A creature living on a large enough cylinder wouldn’t know whether they live on the cylinder or the plane, and so wouldn’t know important facts about the global topology of the space they inhabit. Many physicists and philosophers of physics have the intuition that, in General Relativity, we have access only to local information; from their perspective, it wouldn’t be surprising to learn that global information is hidden from us. Regions are frequently hidden behind horizons, e.g., the event horizon of a black hole or our cosmological horizon, or in some other way.

To put a more technical result on the table, let’s review some vocabulary. A property P is local just in case P can be satisfied within a small region around a given point. In the technical jargon, two spacetimes where, for every point in the first, there is a neighborhood isometric to some neighborhood in the second, and vice versa, satisfy all of the same local properties. For example, all point-wise properties — that is, properties that can be satisfied at a single point — are local properties. A property that is not local is said to be global; for example, the overall topology of a spacetime is a global property. A God-point is a point from which one can see all of spacetime; in technical lingo, a God-point is a point whose past light cone covers all of spacetime. A spacetime A is said to have an observationally indistinguishable counterpart B just in case, for every past light cone in A, there is a point with the “same” past light cone (i.e., an isometric counterpart) in B. When A has an observationally indistinguishable counterpart B, we say that B is a nemesis for A. Notice that this relationship is asymmetric: when B is a nemesis for A, it does not follow that A is a nemesis for B. Here’s the technical result. According to the Malament-Manchak Theorem (MMT), any classical spacetime, sans God-point, satisfying any set of local properties C has an observationally indistinguishable counterpart that also satisfies C but which has a distinct global structure.

The MMT was first conjectured by David Malament in the 1970s and later proved by J.B. Manchak. Manchak’s proof makes use of the so-called clothesline construction. In the clothesline construction, one starts with an arbitrary classical spacetime sans God-point. One then duplicates this spacetime an infinite number of times and bridges together the duplicates. The duplicates are bridged together in a way that ensures that any past light cone in the original spacetime has an isometric counterpart in the clothesline. The clothesline also includes filler regions. In the original clothesline, these fillers are duplicates of the original spacetime; but the fillers can also be “surgically” modified, as can be helpful in proving additional theorems. The MMT-inspired skeptic then holds that, given the MMT and that our spacetime probably doesn’t include a God-point, we likely cannot know the global properties of the spacetime we inhabit.

The original clothesline doesn’t allow us to reach a direct conclusion regarding the beginning of the universe. For example, if the original spacetime includes a Big Bang singularity, so will the clothesline. Nonetheless, I’ve defended the position that the MMT has important corollaries that should make us skeptical concerning the beginning of the universe. For example, while the various singularity theorems require global conditions, spacetimes satisfying those conditions generally have observationally indistinguishable counterparts that do not. Moreover, while I’ve argued that spacetimes with a beginning must satisfy two global properties (everywhere past b-incompleteness and stable causality), spacetimes with those properties have observationally indistinguishable counterparts without them. I’ve also shown that dust FLRW spacetimes satisfying the various standard energy conditions and the Einstein Field Equation and that are both stably causal and everywhere past b-incomplete have observationally indistinguishable counterparts that also satisfy the various standard energy conditions and the Einstein Field Equation but that are not stably causal or everywhere past b-incomplete.

As I’ve said, the MMT helps to confirm an intuition had by many physicists and philosophers of physics: that relativistic physics allows for information to be hidden from us in ways that weren’t possible in pre-relativistic physics, and that this means we cannot know various facts about the global structure of our world. I suspect that this is why the use of the MMT in arguments for various forms of skepticism has largely gone uncriticized. (I know of only one paper launching an objection. In some sense, Gordon Belot’s book also includes an objection, but Belot is quick to point out how his objection can be overcome.) However, in explaining the MMT to philosophers without a background in philosophy of physics, I often encounter a certain degree of resistance to skeptical inferences drawn therefrom. They point out that we have encountered observational indistinguishability before – particularly in the problem of induction – and it’s not usually understood to result in a legitimate reason for drawing skeptical conclusions.

Before proceeding, let’s highlight several ways MMT-inspired skepticism differs from the traditional inductive skepticism. First, in the traditional problem of induction, we imagine a scenario involving a different set of observations than what we expect in the real world. For example, one imagines that the Sun doesn’t rise tomorrow or the discovery of a non-white swan. But in the MMT, the nemesis spacetime includes all of the same observations as in the original spacetime. By definition, there are no observers in the original spacetime who make an observation that fails to appear somewhere or other in the nemesis; there couldn’t be one, because we’ve included every possible observation made at every spacetime point in the original spacetime. In fact, supposing that one had an “eye” located at every spacetime point, there’s no observation that one would gather that rules out the nemesis spacetime. Second, in the traditional problem of induction (as well as Goodman’s new riddle), one imagines a violation of a natural law; for example, the Sun ceasing to exist tomorrow or the Earth careening out of its orbit. But this isn’t so for MMT-inspired skepticism. Recall that the nemesis spacetime preserves all of the local properties from the original spacetime. Most of the laws that appear in our most successful physical theories are stated solely in terms of local properties. While quantum entanglement affords a simple counterexample, we can only know that two particles are entangled upon receiving a signal from both of them. Hence, our knowledge of entangled particles is entirely parasitic on local information, and quantum entanglement cannot help us to reach conclusions about spacetime’s global structure.

There is a further important reason that MMT-inspired skepticism differs from the problem of induction. The traditional problem of induction is a challenge for theory choice, where a collection of past observations is consistent with multiple mutually inconsistent theories. This is distinct from another form of underdetermination, where a set of observations in combination with a fixed choice of theory fails to suffice for making a prediction. For example, suppose that we hold fixed our current theoretical understanding of radioactive decay, and are tasked with predicting the precise time at which a specific Uranium-235 nucleus will decay. (This example is inspired by Norton 2011.) We cannot do so with any precision; our credence that the U-235 nucleus will decay in, say, the next hour is determined entirely by our background theory of radioactive decay. As long as we hold fixed our theoretical understanding of radioactive decay, no additional data, drawn from, e.g., past observations of radioactive decay, should change our credence. The situation with respect to the MMT is similar; given some large pool of data, and holding fixed our background theory of spacetime, spacetime can be extended to other regions in multiple mutually incompatible ways. Just as induction doesn’t allow us to overcome the predictive underdetermination involved in the U-235 example, so, too, induction doesn’t allow us to overcome the underdetermination implied by the MMT.

Sometimes, the suggestion is made that induction can be understood via a theory of intrinsic probability.¹ Bayes’s theorem entails that the probability of a hypothesis h, given evidence e, relative to background knowledge K, is given by Pr(h|e&K) = Pr(h|K)Pr(e|h&K)/Pr(e|K). According to theories of intrinsic probability, the prior probability, Pr(h|K), is determined — at least in part — by the intrinsic probability Pr(h), that is, the probability of h conditioned only on tautological information. While most philosophers of science think that the prior probability is subjective and avoid an objective intrinsic probability altogether, the notion of objective intrinsic probability has been an important minority position; moreover, a number of prominent philosophers of religion — e.g., Richard Swinburne and Paul Draper — have offered theories of objective intrinsic probability. Here, I will consider Draper’s theory, because I consider it to be an advance beyond Swinburne’s. At any rate, similar conclusions will presumably follow for Swinburne’s to those that I offer here.

On Draper’s theory of intrinsic probability, the intrinsic probability depends on two features of h: first, modesty, that is, how much h claims about the world, and, second, coherence, that is, how well the parts of h fit together. On Draper’s view, there are inductive support relations between objectively similar property instances. For example, two white swans are objectively similar, whereas a white swan objectively differs from a black swan. Hence, the hypothesis that all swans are white postulates objective uniformity while the hypothesis that non-Australian swans are white and Australian swans are black postulates objective variety. While objective uniformity cannot be cashed out in purely syntactic terms, this does not mean — pace Goodman — that there is complete symmetry between the two hypotheses. As Draper reminded me in correspondence, the ability to recognize an objective difference in color precedes linguistic abilities; animals without language can recognize the black/white distinction, but do not recognize the wack/blite distinction. In Draper’s view of induction, intrinsic uniformity is more coherent than objective variety, and this explains why hypotheses postulating objective uniformity are more probable than hypotheses postulating objective variety.²

Draperian intrinsic probability generally ranks clotheslines as less probable than the original spacetime from which the clothesline is generated. Why? On the one hand, it is difficult to see why the clothesline would be any less coherent than the original spacetime. When we discuss inductive support relations between objectively similar property instances, we are usually discussing local property instances. Since the clothesline and the original spacetime have the same local properties, it seems to me that we should say they are equally uniform. However, the clothesline fairs worse with respect to modesty. To see this, note that the original spacetime either has finite or infinite size. Supposing that the original spacetime has finite size, the clothesline obviously postulates more, by postulating an infinitude of near-duplicates of the original spacetime. Suppose, instead, that the original spacetime has infinite size. In this case, it might not be immediately obvious that the clothesline postulates anything more than the original spacetime postulates; after all, both are infinitely large. However, Draper would say that the clothesline postulates more. In a crucial footnote, Draper tells us how he thinks about comparing the modesty of two hypotheses with an infinite number of entailments:

One might object that I am measuring immodesty by the number of logical consequences a statement has. This will not do, since every statement has infinitely many logical consequences. That is not, however, what I am doing. When I say that one statement has more content than another, I do not mean that it has a greater number of logical consequences than the other statement, because I use ‘more’ in the “all and then some” sense instead of in the “greater number” sense. Notice also that my using ‘more’ in that sense does not imply that one statement must be entailed by another statement in order to be more modest than that other statement because “all and then some” can apply, not just to whole propositions, but also to the proper parts of a proposition. […] Thus, for example, since ‘purple’ has all of the content that ‘colored’ has and then some, it follows that the statement “this ball is colored” is more modest than the statement “that ball is purple” even though each has infinitely many logical consequences and even though neither statement entails the other. (Draper 2016, p. 53)

Hence, one hypothesis can be less modest than another if one has more entailments than the other in the all-and-then-some sense. Intuitively, that is what’s going on with the clothesline, because the clothesline contains infinitely many copies of the original spacetime. The clothesline literally contains all of the points from the original spacetime and then some. Hence, clothesline spacetimes are generally less intrinsically probable than the original spacetime from which they are constructed.

Nonetheless, it’s not clear to me why it should matter that the clothesline is less intrinsically probable. On the one hand, we consider intrinsic probability in contexts where we leave aside our background knowledge and empirical data. But that’s emphatically not the context in which we consider the MMT. As with the U-235 example, we consider the MMT in the context of a fixed background theory. In other words, it’s not the intrinsic probability that matters for the MMT, but instead the posterior probability. On the other hand, all parties to the discussion already agreed that we probably do not inhabit a clothesline spacetime. In using the clothesline construction to prove the MMT, Manchak is following a long-standing research tradition, where implausible or unrealistic mathematical constructions are used to establish physically significant theorems. As Robert Geroch wrote, unrealistic models can provide “a guide in checking and discovering theorems” (Geroch 1971, p. 72), so that “the mere existence of a [model] having certain global features suggests that there are many models – some perhaps quite reasonable physically – with very similar properties” (ibid, p. 78). Likewise, for Geroch and Horowitz (1979, p. 215), unrealistic models allow us “to gain an understanding of what will work and what will not, of what needs to be and can be ruled out.” The cut-and-paste methodology used to construct the clothesline has a good pedigree: the method is also used extensively by Stephen Hawking and George Ellis in their classic (1973) work on global spacetime structure.

Significantly, the MMT does not make a statement concerning clotheslines. Instead, the MMT states that any classical spacetime, sans God-point, satisfying any set of local properties C has an observationally indistinguishable counterpart, also satisfying C, with a distinct global structure. In this sense, the question is not whether clotheslines are generally improbable, but whether globally distinct nemesis spacetimes are generally improbable. To put the point another way, the clothesline is merely one of the globally distinct nemesis spacetimes. The hypothesis that our spacetime is truly one of the globally distinct nemesi is really the disjunction of all of the possible globally distinct nemesi. Showing that one disjunct is improbable does not suffice for showing that the disjunction is improbable. Hence, even though the clothesline is improbable, very little follows for whether the long disjunction of possible nemesi is improbable.

If we had a probability distribution (or measure) over spacetimes, we could make some statement about how probable or typical nemesi with various global properties might be. Consider the as-yet unsolved measure problem. The measure problem concerns how probable or typical it might be for a specific condition to arise within some proper part of an inflationary multiverse (within, e.g., bubble universes; see, e.g., e.g., Freivogel 2011; Salem 2012; Smeenk 2014). A closely related problem — which I call the generalized measure problem — sets aside inflation (or any other specific hypothesis or mechanism) and asks how probable or typical it might be for specific conditions to obtain within any spacetime region. Physicists pragmatically focus on the measure problem and leave aside the generalized measure problem. But philosophers interested in the totality of physical reality have no such luxury. If a solution to the standard measure problem is so far beyond reach, so much the worse for the generalized measure problem. (For a summary of some of the problems involved in constructing a probability distribution over classes of spacetimes, see (Curiel unpublished).)

We face a fork: either we already know the laws or we do not. Suppose we don’t know the laws. In that case, it may be appropriate for us to consider the intrinsic probability. However, therein looms a different problem for those of us interested in the beginning of the universe. The beginning of the universe cannot be directly observed. Like with other unobservables, the scientific realist has a solution: given some background theory and our observations, we may be committed to something we cannot directly observe. But if we’re supposing that we don’t know the laws, this solution is not available to us. As far as I can tell, we lose all motivation for a beginning. Now suppose we do know the laws. In this case, it’s not appropriate for us to focus on the intrinsic probability; we should instead consider the posterior probability and consider the intrinsic probability only insofar as it affects the posterior probability. In this case, insofar as we should commit to scientific realism, we can use observations and a background theory to infer unobservables. But, in that case, we face the MMT-inspired skepticism, and the corollaries to the MMT that I’ve argued should lead us to skepticism concerning the beginning of the universe.

In a recent video, William Lane Craig has offered a distinct objection. According to Craig, skeptical arguments based on the MMT merely revive the age old skeptical arguments about the “difference between appearance and reality”. I’m not sure what Craig has in mind, because I’m not sure what “appearances” or “reality” Craig is referring to. Perhaps Craig thinks the universe appears to have one set of global properties, but the MMT-inspired skeptic points out that spacetime could have a different set of global properties. At any rate, contra Craig, the MMT-inspired skeptic does not claim that the universe differs from how it appears. Instead, the MMT-inspired skeptic claims that the universe is precisely how it appears; they allow us to have omniscient access to our past light cones. As far as our local reality is concerned, within, for example, our Hubble volume, the MMT-inspired skeptic can grant that appearance precisely matches reality, and does so with more precision than even practicing scientists would usually think reasonable. However, there is a further question about whether we can successfully extrapolate those local appearances in order to infer the global structure, where appearances are not available to us, or to anyone (except, perhaps God, if there is such an entity).

Craig proceeds to claim that if skeptical arguments based on the MMT succeed, then it would “undermine the entire field of cosmology”. But this involves a misunderstanding, either of the MMT or of cosmology. Cosmologists are not usually concerned with spacetime regions arbitrarily far beyond our observational horizon. And that’s for good reason: we have no good way to test hypotheses concerning regions arbitrarily far beyond our cosmological horizon. (This can also explain some of the disinterest that certain physicists have expressed to me concerning the MMT.) Some of the mathematical machinery that cosmologists are routinely concerned with make assumptions concerning such regions, but these assumptions should be understood as idealizations or approximations. For example, no one reasonably thinks we know spacetime to be asymptotically flat or to have a uniform average matter-energy density in regions arbitrarily far beyond those we can observe. For a cosmologist trying to understand the distribution of galaxies, the fluctuation spectrum in the CMB, the acceleration of the visible universe, galaxy filaments, and so on, concerns about regions arbitrarily far beyond our cosmological horizon are best reserved to a late night discussion over beer, or, more charitably, to the philosophers. Such concerns absolutely do not undermine their field. But such concerns are relevant for philosophers who — like Craig — are interested in whether results from physical cosmology can be successfully used as part of a case for a beginning of all of physical reality.

Craig also claims that the MMT-inspired skeptic is countenancing a violation of the Copernican Principle, that is, the principle that we do not occupy a special place in the universe. Craig claims that we need this principle for, without it, distant regions could satisfy a distinct set of laws. At least three points are worth making in reply. First, the MMT concerns nemesi satisfying all of the same local properties as the original spacetime, so that MMT-inspired skepticism is entirely consistent with holding the laws fixed. Second, the MMT does not deny the Copernican Principle. Two observationally indistinguishable but globally distinct spacetimes can both satisfy the Copernican Principle. (In the clothesline construction, where are the special observers that violate the Copernican Principle?) Third, we can distinguish laws from other kinds of general assumptions often used in developing cosmological models. The latter — which we might call mere principles — include not only the Copernican Principle, but also the Cosmological Principle. Cosmology has arguably moved beyond merely assuming such principles and, instead, now pursues hypotheses where those principles are the generic results of almost arbitrary initial conditions.

In a book published in the late 1970s, Amal Kumar Raychaudhuri described physicists who were not happy to simply assume the Cosmological Principle. They wanted “to demonstrate that starting with arbitrary initial conditions, physical processes lead to an evolution of a universe obeying the cosmological principle – inhomogeneities and anisotropies being smoothed out in a time small compared to the age of the universe” (Raychaudhuri 1979, p. 4). There are two paths one could pursue. First, one could start with an almost arbitrary spacetime and, evolving it forward through time, show that one achieves an approximately FLRW spacetime. This path has not borne fruit. Second, one could start with an almost arbitrary spacetime and, evolving it forward through time, show that one achieves a spacetime with a large approximately FLRW region. This is the popular path that, for example, inflationary cosmology pursued in the decade after 1979. As George Ellis, Roy Maartens, and Malcolm MacCallum summarize,

[…] on any reasonable measure the FLRW models are of zero probability within the family of all possible cosmological models. Thus this assumption implies that the universe is of an extremely special, hence highly fine-tuned, nature. From the late 1930s to the 1960s, this was taken as reasonable. However, in the late 1970s, the philosophical tide turned: it became common to assume the universe has a generic rather than special geometric nature. Indeed it was no longer assumed there were any specific cosmological principles at all (Ellis, Maartens, & MacCallum 2012, p. 347).

Of course, this change in cosmological methodology is more grist for the MMT-inspired skeptic’s mill; on this view, spacetime may have almost any global structure, as long as some sufficiently large region has the correct features. In inflationary cosmology, the visible universe is homogeneous and isotropic — and so satisfies the Cosmological Principle — only because a small region — which may have differed almost arbitrarily from the rest of spacetime — underwent inflation. Since spacetime, as a whole, may dramatically differ from that region, it’s difficult to understand how such principles could allow one to infer specific global properties.

Of course, inflation only serves as an example of a broader theme. Inflation is one way of showing how our approximately FLRW universe could result generically, without assuming various cosmological principles. Other research programs, such as ekpyrotic cosmology, pursue the same end without an inflationary mechanism. More generally, whichever mechanism one postulates, the view that the visible universe is a generic result from almost arbitrary initial conditions should make us hesitant to postulate any specific global properties.

Disclaimer: The posts on the Cosmotherium should never be taken as definitive and I am typically not completely convinced of what I post here. This is my place for working out my views without the pressure or rigor of publication.

End Notes

1. In the original version of this post, I wrote the problem of induction is sometimes thought to be overcome via a theory of intrinsic probability. That’s incorrect; the theory of intrinsic probability discussed here presupposes that the problem of induction can be resolved and then assigns intrinsic probabilities based upon a theory of induction.
2. This is where Draper’s theory of intrinsic probability assumes that the problem of induction has a solution. The statement that objective uniformity is more probable than objective variety presupposes induction.

References

Curiel, E. Unpublished. ‘Measure, Topology and Probabilistic Reasoning in Cosmology’. https://arxiv.org/abs/1509.01878

Draper, P. 2016. ‘Simplicity and Natural Theology’. In Michael Bergmann & Jeffrey Brower (Eds.), Reason and Faith: Themes from Richard Swinburne. Oxford University Press, pp. 48-63.

Ellis, G., Maartens, R., & MacCallum, M. 2012. Relativistic Cosmology. Cambridge University Press.

Freivogel, 2011. ‘Making predictions in the multiverse’. Classical and Quantum Gravity 28(20).

Geroch, R. 1971. “Space-Time Structure From a Global Viewpoint”. In B.K. Sachs (Ed.), General Relativity and Cosmology. Academic Press, pp. 71-103.

Geroch, R. & Horowitz, G. 1979. ‘Global structure of spacetime’. In Hawking, S. & Israel, W. (Eds.), General Relativity: An Einstein Centenary Survey. Cambridge University Press, pp. 212-293.

Hawking, S. & Ellis, G. 1973. The Large-Scale Structure of Space-Time. Cambridge University Press.

Norton, J. 2011. ‘Observationally Indistinguishable Spacetimes: A Challenge for Any Inductivist’. In Gregory Morgan (Ed.), Philosophy of Science Matters: The Philosophy of Peter Achinstein, Oxford University Press, pp. 164-176.

Raychaudhuri, A. 1979. Theoretical Cosmology. Clarendon Press.

Salem, M. 2012. ‘Bubble collisions and measures of the multiverse’. Journal of Cosmology and Astroparticle Physics, pp. 1-39.

Smeenk, C. 2014. ‘Predictability crisis in early universe cosmology’. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 46(A), pp. 122-133.