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Does time flow or
lapse or pass? Are the future or the past as real as the present? These
metaphysical questions have been debated for more than two millennia, with no
resolution in sight. Modern physics provides us, however, with tools that
enable us to sharpen these old questions and generate new arguments. Does the
special theory of relativity, for example, show that there is no passage or
that the future is as real as the present? The focus of this entry will be these
new questions and arguments.
Around 500 B. C.
Heraclitus wrote the following:
Everything flows
and nothing abides; everything gives way and nothing stays fixed.
You cannot step twice into the
same river, for other waters and yet others, go flowing on.
Time is a child, moving counters
in a game; the royal power is a child's.[1]
Transience is
basic, and the present is primary. Those things which exist now do not abide.
They slip into the past and non-existence, devoured by time, as all experience
attests.
A generation or so later we have
a classic statement of the opposing view by Parmenides:
There remains, then, but one word by which to express the [true] road: Is. And on this road there are many signs that What Is has no beginning and never will be destroyed: it is whole, still, and without end. It neither was nor will be, it simply is"now, altogether, one, continuous…
Permanence is
basic. No things come to be or, slipping into the past, cease to be. Past,
present, and future are distinctions not marked in the static Is. Time and
becoming are at best secondary, at worst illusory, as our understanding of the
world confirms.
Turn now to modern times and to a
paragraph in Rudolf Carnap's intellectual autobiography (Carnap 1963, pp.
37-38):
Once Einstein said that the problem of the Now worried him seriously. He explained that the experience of the Now means something special for man, something essentially different from the past and the future, but that this important difference does not and cannot occur within physics. That this experience cannot be grasped by science seemed to him a matter of painful but inevitable resignation. I remarked that all that occurs objectively can be described in science; on the one hand the temporal sequence of events is described in physics; and, on the other hand, the peculiarities of man's experiences with respect to time, including his different attitude towards past, present, and future, can be described and (in principle) explained in psychology. But Einstein thought that these scientific descriptions cannot possibly satisfy our human needs; that there is something essential about the Now which is just outside the realm of science. We both agreed that this was not a question of a defect for which science could be blamed, as Bergson thought. I did not wish to press the point, because I wanted primarily to understand his personal attitude to the problem rather than to clarify the theoretical situation. But I definitely had the impression that Einstein's thinking on this point involved a lack of distinction between experience and knowledge. Since science in principle can say all that can be said, there is no unanswerable question left. But though there is no theoretical question left, there is still the common human emotional experience, which is sometimes disturbing for special psychological reasons.
This difference
as expressed here between Einstein and Carnap (that is, between the Heraclitean
and Parmenidean attitude towards time and change) is the subject of this
article, which will use modern physics " especially modern spacetime theory "
as a set of lenses through which it is hoped the riddles of time will come into
sharper focus. There are many ways, however, to approach these questions. Early
in the twentieth century, Anglo-American philosophy turned to consideration of
language as way to clarify philosophical disputes. Philosophers of time debated
the relative primacy of tensed language (concerning the notions of present,
past, and future) or tenseless language (concerning the relations of
simultaneity and temporal precedence). Our considerations of physics will
generally, though not completely, skirt linguistic disputes. The reader
interested in following these debates can find a sophisticated review and discussion
in Tooley (1999).
Other philosophers have been
influenced by analogies between time and modality. The reader interested in
this way of thinking about time should consult the article Temporal Logic.
The present article will focus on time in physics and the relations between
time and space. Other philosophical approaches focus on the primacy of
experience in our understanding of time. The reader interested in these
approaches may wish to consult The Experience and
Perception of Time.
Modern physical
theories are often formulated in a language that permits one to express a
variety of different views with respect to time and its relation to space. One
can, for example, formulate the basic ideas of classical (that is, Newtonian)
physics, the special theory of relativity, and the general theory of relativity
in this language. For a brief introduction to the spacetime view, see the
section on Modern
Spacetime Theories: A Beginner's Guide the entry on The Hole Argument
in this Encyclopedia. For more detail with minimal technical demands the reader
should see the first four chapters of Geroch (1978) or (more demanding) chapter
2 of Friedman (1983).
For our purposes, the defining
feature of a manifold that is a Newtonian spacetime is that the temporal
interval between any two points or events in the spacetime, p and q,
is a well-defined quantity. This quantity is well-defined in that it does not
depend upon point of view, reference frame, coordinate system or “observer”.
This quantity, then, is absolute in the sense of being frame- or
observer-independent. (In the special theory of relativity the temporal
interval between two distinct spacetime points fails to be absolute in this
sense.)
If the temporal interval between
two events is 0, then we say that the two events are simultaneous.
This relation of (absolute) simultaneity is an equivalence relation (That is,
it is reflexive, symmetric, and transitive.) that slices (partitions or
foliates) the spacetime or manifold into mutually exclusive and exhaustive
planes of simultaneity. These planes of simultaneity can then be completely
ordered by the relation ‘is earlier than(tm) or its converse ‘is later than(tm).
The geometrical structure of
Newtonian spacetime reflects the way we ordinarily think about time and is the
proper backdrop for introducing the three major rival metaphysical views of
time, as illustrated below:
Figure 1. Three Metaphysics of Time
The first view, represented on
the left, is the ontologically austere view called presentism, the
view that only the present exists. The past has been but is no longer, while
the future will come to be but is not yet. Note that it is the convention of
these diagrams that one spatial dimension is suppressed. The present is
actually a three dimensional global slice of the spacetime. Moreover, the
illustration necessarily represents the spatial extent of the present as finite
and may suggest that time also has a beginning and/or end. These views are,
however, merely artifacts of the representation and not integral to presentism,
possibilism, or eternalism. The diagram illustrating presentism also has four
arrows pointing up (conventionally, towards the future) attached to the plane
representing the present. These arrows are meant to indicate something that is
integral to presentism, the idea that the present (and hence the existent)
constantly shifts or changes. These arrows represent, then, the dynamic aspect
of time called temporal becoming or passage. The deepest
problem in the metaphysics of time is how to understand passage or becoming and
its relation to existence.
In contrast to the radical
Heraclitean view of presentism, the Parmenidean eternalist picture on the far
right lacks these arrows and indicates that there is no more special about the temporal
present (the now) than the spatial present (the here). Future
and past events at a place, on this view, are no more or less real than distant
events at a time. The now like the here is a function of
one's perspective, one's position in the spacetime, and these positions are
indicated by the line in the spacetime representing the history of spacetime
locations of a particular object or person. Such a line is often called a world
line.
The middle view, possibilism, is
indeed an intermediate view. It is a passage view, but it is less ontologically
sparse than presentism. While on this view the future is still merely possible
rather than actual (hence its name), the past has become and is fully actual.
If one thinks of the future as a branching structure of alternative
possibilities (as the result, for instance, of free human choices or
indeterministic quantum measurements), then one can think of the past and
present as the trunk of that tree, growing as possibilities become actual in
the present.
Possibilism seems to capture much
of the way we think about time and being. While the sparse symmetry of
presentism is attractive, there are many deep asymmetries concerning past and
future that it fails to reflect. I can easily ascertain, for instance, yesterday's
closing number for the Dow Jones Industrial Average, but by no efforts, however
great, can I now ascertain tomorrow's close. And it seems as if my actions (or
certain sorts of quantum measurements) can actualize some future possibilities
as opposed to others, whereas past actions (or the results of past quantum
measurements) seem no longer to admit of alternatives. Even if one allows for
the possibility of retrocausation, for the possibility of an effect preceding
its cause in time, it is generally held that a present cause can not change or
alter the past. It would merely make the past what it was. (See the entry Backwards
Causation for further consideration of this topic.)
Eternalism too, prima facie,
would seem to have trouble accounting for the asymmetries built into
possibilism, in addition to its implausible denial of passage. But the first
topic to which we shall turn is an argument, prominent in twentieth century
philosophy of time, that passage or becoming is a self-contradictory idea. If
the argument is correct, then neither presentism nor possibilism can be correct
metaphysical views of time and being.
At the beginning of the 20th
century, J. M. E. McTaggart (1908) presented an argument which purported to
prove that time is unreal. According to McTaggart (1927, pp. 9-10):
Positions in time, as time appears to us prima facie, are distinguished in two ways. Each position is Earlier than some and Later than some of the other positions… . In the second place, each position is either Past, Present, or Future. The distinctions of the former class are permanent, while those of the latter are not. If M is ever earlier than N, it is always earlier, But an event, which is now present, was future, and will be past.
The first structure of “positions
in time,” McTaggart called the B-series. I will assume that McTaggart
intended the B-series to coincide with the classical spacetime structure
described above. McTaggart noted that there was something static or “permanent”
about the B-series. If, for example, event e1 is earlier
than event e2 at some time or other, then it is earlier
than e2 at all times.
The dynamic element of time must
be represented, in McTaggart's view, by the series of properties of pastness,
presentness, and futurity, which (in contrast to the static B-series) are
constantly changing. A given event becomes less future, becomes present, and
then becomes increasingly past. This latter ever-shifting series McTaggart
called the A-series.
While there are many obscurities
in McTaggart's writing, it seems clear that the argument intended to prove that
time is unreal runs along the following lines:
(1)
there can be no time unless it has a dynamic element (that is, on his view, unless there is an A-series),
(2)
there can be no A-series, because the supposition that there is an A-series leads to contradiction.
The contradiction alleged by
McTaggart is that:
(A1)
every event must have many, if not all, the A-properties (or A-determinations, as they are sometimes called) whereas,
(A2)
since the A-properties are mutually exclusive, no event can have more than one of them.
Near the end of career in which
he spent much time and effort in thinking about McTaggart's argument, C. D.
Broad (1959, p. 765) wrote:
I felt from the first, and still feel, that the difficulty which arises is (a) embarrassing enough prima facie to demand the serious attention of anyone who philosophises about time, and (b) almost certainly due to some purely linguistic source (common, and perhaps peculiar, to the Indo-European verb-system), which it ought to be possible to indicate and make harmless.
Broad's claim (a) was vindicated
by the fact that McTaggart's argument has received serious attention from most
subsequent philosophers who pondered the metaphysics of time. Much of this
debate concerns the relative relations of the two series. Is the A-series
fundamental and the B-series derived from it, or vice versa, or does, perhaps,
one series supervene upon the other? In the formal mode, the questions become
whether the B-series may somehow be reduced to, may be defined in terms of, the
A-series (or vice versa). These debates concern mainly language rather than
physics and will not be considered here.[2]
What emerges from the McTaggart
literature that is relevant to this discussion is, first of all, a tendency to
identify the existence of passage or temporal becoming with the existence of
the A-series (that is, to think of becoming as events changing their properties
of pastness, presentness or nowness, and futurity) and hence the tendency for
debates about the existence of passage to focus on the merits or incoherence of
the A-series rather than examining alternative accounts of becoming. (But Cf.
Fitzgerald, 1985)
There is a tendency amongst those
philosophers who take modern physics seriously to be sceptical of entities like
constantly shifting temporal properties of events, since such properties seem
to play no role in modern physical theory. One view, defended by Paul Horwich
(1987, Chapter 2) and Huw Mellor(1981, 1998), is that even though McTaggart
showed that passage (that is, the A-series) is impossible, the B-series (that
is, static classical spacetime structure) suffices for time.
Before we expand on this theme,
though, first a few words about Broad's (b), his suspicion that there is some
peculiarity of our language(s) that creates or at least reinforces the
credibility of McTaggart's anti-passage argument. Broad suspected that there
was a subtle ambiguity in the copula ‘is(tm) between tensed and tenseless uses,
between the uses in, for instance:
It is raining
and
Seven is prime,
the former
sentence containing a tensed and the latter sentence a non-tensed or tenseless
copula. It has been further suggested (Sellars 1962) that one might understand
a non-tensed copula (indicated by ‘be(tm) rather than ‘is(tm)) after the following
fashion
S be F at t iff (S
was F at t or S is F at t or S
will be F at t),
where the verbs
to the right of the ‘iff(tm) (a logician's abbreviation for ‘if and only if(tm)) are
usual tensed verbs.
Alternatively, one might think of
a tenseless copula as the usual copula stripped of temporal information (Quine,
1960, p. 170, Mellor 1981, 1998, Chapter 7), just as the usual copula carries
no spatial information. If we indicate this tenseless copula by writing ‘BE(tm)
instead of ‘is,(tm) we can say that ‘It BE windy in Chicago(tm) carries information
about the place but not the time of the wind, just as ‘It BE windy at t(tm) tells
us about its time but not its place.
These distinctions will prove helpful
in the subsequent discussion of being in modern physics. For the moment, one
might note that Broad could argue that McTaggart's (A1) seems
plausible if the copula is understood in some tenseless fashion, whereas (A2)
is plausible if the copula is tensed. If, however, the copula is not univocal
in (A1) and (A2), then there is no contradiction involved
in accepting both. (Savitt, 2001)
If McTaggart's argument that
passage is conceptually absurd or self-contradictory fails, philosophers
mindful of modern physics are still left with Einstein's concern that passage
and the now, while deeply embedded in human experience, seem to find no place
in physics. One may agree with Carnap that “all that occurs objectively can be
described in science” and then argue that passage reflects something
perspectival or subjective and so is implicit in the physics or rightly omitted
by it.
The most popular version of this
view holds that now is a token-reflexive or indexical term, like here
(Smart 1963, chapter VII; Mellor 1981, 1998). Physics is not felt to be
incomplete because it fails to treat hereness. Why should its
indifference to nowness be of any greater concern?
Early proponents of this view
often claimed that ‘S is now F(tm) meant ‘S(tm)s being F
is simultaneous with this utterance,(tm) a quite implausible claim. A more
sophisticated version of the view is that the truth-conditions of sentences
like ‘S is now F(tm) can be given solely in terms of the
(tenseless) facts that exist or events that occur at the time of the utterance
or inscription of the given sentence. It should be obvious how to extend the
position to treat past and future in a similar fashion.
Smart claimed that excessive
attention to the tensed notions of now, past, and future
serve to project a “sort of anthropocentric idea on the universe at
large.”(1963, 132) But even if the tensed temporal locutions are
anthropocentric and do locate us in the universe, it may still be
asked whether these temporal locations are in a static structure, “a
four-dimensional continuum of spacetime entities,” (132) or in an unfolding or
dynamic universe. Smart dismisses this latter view because, in his view, it
involves the obscure or mistaken idea that events “become” or “come into
existence.” Becoming and passage are mistakes, and harmful ones at that. Smart
writes: “Our notion of time as flowing, the transitory aspect of time as Broad
has called it, is an illusion which prevents us seeing the world as it really
is.“ (132)
It will be useful to untangle a
couple of ideas that are confounded in these quotes from Smart, with the help
of some arguments of (mostly) Broad's (1938, section 1.22 of Chapter 35). First
is the idea that time “flows” or, more generally, that passage is somehow to be
thought of as like motion. Perhaps time itself somehow moves. Or perhaps, as
Broad wrote in a famous sentence, “[t]he characteristic of presentness is …
supposed to move along this series of event-particles, in the direction from
earlier to later, as the light from a policeman's bullseye [flashlight] might
move along a row of palings.”
Motion is one sort of change,
change of spatial position with respect to time. The motion of time, then, must
be change of time with respect to … What? If the answer, by analogy with
motion, is “time”, one might be rightly puzzled as to how time (or anything
else, for that matter) can change with respect to itself. Furthermore,
if it is just time again, then the ratio of these two quantities expressing the
rate of change is a pure or dimensionless number if the dimensions of the
quantities in this ratio cancel. (See Price 1996, p. 13.) A pure number is not
a rate of change, although it may represent various rates of change (for
instance, 30 meters/second or 30 miles/hour). As Price remarks, “We might just
as well say that the ratio of the circumference of a circle to its diameter
flows at π seconds per second!”
If (in order to avoid this
absurdity) the time in the denominator of the ratio expressing the rate of
time's motion is held to be a different temporal dimension from the one in the
numerator, then for it to be a genuine time there will have to be passage in
it, requiring yet a third temporal dimension. One can see that we are at the
beginning of an infinite regress, unless the third temporal dimension is
identified with the first (as in Schlesinger 1980, Chapter II), leaving us in
the uncomfortable position of having two temporal dimensions. It seems at best
heroic, at worst hopeless, to try to understand passage as a kind of motion.
Broad also thought that trying to
explain or represent passage in terms of qualitative change was “doomed to
failure.” A thing or substance, S, can change in terms of a quality or
property if property P1 and property P2
are determinates under a given determinable and S is P1
at t1 but P2 at t2. The
passage of time, then, is to be thought of as an event's having (say) the
property of presentness and then immediately losing that property but gaining
(and losing in turn) a long and possibly endless series of properties of the
increasing degrees of pastness.
In order for a thing to change it
must evidently persist at least from t1 to t2,
but the events usually supposed in discussions of passage are instantaneous
events, which have no duration at all. They can not undergo qualitative change.
It is sometimes argued that the properties that make up the A-series (and so
change of which represents passage) are special properties, which even
instantaneous events can gain and lose, but this is special pleading. As noted
above, physics has so far no need of such special properties and such special
change and so is unlikely to be sympathetic to this special pleading.
Finally, Broad notes that
(assuming one wishes to think of passage as like qualitative change) the
acquisition and loss of (say) presentness by an event would itself be an event,
a second-order event, in the history of a first-order event. Since the
first-order events are, by hypothesis, durationless, it is tempting to suppose
that this history takes place in a second temporal dimension. We find ourselves
again launched on what looks to be an infinite regress of temporal dimensions.
These are strong arguments
against two perennially tempting ways to construe temporal becoming " as like
motion or qualitative change. They are strong arguments against the existence
of temporal becoming if there is no other way to understand it. Broad thought,
however, that he had a third way. Having pointed out the superficial
grammatical similarity between ‘E became louder(tm) and ‘E
became present(tm), Broad said that our understanding of these two kinds of
assertions need not be dictated by it. He wrote (1938, p. 280-1):
Again, any subject of which we can significantly say that it “became louder” must be a more or less prolonged noise-process, which divides into an earlier phase of less loudness adjoined to a later phase of greater loudness. But a literally instantaneous event-particle can significantly be said to “become present”; and, indeed, in the strict sense of “present” only instantaneous event-particles can be said to “become present”. To “become present” is, in fact, just to “become”, in an absolute sense; i.e., to “come to pass” in the Biblical phraseology, or, most simply, to “happen”. Sentences like “This water became hot” or “This noise became louder” record facts of qualitative change. Sentences like “This event became present” record facts of absolute becoming.
The terminology may be
pretentious, but the idea is simple. Absolute becoming is just the happening of
events. The raison d(tm)être, the very being or existence of events, is
in their happening (at some place and time). If one is willing to embrace this
category of entity at all, then one has the tools for a minimalist understanding
of passage. Given the geometric richness of Newtonian spacetime, we can say
that some events occur at the same time and so form a class of simultaneous
events. If these classes can be, somehow, ordered, then we can say that some
events occur before or after others. The passage of time is just the successive
happening of (simultaneity sets of) events. It may be this picture of passage
that the great logician Kurt Gödel had in mind when he wrote (1949, p. 558):
“The existence of an objective lapse of time … means (or, at least, is
equivalent to the fact) that reality consists of an infinity of layers of ‘now(tm)
which come into existence successively.”
There is an ambiguity in this
last quote, however, that we must note. Did Gödel think that the layers of now
come into existence (as what is to be becomes what is now) and then immediately
cease to exist (as what is now becomes what once was), which is the presentist
metaphysics of time? Or did he think that the layers of now come into existence
and forever stay in existence, as the possibilist picture maintains? If one's
basic ontology consists of the sort of events characterized above and often
invoked in discussions of time, (idealized) instantaneous happenings, then the
presentist picture seems inevitable.
The metaphysics of time is,
however, one of the cross-roads of philosophy where issues intersect. If one
thinks of a basic ontology consisting not of events but of substances or
continuants, then one is apt to wonder what it is that makes sentences marking
episodes in the histories of such substances " sentences like ‘S is Φ
at t(tm) " true. One frequent suggestion is that the “truth-makers” of
such sentences are facts, the fact that at t, S is
Φ. Then one might further note that in the current year, 2001, we can say:
These facts, when compared to
evanescent events, seem to have great stability, the first one lasting (since
it is a fact …) at least from 1980 till the present. The third one is,
however, a special sort of fact, clearly not dependent on human will or choice
and almost certainly not dependent upon any quantum measurements either. Future
facts that do depend upon human choice or quantum measurement, should they be
facts now, would seem to constrain human choice or quantum measurement in ways
that many philosophers find undesirable. It is easy to convince oneself, then,
that future facts of those two sorts can not really be part of the existing.
Perhaps, then, facts like fact 3 above can be argued away as well. The result
of this (lightly sketched) train of thought is, of course, the possibilist
picture of time.
It seems unlikely that a simple
argument will decide between these two metaphysical pictures of time,
presentism and possiblism. Showing that McTaggart's argument is flawed, because
it relies on an ambiguity in the copula ‘is(tm), and that there is a way to
construe passage that side-steps the traditional objections, moreover, does not
show that eternalism is false but only that it is optional. In Newtonian
spacetime it may appear implausible, but it may fare better when we turn to
Minkowski spacetime.
The Special Theory of Relativity
(Einstein, 1905) was presented as a geometric theory of spacetime in Minkowski
(1908).[3] For our purposes, the key change
from Newtonian spacetime to Minkowski spacetime is that in the latter it is no
longer the case that the temporal interval between any two points or events in
the spacetime, p and q, is a well-defined quantity. In fact,
the temporal interval between two points in the spacetime (and hence the
simultaneity of two points in the spacetime) is not defined at all until a
coordinate system or frame of reference (with some arbitrarily chosen spacetime
point as origin of the frame) is chosen. A peculiar feature of special
relativity (as opposed Newtonian physics) is that each coordinate system or
frame of reference defined by an “observer” passing through the chosen origin
and moving with some constant non-zero speed that is less than the speed of
light (as measured in the first frame) picks out a distinct set of
points as simultaneous with the origin. This feature of special relativity is
called the relativity of simultaneity.
The relativity of simultaneity is
a consequence of the even more startling assumption that each of these
“observers”, no matter at what speed or in which direction they or the source
of the light are moving (as long as neither the speed nor the directions
change), must come to the same result (conventionally indicated as c)
when they measure the speed of light. We will not attempt to justify the
assumption of the constancy of the speed of light here, though many standard
texts present the empirical and theoretical background the led to it. Nor is it
obvious that this assumption leads to the relativity of simultaneity, though
one of the joys of even elementary presentations of the subject is that this prima
facie astonishing connection can be convincingly demonstrated to persistent
non-specialists.
A second assumption typically
made in presentations of the special theory is the Principle of Relativity:
All inertial frames of reference are completely equivalent for the formulation
of the laws of physics.[4]
A glance back at Figure 1 reminds
us that presentism and possibilism suppose that one plane of simultaneity is
uniquely metaphysically important. In the former view, it represents all that
exists. In the latter view, it is the locus of becoming, the dividing line
between the merely possible future and the actual past-plus-present. The
special theory of relativity tells us that there is an infinity of planes of
simultaneity passing through any given spacetime point and that no physical
test can distinguish one from amongst the lot. What was metaphysically
distinguished is now physically indistinguishable. Assuming that we humans are
complex physical systems, then we have no way to distinguish the
present from amongst the multitude of presents.
An enthusiast could make much of
this fact. For instance, the mathematician (and science fiction writer) Rudy
Rucker wrote (1984, p. 149):
As it turns out, it is actually impossible to find any objective and universally acceptable definition of “all of space, taken at this instant.”" This follows … from Einstein's special theory of relativity. The idea of the block universe is, thus, more than an attractive metaphysical theory. It is a well-established scientific fact.
On the other
hand, the distinguished philosopher and logician Arthur Prior thought that the
above conclusion showed that special relativity is an incomplete view of
reality (Prior, 1970):[5]
One possible
reaction to this situation, which to my mind is perfectly respectable though it
isn't very fashionable, is to insist that all that physics has shown to be true
or likely is that in some cases we can never know, we can never physically
find out, whether something is actually happening or merely has happened
or will happen.
We shall look at
more nuanced reactions to the relativity of simultaneity below, but first it
will be useful to introduce an argument that plays somewhat the same role in
Minkowski spacetime as McTaggart's argument did in Newtonian spacetime.
Versions of the argument are endorsed in papers by the physicist Cornellis
Rietdijk (1966, 1976) and the philosopher Hilary Putnam (1967), but the presentation
here will be based on an example found in Roger Penrose's book, The
Emperor's New Mind.
Imagine that the Andromeda
galaxy, which is about two million light years or 2-1019 kilometers
from Earth, is at rest with respect to Earth. On Earth two friends walk past
each other, Alice walking along the Earth-Andromeda line towards Andromeda, Bob
walking along that line but away from Andromeda. Each is walking at a
comfortable pace, say 4 km/hour. One can calculate that their planes (or
spaces) of simultaneity at the instant at which they pass each other on Earth
(Call the event of their meeting O) intersect the history or world
line of Andromeda about 5 3/4 days apart. (Call these two events A
and B, respectively. We are idealizing Andromeda as a point,
for the purpose of this example.) Imagine, finally, that during this 5 3/4 day
period between B and A a momentous thing
happens. The Andromedeans launch a space fleet aimed at invading Earth.
Figure 2. The Andromedean Invasion
The launch of the invading fleet
is prior to A and so in some sense in Alice's past.
But since the launch is after B, it is in that same sense in
Bob's future. Penrose comments:
Two people pass each other on the street; and according to one of the two people, an Andromedean space fleet has already set off on its journey, while to the other, the decision as to whether or not the journey will actually take place has not yet been made. How can there still be some uncertainty as to the outcome of that decision? If to either person the decision has already been made, then surely there cannot be any uncertainty. The launching of the space fleet is an inevitability. (p. 303)
This is an odd
situation indeed. An event in Bob's future seems in some way to become fixed or
inevitable by being in Alice's past. But that is not the end of the oddness
here. Imagine that at point A (where Alice's plane of
simultaneity intersects the world line of Andromeda) there is an Andromedean,
Carol, who is walking directly towards Earth at about 4 km/hour. Then Carol's
plane of simultaneity intersects Earth at some point C which
is about 11 1/2 days after O, the meeting of Alice and Bob. If
all events (like A) in Alice's past or present at O
have happened, are fixed, or are real, then the principle of relativity
suggests that we must also extend the same courtesy to Carol; and so
simultaneous with the fixed and real event A (Carol's walking
towards Earth at exactly the point at which Alice's plane of simultaneity
intersects the history of Andromeda) is the event C (and so
fixed and real), the intersection of Carol's plane of simultaneity with Earth,
which is in the future of both Alice and Bob. It is easy to see that,
by adjusting the speeds of Alice and Carol, any event to the future of O
can be shown to be fixed or real or inevitable. But O itself
was just an arbitrarily chosen point in the spacetime. “It begins to seem that
if anything is definite at all,” we might echo Penrose, “then the entire
space-time must indeed be definite! There can be no ‘uncertain(tm) future.” (p.
304)
Roberto Torretti (1983, p. 249)
calls the resulting view of the definiteness or fixity of all events in the
spacetime chronogeometrical determinism. A slightly better name might
be chronogeometrical fatalism, as we will see below. In order to see
more clearly, however, what has gone wrong in the argument above, it will be
useful first to look more closely at the problems attendant upon trying to
import our commonsense or classical intuitions about time into the
understanding of Minkowski spacetime and then to describe briefly the structures
peculiar to that spacetime itself. To begin with the first task, one of the
most notable attempts to bring our time into Minkowski spacetime is to be found
in Sellars (1962), a determined attempt by one of the most profound systematic
metaphysicians of the latter half of the 20th century.
Wilfrid Sellars believed
that the various invariant or observer-independent elements of
Minkowski spacetime (like the light cone structure to be described below) that
are typically given primary consideration in treatments of relativity from a
spacetime perspective are abstractions from and secondary to the ‘perspectival(tm)
pictures, the myriad of coordinate systems or reference frames. When it comes
to time, however, he believed there is something even more fundamental than
these perspectives:
…we must distinguish between a moment, t, and the event of the moment's being present with respect to a given perspective and, above all, between the event of the moment's being present with respect to a given perspective and the event of the moment's being present. The latter, of course, is the essential feature of a temporal picture of the world. (577)
While there is in
Sellars' paper a lengthy and illuminating series of reflections on the relation
between events, facts, and substances, there is no guidance offered on the
relation between a moment's being present with respect to a given perspective
and a moment's simply being present, a concept which is ill-formed from a
relativistic point of view. If this latter is indeed an essential feature of a
temporal picture of the world, then special relativity does not provide us with
a temporal picture of the world. If the world is fundamentally temporal
in the way that Sellars insists it is, then (at least as far a special
relativity as a representation of that world is concerned), Sellars' famous
scientific realism is compromised.
Even though Sellars' conservative
attempt to import pre-relativistic categories into Minkowski spacetime fails,
there are some useful lessons to learn from it. First, Sellars is careful to
distinguish between events as things that happen or occur or take place and the
‘events(tm) (the use of single quotes is Sellars') that are basic in relativity.
The latter are just spacetime points. They do not take place or occur, and they
are not the relata in causal relations, whereas events are. (But cf. Tooley
(1997, Chapter 9 )) While it is not clear what precisely Sellars took the distinction
to be, he is careful to mark a distinction between events and ‘events(tm).
Sellars also presents a
distinction between what he calls (p. 586) categorial existence statements
and what, for lack of a better term, I will call non-categorial existence
statements. The former invoke frameworks, like the framework of substances
or the framework of ‘events(tm), the frameworks that Sellars takes great pains to
compare in his essay. He is inclined towards a view he credits (without source)
to Carnap that to say that, for instance, ‘Things exist(tm) is to make the
metalinguistic claim that there are thing words in our language L now. This use
of ‘exist,(tm) claims Sellars, has no (future or past) tensed contrast.
Non-categorial existence
statements, on the other hand, assert the existence of individuals or less
general kinds in a fully tensed fashion. Sellars would construe them in the
following way (p. 592):
a be existent { before now, now, after now } ≡
∃x(x be { before now, now, after now } and x be Φ1,…,Φn and
‘Φ1(tm),…,‘Φn(tm) be our criteria now for [being] ‘a(tm))
Leaving aside
Sellars' idiosyncratic way of construing existence statements, if a distinction
like the one indicated here can be made, then it would be perfectly coherent to
indicate that one is adopting or working in the framework of ‘events(tm) by
asserting that ‘events exist(tm) (in the categorial sense) without being committed
to the "tenseless existence" of particular ‘events,(tm) which may be
past, present, or future (in the non-categorial sense).
It has sometimes been thought
that commitment to a spacetime framework, as is often explicit in treatments of
special relativity, is tantamount to commitment to eternalism, since to say
that spacetime points exist seems inconsistent with saying that some spacetime
points are future and so do not exist yet or are past and so exist no longer. If
some distinction of the type just sketched can be made between categorial and
non-categorial existence statements, then eternalism is not a straightforward
consequence of adopting the spacetime viewpoint.[6]
Granting Sellars all the
distinctions that he wishes, however, does not give him the tools to avoid the
central problem sketched above. Since the problem is, in one form or another,
the problem that any view that tries to define a notion of becoming in
Minkowski spacetime must address, it is worth examining it a bit more closely.
Sellars wrote (p. 591):
… in the case of an ‘event(tm) framework, a primary temporal picture is a picture with a now. And even if one observer's now is another observer's then, or one observer's simultaneous cross sections of the world are another observer's sets of differently dated ‘events(tm),… each of their now-pictures is a primary picture, and the purely topological picture (which includes the measurements performed by S and S′ as topological facts) which is common to them is not the primary picture of the world construed as a system of ‘events,(tm) but merely a topological abstraction common to the various primary pictures; and the topologically formulated location of individual events in the topological picture is merely the topologically invariant features of the criteria which identify these ‘events(tm) in a primary picture.
In this quote
Sellars is using the term ‘topological(tm) where one would now normally use the
term ‘geometric(tm), and he is forcefully reiterating his view that the spacetime
manifold of ‘events(tm) is merely an abstraction from the infinity of distinct primary
now-pictures of individual observers.
The first question one will
surely want to ask about this view is: how can an infinity of distinct
“now-pictures” each be primary? No answer is forthcoming. The second, and more
troubling question, is: how can this infinity of distinct “now-pictures” be
related to the traditional metaphysical views under discussion? What, in short,
is the connection (if any) between the temporal notions implicit in each of the
pictures and existence of the past, present, and future? The striking fact
about Sellars' schema above for ‘a be existent now(tm) is that it is not
relativized to a reference frame, coordinate system or “observer” and so is not
meaningful relativistically. The definition gives us no guidance as to how to
parcel out existence to elements in the infinity of reference frames that are
admissible at a spacetime point.
If the definition or schema above
were relativized to frames F, F′, etc., so as to connect existence
to relativistically acceptable "primary now-pictures", its
interpretation would be either unhelpful or mysterious. Consider the following
modification of Sellars' schema above:
a be existent now in F ≡
∃x(x be now in F and x be Φ1,…Φn and
‘Φ1,(tm)…,‘Φn(tm) be our criteria now for ‘a(tm))
Suppose it is not
the case that a be existent now in some other frame F′.
It seems as if this difference must result from a's being simultaneous
with some spacetime point O, say, in F while not
being simultaneous with the same point O as coordinatized in F′.
But on this reading Sellars' schema is just a round-about way to indicate that
simultaneity is relative " the point of departure for our metaphysical
questions rather than the answer to any.
The schema looks as if it is
meant to do something more, to connect temporal notions to existence. But if
so, how is existence relative to a frame to be understood? Classical
presentism, for instance, wishes to identify existence with present existence
or existence now. Since the present is relativized to frames in
special relativity, may not existence be relativized to frames as well? This is
a difficult notion to understand or accept. Kurt Gödel (1949, p. 558) said
flatly, “The concept of existence … cannot be relativized without destroying
its meaning completely.” Is the concept of existence, then, like the concept of
truth, which, when relativized (as true-for-me, true-for-you), comes to
something more like belief than truth? Or is it like simultaneity, about which
thoughtful persons a century or so ago might have made pronouncements much like
Gödel's? This difficult and fundamental question has by no means been resolved.
Were this question resolved in
favor of the relativization of existence, what would be the import of a relativized
version of presentism? It would have to hold that what existed changed
radically with one's state of motion. Certain events (say on Mars or a planet
orbiting a distant star) may be existent for you now, sitting at your computer
screen or reading a printout, but other events will replace those as existent
should you decide to walk one way or another. This seems (once again) less like
an interesting metaphysical insight than a restatement of the relativity of
simultaneity. Possibilism is not better off in this regard, for it relies on a
metaphysically distinguished present to separate the real from the potential.
(See the symposium "The Prospects for the Present in Spacetime
Theories" in Howard (2000) for further arguments and references.)
To sum up, then, Sellars' attempt
to tie existence to temporal notions, when properly relativized, is either a
bland re-statement of what special relativity tells us already about
simultaneity or an opaque statement about relativized existence. This dilemma
confronts any attempt to import pre-relativistic notions in Minkowski
spacetime. Let us, then, turn to efforts to understand Minkowski spacetime in a
different way, efforts that will help clarify the puzzling argument about the
Andromedean invasion presented above.
We have said much about the
relativity of simultaneity above but little directly about the invariance of
the speed of light. We must now rectify that situation.
Imagine that at some point O
of the spacetime an idealized point-sized flashbulb flashes for (literally) an
instant. It follows from the invariance of the speed of light that Alice,
passing through O as above, will find herself at the center of
a sphere of photons. The radius of the sphere expands with speed c.
(It follows that Bob, also passing through O but moving with
some constant velocity with respect to Alice, must find himself also at the
center of such a sphere, even though he and Alice are walking away from each
other. Such is relativistic life!) If we try to diagram this situation, it is
helpful to suppress one spatial dimension, as we have in all the figures above,
and so the two-dimensional cut through the expanding sphere looks like an
expanding circle, which becomes a cone when that growth is plotted vertically
up the diagram (and so is called the light cone.) More precisely, this
figure is just half the light cone. If two photons (restricting ourselves to
two dimensions now) converged on point O from opposite
directions, the lines indicating their histories would mark the other half, the
past lobe, of the light cone.[7]
The light cone exists at each
point of the spacetime and is an invariant structure. Since the speed of light
is an invariant quantity, all “observers” agree as to which points of the
spacetime are illuminated by the popping of the flashbulb at O.
Furthermore, as special relativity is standardly understood, the speed of light
is a limiting speed. No material particle can be accelerated from a speed less
that c to a speed equal to or greater than c. Electromagnetic
radiation (including light) always propagates in a vacuum at speed c.
(To see why c is held to be limiting velocity, speed, see Mermin
(1968, Chapter 15) and Nahin (1999, pp. 342-353 and Tech Note 7.) Given these
suppositions, the light cone structure divides all spacetime into three
distinct sorts of regions relative to each spacetime point O.
(See chapters 5 and 6 of Geroch (1978) for a thorough discussion.)
Figure 3. The Light Cone
First, there are the points from
which a photon may travel to O or which may be reached by a
photon from O. We say that these points are lightlike
separated from O. If a photon can travel from O
to A, we can indicate this briefly by writing O
< A. In this case, A lies on the future
light cone of O.
Second, there are the points
inside (rather than on) the future or past light cone of O. We
say that these points are timelike separated from O.
If B is a point in the spacetime timelike separated from O
and future to it (that is, inside O's future light cone), then
a material particle travelling at some relativistically acceptable speed (that
is, less than c) can travel from O to B.
Similarly, a material particle at a point inside the past light cone of O,
can travel at some speed less than c from C to O.
In this case we write C << O; in the
former case, O << B.
Finally, there are the points of
the spacetime that are neither in nor on the light cone of O.
We say that such points are spacelike separated from O.
If D is spacelike separated from O, then no
light signal and no material body can travel from O to D
or vice versa, because such travel would require superluminal speed.
If one makes the natural assumption that information and causal influence are
propagated by electromagnetic signals and material particles, then if D
is spacelike separated from O, events or occurrences at O
can have no causal influence at all on events at D.
We have reached this last
conclusion by means of quite straightforward reasoning from the invariance of
the speed of light. But consider the following observation of Torretti (1983,
p. 247):
Before Einstein … nobody appears to have seriously disputed that any two events might be causally related to each other, regardless of their spatial and temporal distance. The denial of this seemingly modest statement is perhaps the deepest innovation in natural philosophy brought about by Relativity. It has completely upset our traditional views of time, space, and causality …
As one
illustration of how our traditional views of time and causality are upset by
restricting the propagation of causal influence to the light cone structure,
let us revisit the reasoning of the example of the Andromedean invasion that we
used to illustrate and motivate chronogeometrical fatalism. We may be able to
see now that this reasoning is not so compelling as it first seemed, and we may
be able to see why some philosophers have proposed that we look at becoming in
Minkowski spacetime in a way quite different from the traditional way.
To make the exposition easier,
let us add to the story of the Andromedean invasion a fourth observer, Ted, who
is at rest with respect to Earth (and so also Andromeda) at the spot where
Alice and Bob meet. Ted too defines a coordinate system or frame of reference,
and there is a point at Andromeda (We can call it D) that (in
Ted's frame) is simultaneous with the meeting of Alice and Bob and Ted. To make
our exposition easier still, let us suppose that Alice and Bob and Ted all set
their clocks to read 0 at the instant at which they all meet.[8] Let us focus on D.
Ted (at the meeting of Alice and
Bob) assigns to D the time 0, since it is simultaneous (in his
frame) with his time 0. Alice assigns D (roughly) the time -3
days, whereas Bob assigns it time (roughly) +3 days. D is, of
course, spacelike separated from O, and we have been at pains
to explain that from a special relativistic standpoint this spacelike
separation precludes the (physical) possibility that there is any causal
influence upon D of the events at O. Once the
labelling of spacetime points like D with coordinates is
complete, what further content is there, what further could be meant, by adding
that for Alice and Ted D is real or fixed? If there is indeed
no further content, then what possible implications with regard to ‘reality(tm) or
‘fixity(tm) or ‘determinateness(tm) can be drawn from the fact that Bob labels this
point with a positive number, Alice labels it with a negative number, and Ted
labels it with 0?[9]
A good text in special relativity
will sooner or later prove that for any pair of spacelike separated
points (but let us continue to call them O and D)
there is precisely one admissible coordinate system (with O as
origin) in which O and D are simultaneous, an
infinity of admissible coordinate systems in which D is
assigned a positive number (that is, in which O occurs before D),
and an infinity of other admissible coordinate systems in which D
is assigned a negative number (that is, in which D occurs
before O). What metaphysical significance could be gleaned
from the fact that some observers (the usual anthropomorphized way to refer to
admissible coordinate systems) at O must assign positive
times, some negative times, and one time 0 to the distant event D,
which, again, can not be influenced by and can not itself influence the events
at O, according to special relativity at least?
Inability to provide any positive
answer to this question can motivate a different approach to conceptualizing
becoming in Minkowski spacetime, an approach presented by the philosopher
Howard Stein (1968, 1991). The basic idea of this approach is to begin from or
to define concepts in terms of the geometric structure intrinsic to the
spacetime. In the present case, this approach leads one to try to define
‘becoming(tm) in terms of spacetime points and light cones. Pre-relativistically,
‘has become(tm) is defined relative to a plane of simultaneity. We have seen the
limitations of the notion of a plane of simultaneity in special relativity.
Stein begins, then, by proposing that one defines the relation of ‘having
become(tm) or ‘already definite(tm) with respect to spacetime points. A two-place
relation schematically written as Rxy will be intended to capture the
idea that point y has already become or is definite with respect to
point x.
There are two other formal
features that this relation R should possess. It should be transitive
" that is, if z has already become with respect to y and y
has already become with respect to x, then it seems reasonable to
require that z has already become with respect to x. It
should also be reflexive " that is, is seems reasonable to require
that x has become with respect to x itself.
(We can indicate
these conditions briefly as (1) Rzy and Rxz entail Rxy,
for all x, y, z and (2) Rxx, for all x.)
Finally, Stein proposes that the
relation R not hold between every two points in the spacetime. That
is, he proposes that given some choice of spacetime point x, there is
at least one distinct point y that has not become, that is not already
definite, with respect to x. But is there any such relation, any
relation that has all these intuitively desirable characteristics? The answer
is yes. The relation is that between a point x and each point
in or on its past light cone.[10] If one can accept that the relation
Rxy represents in special relativity the notion of becoming (or,
having become), then the existence of the relation specified and found by Stein
is a formal refutation of the Rietdijk-Putnam-Penrose argument for
chronogeometric fatalism.
It is this last issue, of course,
that is controversial. Stein, who wishes to tie his definitions of temporal
concepts to intrinsic geometric structure, holds that “in Einstein-Minkowski
space-time an event's present is constituted by itself alone.” (1968,
p. 15) If one wishes to include even one other event in an event's
present " that is, if one specifies that for each point x there must
be one other distinct point y such that not only Rxy but also
Ryx " then the only relation that satisfies this desideratum and
the other conditions specified by Stein is the universal relation.[11]
Callender (2000, S592) remarks
that requiring that an event‘s present must contain at least one event distinct
from it, which he calls the non-uniqueness condition, “seems the
thinnest requirement one might put on becoming.” He would then not accept
Stein's relation R as representing a genuine relation of becoming
since it fails to meet this condition, but then he also must accept the
conclusion of the Rietdijk-Putnam-Penrose argument, since the only alternative
to R is the universal relation. If one wishes to evade chronogeometric
fatalism, as far as the special theory of relativity is concerned, then it
seems there is no alternative to accepting Stein's relation R as
representing a genuine relation of becoming and to considering that an event's
present is constituted by itself alone. It is a truism that the relativistic
revolution in physics has profound implications for our concepts of space and
time. This last dilemma shows why that truism is true.
There may seem to be an
insuperable obstacle to accepting Stein's relation R as representing a
genuine relation of becoming. R is supposed to represent becoming, but
the light cone structure of Minkowski spacetime, in terms of which it is
defined, is inert. This reaction was voiced, for instance, by Palle Yourgrau,
who wrote that “Stein's mistake is to adduce a structural property as
what ‘justifies the use of our notion of “becoming” in relativistic
spacetime.”(tm) (1999, p. 77) If Yourgrau has put his finger on a “mistake”, then
it is a “mistake” at the very heart of Stein's effort. There are, however, a
few remarks to be made on this score.
First, there have been attempts
to articulate positions like Stein's that try to account for passage in terms
of geometric structure and that seem to incorporate more dynamic elements,
exploiting the fact that persistent objects or substances (including
“observers”) are represented by timelike world lines, rather than by points.
The mathematician G. J. Whitrow (1980, p. 348) wrote:
At a given instant E on the world line of an observer A (who need not be regarded as anything more than a recording instrument), all the events from which A can have received signals lie within the backwards-directed light cone with its vertex at E… . Signals from events [outside the light cone at E] can only reach A after the event E, and when they do reach A they will then lie within A's backward-directed light cone at that instant. The passage of time corresponds to the continual advance of this light cone.
The
physicist-philosopher Abner Shimony, in responding to the claim that special
relativity shows that becoming is subjective or “mind-dependent,” wrote (1993,
p. 284):
Something
fleeting does indeed traverse the world line, but that something is not
subjective; it is the transient now, which as a matter of objective
fact is momentarily present and thereafter is past.
In the felicitous
phrase of Park (1971), we have here two different sorts of animated
Minkowski diagram. Each seems to involve a kind motion, of the light cone
or of the transient now advancing along a world line. Our initial
restrictions on accounts of transience inspired by Broad's arguments should
make us wary of invoking motion to account for passage. Park, moreover, sees no
benefit to adding the animation.
I want now to
make the vital point that the animated diagram may be more intuitive, or more
picturesque, or make better cinema than the atemporal one, but that it contains
no more specific, verifiable information. All of the science of dynamics, that
is, all we know about how complex systems (including ourselves) behave and
interact, is already represented on the atemporal Minkowski diagram.
The non-animated
Minkowski diagram may be “static”, but, as Park points out, the static diagram represents
the evolution in (proper) time of systems along their world lines. The diagram,
if Park is correct, need not itself be animated to represent dynamical
phenomena. If Park is correct, then what Yourgrau called a “mistake” is in fact
a virtue of Stein's account, that he makes no attempt to animate his geometric picture
but leaves whatever transience there may be in what it depicts.
[Please contact
the author with suggestions.]
Broad, Charles
Dunbar | events | facts
| McTaggart, John M. E. | Sellars,
Wilfrid
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