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above the merely relative motions. Equivalently, we can say that Descartes took motion (‘properly speaking’) to be a complete predicate: that is, moves-properly-speaking is a one-place predicate. (In contrast, moves-relative-to is a two-place predicate.) And note that the predicate is complete despite the fact that it ...
bodies to their contiguous surroundings, then x moves-properly-speaking is analyzed as x moves-relative-to contiguous-surroundings(x).) This example illustrates why it is crucial to keep two questions distinct: on the one hand, is motion to be understood in terms of relations between bodies or by invoking something add...
there some ‘true’, privileged notion of motion? Descartes' views show that eschewing absolute motion is logically compatible with accepting true motion; which is of course not to say that his definitions of motion are themselves tenable. There is an interpretational tradition which holds that Descartes only took the fi...
avoid conflict with the Catholic Church. Such conflict was a real concern, since the censure of Galileo's Copernicanism took place only 11 years before publication of the Principles, and had in fact dissuaded Descartes from publishing an earlier work, The World. Indeed, in the Principles (III.28) he is at pains to expl...
because it is swept around the Sun in a giant vortex of matter — the Earth does not move relative to its surroundings in the vortex. The difficulty with the reading, aside from the imputation of cowardice to the old soldier, is that it makes nonsense of Descartes' mechanics, a theory of collisions. For instance, accord...
collision if two equal bodies strike each other at equal and opposite velocities then they will bounce off at equal and opposite velocities (Rule I). On the other hand, if the very same bodies approach each other with the very same relative speed, but at different speeds then they will move off together in the directio...
(Rule III). But if the operative meaning of motion in the Rules is the ordinary sense, then these two situations are just the same situation, differing only in the choice of reference frame, and so could not have different outcomes — bouncing apart versus moving off together. It seems inconceivable that Descartes could...
way. (Additionally, as Pooley 2002 points out, just after he claims that the Earth is at rest ‘properly speaking’, Descartes argues that the Earth is stationary in the ordinary sense, because common practice is to determine the positions of the stars relative to the Earth. Descartes simply didn't need motion properly s...
it has some other significance in his system of thought.) Thus Garber (1992, Chapter 6-8) proposes that Descartes actually took the unequivocal notion of motion properly speaking to be the correct sense of motion in mechanics. Then Rule I covers the case in which the two bodies have equal and opposite motions relative ...
covers the case in which the bodies have different motions relative to those surroundings — one is perhaps at rest in its surroundings. That is, exactly what is needed to make the rules consistent is the kind of privileged, true, sense of motion provided by Descartes' second definition. Insurmountable problems with the...
taking motion properly speaking seriously in Descartes' philosophy clearly gives a more charitable reading. In an unpublished essay — De Gravitatione (Newton, 2004) — and in a Scholium to the definitions given in his 1687 Mathematical Principles of Natural Philosophy (see Newton, 1999 for an up-to-date translation), Ne...
notion in mechanics. (see Stein 1967and Rynasiewicz 1995 for important, and differing, views on the issue.) (This critique is studied in more detail in the entry Newton's views on space, time, and motion.) The most famous argument invokes the so-called ‘Newton's bucket’ experiment. Stripped to its basic elements one co...
the bucket is set spinning about the cord's axis, with - the same bucket and water when they are rotating at the same rate about the cord's axis. As is familiar from any rotating system, there will be a tendency for the water to recede from the axis of rotation in the latter case: in (i) the surface of the
water will be flat (because of the Earth's gravitational field) while in (ii) it will be concave. The analysis of such ‘inertial effects' due to rotation was a major topic of enquiry of ‘natural philosophers' of the time, including Descartes and his followers, and they would certainly have agreed with Newton that the c...
second case demonstrated that the water was moving in a mechanically significant sense. There is thus an immediate problem for the claim that proper motion is the correct mechanical sense of motion: in (i) and (ii) proper motion is anti-correlated with the mechanically significant motion revealed by the surface of the ...
when it is in motion relative to its immediate surroundings — the inner sides of the bucket — but curved in (ii) when it is at rest relative to its immediate surroundings. Thus the mechanically relevant meaning of rotation is not that of proper motion. (You may have noticed a small lacuna in Newton's argument: in (i) t...
at rest and in (ii) in motion relative to that part of its surroundings constituted by the air above it. It's not hard to imagine small modifications to the example to fill this gap.) Newton also points out that the height that the water climbs up the inside of the bucket provides a measure of the rate of rotation of
bucket and water: the higher the water rises up the sides, the greater the tendency to recede must be, and so the faster the water must be rotating in the mechanically significant sense. But supposing, very plausibly, that the measure is unique, that any particular height indicates a particular rate of rotation. Then t...
at any moment implies a unique rate of rotation in a mechanically significant sense. And thus motion in the sense of motion relative to an arbitrary reference body, is not the mechanical sense, since that kind of rotation is not unique at all, but depends on the motion of the reference body. And so Descartes’ change of...
similar reasons, motion in the ordinary sense) is not the mechanically significant sense of motion. In our discussion of Descartes we called the sense of motion operative in the science of mechanics ‘true motion’, and the phrase is used in this way by Newton in the Scholium. Thus Newton's bucket shows that true (rotati...
not identical with, proper motion (as Descartes proposed according to the Garber reading); and Newton further argues that the rate of true (rotational) motion is unique, and so not identical with change of place, which is multiple. Newton proposed instead that true motion is motion relative to a temporally enduring, ri...
course, Descartes also defined motion as relative to an enduring 3-dimensional Euclidean space; the difference is that Descartes space was divided into parts (his space was identical with a plenum of corpuscles) in motion, not a rigid structure in which (mobile) material bodies are embedded. So according to Newton, the...
the rate at which it rotates relative to absolute space. Or put another way, Newton effectively defines the complete predicate x moves-absolutely as x moves-relative-to absolute space; both Newton and Descartes offer the competing complete predicates as analyses of x moves-truly. Newton's proposal for understanding mot...
concepts of constant motion and acceleration that appear in his laws of motion. However, it suffers from two notable interpretational problems, both of which were pressed forcefully by Leibniz (in the Leibniz-Clarke Correspondence, 1715–1716) — which is not to say that Leibniz himself offered a superior account of moti...
that turned out to be empirically inadequate, and are rejected by relativity: Newton's account violates the relativity of simultaneity and postulates a non-dynamical spacetime structure.) First, according to this account, absolute velocity is a well-defined quantity: more simply, the absolute speed of a body is the rat...
But the Galilean relativity of Newton's laws mean that the evolution of a closed system is unaffected by constant changes in velocity; Galileo's experimenter cannot determine from observations inside his cabin whether the boat is at rest in harbor or sailing smoothly. Put another way, according to Newtonian mechanics, ...
this regard absolute velocity is quite unlike acceleration (including rotation); Newtonian acceleration is understood in absolute space as the rate of change of absolute velocity, and is, according to Newtonian mechanics, in general measurable, for instance by measuring the height that the water ascends the sides of th...
the Galilean relativity of his theory is demonstrated in Corollary V of the laws of the Principia, while Corollary VI shows that acceleration is unobservable if all parts of the system accelerate in parallel at the same rate, as they do in a homogeneous gravitational field.) Leibniz argued (rather inconsistently, as we...
were unobservable, they could not be genuine differences at all; and hence that Newton's absolute space, whose existence would entail the reality of such differences, must also be a fiction. Few contemporary philosophers would immediately reject a quantity as meaningless simply because it was not experimentally determi...
and hence of absolute space. The second problem concerns the nature of absolute space. Newton quite clearly distinguished his account from Descartes' — in particular with regards to absolute space's rigidity versus Descartes' ‘hydrodynamical’ space, and the possibility of the vacuum in absolute space. Thus absolute spa...
be part of the physical, not mental, realm. In De Gravitatione, Newton rejected both the standard philosophical categories of substance and attribute as suitable characterizations. Absolute space is not a substance for it lacks causal powers and does not have a fully independent existence, and yet not an attribute sinc...
is a place where there are no bodies in which it might inhere. Newton proposes that space is what we might call a ‘pseudo-substance’, more like a substance than property, yet not quite a substance. (Note that Samuel Clarke, in his Correspondence with Leibniz, which Newton had some role in composing, advocates the prope...
Leibniz objects because of the vacuum problem, Clarke suggests that there might be non-material beings in the vacuum in which space might inhere.) In fact, Newton accepted the principle that everything that exists, exists somewhere — i.e., in absolute space. Thus he viewed absolute space as a necessary consequence of t...
— hence space's ontological dependence. Leibniz was presumably unaware of the unpublished De Gravitatione in which these particular ideas were developed, but as we shall see, his later works are characterized by a robust rejection of any notion of space as a real thing rather than an ideal, purely mental entity. This i...
adherents, but there is something deeply peculiar about a non-material but physical entity, a worry that has influenced many philosophical opponents of absolute space. After the development of relativity (which we will take up below), and its interpretation as a spacetime theory, it was realized that the notion of spac...
inertial frames, but the general idea is that although a spatial distance is well-defined between any two simultaneous points of this spacetime, only the temporal interval is well-defined between non-simultaneous points. Thus things are rather unlike Newton's absolute space, whose points persist through time and mainta...
q are points) is just the distance between p-now and q-now. However, Galilean spacetime has an ‘affine connection’ which effectively specifies for every point of every continuous curve, the rate at which the curve is changing from straightness at that point; for instance, the straight lines are picked out as those curv...
at every point. (Another way of thinking about this space is as possessing — in addition to a distance between any two simultaneous points and a temporal interval between any points — a three-place relation of colinearity, satisfied by three points just in case they lie on a straight line.) Since the trajectories of bo...
affine connection determines the rate of change from straightness at every point of every possible trajectory. The straight trajectories thus defined can be interpreted as the trajectories of bodies moving inertially, and the rate of change from straightness of any trajectory can be interpreted as the acceleration of a...
given a geometric formulation as ‘the rate of change from straightness of a body's trajectory is equal to the forces acting on the body divided by its mass’. The significance of this geometry is that while acceleration is well-defined, velocity is not — in accord with empirically determinability of acceleration but not...
helps see how such a thing is possible: betweenness but not ‘up’ is a well-defined concept in Euclidean space.) Thus Galilean spacetime gives a very nice interpretation of the choice that nature makes when it decides that the laws of mechanics should be formulated in terms of accelerations not velocities (as Aristotle ...
define the complete predicate x accelerates as trajectory(x) has-non-zero-rate-of-change-from-straightness, where trajectory maps bodies onto their trajectories in Galilean spacetime. And this predicate, defined this way, applies to the water in the bucket if and only if it is rotating, according to Newtonian mechanics...
word in this theory. But all of this formulation and definition has been given in terms of the geometry of spacetime, not relations between bodies; acceleration is ‘absolute’ in the sense that there is a preferred (true) sense of acceleration in mechanics and which is not defined in terms of the motions of bodies relat...
this sense of ‘absolute’ is broader than that of motion relative to absolute space, which we defined earlier. In the remainder of this article we will use it in the broader sense. The reader should be aware that the term is used in many ways in the literature, and such equivocation often leads to massive misunderstandi...
this analysis of motion is taken literally then one arrives at a position regarding the ontology of spacetime rather like that of Newton's regarding space: it is some kind of ‘substantial’ (or maybe pseudo-substantial) thing with the geometry of Galilean spacetime, just as absolute space possessed Euclidean geometry. T...
1974). The Galilean substantivalist usually sees himself as adopting a more sophisticated geometry than Newton but sharing his substantivalism (though there is room for debate on Newton's exact ontological views, see DiSalle, 2002). The advantage of the more sophisticated geometry is that although it allows the absolut...
not allow one to define a similar absolute speed or velocity — x accelerates can be defined as a complete predicate in terms of the geometry of Galilean spacetime but not x moves in general — and so the first of Leibniz's problem is resolved. Of course we see that the solution depends on a crucial shift from speed and
velocity to acceleration as the relevant senses of ‘motion’: from the rate of change of position to the rate of rate of change. While this proposal solves the first kind of problem posed by Leibniz, it seems just as vulnerable to the second. While it is true that it involves the rejection of absolute space as Newton co...
with it the need to explicate the nature of an enduring space, the postulation of Galilean spacetime poses the parallel question of the nature of spacetime. Again, it is a physical but non-material something, the points of which may be coincident with material bodies. What kind of thing is it? Could we do without it? A...
some contemporary philosophers believe so. There is a ‘folk-reading’ of Leibniz that one finds either explicitly or implicitly in the philosophy of physics literature which takes account of only some of his remarks on space and motion. The reading underlies vast swathes of the literature: for instance, the quantities c...
Leibniz's view of motion (as Earman acknowledges). But it is perhaps most obvious in introductory texts (e.g., Ray 1991, Huggett 2000 to mention a couple). According to this view, the only quantities of motion are relative quantities, relative velocity, acceleration and so on, and all relative motions are equal, so the...
explicit that other quantities are also ‘real’, and his mechanics implicitly — but obviously — depends on yet others. The length of this section is a measure, not so much the importance of Leibniz's actual views, but the importance of showing what the prevalent folk view leaves out regarding Leibniz's views on the meta...
That said, we shall also see that no one has yet discovered a fully satisfactory way of reconciling the numerous conflicting things that Leibniz says about motion. Some of these tensions can be put down simply to his changing his mind (see Cover and Hartz 1988 for an explication of how Leibniz's views on space develope...
on the fairly short period in the mid 1680-90s during which Leibniz developed his theory of mechanics, and was most concerned with their interpretation. We will supplement this discussion with the important remarks that he made in his Correspondence with Samuel Clarke around 30 years later (1715–1716); this discussion ...
period is one in which he turned to other matters, rather than one in which his views on space were dramatically evolving. Arguably, Leibniz's views concerning space and motion do not have a completely linear logic, starting from some logically sufficient basic premises, but instead form a collection of mutually suppor...
views — concerning the ideality of space, for instance — one is apt to be led in a circle. Still, exposition requires starting somewhere, and Leibniz's argument for the ideality of space in the Correspondence with Clarke is a good place to begin. But bear in mind the caveats made here — this argument was made later tha...
of other relevant writings, and its logical relation to Leibniz's views on motion is complex. Leibniz (LV.47 — this notation means Leibniz's Fifth letter, section 47, and so on) says that (i) a body comes to have the ‘same place’ as another once did, when it comes to stand in the same relations to bodies we ‘suppose’ t...
(more on this later). (ii) That we can define ‘a place’ to be that which any such two bodies have in common (here he claims an analogy with the Euclidean/Eudoxan definition of a rational number in terms of an identity relation between ratios). And finally that (iii) space is all such places taken together. However, he ...
are particular, incapable of being instantiated by more than one individual, even at different times; hence it is impossible for the two bodies to be in literally the same relations to the unchanged bodies. Thus the thing that we take to be the same for the two bodies — the place — is something added by our minds to th...
situation, and only ideal. As a result, space, which is after all constructed from these ideal places, is itself ideal: ‘a certain order, wherein the mind conceives the application of relations’. It's worth pausing briefly to contrast this view of space with those of Descartes and of Newton. Both Descartes and Newton c...
for Descartes it is matter, and for Newton a ‘pseudo-substance’, distinct from matter. And of course for both, these views are intimately tied up with their accounts of motion. Leibniz simply denies the mind-independent reality of space, and this too is bound up with his views concerning motion. (Note that fundamentall...
contemporaneously with his mechanics, everything is in the mind of the monads; but the point that Leibniz is making here is that even within the world that is logically constructed from the contents of the minds of monads, space is ideal.) So far (apart from that remark about ‘unchanged’ bodies) we have not seen Leibni...
of distance between bodies, which is certainly consistent with the folk view of his philosophy. However, Leibniz sought to provide a foundation for the Cartesian/mechanical philosophy in terms of the Aristotelian/scholastic metaphysics of substantial forms (here we discuss the views laid out in Sections 17-22 of the 16...
and Ariew 1989). In particular, he identifies primary matter with what he calls its ‘primitive passive force’ of resistance to changes in motion and to penetration, and the substantial form of a body with its ‘primitive active force’. It is important to realize that these forces are not mere properties of matter, but a...
further that they are not themselves quantifiable. However because of the collisions of bodies with one another, these forces ‘suffer limitation’, and ‘derivative’ passive and active forces result. (There's a real puzzle here. Collision presupposes space, but primitive forces constitute matter prior to any spatial conc...
and Rauzy, 2004.) Derivative passive force shows up in the different degrees of resistance to change of different kinds of matter (of ‘secondary matter’ in scholastic terms), and apparently is measurable. Derivative active force however, is considerably more problematic for Leibniz. On the one hand, it is fundamental t...
is possession of force. But on the other hand, Leibniz endorses the mechanical philosophy, which precisely sought to abolish Aristotelian substantial form, which is what force represents. Leibniz's goal was to reconcile the two philosophies, by providing an Aristotelian metaphysical foundation for modern mechanical sci...
deal with the inherent tensions in such a view. The texts are sufficiently ambiguous to permit dissent, but arguably Leibniz intends that one manifestation of derivative active force is what he calls vis viva — ‘living force’. Leibniz had a famous argument with the Cartesians over the correct definition of this quantit...
effectively as the magnitude of the momentum of a body. Leibniz gave a brilliant argument (repeated in a number of places, for instance Section 17 of the Discourse on Metaphysics) that it was size times speed2 — so (proportional to) kinetic energy. If the proposed identification is correct then kinetic energy quantifie...
at the other way, the quantity of virtus (another term used by Leibniz for active force) associated with a body determines its kinetic energy and hence its speed. As far as the authors know, Leibniz never explicitly says anything conclusive about the relativity of virtus, but it is certainly consistent to read him (as ...
there is a unique quantity of virtus and hence ‘true’ (as we have been using the term) speed associated with each body. At the very least, Leibniz does say that there is a real difference between possession and non-possession of vis viva (e.g., in Section 18 of the Discourse) and it is a small step from there to true, ...
speed. Indeed, for Leibniz, mere change of relative position is not ‘entirely real’ (as we saw for instance in the Correspondence) and only when it has vis viva as its immediate cause is there some reality to it. (However, just to muddy the waters, Leibniz also claims that as a matter of fact, no body ever has zero for...
on the reading proposed means no body is ever at rest, which would be surprising given all the collisions bodies undergo.) An alternative interpretation to the one suggested here might say that Leibniz intends that while there is a difference between motion/virtus and no motion/virtus, there is somehow no difference be...
important to emphasize two points about the preceding account of motion in Leibniz's philosophy. First, motion in the everyday sense — motion relative to something else — is not really real. Fundamentally motion is possession of virtus, something that is ultimately non-spatial (modulo its interpretation as primitive fo...
these lines seems necessary if we aren't simply to ignore important statements by Leibniz on motion — then Leibniz is offering an interpretation of motion that is radically different from the obvious understanding. One might even say that for Leibniz motion is not movement at all! (We will leave to one side the questio...
coherent.) The second point is that however we should understand Leibniz, the folk reading simply does not and cannot take account of his clearly and repeatedly stated view that what is real in motion is force not relative motion, for the folk reading allows Leibniz only relative motion (and of course additionally, mot...
variety of true motion, again contrary to the folk reading). However, from what has been said so far it is still possible that the folk reading is accurate when it comes to Leibniz's views on the phenomena of motion, the subject of his theory of mechanics. The case for the folk reading is in fact supported by Leibniz's...
the tension that we mentioned earlier, between the fundamental role of force/virtus (which we will now take to mean mass times speed2) and its identification with Aristotelian form. Leibniz's way out (e.g., Specimen of Dynamics) is to require that while considerations of force must somehow determine what form of the la...
not to allow one to determine the value of the force (and hence true speed). One might conclude that in this case Leibniz held that the only quantities which can be determined are those of relative position and motion, as the folk reading says. But even in this circumscribed context, it is at best questionable whether ...
Consider first Leibniz's mechanics. Since his laws are what is now (ironically) often called ‘Newtonian’ elastic collision theory, it seems that they satisfy both of his requirements. The laws include conservation of kinetic energy (which we identify with virtus), but they hold in all inertial frames, so the kinetic en...
value. But they do not permit the kinetic energy of a body to take on any values throughout a process. The laws are only Galilean relativistic, and so are not true in every frame. Furthermore, according to the laws of collision, in an inertial frame, if a body does not collide then its Leibnizian force is conserved whi...
in special cases) it does collide then its force changes. According to Leibniz's laws one cannot determine initial kinetic energies, but one certainly can tell when they change. At very least, there are quantities of motion implicit in Leibniz's mechanics — change in force and true speed — that are not merely relative;...
simply missing this obvious fact. That said, when Leibniz discusses the relativity of motion — which he calls the ‘equivalence of hypotheses’ about the states of motion of bodies — some of his statements do suggest that he was confused in this way. For another way of stating the problem for the folk reading is that the...
motions alone suffice for mechanics and that all relative motions are equal is a principle of general relativity, and could Leibniz — a mathematical genius — really have failed to notice that his laws hold only in special frames? Well, just maybe. On the one hand, when he explicitly articulates the principle of the equ...
Specimen of Dynamics) he tends to say only that one cannot assign initial velocities on the basis of the outcome of a collision, which requires only Galilean relativity. However, he confusingly also claimed (On Copernicanism and the Relativity of Motion, also in Garber and Ariew 1989) that the Tychonic and Copernican h...
Sun in an inertial frame (Copernicus), then there is no inertial frame according to which the Sun orbits the Earth (Tycho Brahe), and vice versa: these hypotheses are simply not Galilean equivalent (something else Leibniz could hardly have failed to notice). So there is some textual support for Leibniz endorsing genera...
commentators have suggested solutions to the puzzle of the conflicting pronouncements that Leibniz makes on the subject, but arguably none is completely successful in reconciling all of them (Stein 1977 argues for general relativity, while Roberts 2003 argues the opposite; see also Lodge 2003). So the folk reading simp...
howler regarding his laws, and it is arguable whether it is the best rendering of his pronouncements concerning relativity; it certainly cannot be accepted unquestioningly. However, it is not hard to understand the temptation of the folk reading. In his Correspondence with Clarke, Leibniz says that he believes space to...
of coexistences, as time is an order of successions” (LIII.4), which is naturally taken to mean that space is at base nothing but the distance and temporal relations between bodies. (Though even this passage has its subtleties, because of the ideality of space discussed above, and because in Leibniz's conception space ...
relative distances and times exhaust the spatiotemporal in this way, then shouldn't all quantities of motion be defined in terms of those relations? We have seen two ways in which this would be the wrong conclusion to draw: force seems to involve a notion of speed that is not identified with any relative speed, and (un...
is after all a principle of general relativity) the laws pick out a standard of constant motion that need not be any constant relative motion. Of course, it is hard to reconcile these quantities with the view of space and time that Leibniz proposes — what is speed in size times speed2 or constant speed if not speed rel...
some body or to absolute space? Given Leibniz's view that space is literally ideal (and indeed that even relative motion is not ‘entirely real’) perhaps the best answer is that he took force and hence motion in its real sense not to be determined by motion in a relative sense at all, but to be primitive monadic quantit...
he took x moves to be a complete predicate, but he believed that it could be fully analyzed in terms of strictly monadic predicates: x moves iff x possesses-non-zero-derivative-active-force. And this reading explains just what Leibniz took us to be supposing when we ‘supposed certain bodies to be unchanged’ in the cons...
no force, nothing causing, or making real any motion. It's again helpful to compare Leibniz with Descartes and Newton, this time regarding motion. Commentators often express frustration at Leibniz's response to Newton's arguments for absolute space: “I find nothing … in the Scholium that proves or can prove the reality...
a difference between an absolute true motion of a body and a mere relative change …” (LV.53). Not only does Leibniz apparently fail to take the argument seriously, he then goes on to concede the step in the argument that seems to require absolute space! But with our understanding of Newton and Leibniz, we can see that ...
makes perfect sense (or at least that it is not as disingenuous as it is often taken to be). Newton argues in the Scholium that true motion cannot be identified with the kinds of motion that Descartes considers; but both of these are purely relative motions, and Leibniz is in complete agreement that merely relative mot...
‘entirely real’). Leibniz's ‘concession’ merely registers his agreement with Newton against Descartes on the difference between true and relative motion; he surely understood who and what Newton was refuting, and it was a position that he had himself, in different terms, publicly argued against at length. But as we hav...
to Newton's; true motion was not, for him, a matter of motion relative to absolute space, but the possession of quantity of force, ontologically prior to any spatiotemporal quantities at all. There is indeed nothing in the Scholium explicitly directed against that view, and since it does potentially offer an alternativ...
for Leibniz to claim that there is no deductive inference from true motion to absolute space. The folk reading which belies Leibniz has it that he sought a theory of mechanics formulated in terms only of the relations between bodies. As we'll see presently, in the Nineteenth Century, Ernst Mach indeed proposed such an ...
though certain similarities between Leibniz and Mach — especially the rejection of absolute space — surely helps explain the confusion between the two. But not only is Leibniz often misunderstood, there are influential misreadings of Newton's arguments in the Scholium, influenced by the idea that he is addressing Leibn...

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