Causal Determinism (Stanfor - Rice Universitykm9/Causal Determinism.pdfConceptual Issues in...
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Causal DeterminismFirst published Thu Jan 23, 2003; substantive revision Tue Apr 1, 2008
Causal determinism is, roughly speaking, the idea that every event is necessitated by
antecedent events and conditions together with the laws of nature. The idea is ancient,
but first became subject to clarification and mathematical analysis in the eighteenth
century. Determinism is deeply connected with our understanding of the physical
sciences and their explanatory ambitions, on the one hand, and with our views about
human free action on the other. In both of these general areas there is no agreement over
whether determinism is true (or even whether it can be known true or false), and what
the import for human agency would be in either case.
1. Introduction
2. Conceptual Issues in Determinism
2.1 The World
2.2 The way things are at a time t
2.3 Thereafter
2.4 Laws of nature
2.5 Fixed
3. The Epistemology of Determinism
3.1 Laws again
3.2 Experience
3.3 Determinism and Chaos
3.4 Metaphysical arguments
4. The Status of Determinism in Physical Theories
4.1 Classical mechanics
4.2 Special Relativistic physics
4.3 General Relativity (GTR)
4.4 Quantum mechanics
5. Chance and Determinism
6. Determinism and Human Action
Bibliography
Other Internet Resources
Related Entries
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1. Introduction
In most of what follows, I will speak simply of determinism, rather than of causal
determinism. This follows recent philosophical practice of sharply distinguishing views
and theories of what causation is from any conclusions about the success or failure of
determinism (cf. Earman, 1986; an exception is Mellor 1994). For the most part this
disengagement of the two concepts is appropriate. But as we will see later, the notion of
cause/effect is not so easily disengaged from much of what matters to us about
determinism.
Traditionally determinism has been given various, usually imprecise definitions. This is
only problematic if one is investigating determinism in a specific, well-defined theoretical
context; but it is important to avoid certain major errors of definition. In order to get
started we can begin with a loose and (nearly) all-encompassing definition as follows:
Determinism: The world is governed by (or is under the sway of)
determinism if and only if, given a specified way things are at a time t, the
way things go thereafter is fixed as a matter of natural law.
The italicized phrases are elements that require further explanation and investigation, in
order for us to gain a clear understanding of the concept of determinism.
The roots of the notion of determinism surely lie in a very common philosophical idea:
the idea that everything can, in principle, be explained, or that everything that is, has a
sufficient reason for being and being as it is, and not otherwise. In other words, the
roots of determinism lie in what Leibniz named the Principle of Sufficient Reason. But
since precise physical theories began to be formulated with apparently deterministic
character, the notion has become separable from these roots. Philosophers of science are
frequently interested in the determinism or indeterminism of various theories, without
necessarily starting from a view about Leibniz' Principle.
Since the first clear articulations of the concept, there has been a tendency among
philosophers to believe in the truth of some sort of determinist doctrine. There has also
been a tendency, however, to confuse determinism proper with two related notions:
predictability and fate.
Fatalism is easily disentangled from determinism, to the extent that one can disentangle
mystical forces and gods' wills and foreknowledge (about specific matters) from the
notion of natural/causal law. Not every metaphysical picture makes this disentanglement
possible, of course. As a general matter, we can imagine that certain things are fated to
happen, without this being the result of deterministic natural laws alone; and we can
imagine the world being governed by deterministic laws, without anything at all being
fated to occur (perhaps because there are no gods, nor mystical forces deserving the titles
fate or destiny, and in particular no intentional determination of the “initial conditions” of
the world). In a looser sense, however, it is true that under the assumption of
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determinism, one might say that given the way things have gone in the past, all future
events that will in fact happen are already destined to occur.
Prediction and determinism are also easy to disentangle, barring certain strong theological
commitments. As the following famous expression of determinism by Laplace shows,
however, the two are also easy to commingle:
We ought to regard the present state of the universe as the effect of its
antecedent state and as the cause of the state that is to follow. An intelligence
knowing all the forces acting in nature at a given instant, as well as the
momentary positions of all things in the universe, would be able to
comprehend in one single formula the motions of the largest bodies as well as
the lightest atoms in the world, provided that its intellect were sufficiently
powerful to subject all data to analysis; to it nothing would be uncertain, the
future as well as the past would be present to its eyes. The perfection that the
human mind has been able to give to astronomy affords but a feeble outline of
such an intelligence. (Laplace 1820)
In this century, Karl Popper defined determinism in terms of predictability also.
Laplace probably had God in mind as the powerful intelligence to whose gaze the whole
future is open. If not, he should have: 19th
and 20th
century mathematical studies have
shown convincingly that neither a finite, nor an infinite but embedded-in-the-world
intelligence can have the computing power necessary to predict the actual future, in any
world remotely like ours. “Predictability” is therefore a façon de parler that at best
makes vivid what is at stake in determinism; in rigorous discussions it should be
eschewed. The world could be highly predictable, in some senses, and yet not
deterministic; and it could be deterministic yet highly unpredictable, as many studies of
chaos (sensitive dependence on initial conditions) show.
Predictability does however make vivid what is at stake in determinism: our fears about
our own status as free agents in the world. In Laplace's story, a sufficiently bright demon
who knew how things stood in the world 100 years before my birth could predict every
action, every emotion, every belief in the course of my life. Were she then to watch me
live through it, she might smile condescendingly, as one who watches a marionette dance
to the tugs of strings that it knows nothing about. We can't stand the thought that we are
(in some sense) marionettes. Nor does it matter whether any demon (or even God) can,
or cares to, actually predict what we will do: the existence of the strings of physical
necessity, linked to far-past states of the world and determining our current every move,
is what alarms us. Whether such alarm is actually warranted is a question well outside
the scope of this article (see the entries on free will and incompatibilist theories of
freedom). But a clear understanding of what determinism is, and how we might be able
to decide its truth or falsity, is surely a useful starting point for any attempt to grapple
with this issue. We return to the issue of freedom in Determinism and Human Action
below.
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2. Conceptual Issues in Determinism
Recall that we loosely defined causal determinism as follows, with terms in need of
clarification italicized:
Causal determinism: The world is governed by (or is under the sway of)
determinism if and only if, given a specified way things are at a time t, the
way things go thereafter is fixed as a matter of natural law.
2.1 The World
Why should we start so globally, speaking of the world, with all its myriad events, as
deterministic? One might have thought that a focus on individual events is more
appropriate: an event E is causally determined if and only if there exists a set of prior
events {A, B, C …} that constitute a (jointly) sufficient cause of E. Then if all — or even
just most — events E that are our human actions are causally determined, the problem
that matters to us, namely the challenge to free will, is in force. Nothing so global as
states of the whole world need be invoked, nor even a complete determinism that claims
all events to be causally determined.
For a variety of reasons this approach is fraught with problems, and the reasons explain
why philosophers of science mostly prefer to drop the word “causal” from their
discussions of determinism. Generally, as John Earman quipped (1986), to go this route is
to “… seek to explain a vague concept — determinism — in terms of a truly obscure one
— causation.” More specifically, neither philosophers' nor laymen's conceptions of
events have any correlate in any modern physical theory.[1]
The same goes for the
notions of cause and sufficient cause. A further problem is posed by the fact that, as is
now widely recognized, a set of events {A, B, C …} can only be genuinely sufficient to
produce an effect-event if the set includes an open-ended ceteris paribus clause
excluding the presence of potential disruptors that could intervene to prevent E. For
example, the start of a football game on TV on a normal Saturday afternoon may be
sufficient ceteris paribus to launch Ted toward the fridge to grab a beer; but not if a
million-ton asteroid is approaching his house at .75c from a few thousand miles away,
nor if the phone is about to ring with news of a tragic nature, …, and so on. Bertrand
Russell famously argued against the notion of cause along these lines (and others) in
1912, and the situation has not changed. By trying to define causal determination in
terms of a set of prior sufficient conditions, we inevitably fall into the mess of an
open-ended list of negative conditions required to achieve the desired sufficiency.
Moreover, thinking about how such determination relates to free action, a further
problem arises. If the ceteris paribus clause is open-ended, who is to say that it should
not include the negation of a potential disruptor corresponding to my freely deciding not
to go get the beer? If it does, then we are left saying “When A, B, C, … Ted will then go
to the fridge for a beer, unless D or E or F or … or Ted decides not to do so.” The
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marionette strings of a “sufficient cause” begin to look rather tenuous.
They are also too short. For the typical set of prior events that can (intuitively, plausibly)
be thought to be a sufficient cause of a human action may be so close in time and space
to the agent, as to not look like a threat to freedom so much as like enabling conditions. If
Ted is propelled to the fridge by {seeing the game's on; desiring to repeat the satisfactory
experience of other Saturdays; feeling a bit thirsty; etc}, such things look more like good
reasons to have decided to get a beer, not like external physical events far beyond Ted's
control. Compare this with the claim that {state of the world in 1900; laws of nature}
entail Ted's going to get the beer: the difference is dramatic. So we have a number of
good reasons for sticking to the formulations of determinism that arise most naturally out
of physics. And this means that we are not looking at how a specific event of ordinary
talk is determined by previous events; we are looking at how everything that happens is
determined by what has gone before. The state of the world in 1900 only entails that Ted
grabs a beer from the fridge by way of entailing the entire physical state of affairs at the
later time.
2.2 The way things are at a time t
The typical explication of determinism fastens on the state of the (whole) world at a
particular time (or instant), for a variety of reasons. We will briefly explain some of them.
Why take the state of the whole world, rather than some (perhaps very large) region, as
our starting point? One might, intuitively, think that it would be enough to give the
complete state of things on Earth, say, or perhaps in the whole solar system, at t, to fix
what happens thereafter (for a time at least). But notice that all sorts of influences from
outside the solar system come in at the speed of light, and they may have important
effects. Suppose Mary looks up at the sky on a clear night, and a particularly bright blue
star catches her eye; she thinks “What a lovely star; I think I'll stay outside a bit longer
and enjoy the view.” The state of the solar system one month ago did not fix that that
blue light from Sirius would arrive and strike Mary's retina; it arrived into the solar system
only a day ago, let's say. So evidently, for Mary's actions (and hence, all physical events
generally) to be fixed by the state of things a month ago, that state will have to be fixed
over a much larger spatial region than just the solar system. (If no physical influences can
go faster than light, then the state of things must be given from a spherical volume of
space 1 light-month in radius.)
But in making vivid the “threat” of determinism, we often want to fasten on the idea of
the entire future of the world as being determined. No matter what the “speed limit” on
physical influences is, if we want the entire future of the world to be determined, then we
will have to fix the state of things over all of space, so as not to miss out something that
could later come in “from outside” to spoil things. In the time of Laplace, of course, there
was no known speed limit to the propagation of physical things such as light-rays. In
principle light could travel at any arbitrarily high speed, and some thinkers did suppose
that it was transmitted “instantaneously.” The same went for the force of gravity. In such
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a world, evidently, one has to fix the state of things over the whole of the world at a time
t, in order for events to be strictly determined, by the laws of nature, for any amount of
time thereafter.
In all this, we have been presupposing the common-sense Newtonian framework of
space and time, in which the world-at-a-time is an objective and meaningful notion.
Below when we discuss determinism in relativistic theories we will revisit this
assumption.
2.3 Thereafter
For a wide class of physical theories (i.e., proposed sets of laws of nature), if they can be
viewed as deterministic at all, they can be viewed as bi-directionally deterministic. That
is, a specification of the state of the world at a time t, along with the laws, determines not
only how things go after t, but also how things go before t. Philosophers, while not
exactly unaware of this symmetry, tend to ignore it when thinking of the bearing of
determinism on the free will issue. The reason for this is that we tend to think of the past
(and hence, states of the world in the past) as done, over, fixed and beyond our control.
Forward-looking determinism then entails that these past states — beyond our control,
perhaps occurring long before humans even existed — determine everything we do in our
lives. It then seems a mere curious fact that it is equally true that the state of the world
now determines everything that happened in the past. We have an ingrained habit of
taking the direction of both causation and explanation as being past—-present, even
when discussing physical theories free of any such asymmetry. We will return to this
point shortly.
Another point to notice here is that the notion of things being determined thereafter is
usually taken in an unlimited sense — i.e., determination of all future events, no matter
how remote in time. But conceptually speaking, the world could be only imperfectly
deterministic: things could be determined only, say, for a thousand years or so from any
given starting state of the world. For example, suppose that near-perfect determinism
were regularly (but infrequently) interrupted by spontaneous particle creation events,
which occur on average only once every thousand years in a thousand-light-year-radius
volume of space. This unrealistic example shows how determinism could be strictly false,
and yet the world be deterministic enough for our concerns about free action to be
unchanged.
2.4 Laws of nature
In the loose statement of determinism we are working from, metaphors such as “govern”
and “under the sway of” are used to indicate the strong force being attributed to the laws
of nature. Part of understanding determinism — and especially, whether and why it is
metaphysically important — is getting clear about the status of the presumed laws of
nature.
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In the physical sciences, the assumption that there are fundamental, exceptionless laws of
nature, and that they have some strong sort of modal force, usually goes unquestioned.
Indeed, talk of laws “governing” and so on is so commonplace that it takes an effort of
will to see it as metaphorical. We can characterize the usual assumptions about laws in
this way: the laws of nature are assumed to be pushy explainers. They make things
happen in certain ways , and by having this power, their existence lets us explain why
things happen in certain ways. (For a recent defense of this perspective on laws, see
Maudlin (2007)). Laws, we might say, are implicitly thought of as the cause of everything
that happens. If the laws governing our world are deterministic, then in principle
everything that happens can be explained as following from states of the world at earlier
times. (Again, we note that even though the entailment typically works in the future past
direction also, we have trouble thinking of this as a legitimate explanatory entailment. In
this respect also, we see that laws of nature are being implicitly treated as the causes of
what happens: causation, intuitively, can only go past future.)
It is a remarkable fact that philosophers tend to acknowledge the apparent threat
determinism poses to free will, even when they explicitly reject the view that laws are
pushy explainers. Earman (1986), for example, explicitly adopts a theory of laws of
nature that takes them to be simply the best system of regularities that systematizes all
the events in universal history. This is the Best Systems Analysis (BSA), with roots in the
work of Hume, Mill and Ramsey, and most recently refined and defended by David
Lewis (1973, 1994) and by Earman (1984, 1986). (cf. entry on laws of nature). Yet he
ends his comprehensive Primer on Determinism with a discussion of the free will
problem, taking it as a still-important and unresolved issue. Prima facie at least, this is
quite puzzling, for the BSA is founded on the idea that the laws of nature are
ontologically derivative, not primary; it is the events of universal history, as brute facts,
that make the laws be what they are, and not vice-versa. Taking this idea seriously, the
actions of every human agent in history are simply a part of the universe-wide pattern of
events that determines what the laws are for this world. It is then hard to see how the
most elegant summary of this pattern, the BSA laws, can be thought of as determiners of
human actions. The determination or constraint relations, it would seem, can go one way
or the other, not both!
On second thought however it is not so surprising that broadly Humean philosophers
such as Ayer, Earman, Lewis and others still see a potential problem for freedom posed
by determinism. For even if human actions are part of what makes the laws be what they
are, this does not mean that we automatically have freedom of the kind we think we
have, particularly freedom to have done otherwise given certain past states of affairs. It is
one thing to say that everything occurring in and around my body, and everything
everywhere else, conforms to Maxwell's equations and thus the Maxwell equations are
genuine exceptionless regularities, and that because they in addition are simple and
strong, they turn out to be laws. It is quite another thing to add: thus, I might have chosen
to do otherwise at certain points in my life, and if I had, then Maxwell's equations would
not have been laws. One might try to defend this claim — unpalatable as it seems
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intuitively, to ascribe ourselves law-breaking power — but it does not follow directly
from a Humean approach to laws of nature. Instead, on such views that deny laws most
of their pushiness and explanatory force, questions about determinism and human
freedom simply need to be approached afresh.
A second important genre of theories of laws of nature holds that the laws are in some
sense necessary. For any such approach, laws are just the sort of pushy explainers that
are assumed in the traditional language of physical scientists and free will theorists. But a
third and growing class of philosophers holds that (universal, exceptionless, true) laws of
nature simply do not exist. Among those who hold this are influential philosophers such
as Nancy Cartwright, Bas van Fraassen, and John Dupré. For these philosophers, there is
a simple consequence: determinism is a false doctrine. As with the Humeans, this does
not mean that concerns about human free action are automatically resolved; instead, they
must be addressed afresh in the light of whatever account of physical nature without laws
is put forward. See Dupré (2001) for one such discussion.
2.5 Fixed
We can now put our — still vague — pieces together. Determinism requires a world that
(a) has a well-defined state or description, at any given time, and (b) laws of nature that
are true at all places and times. If we have all these, then if (a) and (b) together logically
entail the state of the world at all other times (or, at least, all times later than that given in
(b)), the world is deterministic. Logical entailment, in a sense broad enough to
encompass mathematical consequence, is the modality behind the determination in
“determinism.”
3. The Epistemology of Determinism
How could we ever decide whether our world is deterministic or not? Given that some
philosophers and some physicists have held firm views — with many prominent examples
on each side — one would think that it should be at least a clearly decidable question.
Unfortunately, even this much is not clear, and the epistemology of determinism turns out
to be a thorny and multi-faceted issue.
3.1 Laws again
As we saw above, for determinism to be true there have to be some laws of nature. Most
philosophers and scientists since the 17th
century have indeed thought that there are. But
in the face of more recent skepticism, how can it be proven that there are? And if this
hurdle can be overcome, don't we have to know, with certainty, precisely what the laws
of our world are, in order to tackle the question of determinism's truth or falsity?
The first hurdle can perhaps be overcome by a combination of metaphysical argument
and appeal to knowledge we already have of the physical world. Philosophers are
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currently pursuing this issue actively, in large part due to the efforts of the anti-laws
minority. The debate has been most recently framed by Cartwright in The Dappled
World (Cartwright 1999) in terms psychologically advantageous to her anti-laws cause.
Those who believe in the existence of traditional, universal laws of nature are
fundamentalists; those who disbelieve are pluralists. This terminology seems to be
becoming standard (see Belot 2001), so the first task in the epistemology of determinism
is for fundamentalists to establish the reality of laws of nature (see Hoefer 2002b).
Even if the first hurdle can be overcome, the second, namely establishing precisely what
the actual laws are, may seem daunting indeed. In a sense, what we are asking for is
precisely what 19th
and 20th
century physicists sometimes set as their goal: the Final
Theory of Everything. But perhaps, as Newton said of establishing the solar system's
absolute motion, “the thing is not altogether desperate.” Many physicists in the past 60
years or so have been convinced of determinism's falsity, because they were convinced
that (a) whatever the Final Theory is, it will be some recognizable variant of the family of
quantum mechanical theories; and (b) all quantum mechanical theories are
non-deterministic. Both (a) and (b) are highly debatable, but the point is that one can see
how arguments in favor of these positions might be mounted. The same was true in the
19th
century, when theorists might have argued that (a) whatever the Final Theory is, it
will involve only continuous fluids and solids governed by partial differential equations;
and (b) all such theories are deterministic. (Here, (b) is almost certainly false; see
Earman (1986),ch. XI). Even if we now are not, we may in future be in a position to
mount a credible argument for or against determinism on the grounds of features we
think we know the Final Theory must have.
3.2 Experience
Determinism could perhaps also receive direct support — confirmation in the sense of
probability-raising, not proof — from experience and experiment. For theories (i.e.,
potential laws of nature) of the sort we are used to in physics, it is typically the case that
if they are deterministic, then to the extent that one can perfectly isolate a system and
repeatedly impose identical starting conditions, the subsequent behavior of the systems
should also be identical. And in broad terms, this is the case in many domains we are
familiar with. Your computer starts up every time you turn it on, and (if you have not
changed any files, have no anti-virus software, re-set the date to the same time before
shutting down, and so on …) always in exactly the same way, with the same speed and
resulting state (until the hard drive fails). The light comes on exactly 32 µsec after the
switch closes (until the day the bulb fails). These cases of repeated, reliable behavior
obviously require some serious ceteris paribus clauses, are never perfectly identical, and
always subject to catastrophic failure at some point. But we tend to think that for the
small deviations, probably there are explanations for them in terms of different starting
conditions or failed isolation, and for the catastrophic failures, definitely there are
explanations in terms of different conditions.
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There have even been studies of paradigmatically “chancy” phenomena such as
coin-flipping, which show that if starting conditions can be precisely controlled and
outside interferences excluded, identical behavior results (see Diaconis, Holmes &
Montgomery 2004). Most of these bits of evidence for determinism no longer seem to cut
much ice, however, because of faith in quantum mechanics and its indeterminism.
Indeterminist physicists and philosophers are ready to acknowledge that macroscopic
repeatability is usually obtainable, where phenomena are so large-scale that quantum
stochasticity gets washed out. But they would maintain that this repeatability is not to be
found in experiments at the microscopic level, and also that at least some failures of
repeatability (in your hard drive, or coin-flipping experiments) are genuinely due to
quantum indeterminism, not just failures to isolate properly or establish identical initial
conditions.
If quantum theories were unquestionably indeterministic, and deterministic theories
guaranteed repeatability of a strong form, there could conceivably be further
experimental input on the question of determinism's truth or falsity. Unfortunately, the
existence of Bohmian quantum theories casts strong doubt on the former point, while
chaos theory casts strong doubt on the latter. More will be said about each of these
complications below.
3.3 Determinism and Chaos
If the world were governed by strictly deterministic laws, might it still look as though
indeterminism reigns? This is one of the difficult questions that chaos theory raises for the
epistemology of determinism.
A deterministic chaotic system has, roughly speaking, two salient features: (i) the
evolution of the system over a long time period effectively mimics a random or stochastic
process — it lacks predictability or computability in some appropriate sense; (ii) two
systems with nearly identical initial states will have radically divergent future
developments, within a finite (and typically, short) timespan. We will use “randomness”
to denote the first feature, and “sensitive dependence on initial conditions” (SDIC) for the
latter. Definitions of chaos may focus on either or both of these properties; Batterman
(1993) argues that only (ii) provides an appropriate basis for defining chaotic systems.
A simple and very important example of a chaotic system in both randomness and SDIC
terms is the Newtonian dynamics of a pool table with a convex obstacle (or obstacles)
(Sinai 1970 and others). See Figure 1:
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Figure 1: Billiard table with convex obstacle
The usual idealizing assumptions are made: no friction, perfectly elastic collisions, no
outside influences. The ball's trajectory is determined by its initial position and direction
of motion. If we imagine a slightly different initial direction, the trajectory will at first be
only slightly different. And collisions with the straight walls will not tend to increase very
rapidly the difference between trajectories. But collisions with the convex object will have
the effect of amplifying the differences. After several collisions with the convex body or
bodies, trajectories that started out very close to one another will have become wildly
different — SDIC.
In the example of the billiard table, we know that we are starting out with a Newtonian
deterministic system — that is how the idealized example is defined. But chaotic
dynamical systems come in a great variety of types: discrete and continuous,
2-dimensional, 3-dimensional and higher, particle-based and fluid-flow-based, and so on.
Mathematically, we may suppose all of these systems share SDIC. But generally they
will also display properties such as unpredictability, non-computability, Kolmogorov-
random behavior, and so on — at least when looked at in the right way, or at the right
level of detail. This leads to the following epistemic difficulty: if, in nature, we find a type
of system that displays some or all of these latter properties, how can we decide which of
the following two hypotheses is true?
1. The system is governed by genuinely stochastic, indeterministic laws (or by
no laws at all), i.e., its apparent randomness is in fact real randomness.
2. The system is governed by underlying deterministic laws, but is chaotic.
In other words, once one appreciates the varieties of chaotic dynamical systems that
exist, mathematically speaking, it starts to look difficult — maybe impossible — for us to
ever decide whether apparently random behavior in nature arises from genuine
stochasticity, or rather from deterministic chaos. Patrick Suppes (1993, 1996) argues, on
the basis of theorems proven by Ornstein (1974 and later) that “There are processes
which can equally well be analyzed as deterministic systems of classical mechanics or as
indeterministic semi-Markov processes, no matter how many observations are made.”
And he concludes that “Deterministic metaphysicians can comfortably hold to their view
knowing they cannot be empirically refuted, but so can indeterministic ones as well.”
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(Suppes (1993), p. 254)
There is certainly an interesting problem area here for the epistemology of determinism,
but it must be handled with care. It may well be true that there are some deterministic
dynamical systems that, when viewed properly, display behavior indistinguishable from
that of a genuinely stochastic process. For example, using the billiard table above, if one
divides its surface into quadrants and looks at which quadrant the ball is in at 30-second
intervals, the resulting sequence is no doubt highly random. But this does not mean that
the same system, when viewed in a different way (perhaps at a higher degree of
precision) does not cease to look random and instead betray its deterministic nature. If
we partition our billiard table into squares 2 centimeters a side and look at which
quadrant the ball is in at .1 second intervals, the resulting sequence will be far from
random. And finally, of course, if we simply look at the billiard table with our eyes, and
see it as a billiard table, there is no obvious way at all to maintain that it may be a truly
random process rather than a deterministic dynamical system. (See Winnie (1996) for a
nice technical and philosophical discussion of these issues. Winnie explicates Ornstein's
and others' results in some detail, and disputes Suppes' philosophical conclusions.)
The dynamical systems usually studied under the label of “chaos” are usually either
purely abstract, mathematical systems, or classical Newtonian systems. It is natural to
wonder whether chaotic behavior carries over into the realm of systems governed by
quantum mechanics as well. Interestingly, it is much harder to find natural correlates of
classical chaotic behavior in true quantum systems. (See Gutzwiller (1990)). Some, at
least, of the interpretive difficulties of quantum mechanics would have to be resolved
before a meaningful assessment of chaos in quantum mechanics could be achieved. For
example, SDIC is hard to find in the Schrödinger evolution of a wavefunction for a
system with finite degrees of freedom; but in Bohmian quantum mechanics it is handled
quite easily on the basis of particle trajectories. (See Dürr, Goldstein and Zhangì (1992)).
The popularization of chaos theory in the past decade and a half has perhaps made it
seem self-evident that nature is full of genuinely chaotic systems. In fact, it is far from
self-evident that such systems exist, other than in an approximate sense. Nevertheless,
the mathematical exploration of chaos in dynamical systems helps us to understand some
of the pitfalls that may attend our efforts to know whether our world is genuinely
deterministic or not.
3.4 Metaphysical arguments
Let us suppose that we shall never have the Final Theory of Everything before us — at
least in our lifetime — and that we also remain unclear (on physical/experimental
grounds) as to whether that Final Theory will be of a type that can or cannot be
deterministic. Is there nothing left that could sway our belief toward or against
determinism? There is, of course: metaphysical argument. Metaphysical arguments on
this issue are not currently very popular. But philosophical fashions change at least twice
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a century, and grand systemic metaphysics of the Leibnizian sort might one day come
back into favor. Conversely, the anti-systemic, anti-fundamentalist metaphysics
propounded by Cartwright (1999) might also come to predominate. As likely as not, for
the foreseeable future metaphysical argument may be just as good a basis on which to
discuss determinism's prospects as any arguments from mathematics or physics.
4. The Status of Determinism in Physical Theories
John Earman's Primer on Determinism (1986) remains the richest storehouse of
information on the truth or falsity of determinism in various physical theories, from
classical mechanics to quantum mechanics and general relativity. (See also his recent
update on the subject, “Aspects of Determinism in Modern Physics” (2007)). Here I will
give only a brief discussion of some key issues, referring the reader to Earman (1986)
and other resources for more detail. Figuring out whether well-established theories are
deterministic or not (or to what extent, if they fall only a bit short) does not do much to
help us know whether our world is really governed by deterministic laws; all our current
best theories, including General Relativity and the Standard Model of particle physics, are
too flawed and ill-understood to be mistaken for anything close to a Final Theory.
Nevertheless, as Earman (1986) stressed, the exploration is very valuable because of the
way it enriches our understanding of the richness and complexity of determinism.
4.1 Classical mechanics
Despite the common belief that classical mechanics (the theory that inspired Laplace in
his articulation of determinism) is perfectly deterministic, in fact the theory is rife with
possibilities for determinism to break down. One class of problems arises due to the
absence of an upper bound on the velocities of moving objects. Below we see the
trajectory of an object that is accelerated so that its velocity increases exponentially so as
to become unbounded in a finite time. See Figure 2:
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Figure 2: An object accelerates so as to reach spatial infinity in a finite
time
By the time t = 0, the object has literally disappeared from the world — its world-line
never reaches the t = 0 surface. (Never mind how the object gets accelerated in this way;
there are mechanisms that are perfectly consistent with classical mechanics that can do
the job. In fact, Xia (1992) showed that such acceleration can be accomplished by
gravitational forces from only 5 finite objects, without collisions. No mechanism is shown
in these diagrams.) This “escape to infinity,” while disturbing, does not yet look like a
violation of determinism. But now recall that classical mechanics is time-symmetric: any
model has a time-inverse, which is also a consistent model of the theory. The
time-inverse of our escaping body is playfully called a “space invader.”
Figure 3: A ‘space invader’ comes in from spatial infinity
Clearly, a world with a space invader does fail to be deterministic. Before t = 0, there
was nothing in the state of things to enable the prediction of the appearance of the
invader at t = 0+.[2]
One might think that the infinity of space is to blame for this strange
behavior, but this is not obviously correct. In finite, “rolled-up” or cylindrical versions of
Newtonian space-time space-invader trajectories can be constructed, though whether a
“reasonable” mechanism to power them exists is not clear.[3]
A second class of determinism-breaking models can be constructed on the basis of
collision phenomena. The first problem is that of multiple-particle collisions for which
Newtonian particle mechanics simply does not have a prescription for what happens.
(Consider three identical point-particles approaching each other at 120 degree angles and
colliding simultaneously. That they bounce back along their approach trajectories is
possible; but it is equally possible for them to bounce in other directions (again with 120
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degree angles between their paths), so long as momentum conservation is respected.)
Moreover, there is a burgeoning literature of physical or quasi-physical systems, usually
set in the context of classical physics, that carry out supertasks (see Earman and Norton
(1998) and the entry on supertasks for a review). Frequently, the puzzle presented is to
decide, on the basis of the well-defined behavior before time t = a, what state the system
will be in at t = a itself. A failure of CM to dictate a well-defined result can then be seen
as a failure of determinism.
In supertasks, one frequently encounters infinite numbers of particles, infinite (or
unbounded) mass densities, and other dubious infinitary phenomena. Coupled with some
of the other breakdowns of determinism in CM, one begins to get a sense that most, if
not all, breakdowns of determinism rely on some combination of the following set of
(physically) dubious mathematical notions: {infinite space; unbounded velocity;
continuity; point-particles; singular fields}. The trouble is, it is difficult to imagine any
recognizable physics (much less CM) that eschews everything in the set.
Finally, an elegant example of apparent violation of determinism in classical physics has
been created by John Norton (2003). As illustrated in Figure 4, imagine a ball sitting at
the apex of a frictionless dome whose equation is specified as a function of radial
distance from the apex point. This rest-state is our initial condition for the system; what
should its future behavior be? Clearly one solution is for the ball to remain at rest at the
apex indefinitely.
Figure 4: A ball may spontaneously start sliding down this dome, with
no violation of 0ewton's laws.
(Reproduced courtesy of John D. 0orton and Philosopher's Imprint)
But curiously, this is not the only solution under standard Newtonian laws. The ball may
also start into motion sliding down the dome — at any moment in time, and in any radial
direction. This example displays “uncaused motion” without, Norton argues, any
violation of Newton's laws, including the First Law. And it does not, unlike some
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supertask examples, require an infinity of particles. Still, many philosophers are
uncomfortable with the moral Norton draws from his dome example, and point out
reasons for questioning the dome's status as a Newtonian system (see e.g. Malament
(2007)).
4.2 Special Relativistic physics
Two features of special relativistic physics make it perhaps the most hospitable
environment for determinism of any major theoretical context: the fact that no process or
signal can travel faster than the speed of light, and the static, unchanging spacetime
structure. The former feature, including a prohibition against tachyons (hypothetical
particles travelling faster than light)[4]
), rules out space invaders and other unbounded-
velocity systems. The latter feature makes the space-time itself nice and stable and
non-singular — unlike the dynamic space-time of General Relativity, as we shall see
below. For source-free electromagnetic fields in special-relativistic space-time, a nice
form of Laplacean determinism is provable. Unfortunately, interesting physics needs
more than source-free electromagnetic fields. Earman (1986) ch. IV surveys in depth the
pitfalls for determinism that arise once things are allowed to get more interesting (e.g. by
the addition of particles interacting gravitationally).
4.3 General Relativity (GTR)
Defining an appropriate form of determinism for the context of general relativistic physics
is extremely difficult, due to both foundational interpretive issues and the plethora of
weirdly-shaped space-time models allowed by the theory's field equations. The simplest
way of treating the issue of determinism in GTR would be to state flatly: determinsim
fails, frequently, and in some of the most interesting models. To leave it at that would
however be to miss an important opportunity to use determinism to probe physical and
philosophical issues of great importance (a use of determinism stressed frequently by
Earman). Here we will briefly describe some of the most important challenges that arise
for determinism, directing the reader yet again to Earman (1986), and also Earman
(1995) for more depth.
4.3.1 Determinism and manifold points
In GTR, we specify a model of the universe by giving a triple of three mathematical
objects, <M, g,T>. M represents a continuous “manifold”: that means a sort of
unstructured space (-time), made up of individual points and having smoothness or
continuity, and dimensionality (usually, 4-dimensional), but no further structure. What is
the further structure a space-time needs? Typically, at least, we expect the time-direction
to be distinguished from space-directions; and we expect there to be well-defined
distances between distinct points; and also a determinate geometry (making certain
continuous paths in M be straight lines, etc.). All of this extra structure is coded into g. So
M and g together represent space-time. T represents the matter and energy content
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distributed around in space-time (if any, of course).
For mathematical reasons not relevant here, it turns out to be possible to take a given
model spacetime and perform a mathematical operation called a “hole diffeomorphism”
h* on it; the diffeomorphism's effect is to shift around the matter content T and the metric
g relative to the continuous manifold M.[5]
If the diffeomorphism is chosen appropriately,
it can move around T and g after a certain time t = 0, but leave everything alone before
that time. Thus, the new model represents the matter content (now h* T) and the metric
(h*g) as differently located relative to the points of M making up space-time. Yet, the
new model is also a perfectly valid model of the theory. This looks on the face of it like a
form of indeterminism: GTR's equations do not specify how things will be distributed in
space-time in the future, even when the past before a given time t is held fixed. See
Figure 5:
Figure 5: “Hole” diffeomorphism shifts contents of spacetime
Usually the shift is confined to a finite region called the hole (for historical reasons). Then
it is easy to see that the state of the world at time t = 0 (and all the history that came
before) does not suffice to fix whether the future will be that of our first model, or its
shifted counterpart in which events inside the hole are different.
This is a form of indeterminism first highlighted by Earman and Norton (1987) as an
interpretive philosophical difficulty for realism about GTR's description of the world,
especially the point manifold M. They showed that realism about the manifold as a part
of the furniture of the universe (which they called “manifold substantivalism”) commits
us to a radical, automatic indeterminism in GTR, and they argued that this is
unacceptable. (See the hole argument and Hoefer (1996) for one response on behalf of
the space-time realist, and discussion of other responses.) For now, we will simply note
that this indeterminism, unlike most others we are discussing in this section, is empirically
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vacuous: our two models <M, g, T> and the shifted model <M, h*g, h*T> are
empirically indistinguishable.
4.3.2 Singularities
The separation of space-time structures into manifold and metric (or connection)
facilitates mathematical clarity in many ways, but also opens up Pandora's box when it
comes to determinism. The indeterminism of the Earman and Norton hole argument is
only the tip of the iceberg; singularities make up much of the rest of the berg. In general
terms, a singularity can be thought of as a “place where things go bad” in one way or
another in the space-time model. For example, near the center of a Schwarzschild black
hole, curvature increases without bound, and at the center itself it is undefined, which
means that Einstein's equations cannot be said to hold, which means (arguably) that this
point does not exist as a part of the space-time at all! Some specific examples are clear,
but giving a general definition of a singularity, like defining determinism itself in GTR, is a
vexed issue (see Earman (1995) for an extended treatment; Callender and Hoefer (2001)
gives a brief overview). We will not attempt here to catalog the various definitions and
types of singularity.
Different types of singularity bring different types of threat to determinism. In the case of
ordinary black holes, mentioned above, all is well outside the so- called “event horizon”,
which is the spherical surface defining the black hole: once a body or light signal passes
through the event horizon to the interior region of the black hole, it can never escape
again. Generally, no violation of determinism looms outside the event horizon; but what
about inside? Some black hole models have so-called “Cauchy horizons” inside the event
horizon, i.e., surfaces beyond which determinism breaks down.
Another way for a model spacetime to be singular is to have points or regions go missing,
in some cases by simple excision. Perhaps the most dramatic form of this involves taking
a nice model with a space-like surface t = E (i.e., a well-defined part of the space-time
that can be considered “the state state of the world at time E”), and cutting out and
throwing away this surface and all points temporally later. The resulting spacetime
satisfies Einstein's equations; but, unfortunately for any inhabitants, the universe comes to
a sudden and unpredictable end at time E. This is too trivial a move to be considered a
real threat to determinism in GTR; we can impose a reasonable requirement that
space-time not “run out” in this way without some physical reason (the spacetime should
be “maximally extended”). For discussion of precise versions of such a requirement, and
whether they succeed in eliminating unwanted singularities, see Earman (1995, chapter
2).
The most problematic kinds of singularities, in terms of determinism, are naked
singularities (singularities not hidden behind an event horizon). When a singularity forms
from gravitational collapse, the usual model of such a process involves the formation of
an event horizon (i.e. a black hole). A universe with an ordinary black hole has a
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singularity, but as noted above, (outside the event horizon at least) nothing unpredictable
happens as a result. A naked singularity, by contrast, has no such protective barrier. In
much the way that anything can disappear by falling into an excised-region singularity, or
appear out of a white hole (white holes themselves are, in fact, technically naked
singularities), there is the worry that anything at all could pop out of a naked singularity,
without warning (hence, violating determinism en passant). While most white hole
models have Cauchy surfaces and are thus arguably deterministic, other naked
singularity models lack this property. Physicists disturbed by the unpredictable
potentialities of such singularities have worked to try to prove various cosmic censorship
hypotheses that show — under (hopefully) plausible physical assumptions — that such
things do not arise by stellar collapse in GTR (and hence are not liable to come into
existence in our world). To date no very general and convincing forms of the hypothesis
have been proven, so the prospects for determinism in GTR as a mathematical theory do
not look terribly good.
4.4 Quantum mechanics
As indicated above, QM is widely thought to be a strongly non-deterministic theory.
Popular belief (even among most physicists) holds that phenomena such as radioactive
decay, photon emission and absorption, and many others are such that only a
probabilistic description of them can be given. The theory does not say what happens in
a given case, but only says what the probabilities of various results are. So, for example,
according to QM the fullest description possible of a radium atom (or a chunk of radium,
for that matter), does not suffice to determine when a given atom will decay, nor how
many atoms in the chunk will have decayed at any given time. The theory gives only the
probabilities for a decay (or a number of decays) to happen within a given span of time.
Einstein and others perhaps thought that this was a defect of the theory that should
eventually be removed, by a supplemental hidden variable theory[6]
that restores
determinism; but subsequent work showed that no such hidden variables account could
exist. At the microscopic level the world is ultimately mysterious and chancy.
So goes the story; but like much popular wisdom, it is partly mistaken and/or misleading.
Ironically, quantum mechanics is one of the best prospects for a genuinely deterministic
theory in modern times! Even more than in the case of GTR and the hole argument,
everything hinges on what interpretational and philosophical decisions one adopts. The
fundamental law at the heart of non-relativistic QM is the Schrödinger equation. The
evolution of a wavefunction describing a physical system under this equation is normally
taken to be perfectly deterministic.[7]
If one adopts an interpretation of QM according to
which that's it — i.e., nothing ever interrupts Schrödinger evolution, and the
wavefunctions governed by the equation tell the complete physical story — then quantum
mechanics is a perfectly deterministic theory. There are several interpretations that
physicists and philosophers have given of QM which go this way. (See the entry on
quantum mechanics.)
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More commonly — and this is part of the basis for the popular wisdom — physicists
have resolved the quantum measurement problem by postulating that some process of
“collapse of the wavefunction” occurs from time to time (particularly during
measurements and observations) that interrupts Schrödinger evolution. The collapse
process is usually postulated to be indeterministic, with probabilities for various
outcomes, via Born's rule, calculable on the basis of a system's wavefunction. The
once-standard, Copenhagen interpretation of QM posits such a collapse. It has the virtue
of solving certain paradoxes such as the infamous Schrödinger's cat paradox, but few
philosophers or physicists can take it very seriously unless they are either idealists or
instrumentalists. The reason is simple: the collapse process is not physically well-defined,
and feels too ad hoc to be a fundamental part of nature's laws.[8]
In 1952 David Bohm created an alternative interpretation of QM — perhaps better
thought of as an alternative theory — that realizes Einstein's dream of a hidden variable
theory, restoring determinism and definiteness to micro-reality. In Bohmian quantum
mechanics, unlike other interpretations, it is postulated that all particles have, at all times,
a definite position and velocity. In addition to the Schrödinger equation, Bohm posited a
guidance equation that determines, on the basis of the system's wavefunction and
particles' initial positions and velocities, what their future positions and velocities should
be. As much as any classical theory of point particles moving under force fields, then,
Bohm's theory is deterministic. Amazingly, he was also able to show that, as long as the
statistical distribution of initial positions and velocities of particles are chosen so as to
meet a “quantum equilibrium” condition, his theory is empirically equivalent to standard
Copenhagen QM. In one sense this is a philosopher's nightmare: with genuine empirical
equivalence as strong as Bohm obtained, it seems experimental evidence can never tell us
which description of reality is correct. (Fortunately, we can safely assume that neither is
perfectly correct, and hope that our Final Theory has no such empirically equivalent
rivals.) In other senses, the Bohm theory is a philosopher's dream come true, eliminating
much (but not all) of the weirdness of standard QM and restoring determinism to the
physics of atoms and photons. The interested reader can find out more from the link
above, and references therein.
This small survey of determinism's status in some prominent physical theories, as
indicated above, does not really tell us anything about whether determinism is true of our
world. Instead, it raises a couple of further disturbing possibilities for the time when we
do have the Final Theory before us (if such time ever comes): first, we may have
difficulty establishing whether the Final Theory is deterministic or not — depending on
whether the theory comes loaded with unsolved interpretational or mathematical puzzles.
Second, we may have reason to worry that the Final Theory, if indeterministic, has an
empirically equivalent yet deterministic rival (as illustrated by Bohmian quantum
mechanics.)
5. Chance and Determinism
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Some philosophers maintain that if determinism holds in our world, then there are no
objective chances in our world. And often the word ‘chance’ here is taken to be
synonymous with 'probability', so these philosophers maintain that there are no non-trivial
objective probabilities for events in our world. (The caveat “non-trivial” is added here
because on some accounts all future events that actually happen have probability,
conditional on past history, equal to 1, and future events that do not happen have
probability equal to zero. Non-trivial probabilities are probabilities strictly between zero
and one.) Conversely, it is often held, if there are laws of nature that are irreducibly
probabilistic, determinism must be false. (Some philosophers would go on to add that
such irreducibly probabilistic laws are the basis of whatever genuine objective chances
obtain in our world.)
The discussion of quantum mechanics in section 4 shows that it may be difficult to know
whether a physical theory postulates genuinely irreducible probabilistic laws or not. If a
Bohmian version of QM is correct, then the probabilities dictated by the Born rule are not
irreducible. If that is the case, should we say that the probabilities dictated by quantum
mechanics are not objective? Or should we say that we need to distinguish ‘chance’ and
‘probabillity’ after all — and hold that not all objective probabilities should be thought of
as objective chances? The first option may seem hard to swallow, given the
many-decimal-place accuracy with which such probability-based quantities as half-lives
and cross-sections can be reliably predicted and verified experimentally with QM.
Whether objective chance and determinism are really incompatible or not may depend on
what view of the nature of laws is adopted. On a “pushy explainers” view of laws such
as that defended by Maudlin (2007), probabilistic laws are interpreted as irreducible
dynamical transition-chances between allowed physical states, and the incompatibility of
such laws with determinism is immediate. But what should a defender of a Humean view
of laws, such as the BSA theory (section 2.4 above), say about probabilistic laws? The
first thing that needs to be done is explain how probabilistic laws can fit into the BSA
account at all, and this requires modification or expansion of the view, since as first
presented the only candidates for laws of nature are true universal generalizations. If
‘probability’ were a univocal, clearly understood notion then this might be simple: We
allow universal generalizations whose logical form is something like: “Whenever
conditions Y obtain, Pr(A) = x”. But it is not at all clear how the meaning of ‘Pr’ should
be understood in such a generalization; and it is even less clear what features the
Humean pattern of actual events must have, for such a generalization to be held true.
(See the entry on interpretations of probability and Lewis (1994).)
Humeans about laws believe that what laws there are is a matter of what patterns are
there to be discerned in the overall mosaic of events that happen in the history of the
world. It seems plausible enough that the patterns to be discerned may include not only
strict associations (whenever X, Y), but also stable statistical associations. If the laws of
nature can include either sort of association, a natural question to ask seems to be: why
can't there be non-probabilistic laws strong enough to ensure determinism, and on top of
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them, probabilistic laws as well? If a Humean wanted to capture the laws not only of
fundamental theories, but also non-fundamental branches of physics such as (classical)
statistical mechanics, such a peaceful coexistence of deterministic laws plus further
probabilistic laws would seem to be desirable. Loewer (2004) argues that this peaceful
coexistence can be achieved within Lewis' version of the BSA account of laws.
6. Determinism and Human Action
In the introduction, we noted the threat that determinism seems to pose to human free
agency. It is hard to see how, if the state of the world 1000 years ago fixes everything I
do during my life, I can meaningfully say that I am a free agent, the author of my own
actions, which I could have freely chosen to perform differently. After all, I have neither
the power to change the laws of nature, nor to change the past! So in what sense can I
attribute freedom of choice to myself?
Philosophers have not lacked ingenuity in devising answers to this question. There is a
long tradition of compatibilists arguing that freedom is fully compatible with physical
determinism. Hume went so far as to argue that determinism is a necessary condition for
freedom — or at least, he argued that some causality principle along the lines of “same
cause, same effect” is required. There have been equally numerous and vigorous
responses by those who are not convinced. Can a clear understanding of what
determinism is, and how it tends to succeed or fail in real physical theories, shed any light
on the controversy?
Physics, particularly 20th
century physics, does have one lesson to impart to the free will
debate; a lesson about the relationship between time and determinism. Recall that we
noticed that the fundamental theories we are familiar with, if they are deterministic at all,
are time-symmetrically deterministic. That is, earlier states of the world can be seen as
fixing all later states; but equally, later states can be seen as fixing all earlier states. We
tend to focus only on the former relationship, but we are not led to do so by the theories
themselves.
Nor does 20th
(21st) -century physics countenance the idea that there is anything
ontologically special about the past, as opposed to the present and the future. In fact, it
fails to use these categories in any respect, and teaches that in some senses they are
probably illusory.[9]
So there is no support in physics for the idea that the past is “fixed”
in some way that the present and future are not, or that it has some ontological power to
constrain our actions that the present and future do not have. It is not hard to uncover the
reasons why we naturally do tend to think of the past as special, and assume that both
physical causation and physical explanation work only in the past present/future direction
(see the entry on thermodynamic asymmetry in time). But these pragmatic matters have
nothing to do with fundamental determinism. If we shake loose from the tendency to see
the past as special, when it comes to the relationships of determinism, it may prove
possible to think of a deterministic world as one in which each part bears a determining
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— or partial-determining — relation to other parts, but in which no particular part (i.e.,
region of space-time) has a special, stronger determining role than any other. Hoefer
(2002) uses these considerations to argue in a novel way for the compatiblity of
determinism with human free agency.
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Related Entries
compatibilism | free will | Hume, David | incompatibilism: (nondeterministic) theories of
free will | laws of nature | Popper, Karl | probability, interpretations of | quantum
mechanics | quantum mechanics: Bohmian mechanics | Russell, Bertrand | space and
time: supertasks | space and time: the hole argument | time: thermodynamic asymmetry in
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Carl Hoefer <[email protected]>
Causal Determinism (Stanford Encyclopedia of Philosophy) http://plato.stanford.edu/entries/determinism-causal/
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