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[ Pobierz całość w formacie PDF ]Relationism and Relativity
by
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.
1
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.
2
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.
3
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
2
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.
1
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
4
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.
5
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
6
M.
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.
2
frames.
7
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, Ε
3
. 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
7
M.
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.
8
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.
9
The Leibniz-Clarke Correspondence
(Alexander Edition), Manchester (1956), 4
th
paper §41, p. 42,
GP VII, 345-440
10
Leibniz-Clarke Correspondence
, Note 8, 5
th
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
11
Leibniz-Clarke Correspondence
, Note 8, 4
th
paper §41, p. 42
12
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
.
13
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.
3
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
14
J. Earman (1989), p. 135, Note 1.
15
J. Earman (1989), p. 12, Note 1; G. Belot (1999), p. 36, Note 5.
16
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.
4
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
3
. 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
3
, 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
17
G. W. Leibniz: “Studies“ (1679), p. 251, Note 12.
5
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