be useful for
clarifying the few difficulties in Maupertuis’s argument 19.
     Although the only text by Leibniz on optics available to Maupertuis would have
been the Unicum opticæ, catoptricæ, et dioptricæ principium (A Unitary Principle of
Optics, Catoptrics, and Dioptrics), published in 1682 on the Acta Eruditorum, I shall
rather focus on the Tentamen anagogicum, or An Anagogical Essay in the
Investigation of Causes, written in 1695, but published in the course of the nineteenth
century. Leibniz advances similar arguments in both papers; the Tentamen
anagogicum, however, is much more detailed on the relationship between mechanical
laws and teleological principles, and for this reason more relevant to my study.
     In the Tentamen anagogicum, Leibniz attempts to reconcile Descartes’s
mechanical account of the laws of optics – particularly the law of refraction – with
Fermat’s teleological account. The Cartesian objections to Fermat’s argument (light
moves faster in denser mediums) are overcome by means of the «Most Determined
Path Principle» (MDPP). 20 A ray of light, Leibniz contends, always moves along the
most determined path, which does not necessarily imply the minimization (or
maximisation) of some quantity. Leibniz’s principle just states that «the easiest path
would prevail among all the possible competing rays» (Leibniz 1989, 479), ease
being defined as «the quantity obtained by multiplying the distance of the path by the
resistance of the medium(s)» (McDonough 2009, 512). By means of this general
principle, Leibniz can easily derive the “Snell–Descartes law” by means of
differential calculus and basic trigonometry21. He thus manages to preserve Fermat’s
teleological approach to optics – MDPP being a teleological principle – adjusting
Fermat’s principle so as to make it consistent with Descartes’s mechanical account of
phenomena.
     From the epistemological point of view, Leibniz acknowledges the fruitfulness of
purely mechanical explanations: efficient causes alone can adequately describe what
goes on in the physical world. Mechanical explanations, however, are per se
insufficient and should be integrated with teleological principles: «The laws of motion
cannot be explained through purely geometric principles or by imagination alone»
(Leibniz 1989, 478). This is because, although mechanical causes can explain all

19
   There is an abundant literature on Maupertuis and Leibniz. With particular reference to the PLA (and
to the debates over this principle), see Kneser 1929, Brunet 1938, Bachelard 1961, Goldenbaum 2016.
20
   See McDonough 2009, 511-512.
21
   For the details of this derivation, see McDonough 2009.


                                                  9
physical phenomena, the ultimate principles of mechanics cannot themselves be
explained mechanically, and need teleological principles in order to be grounded.


       All natural phenomena could be explained mechanically if we understood them well enough,
       but the principles of mechanics themselves cannot be explained geometrically, since they
       depend on more sublime principles which show the wisdom of the Author in the order and
       perfection of his works (Leibniz 1989, 478).


     The teleological account of natural phenomena has for Leibniz a twofold
function 22 . (1) It relates physical nature to God’s intelligent design. In corporeal
nature, alongside the mechanical dimension, which Leibniz calls «the realm of power,
according to which everything can be explained mechanically by efficient causes»,
there is another (and superior) dimension that he calls «the realm of wisdom,
according to which everything can be explained architectonically, so to speak, or by
final causes» (Leibniz 1989, 479). In nature, therefore, finalistic laws operate as a
mark of the divine organization of the world. (2) Leibniz’s argument displays the
inventive role of final causes in physics 23. Teleology is for Leibniz a proper method
for physical investigation: in the case of optics, «we may well believe that we should
not have had this beautiful discovery [the law of refraction] so soon without the
method of final causes» (Leibniz 1989, 480) 24. Optics thus provides an interesting
case study to demonstrate that teleology might work as a powerful explanatory
device.


       So we begin here to show that no other reason can be given for the laws of nature than the
       assumption of an intelligent cause. Or we show also that in the investigation of final causes

22
   On this point, see Duchesneau 1993, 263-264.
23
   In the Unicum opticæ too, Leibniz insists on this very point: «And so those who reject final causes in
physics with Descartes err greatly – not to speak more harshly – since even besides the admiration of
divine wisdom, they would also supply to us the most beautiful principle for discovering some
properties of those things whose interior nature is still not so clearly known to us that we would be able
to use proximate efficient causes and explain the machines which the Creator employed in order to
produce those effects and in order to obtain his ends». Translated by Jeffrey K. McDonough.
http://philosophyfaculty.ucsd.edu/faculty/rutherford/Leibniz/unitary-principle.htm#note1. Accessed 16
January 2018.
24
   Leibniz thinks that Snell had first discovered the law of refraction, and that Descartes contributed
little to the discovery. «It is known that Willebrord Snell, one of the greatest geometricians of his time
and well versed in the methods of the ancients, invented it, having even written a work which was not
published because of its author’s death. But since he had taught it to his disciples, all appearances point
to the conclusion that Descartes, who had come to Holland a little later and who was most interested in
this problem, learned it there. For the way in which Descartes has tried to explain the law of refraction
by efficient causes or by the composition of directions in imitation of the reflection of bullets is
extremely forced and not intelligible enough. To say no more about it here, it shows clearly that it is an
afterthought adjusted somehow to the conclusion and was not discovered by the method he gives».
(Leibniz 1989, 479-480).


                                                    10
         there are cases in which it is necessary to consider the simplest or most determinate, without
         distinguishing whether this is a maximum or a minimum […]. Certain new general theorems
         can be derived which apply equally to refraction and reflection (Leibniz 1989, 484-485).


      As we see, Leibniz insists on the heuristic role of teleological principles. The
inventive power of such principles is well displayed by the case of optics, where
MDPP would have not been found without recourse to teleological reasoning.
      Leibniz, however, wants to go as far as to imply that teleological principles might
have a heuristic function in all fields of knowledge. As he writes in the Definitiones
cogitationesque metaphysicae (1678-80): «All the phenomena of nature can be
explained solely by final causes, exactly as if there were no efficient cause; and all the
phenomena of nature can be explained solely by efficient causes, as if there were no
final cause»25. According to Leibniz, therefore, all mechanical laws might a priori be
rephrased in the vocabulary of teleology. This, however, remains without proof. As
Daniel Garber remarks, «the idea of parallel explanatory structures everywhere in
nature seems a kind of speculative program for a natural philosophy, grounded in a
metaphysical vision rather than in detailed argument, empirical or otherwise» (Garber
2009, 259). 26
      Let us now come back to Maupertuis. What does he retain of Leibniz’s teleological
conception? No doubt that the PLA meets requirement (1). As I shall argue in a
further section, Maupertuis considers the PLA the most universal law of nature and,
as such, the clearest tangible proof of God’s intelligent design. The PLA is the
architectonic law par excellence, through which the Creator has organised the world,
and therefore the cornerstone of the «realm of wisdom». As for requirement (2),
things change significantly. Although Maupertuis vindicates the PLA’s explanatory
power, stating that we should “employ both methods” – «calculate the movements of
bodies, but also consult the design of the Intelligence» (Maupertuis 1744, 426) – to
come to the right conclusions, a careful analysis of his argument shows, however, that
the PLA does not meet requirement (2). The “both-methods” statement remains
without consequence, not only in the Accord, but in further works too. In the 1744
paper, the PLA is formulated on the basis of the scientific inquiry into the laws of
optics (the «realm of power»), but, once formulated, does not seem to have any
particular heuristic function. Rather, the PLA is a generalisation, subsuming already-

25
     Quoted in Garber 2009, 258.
26
     For interesting remarks on this point, see Andrault 2016, 107-109.


                                                     11
known laws under a single formulation. All Maupertuis looks for is in fact the
agreement between particular mechanical laws and a general metaphysical principle,
which is formulated après coup in order to make the picture of physical reality more
consistent and unitary. Of course, this criticism points to the use Maupertuis de facto
makes of the PLA: this, one might say, tells us nothing about the de jure status of the
principle. This I grant. We should, however, note that Maupertuis’s ideas on the de
jure situation, as I showed, are quite confused; the reference to the de jure status of
the PLA might therefore not be an easy way out of the objection.
     In the end, it seems that the main – perhaps the only – application of such a
generalisation is the proof of the existence of God Maupertuis formulates by means of
the PLA 27.


4. The PLA and the Proof of the Existence of God

At the outset of the paper Les lois du mouvement et du repos déduites d’un principe
métaphysique (1746), Maupertuis states that, after discovering the PLA, his present
objective is «to draw from this very source some higher and more important truths»
(Maupertuis 1746, 267). In section 2 of the Lois du mouvement, one finds indeed a
detailed presentation of Maupertuis’s PLA-based demonstration of the existence of
God.
     From the very title of the section, one gathers Maupertuis’s key assumption: «That
the proofs of the existence of God are to be sought in the general laws of nature»
(Maupertuis 1746, 277) 28. This, for a twofold reason: (1) general laws are universal,
therefore they do not suffer any exception, and (2) they are simple, therefore
unequivocal. The criteria of universality and simplicity, however, should be further
specified in order to be actually informative. When can a law be said to be universal?
It does indeed make a difference if one conceives the method of knowledge to be
inductive – in which case, universality is such only par provision – or deductive. And
then: what are the criteria of simplicity for a law? Maupertuis overlooks such
definitional questions, and rather focuses on operative ones: how do we get to
universal and simple laws? He answers by mentioning a discipline that should lead
this research, namely mathematics.

27
   I am only referring to Maupertuis’s own use of the PLA. Other scientists will develop the
mathematical consequences of the PLA and apply it to, e.g., the study of planetary motions (Euler).
28
   An important source for this claim is of course Malebranche.


                                                12
   Why mathematics? Mathematical laws are universal and simple, but also
absolutely certain (although not necessary 29): «Such [mathematical] certainty does not
depend on the method the geometers follow, but on the simplicity of the objects they
consider» (Maupertuis 1746, 278). While not claiming the irrelevance of method,
Maupertuis emphasizes the nature of the object as the decisive criterion to attain
certainty – universality, although not mentioned, is closely related to simplicity. The
question thus arises of what role the appeal to mathematics plays in this context.
There are a few theoretical options on the table. (A) Mathematics might be a mere
example from which to draw inspiration: mathematicians produce certain proofs, so
should philosophers. (B) The method of mathematics should likewise be used in
metaphysics. Although prima facie plausible – Christian Wolff, and many eighteenth-
century philosophers with him, hold this very position – this option is explicitly
discarded by Maupertuis when he says that «[mathematical] certainty does not depend
on the method the geometers follow». (C) Mathematical demonstrations and theorems
should play some direct role in metaphysics. This is the option Maupertuis endorses.
His choice is grounded on a specific epistemological assumption:


       Let us see if we can make a better use of this science [mathematics]. The proofs of the
       existence of God it will provide should have over all others the advantage of being
       evident, which is typical of mathematical truths. Those who do not trust metaphysical
       reasoning will be more at ease with these kind of proofs (Maupertuis 1746, 278).


   When Maupertuis states that the existence of God should be proved through
mathematics, he seems to presuppose an axiom, one might call the «transitivity of
certainty (TOC) axiom». If the demonstration proving that proposition x
(mathematical) is true can be used to prove also that proposition y (non-mathematical)
is true, then proposition y will hold the same degree of certainty that proposition x
does. The axiom works well in all cases where mathematical models are used to
demonstrate non-mathematical statements (typically, physical laws). The question
Maupertuis leaves unasked, however, is whether the TOC axiom holds even when y is
of a wholly different nature than x, e.g., for metaphysical and theological
propositions. This appears indeed harder to believe.




29
   Later on in his career, Maupertuis will further reflect on the notions of contingency and necessity:
see Charrak 2006.


                                                  13
     Leaving aside the problem raised by the TOC axiom, let us come back to
Maupertuis’s argument, and see how it continues. Maupertuis specifies which
mathematical laws attention shall be focused on:


           The Supreme Being is everywhere, but is not everywhere visible to the same extent. We shall
           better see Him in the simplest objects: let’s search for Him in the first laws he imposed on
           nature […] according to which movement is conserved, distributed, or destroyed (Maupertuis
           1746, 279).


     This is not surprising, since the PLA is a universal law overruling motion in
physical nature. Surprising, however, is what Maupertuis states immediately
afterwards, concerning method.


           I could have started from these laws such as they are given by mathematicians, and confirmed
           by experience, searching therein the marks of God’s wisdom and power. However, as those
           who formulated these laws relied on hypotheses that were not purely geometrical, and
           therefore their certitude seems not to be established in strict demonstrations, I thought it better
           and more useful to deduce such laws from the attributes of the almighty and all wise Being. If
           the laws I find in this way are the same as those we observe in the universe, wouldn’t this be
           the strongest proof that this Being exists, and that He is the author of these laws? (Maupertuis
           1746, 279)


     The laws of motion, insofar as they are grounded on experience, are not certain in
an absolute sense. Mathematics, as all other natural sciences, relies on ancillary
hypotheses that are – at least to some extent – arbitrary. Maupertuis insists on this
very point in other texts as well, most notably in the Réflexions philosophiques sur
l’origine des langues et la signification des mots (1740). Here he says: «What we call
‘our sciences’ depend so closely on the ways in which we first designated
perceptions, that I think the questions and propositions would all be different if we
had established other expressions out of our first perceptions» (Maupertuis 1768a,
268) 30.
     Maupertuis’s claim on the uncertainty of the laws of motion, formulated
inductively, poses some problems. Mathematical demonstrations would indeed be less
strict than purely deductive demonstrations, namely those solely relying on a priori
principles. It is therefore hard to see how mathematics could be the norm for the


30
  And also: «I do not think the diversity of their [people of remote countries] philosophy derives from
any difference in their first perceptions, but rather from the accustomed language of every country,
from this destination of signs to the different parts of perceptions: destination which is very much
arbitrary, and that the first men could have done in so many different ways […]» (Maupertuis 1768a,
275-276).


                                                      14
strictness of metaphysical demonstrations. (Although indeed Maupertuis discards
option (B), options (A) and (C) are still on the table.)
     Of particular interest is the method Maupertuis claims he will follow: «I thought it
better and more useful to deduce such laws from the attributes of the almighty and all
wise Being. If the laws I find in this way are the same as those we observe in the
universe […]». Since the inductive formulation of the laws of motion is not absolutely
certain, one should rather deduce them top-down, from the divine attributes. If the
results of the deduction coincide with those of the induction, then one gets the
strongest confirmation of their accuracy. Far from being exhaustive, Maupertuis’s
explanation creates more confusion than clarity. Indeed, if (bottom-up) induction, in
the context of the eighteenth-century Newtonian-inspired empiricist philosophy, is a
univocally defined concept, (top-down) deduction is not – this is a fortiori the case if
deduction is to be made from the attributes of God.
     Maupertuis starts with a historical survey of the philosophers’ ideas on motion.
Some ancient philosophers, confronted with the difficulty of making sense of
movement, denied its reality. This is because, Maupertuis says, the only source of
knowledge we have about movement is experience, and experience can always be
doubtful. Of some propositions, however, we can acquire reliable knowledge by
means of reiterated experiences or relevant counterexamples: e.g., motion is not
essential to matter, and matter cannot move itself if not impelled by some external
cause. The question is not therefore whether matter moves or not, nor whether motion
is essential to it, nor whether it can move itself, but what is the external cause causing
it to move. Philosophers, Maupertuis says, variously tried to answer the question –
appealing to an unmoved mover (Aristotle) or to God (Malebranche) – though
unsuccessfully. This because they lacked a fundamental principle, which the laws of
motion obey: «It was necessary to know that all laws of motion and rest are grounded
on the most suitable principle (le principe le plus convenable) 31 to see that they had
been established by an all wise and almighty Being» (Maupertuis 1746, 282). In an
updated version of the 1746 paper, namely the first part of the Essai de cosmologie
(1750) 32, Maupertuis further comments: «Therefore, I shall not look for these laws [of
motion] in mechanics, but in the wisdom of the Supreme Being» (Maupertuis 1768a,


31
   In the Essai de Cosmologie (1750), Maupertuis rather says: «on the principle of the best (le principe
du mieux)» (Maupertuis 1768a, 35).
32
   Which is otherwise the same text as Maupertuis 1746.


                                                  15
35). Although the only source of knowledge of motion is experience, the principle
underlying the very structure of motion is a priori. Maupertuis does not justify this
assumption. To do so, the only argument he could adduce is a circular one: laws of
motion are grounded on the principle of the best because God is their author, the
strongest proof that God is their author being that they are grounded on the principle
of the best.
      Pursuing his discussion, Maupertuis notes that modern authors – Descartes,
Huygens, and Leibniz – were not much bothered by discussions on the nature of
motion, but rather focused on the laws according to which «it is distributed,
conserved, and destroyed» (Maupertuis 1746, 283). After mentioning the debate over
the existence of hard bodies 33 , Maupertuis discusses Descartes’s and Leibniz’s
opposing views on the conservation of movement. Descartes argued that the quantity
of movement is constant in nature, whereas Leibniz held that living force (measured
by the product of the mass and the square of velocity) is a constant. Yet, Maupertuis
contends, «conservation of movement is true only in some cases, [and] conservation
of force takes place only for certain bodies. Neither of them can be considered a
universal principle, nor a general result (un résultat général) of the laws of motion»
(Maupertuis 1746, 285). Maupertuis himself is the first who has in fact discovered the
truly universal principle of motion, namely «the principle of least action: so wise a
principle, and so worthy of the Supreme Being» (Maupertuis 1746, 286). From here,
Maupertuis concludes: «The laws of movement and rest deduced from this principle
are precisely the same as those we observe in nature» (Maupertuis 1746, 286).
      The conclusion of the argument is therefore that the laws of movement one finds
through empirical investigation are the same as those one can deduce from the PLA.
The PLA thus plays the role of the explanans, rather than of the explanandum. In
other words, the PLA is not the law that is to be deduced, but a most general principle
from which to deduce particular laws. But how is this principle found? This
Maupertuis leaves unsettled. To begin with, no deduction from the attributes of God –
whatever this might mean – takes place. Two alternatives then remain: if the PLA is
found intuitively, it might count as a first-type principle, and be the cornerstone of
some «superior science»; if on the contrary it is found inductively, it would be a
second-type principle, more general indeed than all other second-type principles – not


33
     On this debate, see Shea 1988.


                                           16
absolutely general and universal, though. Maupertuis hesitates over these two
alternatives, none of which is fully and satisfactorily argued.


Conclusion

In this paper, I have tried to highlight some of the difficulties to be found in
Maupertuis’s writings (1740-1746) on PLA. It emerges from my analysis that the
epistemological status of PLA is problematic insofar as Maupertuis tends to present it
as a universal principle of nature even though this claim is incompatible with the
empiricist inspiration of his philosophy. While Maupertuis’s idea of scientific method
is mostly borrowed from Newton, the influence of the Leibnizian tradition pushes him
toward architectonic reasoning: this creates a tension in his argument that cannot be
easily resolved.




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