Aug 1, 2015 05:53 PM
http://plato.stanford.edu/entries/spacetime-holearg/
EXCERPTS: Virtually all modern spacetime theories are now built in the same way. The theory posits a manifold of events and then assigns further structures to those events to represent the content of spacetime. A standard example is Einstein's general theory of relativity. As a host for the hole argument, we will pursue one of its best known applications, the expanding universes of modern relativistic cosmology. [...] Here are the two, basic building blocks of modern, relativistic cosmology: a manifold of events and the fields defined on it.
[...] Recall our original concern: we want to know whether we can conceive of spacetime as a substance, that is, as something that exists independently. To do this, we need to know what in the above structures represents spacetime. One popular answer to that question is that the manifold of events represents spacetime. We shall see shortly that this popular form of the answer is the one that figures in the hole argument. This choice is natural since modern spacetime theories are built up by first positing a manifold of events and then defining further structures on them. So the manifold plays the role of a container just as we expect spacetime does....
[....] In sum the hole argument amounts to this:
1. If one has two distributions of metric and matter fields related by a hole transformation, manifold substantivalists must maintain that the two systems represent two distinct physical systems.
2. This physical distinctness transcends both observation and the determining power of the theory since: The two distributions are observationally identical. The laws of the theory cannot pick between the two developments of the fields into the hole.
3. Therefore the manifold substantivalist advocates an unwarranted bloating of our physical ontology and the doctrine should be discarded.
[....] The notion that the manifold represents an independently existing thing is quite natural in the realist view of physical theories. In that view one tries to construe physical theories literally. If formulated as above, a spacetime is a manifold of events with certain fields defined on the manifold. The literal reading is that this manifold is an independently existing structure that bears properties.
So far we have characterized the substantivalist doctrine as the view that spacetime has an existence independent of its contents. This formulation conjures up powerful if vague intuitive pictures, but it is not clear enough for interpretation in the context of physical theories. If we represent spacetime by a manifold of events, how do we characterize the independence of its existence? Is it the counterfactual claim that were there no metric or matter fields, there would still be a manifold of events? That counterfactual is automatically denied by the standard formulation which posits that all spacetimes have at least metrical structure. That seems too cheap a refutation of manifold substantivalism. Surely, there must be an improved formulation. Fortunately, we do not need to wrestle with finding it. For present purposes we need only consider a consequence of the substantivalist view and can set aside the task of giving a precise formulation of the substantivalist view.
In their celebrated debate over space and time, Leibniz taunted the substantivalist Newton's representative, Clarke, by asking how the world would change if East and West were switched. For Leibniz there would be no change since all spatial relations between bodies would be preserved by such a switch. But the Newtonian substantivalist had to concede that the bodies of the world were now located in different spatial positions, so the two systems were physically distinct.
Correspondingly, when we spread the metric and matter fields differently over a manifold of events, we are now assigning metrical and material properties in different ways to the events of the manifold. For example, imagine that a galaxy passes through some event E in the hole. After the hole transformation, this galaxy might not pass through that event. For the manifold substantivalist, this must be a matter of objective physical fact: either the galaxy passes through E or not. The two distributions represent two physically distinct possibilities.
[...] It is important to see that the unhappy consequence does not consist merely of a failure of determinism. We are all too familiar with such failures and it is certainly not automatic grounds for dismissal of a physical theory. The best known instance of a widely celebrated, indeterministic theory is quantum theory, where, in the standard interpretation, the measurement of a system can lead to an indeterministic collapse onto one of many possible outcomes. Less well known is that it is possible to devise indeterministic systems in classical physics as well. Most examples involves oddities such as bodies materializing at unbounded speed from spatial infinity, so called “space invaders.” (Earman, 1986a, Ch. III; see also determinism: causal) Or they may arise through the interaction of infinitely many bodies in a supertask. More recently an extremely simple example has emerged in which a single mass sits atop a dome and spontaneously sets itself into motion after an arbitrary time delay and in an arbitrary direction (Norton, 2003, Section 3).
The problem with the failure of determinism in the hole argument is not the fact of failure but the way that it fails. If we deny manifold substantivalism and accept Leibniz equivalence, then the indeterminism induced by a hole transformation is eradicated. While there are uncountably many mathematically distinct developments of the fields into the hole, under Leibniz Equivalence, they are all physically the same. That is, there is a unique development of the physical fields into the hole after all. Thus the indeterminism is a direct product of the substantivalist viewpoint. Similarly, if we accept Leibniz equivalence, then we are no longer troubled that the two distributions cannot be distinguished by any possible observation. They are merely different mathematical descriptions of the same physical reality and so should agree on all observables.
We can load up any physical theory with superfluous, phantom properties that cannot be fixed by observation. If their invisibility to observation is not sufficient warning that these properties are illegitimate, finding that they visit indeterminism onto a theory that is otherwise deterministic in this set-up ought to be warning enough. These properties are invisible to both observation and theory; they should be discarded along with any doctrine that requires their retention....
EXCERPTS: Virtually all modern spacetime theories are now built in the same way. The theory posits a manifold of events and then assigns further structures to those events to represent the content of spacetime. A standard example is Einstein's general theory of relativity. As a host for the hole argument, we will pursue one of its best known applications, the expanding universes of modern relativistic cosmology. [...] Here are the two, basic building blocks of modern, relativistic cosmology: a manifold of events and the fields defined on it.
[...] Recall our original concern: we want to know whether we can conceive of spacetime as a substance, that is, as something that exists independently. To do this, we need to know what in the above structures represents spacetime. One popular answer to that question is that the manifold of events represents spacetime. We shall see shortly that this popular form of the answer is the one that figures in the hole argument. This choice is natural since modern spacetime theories are built up by first positing a manifold of events and then defining further structures on them. So the manifold plays the role of a container just as we expect spacetime does....
[....] In sum the hole argument amounts to this:
1. If one has two distributions of metric and matter fields related by a hole transformation, manifold substantivalists must maintain that the two systems represent two distinct physical systems.
2. This physical distinctness transcends both observation and the determining power of the theory since: The two distributions are observationally identical. The laws of the theory cannot pick between the two developments of the fields into the hole.
3. Therefore the manifold substantivalist advocates an unwarranted bloating of our physical ontology and the doctrine should be discarded.
[....] The notion that the manifold represents an independently existing thing is quite natural in the realist view of physical theories. In that view one tries to construe physical theories literally. If formulated as above, a spacetime is a manifold of events with certain fields defined on the manifold. The literal reading is that this manifold is an independently existing structure that bears properties.
So far we have characterized the substantivalist doctrine as the view that spacetime has an existence independent of its contents. This formulation conjures up powerful if vague intuitive pictures, but it is not clear enough for interpretation in the context of physical theories. If we represent spacetime by a manifold of events, how do we characterize the independence of its existence? Is it the counterfactual claim that were there no metric or matter fields, there would still be a manifold of events? That counterfactual is automatically denied by the standard formulation which posits that all spacetimes have at least metrical structure. That seems too cheap a refutation of manifold substantivalism. Surely, there must be an improved formulation. Fortunately, we do not need to wrestle with finding it. For present purposes we need only consider a consequence of the substantivalist view and can set aside the task of giving a precise formulation of the substantivalist view.
In their celebrated debate over space and time, Leibniz taunted the substantivalist Newton's representative, Clarke, by asking how the world would change if East and West were switched. For Leibniz there would be no change since all spatial relations between bodies would be preserved by such a switch. But the Newtonian substantivalist had to concede that the bodies of the world were now located in different spatial positions, so the two systems were physically distinct.
Correspondingly, when we spread the metric and matter fields differently over a manifold of events, we are now assigning metrical and material properties in different ways to the events of the manifold. For example, imagine that a galaxy passes through some event E in the hole. After the hole transformation, this galaxy might not pass through that event. For the manifold substantivalist, this must be a matter of objective physical fact: either the galaxy passes through E or not. The two distributions represent two physically distinct possibilities.
[...] It is important to see that the unhappy consequence does not consist merely of a failure of determinism. We are all too familiar with such failures and it is certainly not automatic grounds for dismissal of a physical theory. The best known instance of a widely celebrated, indeterministic theory is quantum theory, where, in the standard interpretation, the measurement of a system can lead to an indeterministic collapse onto one of many possible outcomes. Less well known is that it is possible to devise indeterministic systems in classical physics as well. Most examples involves oddities such as bodies materializing at unbounded speed from spatial infinity, so called “space invaders.” (Earman, 1986a, Ch. III; see also determinism: causal) Or they may arise through the interaction of infinitely many bodies in a supertask. More recently an extremely simple example has emerged in which a single mass sits atop a dome and spontaneously sets itself into motion after an arbitrary time delay and in an arbitrary direction (Norton, 2003, Section 3).
The problem with the failure of determinism in the hole argument is not the fact of failure but the way that it fails. If we deny manifold substantivalism and accept Leibniz equivalence, then the indeterminism induced by a hole transformation is eradicated. While there are uncountably many mathematically distinct developments of the fields into the hole, under Leibniz Equivalence, they are all physically the same. That is, there is a unique development of the physical fields into the hole after all. Thus the indeterminism is a direct product of the substantivalist viewpoint. Similarly, if we accept Leibniz equivalence, then we are no longer troubled that the two distributions cannot be distinguished by any possible observation. They are merely different mathematical descriptions of the same physical reality and so should agree on all observables.
We can load up any physical theory with superfluous, phantom properties that cannot be fixed by observation. If their invisibility to observation is not sufficient warning that these properties are illegitimate, finding that they visit indeterminism onto a theory that is otherwise deterministic in this set-up ought to be warning enough. These properties are invisible to both observation and theory; they should be discarded along with any doctrine that requires their retention....