Leibniz on Metaphysical Perfection, Physical Optimality, and Method in
Physics; or, a real tour de force*
George Gale
University of Missouri–Kansas City
Kansas City MO 64110
*Presented at The North American Leibniz Society meeting, Chicago, April
2002.Laurence Carlin provided some careful and very useful comments in
response to the presentation at the NALS meeting, for which I am grateful. In
general, Carlin finds himself in agreement with others, e.g., Blumenfield (1995),
who hold that the “quantity of essence” interpretation of Leibnizian perfection is
to be preferred to the “ratio” interpretation. Nonetheless, Carlin finds himself
mostly in agreement with my view that Leibniz’ perfection (whichever the
interpretation), plays a role in justifying certain fruitful physical methods,
especially in optics.


 George Gale 2002
1. Introduction: Leibniz the Great System-Building Mathematician
From this it is now marvelously understood
how in the very origin of things a sort of divine
mathematics or metaphysical mechanics was
employed...
——Ultimate Origin of Things
(GP VII 304)
Leibniz was one of the greatest of the systematic thinkers. Trying to
disentangle and isolate one of his ideas—for example, the idea that God
chose to create the best, most perfect world—is almost impossible. Grab
hold of one end of the idea, for example, perfection-in-the-large—
perfection as it is exhibited on a world-wide basis—and almost instantly
one confronts the other end of the scale, grappling with perfection-in-thesmall, monadic perfection. Unfortunately, grappling would seem to be my
fate today: I want to examine in some detail Leibniz’ thoughts about
method in physics, focusing most especially upon his discovery and use
of extremal principles, and his views about their relation to teleological
explanations. But before one can move into Leibniz’ physics, his
metaphysics-qua-foundations-of-physics must be dealt with, at least
briefly. But this means, in turn, that perfection, both large and small,
must be reckoned with, again at least briefly. Finally, as if this weren’t
enough already, since Leibniz’ variational method proved to be both
powerful and fruitful, it has an established legacy in physics even unto
our own day and I must thereby end my work here today by discussing
some high points of the current nature, role and status of variational
principles in physics.
The theme of my exploration is Leibniz-as-mathematician. Since
Leibniz did many things in addition to his philosophizing, it has proven
fruitful to look at his philosophy from the perspective of some of these
other activities of his. Thus much ado has been made about Leibniz-aslogician (Russell and Couturat come immediately to mind), and Leibnizas-physicist (here one thinks of Garber and Gale, for example). But it
strikes me that not enough has yet been made about Leibniz-asmathematician. I hope to begin to fix this deficiency here.
My goal in each section will be to reveal the essentially
mathematical form of Leibniz’ response to the philosophical problem at
issue. For example, one leitmotif that runs throughout Leibniz’ physical
and metaphysical thinking is that, among the varied possible paths a
process might have taken, that one is actual which maximizes or
minimizes some essential property of the targeted process. Even more
markedly, in some areas, for example the analyses of perfection for a
world and perfection for a monad, both maxima and minima occur, put
in a proportion or ratio of the former to the latter.
The crescendo of this leitmotif occurs in Leibniz’ actual use of his
invention—the analysis of variations—in physics, where he argues that,
because the universe was chosen by God to have a certain mathematical
form, physicists know beforehand that it will be useful to look for this
form in their researches. My conclusion is the not entirely surprising one
that mathematical physicists succeed in their discoveries this way simply
because they are mimicking precisely the activities God, the Original
Mathematical Physicist, went through when He was discovering which
world to create.
2. All the World is in Three Parts Divided
It is frequently possible—indeed, sometimes it is necessary—to recognize
that Leibniz’ metaphysics has three levels of focus: the genuinely
metaphysical, fundamental, foundational level, the level of the monads.
Less fundamental than this is the level of the metaphysics of physics, the
level of corporeal substances. Finally, there is the level of everyday
experience, the level of bodies. Cf. Figure 1. Perfection has a homologue
at each of the three levels. This homology is in great part what Leibniz’
famous “well-founding” comes down to. That Leibniz’ system is in three
parts divided is no longer controversial, although it was when I first
proposed it in 1970. (Gale 1970) What remains controversial is where
precisely to do the dividing. For example, Garber initially agreed with my
division (Garber 1985, p. 127), but has since changed his mind a bit.
(Garber 1995, p. 342) As we shall see, it is the fact that perfection is
homologous across all levels in Leibniz’ system which guarantees his
method in physics.
Figure 1 will be supplied anon. Sorry!
3. In the Beginning there was Perfection: On what God chose
According to Leibniz, God chose to create a world. The chosen world
would be of a certain sort: it would be the most “perfect” world. It is clear
that God’s decision wasn’t specifically about which world to create; rather,
his decision involved what “perfection” meant. This follows from the fact that,
given any criterion P, there just is, automatically, a rank-ordered series of
worlds differing by the degree to which they satisfy P. There are no ties: “In
my opinion, unless there were an optimal series, God clearly would have
created nothing.” (GP II 424) As will be noted later, this is precisely what
happens during a variational analysis: although many of the possibilities are
similar, or even tied —“twinned”—the best one is unique.
Unfortunately, although Leibniz seems to think he is clear enough
himself about what P comes to, not all of us commentators agree that he’s
been clear enough. Thus, there is disagreement about what “perfection”
means, at least to a degree. What I’ll do here below is try to pretty much
stick to the points of agreement among most of us.
According to a well-known passage, God chose the world which
exhibited a certain ratio: “God, however, has chosen the most perfect, that is
to say, the one which is at the same time the simplest in hypotheses and the
richest in phenomena, as might be a geometric line whose construction
would be easy but whose properties and effects would be very remarkable
and of wide reach.” (Discourse on Metaphysics §VI; GP IV 131; L. 306)
Typically, as Leibniz relates it, God’s choice depends on one version or
another of this kind of ratio. For example, at Monadology §58, the ratio puts
variety and order in direct proportionality. The point Leibniz wishes to make
is frequently illustrated by his analogy between the Divine construction and
human construction: just as the best human architect is that one who gets
the best bang for the buck, so also does God. In terms of the proportionality
of bang to buck, of richness to simplicity
“As for the simplicity of the ways of God, this is shown especially in the
means which he uses, whereas the variety, opulence, and abundance
appears in regard to the ends or results. The one ought thus to be in
equilibrium with the other, just as the funds intended for a building
should be proportional to the size and beauty one requires in it.” (DM
§V; L 306)
According to this, then, God chose to create that one world which showed a
particular end::means proportion, an optimum ratio of benefits to costs.
Everyone agrees that Leibniz in fact says these things above. And most
everyone agrees in a fairly straightforward way about the ordinary way to
interpret Leibniz’ talk about simplicity, richness, variety, order, (and maybe
even harmony) using the guide furnished by the Architectural Model. Yet,
even given this general agreement, there are serious particular
disagreements, especially about whether there is a trade-off between end and
means.1 But in what follows, I intend to concentrate upon what follows from
pursuit of this notion of a ratio or proportion, since these positive results
seem to me to far outweigh the negatives of this view.
1Cf.
(Brown 1987) especially pp. 197ff; (Blumenfield 1995) especially p. 386ff; probably
the most nuanced discussion of the issue is in (Rutherford 1995), pp. 26ff. I don’t see
how Brown and Blumenfield can get by the architect example without denying that
Leibniz didn’t mean what he obviously means through its use. That is, if they’re right,
God is unlike human architects. Rutherford seems to accept the architecht example—
he speaks of a “craftsman” (p. 14), he later seems to me to confuse the situation by
attempting to make the cost/benefit nature of the Craftsman’s Analysis go away. (p.
23-6)
4. The Law of the Series and the Cosmic Equation: perfection writ both large
and small
As we have seen, perfection, a metaphysical property involved in God’s
choice of the world, is expressed as a mathematical entity. This is not
unusal; after all, Leibniz was a mathematician. In the textual descriptions
seen above, worldwide perfection, as an object of God’s analysis, is
represented as a certain sort of a ratio. As well as worldwide perfection, it is
possible to rationally re-construct the ratio form Leibniz had in mind with
respect to the perfection of individual monads. Given that “activity and
passivity are mutual among creatures” and that monadic activity and
passivity is comprised by the distinct and confused perceptions of a monad
at an instant, instantaneous monadic perfection may be defined as:
Perfection Monad Xt1 =
Distinct Perceptionst 1
Ct 1

Kt 1
Confused Perceptionst1
An individual monad’s total perfection may be summed as
Total Perfection Monad X =
Cti
 Kt
i
i
It should be noted that perfection is definable at the other two levels of
Leibniz’ system—corporeal substances and phenomenal bodies—in a fashion
analogous to the definition for individual monads. For example,
Perfection Corporeal Substance CS Xt1 =
Total Perfection CS X = 
i
Primitive Active Forcet1
Primitive Passive Forcet1
Primitive Active Forceti
Primitive Passive Forceti
But ratios are not the only mathematical form Leibniz depended upon
to talk about how God made his choice of a world to create. Needless to say,
these other mathematical forms are more complicated than simple ratios;
yet, for all their complication, they still preserve the fundamental notion of a
proportion, a comparative index of variations among the possibles. Consider,
for a first example, the so-called Law of the Series.
Leibniz’ thoughts about the ‘law of the series’ go back at least to 1676.
He has in mind certain mathematical formulae, written in variables, which
generate a series of numbers, the order of which corresponds to the order
determined by the relationships among the variables. The variable
expression is what Leibniz calls ‘the law of the series.’ It is the relationship
between this variable expression and the series of numbers that it generates
that Leibniz uses to undergird the relationship between primitive active and
passive force (which is perfection on the level of the corpuscular substances)
and derivative active and passive force (which is perfection on the level of
phenomenal bodies). 2Clearly enough, at these two levels, what is involved is
a particular conservation law true of the entities whose laws they are.
On the fundamental metaphysical level, the level of the monads, the
situation is identical. In Leibniz’ view, each individual substance, each
monad, has a law of the series, its series, which expresses the sequence of
ongoing states that constitute the experience of the monad. (to De Volder, GP
II, 171, L. 534) Obviously, the law is about perceptual states, constituted at
each juncture by a definite proportion of clear vs. confused perception. The
individual’s law expresses its essence; it is, to use Rescher’s happy term, the
program for the monad in question. “The succeeding substance will be
considered the same as the preceeding as long as the same law of the series
or of simple continuous transition persists...This is the very fact, I say,
which constitutes the enduring substance.” (to De Volder, GP II, 171, L. 535)
The “continuous transition” referred to here can be nothing other than the
movement among successive perceptual states.
This mathematical entity—the law of the series—must be, in some
significant sense, equivalent to the logical entity comprised by a monad’s
complete individual concept. After all, the complete individual concept tells
all and only the events in a substance’s life, which, in the case of a monad,
comes to all and only its perceptual events.
Since every monad has its own law of the series, just as one would
expect, there is a corresponding ‘law of the series’ for the universe as a
whole: “For me, nothing is permanent in things except the law itself...and
which corresponds, in individual things, to that law which determines the
2This
relationship is precisely what “well-founding” comes to.
whole world.” (to De Volder, GP II, 171, L. 535) This law of “the whole world”
has been called “Leibniz’ cosmic equation”. (Grene and Ravetz 1962) In a
letter to Varignon in 1702, Leibniz remarks that if “we could express by a
formula of a higher Characteristic some essential property of the universe,
we could read from it all the successive states of every part of the universe at
all assigned times.” ( E. Cassirer 1924, II, p. 556-7, Wiener, p. 185) Although
this initial thought is couched conditionally, Leibniz in fact almost
immediately asserts it:
Therefore I think I have good reasons for believing that all the different
classes of beings whose assemblage forms the universe are, in the
ideas of God who knows distinctly their essential gradations, only like
so many ordinates of the same curve. (Ibid. 558-9, W. 186-7)
Since each of these beings—the monads—(which are the ordinates of the
cosmic formula) is itself expressed by a law of the series, it follows that the
cosmic expression is a law of the series of laws of the series. Both the cosmic
law and the laws of the monads are infinite; hence the cosmic law contains
an infinite number of infinite terms.
5. Perfection Writ Everywhere Densely: the brachistochrone
So far, we have seen that perfection is described by various mathematical
forms, most fundamentally by a ratio; and that these descriptions hold at every
level of Leibniz’ system, from the individual monad’s law of the series to the
whole world’s cosmic law. In the present section, we will see the ultimate
mathematical impetus—indeed, the ultimate paradigm—for Leibnizian
perfection. As I shall note, it is the fact that there is a mathematical property
which can be everywhere homologous in form that seems to have justified
Leibniz’ own beliefs that he had found the correct interpretation of perfection.
Let us begin with Leibniz’ claims about the the scale-invariance of his
perfect property:
The most beautiful thing about this view seems to me to be that
the principle of perfection is not limited to the general but
descends also to the particulars of things and of phenomena and
that in this respect it closely resembles the method of optimal
forms, that is to say, of forms which provide a maximum or minimum
as the case may be—a method which I have introduced into
geometry. (Tentamen Anagogicum, G VII 270-9; L. 477)
Two things are clear here. First, perfection applies equally to the general
and to the particular. That is, it is invariant at any scale. Secondly, the
paradigm of this kind of a property is that upon which depends his
method of optimal forms in mathematical analysis. Once again, then, it
is the mathematical model which is leading the metaphysical thinking. A
little further along, Leibniz adds a bit more detail to what he has just
said:
For in these forms or figures the optimum is found not only in the whole
but also in each part, and it would not even suffice in the whole without
this. For example, if in the case of the curve of shortest descent between
two given points, we choose any two points on this curve at will, the part
of the line intercepted between them is also necessarily the line of
shortest descent with regard to them. It is in this way that the smallest
parts of the universe are ruled in accordance with the order of
greatest perfection; otherwise the whole would not be so ruled. (Ibid.
L. 478-9, emphasis added)
Pefection, as a metaphysical property, is a function of the perceptions of
monadic substances, described in the form of a ratio. Perfection, as a physical
property, an optimal property, is a function of physical systems at all scales,
described by the mathematics of optimal forms. The demand that Leibniz puts
upon his notion of perfection—that it be oblivious to scale—is severe; yet, the
figure that he finds, the brachistochrone, is just that: oblivious to scale.
Moreover, it, like metaphysical perfection, exhibits the form of a sort of
optimizing proportion: what is the path of quickest/shortest descent with
respect to the gravitational force?3
From the methodological point of view, this property is to be discovered
and described by the method of optimal forms, an application of the calculus of
variations.
3Johann
Bernoulli set the original problem thusly: “If in a vertical plane two points A
and B are given, indicate for a mobile point M moving under its own weight, a trajectory
from A to B in the shortest time.” (quoted in Stöltzner 2000, p. 627)
6. The Two Kingdoms: Efficient Causes and Final Causes
Leibniz held that to exist meant to act.(G. VII, 326-7, L. 271) But the
purely material physical substances of Descartes and the other moderns were
completely passive, consisting, as they did, of extensional properties only. Hence,
since they could not act, they apparently could not exist, at least by Leibniz’
criterion. Hence, since physical substances obviously do exist, Descartes has it
wrong. To purely passive matter, an active power, an active force, an agent, must
be added. (Ibid.) Thus Leibniz makes what he later calls his “addition of
something metaphysical to physics.” He describes his move here:
Motion is a transient thing which never exists strictly speaking, seeing
that its parts are never all together. But it is force (which is the cause of
motion) that truly exists; so, in addition, apart from mass, shape and
change (which is motion), there is something else in corporeal nature:
namely, force. Consequently, we must not be surprised if nature (that is to
say, the Divine Wisdom), established its laws on that which is most real.
(Essay on Dynamics, Costabel p. 131)
It is this additional entity, this force, which justifies a role for final causes
in physics: “the distinction between force[ mv2 ] and quantity of motion[ mv ]4...is
important to show that we must have recourse to metaphysical considerations
apart from extension in order to explain the phenomena of bodies.” (Discourse on
Metaphysics, §18, L. 315) Exactly how Leibniz’ choice of force imported
metaphysics into physics is an interesting story, much too complicated to relate
in any detail at this point. (Cf. Gale 1988 for the full story) The main highlights
are these. As is well known, space is not a genuine metaphysical reality for
Leibniz; moreover, motion, as something which occurs within space, is not
genuinely real either. But force is genuinely real, for three reasons. First, while
identifying which member of a swarm of particles moves varies according to
choice of framework, force, as cause of the motion, cannot be similarly varied.
Secondly, since v2 is a scalar, whereas Descartes mv is vectorial, force is
absolute, not relative to a particular co-ordinate system, and, importantly, it
4Most
physicists agreed that some quantity relating to motion was conserved in
interactions. Decartes argued that it was the factor (speed x mass). Leibniz showed that
Descartes’ selection violated the stricture against perpetual motion. Leibniz selected
kinetic energy ( mv2/2) as his conserved quantity. Cf. Brief Demonstration, L. 296, and
Gale 1981.
thereby cannot be negative. Finally, its method of estimation is via its future
effect, that is, as Leibniz puts it,
Force must not be estimated by the product of speed and mass, but by
its future effect. But it seems that force or power is something of the
present reality, but that the future effect is not.(G III 48, Gueroult p.
47)
Leibniz got this method from Huygens. As Gueroult describes it, present force is
estimated by a future state, “that is to say, the future effect, or the actual
capacity of the body animated by the uplifting force to re-elevate itself at a later
time to the height which will exhaust this force.”5 (Gueroult 47) Thus, the
present intrinsically involves the future; it is, to coin a phrase, ‘pregnant with
the future.’
At precisely this point, final causality and teleological explanation make
their entry. Since force is always directed toward some future effect, it behaves
as if it were acting toward some end. Moreover, it acts as if it had certain goals,
namely, conserving itself, causing optimal pathways to be followed, and so on.
Leibniz’ ultimate conclusion from these considerations is that physics
must necessarily allow two distinct types of explanation: efficient causal
explanation, mechanical explanation, proceeding along the normal time line; and
final causal explanation, teleological explanation, in which the future in some
real sense influences the present. He frequently refers to these two explanatory
situations as ‘kingdoms’:
I usually say that there are, so to speak, two kingdoms even in corporeal
nature, which interpenetrate without confusing or interfering with each
other—the realm of power, according to which everything can be explained
mechanically by efficient causes when we have sufficiently penetrated into
its interior, and the realm of wisdom, according to which everything can be
explained architectonically§, so to speak, or by final causes when we
understand its ways sufficiently. (Tentamen Anagogicum, G VII 270-9; L.
479)
§Architechtonic, from Aristotle, provides that final causes or ends are to be used to explain
subordinate ends or means.
5Leibniz
typically refers to a pendulum, or something like a teeter-totter. Cf. Gale 1973.
One of the very first discussions of force, Discourse on Metaphysics, already
argues in favor of the equivalence of the two kingdoms:
A reconciliation of two methods of explanation, one of which proceeds by
final causes, the other by efficient causes; to satisfy both those who
explain nature mechanically as well as those who have recourse to
incorporeal natures. (Discourse on Metaphysics, §22, L. 317)
In his detailed work in physics, Leibniz uses telological reasoning to
analyze situations in optics, as we shall see in the next section. Further, Leibniz’
claim that efficient and telological explanations are co-extensive, is an axiom of
our contemporary physical methodology, as we shall see in §8.
7. Doing Things With Perfection : Minimaxima in Optics
The properties of the brachistochrone and the use of final causes
come together in Leibniz’ method in optics. Given the model of the
brachistochrone, whose formal properties are scale-invariant, and the
view that final cause and efficient cause explanations are everywhere coextensive, Leibniz argues that optics may be done in a straightforward
manner. He begins by noting that, although efficient causal explanations
are prized by the mechanists, “the way of final causes, however, is easier
and is often useful for understanding important and useful truths, which
one would be a long time seeking by the other more physical route.” (DM
§22, L. 317) He immediately provides an example: “I believe, too, that
Snell, who first discovered the rules of refraction, would have waited a
long time to find them if he had sought first to discover how light is
formed.” (Ibid.) Six years later, Leibniz himself provides an analysis of
light of his own. He introduces the effort plainly and simply:
The inquiry into final causes in physics is precisely the application
of the method which I think ought to be used, and those who have
sought to banish it from their philosophy have not adequately
considered its usefulness. (Tentamen Anagogicum, G VII 270-9; L.
477)
The analysis begins by remarking what we have seen before, namely,
that “the principle of perfection is not limited to the general, but
descends also to the particulars of things”. That is, “the optimum is found
not only in the whole but also in each part.” Leibniz’ analysis begins by
interpreting what perfection comes to in optics. It then continues by
noting that this new interpretation of perfection serves architechtonic
principles, that is, the system behaves in such ways as to achieve a final
goal. Put simply, light rays act teleologically.
I shall propose as a general principle of optics that a ray of light
moves from one point to another by the path which is found to be
easiest in relation to the plane surfaces which must serve as the
rule for other surfaces. (Tentamen Anagogicum, L. 478)
Other ways of describing the action of light rays include:
in the absence of a minimum it is necessary to hold to the most
determined, which can be the simplest even when it is a maximum.
(Ibid. L. 479)
Moreover, this action demands a certain methodological principle:
according to architetonic* principles, curved surfaces must be
ruled by the planes tangent to them, I shall now explain how it
remains always universally true that the ray is directed in the most
determined or unique path, even in relation to curves. (Ibid. L. 479)
Here Leibniz relativizes his notion of perfection to optics: the path
of light is the “easiest”, a meaning which he variously interprets as
“shortest”, “fastest”, “simplest”, “unique”, and “most determined.”
Obviously, these terms have strong teleological connotations. Further,
these meanings are relative to the “plane surfaces which must serve as
the rule for other surfaces.” What all this means isn’t prima facie clear.
But some diligent analytical work on our parts can make quite a bit of
sense out of it.
In the first place, “easiest” is a physically-referring term. It refers to
a certain property of a physical system, namely, a system including a ray
of light. Variations on “easiest”—”unique”, “most determined”, etc.—have
identical ontological import.
But the rule regarding plane surfaces is a methodological rule: it
refers to how the physicist is to discover and describe the actual path of
the ray of light.
Here’s how the scheme works, beginning with the least
complicated physical system. Suppose a source for a light ray, and a
sink, and a plane mirror. Now let us ask ourselves “What will be the path
of a light ray from source to sink, via a bounce off the mirror?” If Leibniz
is right that physical systems of this most metaphysically perfect world
are physically perfect as well, it will be possible to answer this question
forthwith, using his methods. Which it is.
Suppose a plane mirror P off which bounce light rays travelling
between the source and the sink.
sink
source
T
T'
M
P
In general, all light rays have a twin, e.g., T and T’, which is their
physical equivalent in terms of distance travelled, angles of bounce,
travel time, energy disipated, etc. From the point of view of method, each
equation involving the twins has two solutions. In terms of Leibniz’
variational analysis, all the paths of light provide the variations, or
possible paths, that the light may take.
But there is one path, M, which is not twinned. It is in that sense
the unique path; moreover, it is the “most determined” metaphysically,
“easiest (and fastest)” physically, and “simplest” methodologically. To
speak metaphysically, God would have needed to insert into the causal
laws of the world a term which would determine each light ray to take
one of the twin pathways open to it, rather than the other. For example,
“always take the northern (or left, or top, etc.) of the twin pathways”
would provide such a rule. Physically, the untwinned pathway is the
shortest, fastest, and most energy conservative. Methodologically, the
equation for the trajectory has only one solution, and thus is the easiest
to find. Among the huge set of variations, this is the path of the true ray.
Finally, and this will be important later, the physicist knows in
advance that this situation will occur, and can look for it from the start,
thereby optimizing discovery .
Leibniz uses this basic analysis of light behavior as the tool for
analysis of reflection from all other curved surfaces. For example
consider the case of the parabola.
P
M
F
In this case, Leibniz stipulates that the curve may be considered to be
composed of straight lines. At any point, the straight line is represented
by the same plane mirror P from our earlier example. The incoming ray
M bounces from P and intersects a point F. This behavior is true for all
incoming rays parallel to M.
This analysis works for all conic sections.
Leibniz held that this sort of perfection-based analysis, which he
called “optimal forms”, would work anywhere in physics, simply because
perfection was a scale-invariant property and, moreover, final and
efficient causality interpenetrate everywhere. Nicely enough, today’s
physicists agree with him.
______________________________
8. Doing Things With Perfection (today): Variational Analysis
We have already seen a rough-and-ready version of variational
analysis. Attempting to answer the question “which path will the light take
when it bounces off the mirror?” required us to consider the alternate
possible paths, the variations, available to the light rays. All problems in
variational analysis are set up in the same way: a physical system is
stipulated which has alternative ‘paths’ that some process in the system can
follow. Then, using calculus, the paths are compared in such wise that one
of them is singled out as being unique with respect to some property or other
of the system. This path is then selected, from among all the possible
variants, as ‘the true path.’ Typically, the unique property—as with light—is
time-of-travel; another extremely important property is energy-use. As
Leibniz noted, the property’s uniqueness consists in its extremal nature, that
is, its value is either a maximum or a minimum in the system. Extremality is
quite general, as Euler notes: “all the actions of nature follow a particular
law of maximum or minum.” (Euler 1744 p. 298)
But “in the application of variational principles to natural science, the
choice of which property F to vary is of central importance.” (Stöltzner 2000
p. 629) Euler finds that the choice is, most likely, a philosophical one: “what
this property is, this does not seem to be able to be made easily without
metaphysics.’ (Euler, Ibid.) One of the ultimate statements of the issue was
Maupertuis’ “solemn announcement” (Stöltzner, Ibid.) that he had finally
found this long-sought property, and it was nothing other than “action”,
defined as the minimum when
 vds   v dt
2
According to Maupertuis, “this is the principle of least quantity of action: a
principle so wise, so worthy of a Supreme Being, and to which nature so
constantly clings.” (Schramm 1985 p. 205) Although Maupertuis’ claim was
most certainly controversial at the time he made it, the principle of least
action is now taken to be one of the fundamental rules in physics. Once it
was generalized by Lagrange and Hamilton,6 it became an extremely
powerful tool, producing “the correct equations of motion even on occasions
when the energy of the particle or system of particles is not conserved yet
still can be represented in terms of a potential function!” (Lemons 1997 p.
96)
As Leibniz first noted, there are two highly significant features of
variational analysis: first, the method effectively captures the notion of final
causal—teleological—explanation; secondly, it can be shown that wherever
there is a relevant efficient causal explanation for a process in a physical
system, there will also be a relevant final causal explanation for the same
process. This claim has been controversial. Here is how it works out.
Mathematically, there is a structural difference between the
explanations of the path of a thrown baseball and a reflected light ray. The
path of the ray “is determined by the Principle of Least Time plus boundary
values.” (Lemons 1997 p. 12) Here, “boundary” refers to the source and sink
locations, the endpoints of the ray. Typical Newtonian mechanics problems
are different: The baseball’s trajectory is not a “boundary values problem“,
rather it is an “initial values problem”; that is, the trajectory “is determined
by Newton’s Second Law plus the baseball’s initial position and velocity.”
(Lemons, Ibid.)
Since Leibniz, an explanation relying upon {variational principle +
boundary conditions} has been “said to account for pheomena...by
postulating a final cause.” A final cause is said to operate
when the result of a natural process (the ray endpoints and a
minimum propagation time) determines the means by which that
result is achieved (the true ray path).7 (Lemons, Ibid.)
It will be noted that this definition precisely satisfies Leibniz’ notion of
architechtonic.
6Lemons
recounts the delightful story of how the nineteen-year-old Italian Lagrange
sent his stunning results to Euler, who immediately forwarded them to his official chief,
Maupertuis, then president of the Berlin Academy. Maupertuis immediately recognized
an ally in the fight to defend variational analysis against the Philistines, and caused a
Prussian mathematics chair to be offerred to Lagrange, who. out of shyness, turned it
down. Hamilton was so impressed with Lagrange that he called it “a kind of scientific
poem.” (Lemons 1997 p. 89)
7“True” here means actual, and contrasts with all the possible variant paths which were
not taken.
Our usual sense of causality in physics “is that of efficient cause in
which a state of motion (the baseball’s parabolic trajectory) is caused by the
immediate prior condition (the throwing hand and the earth’s pull).”
(Lemons, Ibid.) Lemons remarks that Leibniz was “greatly attracted to
explanation by final causes” because they were frequently easier than
efficient causes to discover, and, as Leibniz himself notes, “important and
useful truths” were frequently discerned, “which one would be a long time in
finding by the other more physical route.”(Leibniz DM) Lemons’ conclusion is
worth quoting at length:
Modern physics has largely, if not explicitly, adopted Leibniz’ point of
view. Variational principles such as Fermat’s are mathematical
expressions of final cause. Much, if not all, of modern physics can be
written in terms of variational principles. They are useful not only
because they are simply stated, have wide application, and solve
problems easily, but also because their form serves as a template out
of which new laws, such as those composing General Relativity, have
been and can be generated. (Lemons, Ibid. pp. 12-13)
Lemons here introduces our final topic, what Leibniz called the
‘interpenetration’ of final and efficient causal explanation. Two hundred and
fifty years ago, Euler’s conclusion on the use of final causality was as
sweeping as Lemons’ here, even as it went on to make the further point that
the two types of explanation are everywhere equivalent:
there is absolutely no doubt that every effect in the universe can be
explained as satisfactorily from final causes, by the aid of the method
of maxima and minima, as it can from the effective causes. (Euler
1744 pp. 76-77)
Probably the most sympathetic and accurate modern portrayal of Leibniz’
views on final causes came from German astrophysicist C.F. von Weizsäcker.
According to von Weizsäcker, “the variational principles of physics were for
Leibniz not mere analogies, but definitive consequences of the optimal
character of the real world.” Thus, “in the best possible world variational
principles must be valid, and that such principles are valid in the real world
confirms the fact that it is the best.” (von Weizsäcker 1952, p. 188) This
argument seems to me to capture a genuine Leibnizian insight, and quite
economically. Deductively, analytically, and most likely a priori, were a world
organized top to bottom (remember the brachistochrone!) in optimal fashion,
it would comprise variational principles. Moreover, from an inductive point of
view, were we to conclude that our world did not support variational
principles—assuming we discovered them at all—in physics, then that would
certainly count against our world’s being the best possible.
von Weizsäcker remarks the essential teleological nature of variational
principles in that “the path of light is determined through the end point
which the light ray is to reach only after this path has been traversed” which
very much seems “to be not a ‘mechanical’ cause, but a goal to be reached in
the future.” (von Weizsäcker p. 189) Even in the most generalized version of
the principle (Hamilton’s Principle) “the real path is still distinguished from
all possible paths in mathematically transparent fashion” while all the time
retaining “the teleological anticipation of the future.” (von Weizsäcker, Ibid.)
Of course, as von Weizsäcker—writing sixty years ago now—admits,
there are those who find such teleological reasoning repugnant in physics,
who feel that the final causal explanation somehow ‘contradicts’ the efficient
causal explanation. But “it is a decisive cognition of modern
mathmatics...that this contradiction between the causal and final
determination of events does not in fact exist.” (von Weizsäcker p. 191) Since
“the final ‘goal’ and the causal ‘law’ are only different ways of expressing the
same principles”, the “law is set up in precisely such a way that the effects
governed by it must realize the goal”, and, equivalently, “the goal only states
the consequence which must necessarily result according to the law.” And
thereby follows necessarily Leibniz’ total ‘interpenetration’ of efficient and
final causes, the kingdoms of power and wisdom.
von Weizsäcker concludes with a strong seconding of Leibniz’ justifying
the variational principle in God’s choice of a particular sort of world-wide
perfection:
And the perfection of a world in which variational principles hold,
consists in the fact that it unites the greatest richness in phenomena
by a law as simple, and as transparent for the mind as possible; it
consists in the fact that such a world possesses the greatest
intellectual beauty.
For Leibniz, it’s perfection all the way down.
§9. Conclusion
‘It is now marvelously clear how in the origin of physics a sort of a
divine mathematics is employed,’ to closely paraphrase Leibniz in
Ultimate Origin of Things. When God sought to discover which world to
create, He looked over the field of infinite variations, all the possible ways
worlds might exist, and then, willing as his final cause creation of the
most perfect, the optimal, among those worlds, he used a divine
mathematics, a godly analysis of variations, to discover us.
Because God used this divine mathematics, when physicists seek
to discover which path in a system is the true path, they look over the
field of infinite variations, all the possible paths the process might take,
and then, willing as final cause discovery of the most perfect, the
optimal, among those paths, they can use the human homologue of that
same divine mathematics to reach their goal.
Thus, physicists can use the same method God did to find the
property he created, His perfection, simply because that property is what
he found using this method. For Leibniz, it’s perfection all the way down.
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