Relationism and Relativity

Relationism and Relativity

Friedel Weinert (University of Bradford, UK)

I. Introduction

Leibniz’s relational view states that space is the order of coexisting things and

time is the order of successive events.

Leibniz makes time and space relative to

material events in the universe. Without material happenings time and space can only
be ideal. In a physical sense, time and space become relational properties: for time
and space to exist the universe must be filled with matter and changing material
events. Time and space become relations between spatio-temporal locations.

Note

that Leibniz does not require the presence of observers in the universe. Time and
space are constituted by the existence of material events. Human observers construct
the notions of time and space from the observations of material changes in the
universe. Several notions of time are embedded in Leibniz’s theory. What is known as
his relational view is based on an earlier-later relation between material events in the
universe, for which no human observer is required. There will be a succession of
events, without the presence of human observers. This succession constitutes what
may be called empirical time.

But, as we shall see, the question arises of how events

succeed each other. If events succeed each other in a regular, measurable fashion,
rather than in a random, chaotic fashion, then empirical time becomes physical or
clock time, as it is used in physics. Finally, in introducing human observers, Leibniz
also refers to a richer human notion of time. Such a human notion of time is a mixture
of both physical and conventional aspects of time. Humans construct calendars and
other temporal metrics from the observation of the regular succession of physical
events. A tensed view of time, with its predicates of past, present and future, is
introduced. In the writer's assessment this is the real impact of recent attempts to
show that time and space are 'ideal' for Leibniz.

In this paper we will mostly be concerned with the empirical and physical

notions of time. Compared to Newton, the relational view accepts that both time and
space are universal (setting aside any relativistic correction of this feature of time) but
not absolute.

This article was published in the conference Reader: VIII. International Leibniz Congress, Einheit in

der Vielfalt, eds. H. Breger/J. Herbst/S. Erdner (Hannover 2006), 1138-46; please quote this reference;
by permission of the publisher.

Stating Leibniz’s view in phenomenological terms does not take into account his profound

metaphysics of monads. This restriction seems justified, since relations between phenomena are at least
derivative of relations amongst monads. Richard Arthur has made a convincing case for monads to
entertain relations and for Leibniz’s principle of interrelatedness to cover both phenomena and monads.
See R. Arthur: “Leibniz's Theory of Time”, in: K. Okruhlik/J.R.Brown (eds): The Natural Philosophy
of Leibniz, Dordrecht 1985, pp. 263-313. Most discussions of the ontology of space-time theories only
consider the phenomenological aspect of Leibniz’s argument. See J. Earman: World Enough and
Space-Time, Cambridge (Mass.) 1989, Ch. I.

In terms of McTaggart’s distinction beween the A-theory and the B-theory of time, this means that

Leibniz’s theory of time is more compatible with a tenseless earlier-later view of time than a tensed
view of time (past-present-future). It is important to realize, as several commentators have already
established, that tenselessness (earlier-later) is not to be equated with changelessness or a block
universe. See J. Smart: Philosophy and Scientific Realism, London 1963, pp. 131-48; A. Grünbaum:
Philosophical Problems of Space and Time, Dordrecht

1973, pp. 314-29; M. J. Futch: “Leibniz’s

non-tensed theory of time”, in: International Studies in the Philosophy of Science 16 (2002), pp. 125-
39.

Newton had posited time and space as physical properties of the universe in a dual

sense:
Time and space were universal properties of the cosmos in the sense that they were

not dependent on specific reference frames or measurement acts. Every measuring
instrument, irrespective of its space-time location or state of motion (rest or uniform
velocity), would measure the same temporal and spatial extensions between events.

Time and space were absolute properties of Newton’s universe in the sense that they

were not dependent on material events happening in the universe. Although the
modern literature on space-time theories has developed several senses of
absoluteness, the most apposite characterization for present purposes is that Newton
postulated the existence of a substratum of space-time points, which need not be
occupied by material bodies.

Leibniz did not object to Newton’s characterization of

time and space as universal properties of the universe. In fact, a questioning of this
aspect of time had to await the advent of the Special theory of relativity (STR). If
we pose observers into the material universe, Leibniz would agree with Newton
that, given an event E, all observers throughout the universe would measure the
same temporal length for this event. Equally for spatial considerations: a physical
object O, placed anywhere in the universe will be assigned the same spatial
extensions by all observers, from whichever space-time location they observe it.

It was the absoluteness of time and space, postulated in Newton’s mechanics, to

which Leibniz objected. Leibniz employed his Principle of Sufficient Reason to
criticize a notion of time and space, which was not tied to any material processes in
the universe.

In recent times, both the relational and the substantival view have been

reconstructued as space-time theories. But Leibnizian relationism has been regarded
as deficient on two accounts:

Relationism about Ontology. It is often suggested in the literature that Leibniz

makes the relational view too dependent on the existence of material processes or
entities in the universe. John Earman characterizes Leibnizian relationism as the
view that “spatiotemporal relations among bodies and events are direct.” That is,
there is no underlying substratum of space-time points, which physical events would
merely occupy.

Michael Friedman holds that Leibnizian relationism “wishes to

limit the domain over which quantifiers of our theories range to the set of physical
events, that is, the set of space-time points that are actually occupied by material
objects and processes.”

In Friedman’s formulation, relationism constructs spatio-

temporal relations between bodies as embeddable in a fictional space-time. This
fictional space-time acts as a representation of the properties of concrete physical
objects and the relations between them.

Relationism about Motion. A major drawback of relationism, according to

Friedman, is that there are no inertial trajectories to be found amongst the material
bodies in the universe. But the relationist cannot postulate unoccupied inertial

J. Earman/M. Friedman: “The Meaning and Status of Newton's Law of Inertia and the Nature of

Gravitational Forces”, Philosophy of Science 40 (1973), pp. 329-59; J. Earman (1989), pp.11-2, Note 1;
M. Friedman: Foundations of Space-Time Theories. Princeton (N.J.) 1983.

J. Earman (1989), pp. 12, 114, Note 1; G. Belot: “Geometry and Motion”, in: British Journal for the

Philosophy of Science 51 (2000), 561-85

Friedman (1983), p. 217, Note 3; G. Belot: “Rehabilitating relationism”, in: International Studies

in the Philosophy of Science 13 (1999), pp. 35-52

for an overview of similar formulations.

frames.

The lack of inertial trajectories in the material world and the prohibition of

unoccupied inertial frames deprive the relationist of the possibility of defining
inertial frames of references. The general consensus is that Leibnizian space-time
amounts to no more than a topology of time and therefore fails to support a proper
theory of motion.

The purpose of this paper is to assess these claims by focusing on Leibniz’s

discussion of the notions of ‘order’ and the ‘geometry of situations’. In the final part
the ‘geometry of situations’ will offer a transition to a space-time relationism.

II. The Geometry of Situations

In an important sense, Leibniz makes time derivative of spatial relations.

For

Leibniz defines time as the order of succession of simultaneous events. Events, in a
primary sense, are changes that happen to material bodies. But it is not the particular
situation of bodies that constitutes space. Rather, it is the geometric order, in which
bodies are placed that constitutes space. Time is “that order with respect to (the)
successive position” of bodies

. In many of his formulations Leibniz insists on the

term ‘order’. Space is not identical with bodies. Space is “nothing else but an order of
the existence of things.”

Leibniz agrees with Clarke that “space does not depend

upon the situation of particular bodies”; rather it is the order, which renders bodies
capable of being situated, and time is that order with respect to the successive position
of things.

Leibniz even contemplates the possibility of unoccupied spatial locations

when he says that space is nothing but “the possibility of placing” bodies.

An important aspect in a consideration of relationism about ontology is

Leibniz’s method of the geometry of situations.

In these writings Leibniz criticizes

the Cartesian focus on extension alone, i.e. algebra, which is concerned with
magnitudes. Leibniz endeavours to introduce a geometrical analysis – a consideration
of situations – which give rise to an analysis of congruences, equalities, similarities
and loci of geometrical shapes. The reflections are important because they take the
key notion of order beyond the analogy of the genealogical tree. (There is a
genealogical relation between family members but the genealogical tree does not exist
over and above the family members and their relations.) Leibniz’s geometry of
situations can without difficulty be described as a set of constant 3 dimensional
Euclidean space-time slices (since no gravitational effects are considered). Objects
existing at the same time exist on a simultaneity plane,

. Any object, A, existing

simultaneously with an object, B, exist on the same simultaneity plane perpendicular
to a time axis, on which for present purposes, no values need to be inscribed. Such a

Friedman (1983), Ch. VI, Note 3; T. Maudlin: “Buckets of Water and Waves of Space: Why

Spacetime is probably a Substance”, in: Philosophy of Science 60 (1993), pp. 183-203. This argument
was already used by Newton.

R. Arthur: “Space and Relativity in Newton and Leibniz”, in: British Journal for the Philosophy of

Science 45 (1994), pp. 219-40; R. Arthur (1985), Note 1.

The Leibniz-Clarke Correspondence (Alexander Edition), Manchester (1956), 4

paper §41, p. 42,

GP VII, 345-440

Leibniz-Clarke Correspondence, Note 8, 5

paper §29, p. 63; G.W. Leibniz, “An Example of

Demonstrations about the Nature of Corporeal Things” (1671), transl. L. Loemker (ed.): Leibniz:
Philosophical Papers and Letters, Dordrecht 1970, p. 144

Leibniz-Clarke Correspondence, Note 8, 4

paper §41, p. 42

Leibniz-Clarke Correspondence, Note 8, 3rd paper §4, p. 26; G. W. Leibniz, “On Body and Force”

(1702), GP IV, 393-400, GM VI, 98-106, transl. R. Ariew/D. Garber

(eds.): G. W. Leibniz:

Philosophical Essays, Indianapolis & Cambridge 1989, p. 251

G. W. Leibniz: “Studies in a Geometry of Situations” (1679), GM II, pp. 17-20, transl. Loemker

(1970), pp. 248-58, Note 9; cf. R. Arthur (1994), §V, Note7.

simultaneity plane will be called Now. For the sake of convenience we can use an
analogy: the 3 dimensional simultaneity slice is like a billiard ball table on which a
number of billiard balls rest. These bodies entertain geometric relations. Body A is at
a distance ‘x’ from body B. A may be at rest and B rotate around it. If we place a third
object, C, on the simultaneity plane, for instance, in the path of B, B and C will
collide. The collision will be governed by conservation principles. Objects therefore
entertain physical relations. It is not important whether this distance is expressed in
numerical figures. One object could be ‘some portion of its own size’ away from its
sister object. Three-dimensional macro-objects can coexist on a simultaneity plane
and entertain geometric and physical relations.

What does it mean, then, to say that Leibnizian relationism only admits space-

time points, if they are constituted by the presence of material objects and processes?
Recall that Leibniz calls space, in terms of possibility, an order of coexisting things.
This order must be, as the ‘geometry of situations’ shows, an order of physical and
geometric relations. These are lawlike relations so that the order itself must be
lawlike. So when Leibniz calls space, in terms of possibilities, an order of coexisting
things, it is the existence of material objects in the universe and the intrinsic physico-
geometric relations between them, which denote space in terms of possibilities.
Without the existence of any material things, there would be no space, no physico-
geometric relations – space would be ‘ideal’. The existence of things 'creates' absolute
simultaneity planes. The existence of things ‘creates’ a space of possibilities.
Possibilia may be construed as bodies standing in “Euclidean relations to one another
in many different configurations” or as the structure of the set of spatial relations.

What needs to be added is that lawlike physical relations also obtain between bodies.
The existence of things constrains the space of possibilities but does not exhaust it.
According to Leibnizian relationism, there is no underlying substratum of space-time
points. In this sense “spatio-temporal relations among bodies and events are direct.”

But in another sense, this formulation is prone to misleading characterizations of
relationism as the view, which wishes to limit the set of space-time points to those
occupied by material processes or events. We have just seen that this characterization
is incorrect, by the standards of the ‘geometry of situations’. The geometry of
situation gives room to actual and possible relations between bodies. These bodies can
be represented in idealized geometric shapes. As we shall see it gives rise to an
inertial structure.

A number of authors have suggested that the Leibnizian view of the

ontological status of space-time satisfies a modern supervenience relation.

The

simultaneity planes are supervenient on the physico-geometric relations of coexisting
bodies. Supervenience requires a) a co-variation of the properties of one domain, the
physical base (as constituted here by bodies and their physico-geometric relations),
with a supervenient domain and b) the dependence of the supervenient domain (the
simultaneity slices constituted by the geometry of situations and their endurance in
time) on the base domain. The base constrains the supervenient domain.

In this sense, a relationist can claim that space-time is ontologically a

supervenient phenomenon, without having to admit that it is purely fictional. But it

J. Earman (1989), p. 135, Note 1.

J. Earman (1989), p. 12, Note 1; G. Belot (1999), p. 36, Note 5.

J. T. Roberts: “Leibniz on Force and Absolute Motion”, in: Philosophy of Science 70 (2003), p. 571;

P. Teller: “Substance, Relations and Arguments About the Nature of Space-Time”, in: The
Philosophical Review, Vol. C (1991), p. 396; J. Earman (1989), p. 135, Note 1.

would be wrong to say that every variation in the physical base will lead to variations
in space-time structure. A change in geometric relations between bodies does not
change the structure of E

. We therefore require invariance conditions in two respects.

The first respect (a) refers to geometric symmetries. The physico-geometric relations
of objects are invariant under space translation, rotation and reflection on the
simultaneity planes. Leibniz’s geometry of situations reflects this invariance
condition. For instance, two triangles can be congruent “with respect to the order of
their points, (…) they can occupy exactly the same place, and (...) one can be applied
or placed on the other without changing anything in the two figures except their
place.”

The second aspect (b) refers to time translation symmetries, i.e. relationism

about motion. The relations between simultaneity planes should be Galileo-invariant
in the sense that two such planes can be joined by inertial trajectories. While (b) is
uncontroversial for Galilean space-time, it has often been regarded as the sticking
point for Leibnizian space-time. In the following sections we will argue that
Leibnizian space-time is not geometrically weaker than Galilean space-time.

Relationism about ontology has advantages over substantival space-time. The

material things in the universe have no effect on the nature of time and space, in
Newton’s view. Not so on the relational view: the space of possibilities is constrained
by the prior existence of material things and events and their (physico-geometric)
relations. It remains a question of empirical study to determine in which way the
matter in the universe constrains the relations. Relationism, on the level of the
simultaneity planes, differs from Galilean space-time, not in its mathematical
structure, E

, but its ontological import. It differs ontologically, not geometrically. But

can relationism secure enough inertial structure to present a viable view of motion?

Figure I: Geometry of Situations. Time as the succession of spatial order, according to Leibniz

III. The Order of Succession

Leibniz characterizes time as the order of the succession of events. Events are

made dependent on the coexistence of things. It is the physico-geometric order, in

G. W. Leibniz: “Studies“ (1679), p. 251, Note 12.

which bodies are placed that constitutes space. Now time is ‘that order with respect to
(the) successive position’ of bodies. Time is therefore, in a certain sense, derivative of
space. This amounts to relationism about motion. Not only is the motion of bodies to
be determined in relation to other bodies, Leibniz also adopts the Galilean relativity
principle whereby it was not possible to define sameness of place at different times.
Remaining in the pre-relativistic era, consider an arrangement of bodies coexisting
with each other at a particular moment in time. We have already seen that this
collection of bodies constitutes a lawlike space of possibilities or a plane of
simultaneity. Now such an arrangement of bodies may endure without change to its
geometry of situations (Figure I).

According to Leibniz the transition from Stage I to Stage II constitutes time,

for time is the spatial order of things with respect to their successive positions. The
enduring structure of these bodies in their respective positions, and of their physico-
geometric relations, will have to be considered as the succession of events. The
objection against relationism is that the actual bodies in the universe may not be
regular enough to constitute a metric of time. Leibniz himself seems to admit this
weakness. He states that “motion belongs to the class of relative phenomena” and that
“no body ever preserves exactly the same distance from another for any length of
time.”

Assume they do: how do we determine the motion of these bodies if there is

no inertial motion? Given the relativity of all motion, which Leibniz accepts, it seems
that there is no appropriate kinematic connection between their successive planes of
simultaneity. This is the problem of the affine connection: many commentators have
claimed that Leibnizian space-time lacks an inertial structure.

The general

consensus is that Leibnizian space-time is much weaker than Galilean space-time. So
even if the geometry of situations allows actual and possible relations, the lack of any
inertial structure seems to vitiate any serious attempt to construct a theory of motion
on Leibnizian grounds. But we should consider more clearly what the geometry of
situations implies for Leibnizian space-time.

IV. Leibnizian Space-Time

Leibnizian space-time is often regarded as weaker than Galilean space-time. In

Galilean space-time absolute space has disappeared. The relativity principle states that
the laws of motion are the same whether they are expressed in terms of a system at
rest or in constant motion. Absolute rest and absolute velocity cannot be detected.
With absolute space vanished, there is no longer an absolute spatial measure between
the layers of Now. Inertial lines (for systems at rest or in motion) connect the
simultaneity planes. These inertial connections are privileged in Galilean space-time
since they mark world lines of unaccelerated particles. There is one time axis, against
which the motion of inertial systems can be measured. Simultaneity is absolute:
events happen at the same time for all observers. Space-time is now made up of
simultaneity planes, glued together by inertial lines. There is no frame-independent
way of defining sameness of place at different times; but it is possible to refer
different spatial locations to the same time axis.

Leibnizian space-time is characterized as a position, which besides temporal

relations only admits spatial relations between simultaneous events.

It is said that

according to Leibnizian space-time the universe still carves uniquely into slices

G. W. Leibniz: “Specimen Dynamicum (1695), GM VI, pp. 234-54; transl. Loemker (1970), pp. 446,

449, italics in original, Note 9.

J. Earman (1989), p. 92, Note 1; Arthur (1985), Note 1.

T. Maudlin (1993), p. 267, Note 6.

carrying Euclidean 3-metrics, without a precise temporal metric between the slices.
Nothing privileged ‘stitches’ the slices together any longer.

In the stronger Galilean

space-time, the privileged stitching is guaranteed by the inertial lines, provided by
idealized trajectories taken by idealized objects at rest or in constant motion. In
Leibnizian space-time, because of its preoccupation with material events and
occupied space-time locations, no succession of events presumably stands out, which
could serve as ‘privileged’ stitching. Although there is a topology of (occupied)
sequential space-time points there is no temporal metric. There is simply no invariant
notion of straight-line motion, since all motion is relative motion

True, time is a

succession of events, so there is some temporal filament, which connects the
hyperplanes. But this is insufficient to establish any privileged inertial frames and a
theory of motion. As Clarke objected to Leibniz in his fourth reply: events may
succeed each other at different rates. How can we measure this rate if there is no
available yardstick? It is commonly agreed that the fatal flaw in Leibniz’s theory is its
“inability to sustain a definition of (…) affine connection.”

This interpretation of Leibnizian space-time ignores Leibniz’s insistence on

the term ‘order’ in his characterization of time and the geometry of situations. The use
of the term ‘order’ – time as the order of successive events – makes Leibnizian space-
time similar to Galilean space-time.

This is a consequence of Leibniz’s method of the geometry of situations.

Leibniz establishes relations of congruence between Euclidean bodies on a Euclidean
plane. In establishing these relations he considers what today we would call
transformations rules (rotation, translation). Equally important for a consideration of
inertial connections in Leibnizian space-time is his acceptance of ‘natural inertia’: “a
body retains an impetus and remains constant in its speed or that is has a tendency to
preserve in the series of changes which it has once begun.”

The geometry of

situation contains an inertial structure, i.e. systems of actual and possible lawlike

M. Wilson: “There’s a Hole and a Bucket, Dear Leibniz”, in P. A. French et al. (eds.): Midwest

Studies in Philosophy XVIII, Notre Dame 1993, pp. 202-41.

J. Earman (1989), p. 72, Note 1.

R. Arthur (1985), p. 307, Note 1.

A number of suggestions have been made in the literature, which suggests ways, in which Leibnizian

space-time may sustain affine connections.
R. Arthur (1994, p. 230, Note 7) has suggested that the affine connection may have ‘its origin in the
laws governing relations of bodies.’ In his Fifth Paper to Clarke, Leibniz does indeed suggest that out
of constantly moving bodies, we can construct or feign that some bodies remain in a fixed position.
These ‘fixed existents’ may be used to construct coordinate systems with respect to which the laws of
motion and inertial trajectories can be formulated. He characterizes ‘fixed existents’ as those ‘in which
there has been no cause of any change of the order of their coexistence with others.’ But we have to
feign that among those co-existents ‘there is a sufficient number of them, which have undergone no
change (Leibniz-Clarke Correspondence, Note 8, 5

paper §47, PP. 69-70). This proposal shows in

terms of our earlier distinction that inertial frames can be constructed for human time. The human mind
can abstract order from the flux of phenomena. But it does not show whether inertial frames exist for
empirical time. So this suggestion does not help the relationist, for the substantivalist can simply ask
whether inertial frames exist in a world deprived of human observers, as they do for Newton.
A second proposal is to refer to Leibniz’s views on motive force (Roberts 2003, Note 15). Leibniz
regarded motive force as real, inherent in bodies, and absolute. Through his insistence on motive force
Leibniz tries to connect his views on physics with his metaphysical principles. This connection is not
very helpful in a reconstruction of Leibnizian space-time since the problem of absolute motion can only
solved metaphysically. See H. Reichenbach: “Die Bewegungslehre bei Newton, Leibniz und
Huyghens”, in: Kantstudien 29 (1924), 434.

G. W. Leibniz: “On Nature Itself” (1698), GP IV, 504-16, transl. Loemker (1970), pp. 503, 506,

Note 9; cf. G. W. Leibniz, “Specimen Dynamicum” (1695), Note 17, transl. Loemker (1970), pp. 437,
449; cf. Teller (1991), Note 15.

trajectories. This includes a measure of temporal invariance between the simultaneity
planes. Leibniz explicitly states that all motion is in straight lines and that time is
measured by uniform motion in the same line.

Although Leibniz justifies the

inertia of motion through metaphysical arguments, the geometry of situation already
gives rise to the ideal of inertial motion.

At some stage of the argument Newton was in a similar position to Leibniz.

Relationists and substantivalists have to start empirically from relational motions.
Newton justified the need for absolute space not only with respect to the inertial
effects of rotation (bucket experiment) but by pointing out that all motion in the
universe may be relative and non-constant. Nevertheless in his Principia Newton
granted that there are approximations in the physical universe to his notion of absolute
time, i.e. the motion of Jupiter’s moons and the existence of fixed stars. Furthermore
from the observed motion we can abstract ‘idealized’ motions.

While Newton

infers from this situation the need for absolute space and time, without reference to
physical bodies, Leibniz rejects absolute space and time. To secure the idealization
the Leibnizian relationist can invoke the geometry of situations, so can the modern
relationist.

V. Space-time Relationism

There is an important link which connects the classical with the modern

relationist: it is the notion of the geometry of situations or the geometric order of
events. As we have seen Leibniz rejected the idea of absolute space and time but
retained the idea of universal time. The STR has destroyed the notion of universal
time: there are as many times as there are moving reference frames. The relational
view of time needs to be adapted to the constraints, which the STR has imposed. Let
us dub the alliance between relativity and relationism space-time relationism.
Consider first, relationism about ontology. The geometry of situation on the
simultaneity planes is quasi-Euclidean; it still comprises actual and possible relations.
The observers become aware that there is no longer any notion of absolute
simultaneity. Two events, which are simultaneous for one observer, confined to one
reference frame, may not be simultaneous for observers in a reference frame moving
with constant velocity relative to the first. This is due to the finite and invariant speed
of light. As a consequence, there are no instantaneous signals between events on a
simultaneity plane and no perfectly rigid bodies. Nor do two moving observers agree
on the time, which their respective clocks indicate. This has consequences for
relationism about motion. The proponent of space-time relationism will have to rely
on a relational view of time as a succession in events in Minkowski space-time.
Minkowski has geometrized space and time. That is, Minkowski formulated the STR
in terms of a geometric space-time structure. This space-time structure no longer
separates events neatly into spatial and temporal components. It is a four-dimensional
structure, which merges space and time into space-time. According to Minkowski, the
space-time world consists of a collection of space-time events. Every such event can
be described by a system of temporal and spatial axes (a reference frame). These are
no longer invariant due to the behaviour of clocks and rods in moving systems. The
trajectory of a reference system through space-time is a world line. The world line
describes the motion of the reference frame, according to its clocks and rods. The
spatial and temporal measurements of observers in these reference systems have no

G. W. Leibniz : “Studies in Physics” (1671), GP IV, 228-32; transl. Loemker (1970), 140-1, Note 9;

“Specimen Dynamicum” (1695), Note 17, transl. Loemker (1970), p.449, Note 9.

J. Earman (1989), Chap. 9.10, Note 1.

universal validity. For Leibniz the geometry of situations on a simultaneity plane was
based on physico-geometric Euclidean relations between bodies, connected by a
universal time axis.

In Minkowski space-time there is no universal time axis, no

absolute simultaneity. Inertial motion corresponds to straight, unaccelerated world
lines. A vertical line represents a particle at rest. A straight, inclined slope represents a
particle in uniform motion. The inclination of a world line away from the vertical
world line indicates the speed of the particle with respect to a stationary particle. Each
inertial frame has its own temporal metric. As sameness of place and sameness of
time can no longer be defined, two inertial frames are related by the Lorentz
transformations. There exists, however, an invariant relation between the space-time
events, which is the space-time interval, I. It is a measure of the distance between
space-time events and is the same for all observers in space-time. It is similar to the
invariance of spatial distance between two points according to Pythagoras’s theorem
but it contains a temporal element. Furthermore, space-time events and world lines are
confined within isotropic light cones, which emanate from each event. The invariant
speed of light, c, confines material happenings within the light cones, which define
the limits of causal connectibility.

For Leibniz the geometry of situations is defined by the geometric order of

idealized shapes on a Euclidean plane, which undergo active physical transformations
(rotation, translation). For the space-time relationist the geometry of situations must
be constrained by the 4-d structure of Minkowski space-time. The space-relationist
operates with space-time events, world lines, light signals and the invariant space-
time interval, I. But these devices are as much models of physical systems, as
Leibniz’s geometric shapes were of real bodies. Leibniz credits the geometry of
situations with the possibility of discovering the ‘structure of plants and animals’.
Leibniz sees in this ‘geometrical’ analysis a ‘method of abridgment’ to reveal the
structure of complex things in nature. Minkowski used a similar approach to describe
the structure of relativistic space-time. The geometry of situations is a powerful
method, more powerful than Leibniz could have anticipated.

Leibniz’s correspondence shows that he was aware of the importance of Roemer’s discovery of the

finite speed of light, although this does not suggest that he accepted the invariance of the speed of light.
On the geometry of situations for relativistic space-time, see D. E. Liebscher: The Geometry of Time,
Wienhelm 2005

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