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Are Cosmological Theories Compatible With All Possible Evidence? A Missing Epistemological Link
Draft
Dr. Nick Bostrom
Department of Philosophy
Yale University
New Haven, Connecticut 06520
U. S. A.
Homepage: http://www.nickbostrom.com
Email: nick@nickbostrom.com
Are Cosmological Theories Compatible With All Possible Evidence? A Missing Epistemological Link*
I.
Space is big. It is very, very big. On the currently most favored cosmological theories we are living in an infinite world, a world that contains an infinite number of planets, stars, galaxies and black holes. This is an implication of most multiverse theories, according to which our universe is just one in a vast ensemble of physically real universes. But it is also a consequence of the standard Big Bang cosmology, if combined with the assumption that our universe is open, as recent evidence strongly suggests it is. An open universe assuming the simplest topology is spatially infinite an every point in time, and contains infinitely many planets etc.
Philosophical investigations relating to the vastness of cosmos have focused on the finetuning of our universe. Finetuning refers to the alleged fact that the laws of physics are such that if any of several physical constants had been even slightly different, then life would not have existed. A philosophical cottage industry has arisen out of the controversies surrounding issues such as whether finetuning is in some sense improbable, whether it should be regarded as surprising ( ADDIN ENRfu [3], ADDIN ENRfu [4]), whether it calls out for explanation (and if so whether a multiverse theory could explain it ( ADDIN ENRfu [5], ADDIN ENRfu [6])), whether it suggests ways in which current physics is incomplete ADDIN ENRfu [7], or whether it is evidence for the hypothesis that our universe resulted from design ADDIN ENRfu [8].
Here I wish instead to address a more fundamental problem: How can vastworld cosmologies have any observational consequences at all? I will show that these cosmologies imply (or give a very high probability to) the proposition that every possible observation is in fact made. This creates a challenge: if a theory is such that for any possible human observation that we specify, the theory says that that observation will be made, then how do we test the theory? I call this a challenge, because current cosmological theories clearly do have some connection to observation. Cosmologists are constantly modifying and refining theories in light of empirical findings, and they are presumably not irrational in doing so. And in fact, I think that the challenge can be met. A later section proposes a new methodological principle that I argue can provide the needed link between cosmological theory and observation.
II.
Consider a random phenomenon, for example Hawking radiation. When black holes evaporate, they do so in a random manner such that for any given physical object there is a finite (although astronomically small) probability that it will be emitted by any given black hole in a given time interval. Such things as boots, computers, or ecosystems have some finite probability of popping out from a black hole. The same holds of course for human bodies, or human brains in particular states. Assuming that mental states supervene on brain states, there is thus a finite probability that a black hole will produce a brain in a state of making any given observation. Some of the observations made by such a brains will be illusory, and some will be veridical. For example, some brains produced by black holes will have the illusory of experience of reading a measurement device that does not exist. Other brains, with the same experiences, will be making veridical observations a measurement device may materialize together with the brain and may have caused the brain to make the observation. But the point that matters here is that any observation we could make has a finite probability of being produced by any given black hole.
The probability of anything macroscopic and organized appearing from a black hole is, of course, minuscule. The probability of a given conscious brainstate being created is even tinier. Yet even a lowprobability outcome has a high probability of occurring if the random process is repeated often enough. And that is precisely what happens in our world, if cosmos is very vast. In the limiting case where cosmos contains an infinite number of black holes, the probability of any given observation being made is one.
There are good grounds for believing that our universe is open and contains an infinite number of black holes. Therefore, we have reason to think that any possible human observation is in fact instantiated in the actual world. Evidence for the existence of a multiverse would only add further support to this proposition.
It is not necessary to invoke black holes to make this point. Any random physical phenomenon would do. It seems we dont even have to limit the argument to quantum fluctuations. Classical thermal fluctuations could, presumably, in principle lead to the molecules in a cloud of gas, which contains the right elements, to spontaneously bump into each other to form a biological structure such as a human brain.
The problem is that it seems impossible to get any empirical evidence that could distinguish between various Big World theories. For any observation we make, all such theories assign a probability of one to the hypothesis that that observation be made. That means that the fact that the observation is made is no reason whatever to prefer one of these theories to the others. Experimental results appear totally irrelevant.
We can see this formally as follows. Let B be the proposition that we are in a Big World, defined as one that is big enough and random enough to make it highly probable that every possible human observation is made. Let T be some theory that is compatible with B, and let be E some proposition asserting that some specific observation is made. Let P be an epistemic probability function. Bayess theorem states that
P(TE&B) = P(ET&B)P(TB) / P(EB).
In order to determine whether E makes a difference to the probability of T (relative to the background assumption B), we need to compute the difference P(TE&B)  P(TB). By some simple algebra, it is easy to see that
P(TE&B)  P(TB) EMBED Equation.3 0 if and only if P(ET&B) EMBED Equation.3 P(EB).
This means that E will fail to give empirical support to E (modulo B) if E is about equally probable given T&B as it is given B. We saw above that P(ET&B) EMBED Equation.3 P(EB) EMBED Equation.3 1. Consequently, whether E is true or false is irrelevant for whether we should believe in T, given we know that B.
Let T2 be some perverse permutation of an astrophysical theory T1 that we actually embrace. T2 differs from the T1 by assigning a different value to some physical constant; to be specific, let us suppose that T1 says that the temperature of the cosmic microwave background radiation is about 2.7 Kelvin (which is the observed value) whereas T2 says it is, say, 3.1 K. Suppose furthermore that both T1 and T2 say that we are living in a Big World. One would have thought that our experimental evidence favors T1 over T2. Yet, the above argument seems to show that this view is mistaken. Our observational evidence is supports T2 just as much as T1. We really have no reason to think that the background radiation is 2.7 K rather than 3.1 K.
III.
At first sight, it could seem as if this simply rehashes the lesson, familiar from Duhem and Quine, that it is always possible to rescue a theory from falsification by modifying some auxiliary assumption, so that strictly speaking no scientific theory ever implies any observational consequences. The above argument would then merely have provided an illustration of how this general result applies to cosmological theories. But this would miss the point.
If the argument given above is correct, it establishes a much more radical conclusion. It purports to show that all Big World theories are not only logically compatible with any observational evidence, but they are also perfectly probabilistically compatible. They all give the same conditional probability (namely one) to every observation statement E defined as above. This entails that no such observation statement can have any bearing, whether logical or probabilistic, on whether the theory is true. If that were the case, it would not seem worthwhile to make astronomical observations if what we are interested in is determining which Big World theory to favor. The only reasons we could have for choosing between such theories would be either a priori ones (simplicity, elegance etc.) or pragmatic ones (such as ease of calculation).
Nor is the argument making the ancient statement that human epistemic faculties are fallible, that we can never be certain that we are not dreaming or are brains in a vat. No, the point here is not that such illusions could occur, but rather that we have reason to believe that they do occur, not just some of them but all possible ones. In other words, we can be fairly confident that the observations we make, along with all possible observations we could make in the future, are being made by brains in vats and by humans that have spontaneously materialized from black holes or from thermal fluctuations. The argument would entail that this abundance of observations makes it impossible to derive distinguishing observational consequences from contemporary cosmological theories.
IV.
Most readers will find this conclusion unacceptable. Or so, at least, I hope. Cosmologists certainly appear to be doing experimental work and modify their theories in light of new empirical findings. The COBE satellite, the Hubble Space Telescope, and other devices have are showering us with a wealth of data, causing a minor renaissance in the world of astrophysics. Yet the argument described above would show that the empirical import of this information could never go beyond the limited role of providing support for the hypothesis that we are living in a Big World, for instance by showing that the universe is open. Nothing apart from this one fact could be learnt from such observations. Once we have established that the universe is open and infinite, then any further work in observational astronomy would be a waste of time and money.
Worse still, the leaky connection between theory and observation in cosmology spills over into other domains. Since nothing hinges on how we defined T in the derivation above, the argument can easily be extended to prove that observation does not have a bearing on any scientific question as long as we assume that we are living in a Big World.
This consequence is absurd, so we should look for a way to fix the methodological pipe and restore the flow of testable observational consequences from Big World theories. How can we do that?
V.
It may seem as our troubles originate from the somewhat technical point: that in a large enough cosmos, every observation will be made by some observers here and there. It remains the case, however, that those observers will exceedingly rare and far between. For every observation made by a freak observer spontaneously materializing from Hawking radiation or thermal fluctuations, there are trillions and trillions of observations made by regular observers that have evolved on planets like our own, and who make veridical observations of the universe they are living in. Why can we not solve the problem, then, by saying that although all these freak observers exist and are suffering from various illusions, it is highly unlikely that we are among their numbers? Then we should think, rather, that we are very probably one of the regular observers whose observations reflect reality. We could safely ignore the freak observers and their illusions in most contexts when doing science.
In my view, this response suggests the right way to proceed! Because the freak observers are in such a tiny minority, their observations can be disregarded for most purposes. It is possible that we are freak observers: we should assign to that hypothesis some finite probability but such a tiny one that it does not make any practical difference.
If we want to go with this idea, it is crucial that we construe our evidence differently than we did above. If our evidence is simply Such and such an observation is made. then the evidence has probability one given any Big World theory and we ram or heads straight into the problems we described. But if when we construe our evidence in the more specific form We are making such and such observations., we have a way out. For we can then say that although Big World theories make it probable that some such observations be made, they need not make it probable that we should be the ones making them.
Let us therefore define:
E := Such and such observations are made by us.
E contains an indexical component that the original evidencestatement we considered, E, did not. E is thus logically stronger than E. The rationality requirement that one should take all relevant evidence into account dictates that in case E leads to different conclusions than does E, then it is E that determines what we ought to believe.
A question that now arises is, how to determine the evidential bearing that statements of the form of E have on cosmological theories? Using Bayess theorem, we can turn the question around and ask, how do we evaluate P(ET&B), the conditional probability that a Big World theory gives to us making certain observations? The argument in foregoing sections showed that if we hope to be able to derive any empirical implications from Big World theories, then P(ET&B) should not generally be set to unity or close to unity. P(ET&B) must take on values that depend on the particular theory and the particular evidence that we are we are considering. Some theories T are supported by some evidence E; for these choices P(ET&B) is relatively large. For other choices of E and T, the conditional probability will be relatively small.
To be concrete, consider the two rival theories T1 and T2 about the temperature of the cosmic microwave background. Let E be the proposition that we have made those observations that cosmologists innocently take to support T1. E includes readings from radio telescopes etc. Intuitively, we want P(ET1&B) > P(ET2&B). That inequality must be the reason why cosmologists believe that the background radiation is in accordance with T1 rather than T2, since a priori there is no ground for assigning T1 a substantially greater probability than T2.
A natural way in which we can achieve this result is by postulating that we should think of ourselves as being in some sense random observers. Here we use the idea that the essential difference between T1 and T2 is that the fraction of observers that would be making observations in agreement with E is enormously greater on T1 than on T2. If we reason as if we were randomly selected samples from the set of all observers, or from some suitable subset thereof, then we can explicate the conditional probability P(ET&B) in terms of the expected fraction of all observers in the reference class that the conjunction of T and V says would be making the kind of observations that E says that we are making. As we shall see, this postulate enables us to conclude that P(ET1&B) > P(ET2&B).
Let us call this postulate the Self Sampling Assumption:
(SSA) Observers should reason as if they were a random sample from the set of all observers in their reference class.
The general problem of how to define the reference class is complicated, and I shall not address it here. For the purposes of this paper, we can think of the reference class as consisting of all observers who will ever have existed. We can also assume a uniform sampling density over this reference class. Moreover, it simplifies things if we set aside complications arising from assigning probabilities over infinite domains by assuming that B entails that the number of observers is finite, albeit such a large finite number that the problems described above obtain. Making these assumptions enables us to focus on basic principles.
Here is how SSA supplies the missing link needed to connect theories like T1 and T2 to observation. On T2, the only observers who observe an apparent temperature of the cosmic microwave background CBM EMBED Equation.3 2.7 K, are those that have various sorts of rare illusions, for example because their brains have been generated by black holes and are therefore not attuned to the world they are living in. On T1, by contrast, every observer who makes the appropriate astronomical measurements and is not deluded will observe CBM EMBED Equation.3 2.7 K. A much greater fraction of the observers in the reference class observe CBM EMBED Equation.3 2.7 K if T1 is true than if T2 is true. By SSA, we consider ourselves as random observers; it follows that on T1 we would be more likely to find ourselves as one of those observers who observe CBM EMBED Equation.3 2.7 K than we would on T2. Therefore P(ET1&B) >> P(E T2&B). Supposing that the prior probabilities of T1 and T2 are roughly the same, P(T1) EMBED Equation.3 P(T2), it is then trivial to derive via Bayess theorem that P(T1E&B) > P(T2E&B). This vindicates the intuitive view that we do have empirical evidence that favors T1 over T2.
The job that SSA is doing in this derivation is to enable the step from a proposition about fractions of observers to propositions about corresponding probabilities. We get the propositions about fractions of observers by analyzing T1 and T2 and combining them with relevant background information B; from this we conclude that there would be an extremely small fraction of observers observing CBM EMBED Equation.3 2.7 K given T2 and a much larger fraction given T1. We then consider the evidence E, which is that we are observing CBM EMBED Equation.3 2.7 K. SSA authorizes us to think of the we as a kind of random variable ranging over the class of actual observers. From this it then follows that E is more probable given T1 than given T2. But without assuming SSA, all we can say is that a greater fraction of observers observe CBM EMBED Equation.3 2.7 K if T1 is true; at that point the argument would grind to a halt. We could not reach the conclusion that T1 is supported over T2. It is for this reason that I propose that SSA, or something like it, be adopted as a methodological principle.
It may seem mysterious how probabilities of this sort can exist how can we possibly make sense of the idea that there was some chance that we might have been other observers than we are? However, what I am suggesting here is not the existence of some objective, or physical, chances. I am not suggesting that there is a physical randomization mechanism, a cosmic fortune wheel, as it were, that assigns souls to bodies in a stochastic manner. Rather, we should think of these probabilities as epistemic. They are part of a proposal explicating the epistemic relations that hold between theories (such as T1 and T2) and evidence (such as E) that contains an indexical component. The status of SSA could be regarded as in some respects akin to that of the David Lewiss Principal Principle.
VI.
It may be worth illustrating how SSA works by considering a simple thought experiment.
Blackbeards and Whitebeards.
In an otherwise emtpy world there are three rooms. God tosses a fair coin and creates three observers as a result, placing them in different rooms. If the coin falls heads, He creates two observers with black beards and one with white beard. If it falls heads, it is the other way around: He creates two whitebeards and one blackbeard. All observers are aware of the setup. There is a mirror in each room, so the observers know what color their beard is. You find yourself in one of the rooms, as a blackbeard. What credence should you give to the hypothesis that the coin fell heads?
The situation is depicted in Figure 1.
Because of the direct analogy to the cosmology case, we know that the answer must be that you should assign a greater credence to Heads than to Tails. Let us apply SSA and see how we get this result.
From the setup, we know that the prior probability of Heads is 50%. This is the probability you should assign to Heads before you have looked in the mirror and thus before you know what beard color you have. (That this probability is 50% follows from the Principal Principle together with the fact you know that the coin toss was fair.) We thus have
P(Heads) = P(Tails) = 1/2.
Next we consider the conditional probability of you observing that you have black beard given a specific outcome of the toss. If the coin fell heads, then two out of three observers observe having black beard. If the coin fell tails, then one out of three observe having black beard. By SSA, you reason as if you were a randomly sampled observer. This gives
P(Black  Heads) = 2/3
P(Black  Tails) = 1/3.
Using Bayess theorem, we can then calculate the conditional probability of Heads given that you have black beard:
EMBED Equation.3
EMBED Equation.3 .
After looking in the mirror and learning that your beard is black, you should therefore assign a credence of 2/3 to Heads and 1/3 to Tails.
This result mirrors that of the cosmology example. Because one theory (T1, Heads) entails that a greater fraction of all observers are observing what you are observing (E, Black) than does another theory (T2, Tails), the former theory obtains preferential support from your observation.
VII.
Big World theories are popular in contemporary cosmology. This paper has attempted to show that such theories give rise to a peculiar methodological problem: because the theories say that the world is so big, they imply (or make highly probable) that every possible human observation is in fact made. The prima facie difficulty is that if the theory assigns a very high conditional probability to every observation we could make, then how could such a theory possibly be refuted by our observations? How could we ever have empirical reasons for preferring one such theory over another, if they all fit equally well with whatever we observe? This problems is different from and more radical than the problem of underdetermination of theory by data associated with Duhem and Quine. And it spills over into other domains than cosmology.
After describing this problem, I proposed a solution. By introducing a new methodological principle, the SelfSampling Assumption, we saw that it is possible to connect Big World theories to observation in an intuitive way that vindicates the practices of scientists who are using experimental findings to evaluate cosmological hypotheses.
We can explore implications of the SelfSampling Assumption in variations of the Blackbeards and Whitebeards thought experiment, suggesting avenues for further research. Some questions for future studies to address are: how SSA relates to the various socalled anthropic principles; whether it can be used to develop an epistemology for selflocating belief and decision rules for game theoretic problems involving imperfect recall (see e.g. ADDIN ENRfu [14], ADDIN ENRfu [15]); and whether we can use SSA as a building block in a general theory of observational selection effects, which may be responsible for subtle biases in some fields such as evolutionary biology ADDIN ENRfu [16].
References
ADDIN ENBbu 1. LachizeRey, M. and J.P. Luminet. 1995. "Cosmic Topology". Physics Reports, 254(3): 135214.
2. Martin, J.L. 1995. General Relativity. 3rd ed., London: Prentice Hall.
3. Earman, J. 1987. "The SAP also rises: a critical examination of the anthropic principle". Philosophical Quarterly, 24(4): 30717.
4. Leslie, J. 1989. Universes. London: Routledge.
5. Smith, Q. 1994. "Anthropic explanations in cosmology". Australasian Journal of Philosophy, 72(3): 371382.
6. Hacking, I. 1987. "The inverse gambler's fallacy: the argument from design. The anthropic principle applied to wheeler universes". Mind, 76: 33140.
7. McMullin, E. 1993. "Indifference Principle and Anthropic Principle in Cosmology". Stud. Hist. Phil. Sci., 24(3): 359389.
8. Swinburne, R., Argument from the finetuning of the universe, in Physical cosmology and philosophy, J. Leslie, Editor. 1990, Collier Macmillan: New York. p. 15473.
9. Vilenkin, A. 1998. "Unambiguous probabilities in an eternally inflating universe". Phys. Rev. Lett., 81: 55015504.
10. Linde, A. and A. Mezhlumian. 1996. "On Regularization Scheme Dependence of Predictions in Inflationary Cosmology". Phys. Rev. D, 53: 42674274.
11. Lewis, D. 1986. Philosophical Papers. Vol. 2., New York: Oxford University Press.
12. Lewis, D. 1994. "Humean Supervenience Debugged". Mind, 103(412): 47390.
13. Mellor, H. 1971. The matter of chance. Cambridge: Cambridge University Press.
14. Piccione, M. and A. Rubinstein. 1997. "On the Interpretation of Decision Problems with Imperfect Recall". Games and Economic Behaviour, 20: 324.
15. Bostrom, N. 2000. "Observerrelative chances in anthropic reasoning?". Erkenntnis, 52: 93108.
16. Carter, B. 1983. "The anthropic principle and its implications for biological evolution". Phil. Trans. R. Soc., A 310: 347363.
* Im grateful for comments to Elliott Sober, Peter Milne, Colin Howson, Craig Callender, Milan Cirkovic, and audience members of the British Society for the Philosophy of Science Annual Conference 2000.
I.e. that space is simply connected. There is a recent burst of interest in the possibility that our universe might be multiply connected, in which case it could be both finite and hyperbolic. A multiply connected space could lead to a telltale pattern consisting of a superposition of multiple images of the night sky seen at varying distances from Earth (roughly, one image for each lap around the universe which the light has traveled). Such a pattern has not been found, although the search continues. For an introduction to multiply connected topologies in cosmology, see ADDIN ENRfu [1].
A widespread misconception is that the open universe in the standard Big Bang model becomes spatially infinite only in the temporal limit. The observable universe is finite, but only a small part of the whole is observable (by us). One fallacious intuition that might be responsible for this misconception is that the universe came into existence at some spatial point in the Big Bang. A better way of picturing things is to imagine space as an infinite rubber sheet, and gravitationally bound groupings (such as stars and galaxies) as buttons glued on. As we move forward in time, the sheet is stretched in all directions so that the separation between the buttons increases. Going backwards in time, we imagine the buttons coming closer together until, at time zero, the density of the (still spatially infinite) universe becomes infinite everywhere. See e.g. ADDIN ENRfu [2].
In fact, there is a probability of unity that infinitely many such observers will appear. But one observer will suffice for our purposes.
I restrict the assertion to human observations in order to avoid questions as to whether there may be other kinds of possible observations that perhaps could have infinite complexity or be of some alien or divine nature that does not supervene on stuff that is emitted from black holes such stuff is physical and of finitely bounded size and weight.
Some cosmologists are recently becoming aware of the problematic that this paper describes (e.g. ADDIN ENRfu [9], ADDIN ENRfu [10]).
The Principal Principle purports to express the connection between physical chance to epistemic credence; see e.g. ADDIN ENRfu [11], ADDIN ENRfu [12]. A similar principle had earlier been introduced by Hugh Mellor ADDIN ENRfu [13].
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