The second installment of a series dedicated to pondering one or two ideas from each chapter of Nick Bostrom’s 2002 book, Anthropic Bias: Observation Selection Effects in Science and Philosophy. The first installment is here; it includes a brief preface about my goals and guidelines for the series.
The Chapter 2 survey will span three blog posts:
Part I discusses attempts at relating Ian Hacking’s Inverse Gambler’s Fallacy to subjective probabilities about cosmology. I go through the text quickly, hitting key points and pausing here and there to ponder the arguments. I also spend some time trying to be sure I understand those arguments, while explaining notation and background technical concepts as best I can (I’m not sure if this helps or hurts). I can’t be sure I’m getting anyone’s claims entirely correct. What you get here is mostly a thinking-out-loud, real-time reading, for better or worse. Corrections are encouraged.
Part II will focus on the notion of surprising vs. unsurprising improbable events, and will include mostly my own ruminations, as I have for some time found that to be an exceedingly thought-provoking and supremely confusing notion.
Part III will return to Bostrom’s book to finish surveying up the chapter.
Chapter 2: Fine-Tuning in Cosmology
The chapter begins with some definitions:
“Fine-tuning” refers to the supposed fact that there is a set of cosmological parameters or fundamental physical constants that are such that had they been very slightly different, the universe would have been void of intelligent life. … Our universe, having just the right conditions for life, appears to be balancing on a knife’s edge. (Bostrom 11)
The design hypothesis states that our universe is the result of purposeful design. The “agent” doing the designing need not be a theistic God, although this is of course one archetypal version of the design hypothesis. (Bostrom 11)
[The ensemble hypothesis] states that the universe we observe is only a small part of the totality of physical existence. This totality need not be fine-tuned. If it is sufficiently big and variegated, so that it was likely to contain as a proper part the sort of fine-tuned universe we observe, then an observation selection effect can be invoked to explained why we see a fine-tuned universe. The usual form of the ensemble hypothesis is that our universe is but one in a vast ensemble of actually existing universes, the totality of which we can call “the multiverse.” (Bostrom 12)
Following some brief elaboration, Bostrom notes that, “in contrast to some versions of the design hypothesis, the meaningfulness of the ensemble hypothesis is not much in question” (Bostrom 13). The long and short of it, unsurprisingly, is that unless we’re already convinced of the existence of a universe-designer, we should put much higher credence in the ensemble hypothesis than in the designer hypothesis; and so most of our serious thinking will happen in the neighborhood of the ensemble hypothesis. That’s fine by me.
Bostrom then asks whether assuming the universe is fine-tuned needs explaining, given that it “is an empirical assumption that is not trivial” (Bostrom 13). Some careful thought follows, during which he notes that “the case for fine-tuning is quite strong” (Bostrom 14) and ultimately develops three questions that will serve as the rest of the chapter’s focus:
…is the fact that the universe would have been lifeless if all the values of fundamental constants had been very slightly different (assuming this is a fact) relevant in assessing whether an explanation is called for of why the constants have the values they have? And does it give support to the multiverse hypothesis? Or, alternatively, to the design hypothesis? (Bostrom 16)
As alluded to above, the first two questions will be emphasized over the third. To get us started, Bostrom introduces anthropic reasoning into the mix…
Can an anthropic argument based on an observation selection effect together with the assumption that an ensemble of universes exists explain the apparent fine-tuning of our universe? (Bostrom 16)
…then directs us to Hacking’s 1987 answer in the affirmative, conditional on the ensemble consisting of all possible big-bang universes (a position Hacking attributes to Brandon Carter):
Why do we exist? Because we are a possible universe [sic], and all possible ones exist. Why are we in an orderly universe? Because the only universes that we could observe are orderly ones that support our form of life… nothing is left to chance. Everything in this reasoning is deductive. (Bostrom 16)1
Bostrom notes that Hacking contrasts this with a (John Archibald) Wheeler-type multiverse explanation, in which “there is a never-ending sequence of universes each of which begins with a big bang and ends with a big crunch which bounces back in a new big bang, and so forth” (Bostrom 16–17). Anthropic reasoning in that case would be that, given enough time, a universe like ours would eventually instantiate. Bostrom doesn’t see much difference in this line of reasoning and that of the Carter-type case (I agree, provided we have good reason to believe that every possible permutation will instantiate). Hacking, however, thinks Wheeler commits the “Inverse Gambler’s Fallacy”:
[A gambler] enters the room as a roll is about to be made. The kibitzer asks, ‘Is this the first roll of the dice, do you think, or have we made many a one earlier tonight?’ … slyly, he says, ‘Can I wait until I see how this roll comes out, before I lay my bet with you on the number of past plays made tonight?’ The kibitzer … agrees. The roll is double-six. The gambler foolishly says, ‘Ha, that makes a difference—I think there have been quite a few rolls.’ (Bostrom 17)2
The gambler’s mistake is obvious, but I’ll ponder it for a moment.
On the one hand, double-six has the same probability as any other two-die outcome: 1/36. But the roll has special meaning, even beyond its being a double (would the gambler have been as impressed by a double-four, i.e., “hard eight”?). Namely, double-six is the only roll that adds to twelve. Most rolls add to seven: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}. There is no “hard seven.” (Note that there are five ways to sum to eight, but of course only one hard eight.) The gambler—whether good or bad at his vocation—need not explicitly have all this in mind to assign special meaning to a given outcome. In this case the outcome is the high side of a “horn bet” (the low side being a double-one, which is the only way to sum to two). Our probability intuitions tend to go out of whack when events have special meaning, a theme that will come up a lot during my exploration of this book.
Notice that there are other properties that could be probabilistically significant for—i.e., generate yet more meaning for—a given roll; for example, which numbers on the dice directly face the gambler (while the sixes face the ceiling).
Looking now at frequencies: An experienced gambler will have seen relatively few double-sixes. On average, you’d expect to see that roughly once per 36 tosses (as with any other two-die outcome; but again, not all outcomes are equally meaningful). And to feel 99% confident in hitting a double-six, you’d need to expect to roll the dice 164 times. Part of what leads us to fallacy is that these are averages. So, if you chuck the dice once and don’t get double-six (there’s a 97.2% chance of that), you should reset and still commit to 164 throws to maintain 99% confidence for seeing a double-six.
In other words, if in the unlikely event you throw 163 fair rolls and haven’t yet seen a double-six, it’d be fallacious to think you’re now due. This is, in fact, the classic gambler’s fallacy. Hacking’s gambler commits the inverse of this by seeing double-six and inferring something about what came before.
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Sidebar
In case you’d like to see the math for the above (using the Desmos and WolframAlpha calculators):
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(*Note that this is not the same as the number of expected rolls of a single die to get two sixes in a row, which is 42. Perhaps at some point I’ll attempt an intuitive explanation of how to calculate such examples. An intuitively neutral approach is to use this formula, where [1/6] is the probability of rolling a six: [1 + (1/6)]÷[1/6]^2. I prefer an approach that reflects the sorts of stories we might tell about how we got two sixes in a row, and that what we are calculating is the weighted average of those stories [e.g., 1/36th of the time, the story will be, “My first two rolls were sixes, so it only took me two rolls!”], but the formula works for now.)
Back to Bostrom…
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Bostrom claims that, while the gambler makes a mistake, so does Hacking in relating this analogy to fine-tuning. Namely, the example lacks the essential ingredient of observation selection effect. A closer example, Bostrom notes, would be to have the gambler wait outside until a double-six is rolled. The gambler, who’s aware of this arrangement, would now have reason to think many rolls preceded the “fine-tuned” one he sees, and he’d have an explanation for why he’s seeing that roll rather than a more likely one. This correction makes the example friendly to the Wheeler’s reasoning. Maybe Hacking’s analogy could be nudged closer to an example of anthropic bias, without making it supportive of Wheeler’s reasoning, by blocking the gambler’s awareness of time; but fair enough.
Our attention is then turned to a more recent attack on the anthropic explanation of fine-tuning from Roger White and Phil Dowe3, who home in on a particular point of Hacking’s about where the gambler goes wrong; namely, in failing to take into account the most specific version of the event that needs explaining, i.e., the double-six he just saw, rather than “some double-six.” From Hacking:
…there is another fact which one seeks to explain. This is the fact that double-six occurred somewhere or other in a sequence of trials … it may correctly be urged that since we know that double-six has just occurred, we know the disjunctive fact that double-six occurred at least once in a sequence of one trial, or of two trials, or of three trials … or of many trials. …
Here we must make a remark about inference to the best explanation. If F is known, and E is the best explanation of F, then we are supposed to infer E. However, we cannot give this rule carte blanche. If F is known, then F-or-G is known, but E* might be the best explanation of F-or-G, and yet knowledge of F gives not the slightest reason to believe E*. (John, an excellent swimmer, drowns in Lake Ontario. Therefore he drowns in either Lake Ontario or the Gulf of Mexico. At the time of his death, a hurricane is ravaging the Gulf. So the best explanation of why he drowned is that he was overtaken by a hurricane, which is absurd.) We must insist that F, the fact to be explained, is the most specific version of what is known and not a disjunctive consequence of what is known. In the case of double-six the most specific version is that double-six has just occurred. That is the fact to be explained, and a long run of previous trials is of no value in explaining that fact. (Hacking 335–336)4
I’ll reference this quote again below, after considering some criticisms from Bostrom. For now, I’ll summarize the takeaway her as: the “many trials” inference appears (to the gambler) to be the best explanation for the double-six that “just occurred” given the reasonable assumption that “rare events are more likely to occur somewhere or other in a long run of trials than in a short one” (Hacking 334); but this is a fallacious inference given that this reasoning only works for a double-six occurring “somewhere or another.” Hacking does a nice job of developing this idea, which he then goes on to apply to fine-tuning, as do White and Dowe. Their essential claim, as illustrated by the gambler’s error, is that postulating many universes doesn’t get us closer to explaining why this universe is fine-tuned.
To investigate this claim more closely, Bostrom introduces some formal notation borrowed from White (whose argument will be emphasized from here on):
There are n possible big bang universe configurations, represented as: {T1, T2, T3, …, Tn}.
Each big bang universe configuration is equally probable: P(Ti) = 1/n. (It’s like rolling a fair, n-sided die.)
Assume that only configuration T1 permits life to evolve.
Let x be a variable that ranges over the set of actually existing universes. (So, of David Bowie’s birth universe[s], we can say “all x in which David Bowie was born.”)
Assume ∀x∃!i(Tix), where “∀” (or the “universal quantifier”) means “all” or “each” or “every”; and “∃” (the “existential quantifier”) means “there exists” or “some”; and “∃!” (the “unique existential quantifier”) means “there exists exactly one.”5 Bostrom and White verbalize this as “Each universe instantiates a unique Ti.” Colloquially, that might seem to suggest that no two universes can be identically configured, which isn’t the case. Rather, it’s more like, “For each universe, there is exactly one configuration that universe instantiates,” or “Each universe takes on exactly one configuration from the set of possible configurations.” It basically describes a function such that, for any given big bang you input, you’ll get an output of exactly one universe configuration.
Let m be the number of actually existing universes.
Let α represent our universe. Another way to put this is that α represents whichever universe happens to be ours.
We then get three definitions (where “:=” means “equal by definition”):
E := T1α (“α is life-permitting.”)
E’ := ∃x(T1x) (“Some universe is life-permitting.”)
M := m≫0 (“The number of universes is much greater than zero.” or “There are many universes.” Essentially, this is the multiverse hypothesis)
According to White, while it’s true that M being true increases the probability of E’ being true, it’s false that M being true increases the probability of E being true. Formally:
P(E’|M) > P(E’|¬M) (“The probability of E’ given M is greater than the probability of E’ given not-M.”
P(E|M) = P(E|¬M) = 1/n (“The probability of E given M or not-M is 1/n.”)
These last two statements are analogous to the sensible claim that the probability of seeing at least one double-six is higher in many throws than it is in not-many throws, while the quantity of throws has no bearing at all on the probability of any given throw landing double-six. Note that the analogy relies on the assumption that “the events which give rise to universes are not causally related in such a way that the outcome of one renders the outcome of another more or less probable. They are like independent rolls of a die” (Bostrom 19–20)6; this assumption will be brought into question shortly, but first Bostrom considers White’s argument in its own terms.
Bostrom says the argument has some initial plausibility, and I agree. But he sees a fallacy. Before looking at that, a sidebar with some quick notes about conditional probability, (in)dependence, and subjective probability for those who aren’t familiar with these topics or could use a reminder.
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We say two events are independent when knowledge about one event gives no evidence for updating our beliefs about the other event. If A and B are independent, then the probability of A is the same whether or not B has happened, and vice versa. Dependence is simply the opposite of this. If A and B are dependent, then new information about A implies new information about B.
Some examples and clarification:
A fair die and coin are tossed. Their outcomes are independent. You learn the coin landed heads. You assign probability 1/6 to the die having landing six.
The outcomes of a single coin toss are dependent. If you learn a coin landed heads, you assign probability zero to its having landed tails.
The outcomes of a single die throw are dependent. A fair die is thrown. You assign probability 1/6 of its having landed four. More formally: P(4) = 1/6. You learn it landed on an even number. So you update the probability of its having landed four from 1/6 to 1/3 (as it now has an equal probability of having landed 2, 4, or 6). More formally, we’d say that the probability of the die landing four given that it landed even is P(4|even) = 1/3.
P(A|B) might be verbalized as “the probability of A given B” or “the probability of A conditional on B” or “conditionalize the probability of A on B” or “the probability of A happening when B happens” or “the probability that A is true when B is true” or “the probability hypothesis A is true given evidence B” or some similar wording, depending on context.
If A and B are dependent, then the P(A|B) ≠ P(A). And the P(B|A) ≠ P(B).
Similarly, if A and B are independent, then the P(A|B) = P(A). And the P(B|A) = P(B).
Note that P(A∩B) = P(B)P(A|B). The “∩” symbol stands for “intersection,” but can also be thought of as “and.” We can say this as, “the probability of A and B occurring equals the probability of B occurring times the probability of A occurring given that B has occurred.”
Other useful formulations can be derived by moving things around. For example, divide both sides of P(A∩B) = P(B)P(A|B) by P(B) to get: P(A|B) =P(A∩B)/P(B). Think of this as saying, “the probability of A being true given that we know B is true, is the ratio of the times when A and B are both true to the times when B is true, whether or not A is true.”
Example: The probability of having rolled a four given that we know an even number was rolled, is the ratio of {all the possible outcomes in which a four and an even number is rolled} to {all the possible outcomes in which an even number is rolled}; there is {one possible outcome in which a four and an even number is rolled (i.e., when you roll a four)}, and there are {three possible outcomes in which an even number is rolled (i.e., when you roll a two, four, or six)}. So the ratio is 1/3, and that’s the probability of having rolled a four given an even outcome.
I’ll discuss conditional probability a little more below, when I consider Bayes’ theorem.
Finally, many of our dealings here involve subjective probability. Think of this is as your degree of belief or degree of confidence that some proposition is true. Some evidence might increase your degree of confidence, even if you don’t want to put an exact number on it. Finding out that two dice were thrown a “whole lot” of times rather than “just a few” times should increase your belief that a double-six was thrown. You might venture a rough estimate just to have a number to work with, but this will come down to your judgment (maybe the people in the gambling hall appear too sober for a “whole lot” to mean more than 30). Unlike when you learn there were exactly 22 throws of the dice; then you’re dealing with an objectively calculable probability for double-six. Until, that is, you find out they were thrown by someone who possesses the distinct air of a cheater (according to you, at least).
Back to Bostrom’s analysis…
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To challenge White and friends, Bostrom develops yet more formalisms. Recall that α means “our universe.” Bostrom creates a proposition Ei, “if some universe is life-permitting, i is life-permitting” (Bostrom 20), where i denotes any actually existing universe α, β1, … , β(m–1). From this, he notes that the conjunction of E’ (“some universe is life-permitting”) and Eα (“if some universe is life-permitting, α is life-permitting”) is equivalent to E (“α is life-permitting”). Bostrom then notes that White is committed to:
P(M|E’) > P(M) (I.e., “The probability that {the multiverse hypothesis is true given that some universe is life-permitting} is greater than the probability that {the multiverse hypothesis is true given no extra information}.” This is implied by White’s above claim about the dependence of M and E’; it also shows up in [White 264]. If you’d like to see more form work to motivate the implication of this claim, see the below sidebar.)
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Sidebar
White earlier claimed that P(E’|M) > P(E’|¬M). Some work can be done to show that this suggests P(E’|M) > P(E’), but I think this intuitive enough to verbally motivate as follows: Your confidence in {a double-six happening at some point} given {many dice throws} should be higher than your confidence in {double-six happening at some point} given {not many dice throws}. If that’s true, then your confidence in {a double-six happening at some point} given {many dice throws} should be greater than your confidence in {a double-six happening given {no evidence at all about how many dice throws there’ve been}. If you accept this, then we can use the conditional probability formulations noted in the previous sidebar, and a little rearranging, to get:
(1) start with P(E’|M) > P(E’); our goal is to get to P(M|E’) > P(M).
(2) recall that P(A|B) = P(A∩B)/P(B); apply this to the right side of the inequality:
(3) P(E’∩M)/P(M) > P(E’)
(4) multiply both sides by P(M) to get: P(E’∩M) > P(E’)P(M)
(5) divide both sides by P(E’) to get: P(E’∩M)/P(E’) > P(M)
(6) this step is just for clarification, but since (A∩B) = (B∩A), you can rearrange (5): P(M∩E’)/P(E’) > P(M)
(7) use (2) again to get: P(M|E’) > P(M)
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P(M|E) = P(M) (Due to White’s claim about the dependence of M and E; also stated in [White 264].)
From Bostrom’s claim that P(M|E’Eα) := P(M|E), it follows by substitution that P(M|E’) > P(M|E’Eα) (White himself expresses this as: P(M|E’ & E) = P(M). [White 5].)
By symmetry, the P(M|E’Eβj) equals some constant, given that there’s no reason to think any given β‘s being fine-tuned provides any more evidence for M than any other β‘s being fine-tuned. In other words, notes Bostrom: E’ (“some universe is life-permitting”) implies the disjunction E’Eα ∨ E’Eβ1 ∨ E’Eβ2 ∨ … ∨ E’Eβ(m–1). I read this as: “some universe is life permitting and it’s our universe, or some universe is life permitting and it’s universe β1, or some universe is life permitting and it’s universe β2, or…”
Now for Bostrom’s conclusion. Recall that P(M|E’Eα) < P(M|E’). That, along with the point noted in the previous paragraph, implies that P(M|E’Eβj) > P(M|E’) for every βj. The strange result here (according to Bostrom) is that, assuming some universe is known to be life-permitting, finding out that that universe is a given β-universe increases the probability of M, while finding out that it is α decreases that probability. Strung together: P(M|E’Eβj) > P(M|E’) > P(M|E’Eα).
Before commenting on Bostrom’s reasoning about White’s claims and commitments, I’d like to make sure I understand it. E’ (“some universe is life-permitting”) is weightier evidence for M than is the conjunction of E’ and Eα (i.e., E’Eα). In fact, E’Eα neither increases of decreases your confidence in M. Which is to say that if you know some universe is life-permitting, knowing that that universe could turn out to be {α ∨ β1 ∨ β2 ∨ … ∨ β(m–1)} weighs more than knowing that the universe in question is indeed α. This also means {E’ minus E’Eα} adds more weight to your confidence in M than does just E’.
In other words, if you remove α from the set of possible outcomes that may make E’ true, you’ve learned that the universe in question is {β1 ∨ β2 ∨ … ∨ β(m–1)}. This should increase your confidence in M, you no longer have α in the set (had α turned out to be the universe in question, any confidence E’ had given you in M would have been completely undone). Any one of those β-universes should increase your confidence in M by the same amount, should it turn out to be the life-permitting one. There’s no reason to think, for example, that β7 being the life-permitting universe should add less weight to your confidence in M than would, say, β1.
So, provided you know some universe is life-permitting, learning that it’s α should decrease your confidence in M, while learning that it’s any given β-universe should increase your confidence in M.
In short, using β7 as an example: P(M|E’Eβ7) > P(M|E’) = P(M|E’E({α ∨ β1 ∨ β2 ∨ … ∨ β(m–1)}) > P(M|E’Eα) = P(M|E) = P(M)
Provided I’m getting this right, I think I follow, and it seems to make good technical sense. And I agree that it would be strange for E’Eβ7 to raise the probability of M while E’Eα has no effect at all. I’m concerned, however, that there are some significant disconnects between this analysis and White’s grounding intuition. Namely, I think it miscasts White’s argument to commit him this result; at least, it’s contrary to the intuition he aims to bear out. Perhaps White’s formalisms infelicitously imply the result. But I don’t think so. I’ll look at this more closely in a moment, by way of applying Bostrom’s analysis to the gambler example. First, some further general comments about the disconnect.
The first thing to notice is the roll of subjectivity in developing our degrees of confidence here. For example, on learning that some universe is life-permitting, your degree of confidence might go up more or less depending on whether you think you’re more or less likely to learn “some universe” refers to your own. The phrase “likely to learn” is important here, too. It may be reasonable to assume that any existing universe is just as likely as any other to be life-permitting, but you simply don’t know how many existing universes there are (indeed, this is precisely the thing you’re looking for evidence about). You know for sure that your universe exists, however. So maybe it’d be reasonable to be more confident in learning that “some” refers to your universe rather than some other universe.
These questions about the subjective features of the example suggest two potential disconnects with White’s treatment of the gambler example. The first is: How do you learn E’? It would be trivial to learn it simply from observing that you live in a life-permitting universe. To carry any weight on its own, E’ must include the β-universes; indeed, this is how Bostrom represents E’. In other words, it seems you first learn E’, and then are waiting to learn whether E’ is made true—at least in the statement of E’ that you’ve heard—by α or by some β-universe. The potential disconnect is in the fact that Bostrom seems to take issue with the idea of you having pre-knowledge that α might be what you learn makes E’ true. In other words, he’s sliced up {α ∨ β1 ∨ β2 ∨ … ∨ β(m–1)} equally so that you’re just as likely to learn any one of them, and so that each, if you learn it, should carry the same weight as evidence for M as does any other. But, again, this is contrary to White’s subjective setup. Which leads to problems—namely, that Eβ7 makes E’ heavier evidence, while Eα makes it lighter.
But White doesn’t slice things up this way. On his view, E’ is borne out either by α or by ¬α (i.e., by some β-universe). That in mind, consider applying some arbitrary probabilities. Suppose you gauge P(M) to be 5%. You find out some universe is life-permitting. Your confidence in M increases from 5% to 25%. It would go up higher, except you know it’s possible you might learn that the life-permitting universe in question is your own, a fact that would have no bearing on your confidence in M.
This is where the strangeness enters for Bostrom, who seems to insist that you shouldn’t have such a perspective on your own [or any other universe] before learning which universe is life-permitting. Increasingly, it strikes me that this insistence—which I’m not sure if Bostrom believes White indeed shares or is failing to share despite his argument’s grounding—is precisely what accounts for the disconnect between Bostrom’s analysis and White’s argument: White does seem to wish for us to have that mitigating worry about learning that the life-permitting is our own. This wish seems sensible to me: “Hey, I already knew the universe I’m in must be life-permitting by virtue of my being alive and being in it and observing myself in that condition. Tell me something I didn’t know!”
To be clear, one need not characterize pre-knowledge is occurrent. It could be dispositional. A crude analogy comes to mind: “You’re going to the dance with someone! You want to know who? Ok. You’re going with yourself.”
Despite all this Bostrom’s analysis leads him to a competing intuition: “it’s hard to see how the fact that we are in this universe could justify treating its being life-permitting as giving a lower probability to the multiverse hypothesis than any other universe’s being life-permitting would have given it” (Bostrom 21).
So is the problem with Hacking’s gambler analogy (are there any perfect analogies?), White’s formal expression of that analogy, the Hacking/White grounding intuition, or with Bostrom’s analysis?
To approach these questions, I’ll look more closely at the intuition grounding the gambler example by holding it up to Bostrom’s analysis, keeping in mind the aforementioned concern about whether the example truly captures the element of anthropic bias. Similarly, I worry something is lost in how much easier it is to slice up a dice example into objective probabilities (a potential liability for the analogy to cosmology); but something is gained by being able to contrast those elements with the subjective features of the example, thus bringer into starker relief the intuition in question. Here’s White’s intuition in his own words:
Suppose on being asked how many times the pair of dice have been rolled, the gambler asks if a double-six has been rolled. Upon learning that one has, he is more confident than he was that the dice have been rolled a number of times. Here his reasoning is sound, for the more times the dice have been rolled, the greater the chance that a double-six has been rolled. However, when the gambler witnesses a single roll and is then more confident that the dice have been rolled before, he is clearly making a mistake. The difference is that in the first case, the information he has gained is just that some roll or other landed double-six, in the second case, he witnesses a specific roll. Compare this with the case where astronomers discover that one or more other big bangs have occurred, and ask us to guess if there have been one or many. We might ask whether any had produced a universe containing life, and on learning that one did, be more inclined to suppose that there have been many. This reasoning would be correct. But this is not our situation. Like the gambler in the second case we have simply witnessed a single big bang producing this universe. And no number of other big bangs can affect the probability of the outcome we observed. (White 264–265)
Again, this seems like fine reasoning to me (assuming life-sustaining universes are rare and independent). Key points lost in Bostrom’s interpretation are that the gambler doesn’t learn about a specific roll when he asks whether a double-six has been rolled, but rather that some roll landed double-six; but he does learn about a specific roll when he sees a double-six rolled. The latter gets at Bostrom’s question about what it is that makes the universe we’re in precisely—it is the one and only roll of the dice we’ve observed, the only one we’re promised to observe, and it comes down to learning about one specific event. Bostrom may claim that this is a fine intuition, but White’s formal expression of the argument implies an unacceptable strangeness.
So, let’s see if I can formalize this in a way that protects that grounding intuition against Bostrom’s analysis. I’ll do this by assigning arbitrary numbers to the gambler scenario; this’ll get a bit hand-wavey, but I think that’s fine:
M = “many rolls before you got here” = 100. (I take M in the cosmology case to mean infinitely many. One hundred seems reasonable here. Assume 100 is also the maximum number of rolls ever rolled at a gambling hall on a given day.)
P(M) = 1/2. (Suppose this is the gambler’s subjective probability, and it’s a solid guess given his familiarity with the sporadic hours kept by gambling halls.)
¬M = “not many rolls before you got here” = 1. (An oversimplification, given that anything from 1 to 99 could be reasonably argued to count as “not many,” each occurring with a probability of (1/2)(1/99); an unnecessary complication I avoid by assuming ¬M = 1. Notice that I didn’t include zero as a possibility. For one thing, zero wouldn’t work well when translated to the cosmological case, in which E’ tells us some universe is indeed life-permitting and we at the very least know E’ due to actually observing a life-permitting universe. So, I assume the gambler’s story contains at least roll. More importantly, I presume the gambler learns of at least one instance of double-six, if only the one he’s seen rolled.)
P(¬M) = 1/2. This is just 1–P(M).
D = “double-six was rolled at some point since the gambling hall’s been open”
P(D) =”the probability of D being true given that there have been 100 rolls tonight + the probability of D being true given that there’s been 1 roll tonight.” This may vary depending on the scenario. For example, if the 100 rolls does not include the one you just saw, then: P(D|M)P(M) + P(D|¬M)P(¬M) = (1–(35/36)^100)(1/2) + (1/36)(1/2). (This is approximately .484, but I’ll leave it precise. I’ll also assume the 100-roll formulation for now (for reasons that will hopefully make sense as I go along), but will revisit the point below when I consider Bostrom’s analysis.)
D* = “the roll you’re about to see lands double-six” or “the roll you just saw landed double-six”
P(D*) = 1/36 (The usual probability for double-six given two fair dice.)
We can put all this into Bayes’ theorem, which is a derivation of the conditional probability we’ve been doing so far*, to show that the P(M|D) > P(M), while the P(M|D*) = P(M):
(*Specifically, Bayes’ theorem says that the probability of a hypothesis H given some evidence E is equal to that evidence showing up when the hypothesis is true, divided by all the possible scenarios in which that evidence shows up (including those in which the hypothesis is false): P(H|E) = [P(E|H)P(H)]/[P(E|H)P(H) +P(E|¬H)P(¬H)]. This is really just the usual sort of basic probability we’re used to, where we ask what fraction of all possible outcomes are the desired outcomes.)
This all seems fine to me, and I think a well-translated application to the cosmological scenario should bear it out. In other words, a redefinition of D* as, say, DD* = “some roll tonight was a double-six and it was the one you just saw,” shouldn’t change any of the above math. The gambler is being asked about the (potentially many) rolls that preceded the one he saw, just as the universe-observer is being asked about all the (potentially infinitely many) universes the observer can’t see.
This points to a distinction between the gambler and cosmology cases that may be more significant that it first appears—that is, if “life-permitting” corresponds to “double-six,” and “big bang” corresponds to “dice roll,” the distinction is that dice-roll observers know that the dice roll they’re seeing is the most recent, but big–bang–observers of course have no idea whether the universe they see is a result of the most recent (and only) big bang or the first big bang (of many) that may exist simultaneously or sequentially. In a certain sense, this seems clearly insignificant: when you only learn about one universe, however you learn about it, you cannot infer M without committing the Inverse Gambler’s Fallacy.
Of the author’s I’ve discussed here, it seems all but Hacking recognize this. Recall that Hacking introduced the gambler example in order to expose a fallacy with “respect to J. A. Wheeler’s oscillating universe theory, according to which our universe is the latest of a long temporal sequence of universes” (White 5), though Hacking didn’t seem to find anthropic bias a problem given a Carter-type universe, prompting Bostrom to note that “Hacking is probably alone in thinking [anthropic reasoning] works in one but not the other” (Hacking 18). Similarly, White notes, “Although Hacking does not mention them, similar points apply to models of coexisting universes” (White 263) and refers to Hacking’s endorsement of Carter’s hypothesis as “puzzling” (White 265): “Since Hacking, correctly in my view, finds fault with the argument for Wheeler’s hypothesis, he should likewise find fault with the argument for Carter’s (White 265)”.
I think this observation worth noting as it emphasizes that how we learn some universe exists matters. If what we’ve learned is anything that tells us this universe is life-permitting, where “this” is the universe we would know about whether or not we’re explicitly told E’, this is unhelpful for our confidence in M. On the other hand, we could learn something about some other specific universe in a way that does give information about M, namely due to the fact that the specificity must still be vague enough so that it only implies “some universe” (more on this in a moment). Similarly, simply naming some random dice roll that’s happened today may give information about a specific roll, but it’s only some specific roll. it isn’t this roll. Suppose the dice-roller names each roll, and you learn not that “some roll landed double-six,” but that “Orff landed double-six.” Even if you were told that Orff is not the roll you just saw, you haven’t learned anything more than when you are told “some roll landed double-six.”
This strikes me as essentially the situation Bostrom finds strange. Otherwise, we really are learning about a specific roll in the sense of its being this roll, which gives us no information about M. It’s as if it doesn’t matter when you show up to the hall. This needs more picking apart, however, as I do see a threat to White’s argument here.
If we should view being told that Orff landed double-six as being essentially like finding yourself randomly plopped down at some point in the sequence of the day’s throws (you don’t know which), so that you have learned something about this roll, then to increase your confidence in M would be to commit the Inverse Gambler’s Fallacy, just as when you’ve seen the roll. As I understand White’s intuition, White would agree with this. And yet Bostrom claims White’s basic argument entails a commitment to committing the fallacy in the Orff-type case.
This strikes me as wrong, however, as learning that Orff landed double-six is significantly distinct from learning that the roll you just saw landed double-six. In fact, “Orff” is interchangeable with “some.” So, whether the roll you see happens at the same time, in the middle of, or after many rolls, what matters is that you learn nothing about wether there are many other rolls merely by virtue of the roll you see. But if you’re told “a specific roll landed double-six, but not the one you just saw,” then it’s reasonable for your confidence in M to increase.
The phrase “but not the one you just saw” is significant here. Without that, you’ve really just learned E’ when you learn of Orff. So this all adds up as far as I can tell. And it leads me to underscore that, in order to preserve White’s gambler intuition in its cosmological analog, we must assume that we already know our universe is life-permitting when we learn that “some universe” is life-permitting. What we don’t know, however, is whether “some universe” refers to our own. Perhaps Bostrom allows for this, with the idea in mind that we learn “some universe” is life-permitting upon simply coming on the realization that one lives in a universe; but, as I’ve already noted, I think this underserves White’s reasonable intuition. Bostrom may simply think White’s grounding intuition doesn’t hold up to scrutiny:
It is true that we are in this universe and not any of the others—but that fact presupposes that this universe is life-permitting and us being in it. So it’s hard to see how the fact that we are in this universe could justify treating its being life-permitting as giving a lower probability to the multiverse hypothesis than any other universe’s being life-permitting would have given it. (Bostrom 21)
That in mind, I’ll wrap up these considerations of Bostrom’s call to strangeness—i.e., that learning about our universe should lower our confidence in M while learning about some other universe raises it—with a more direct application of Bostrom’s critique to DD* (“some roll tonight was a double-six and it was the one you just saw”).
First, recall that White contrasted Hacking’s gambler situation with one in which the gambler has not seen a double-six rolled, but instead is told some roll had landed double-six; White emphasizes that we Earthlings who wonder about multiverse theory are in the Hacking situation. Ok. You enter the gambling hall and are asked to guess if there’ve so far tonight been many or not many rolls. You ask to see the dice rolled before deciding. A double-six is rolled. You’ve now trivially learned that some double-six was rolled (i.e., the one you just saw). According to White, however, if your confidence in M is affected by what you’ve (trivially) learned, you commit the Inverse Gambler’s Fallacy.
Bostrom and the math I showed above agree with this. But this isn’t in line with Bostrom’s analysis. We get closer to that with the following example.
Rewind the tape. You’re asked to guess if there’ve been many or not many rolls so far tonight. You ask to see the dice rolled before deciding. The dice are rolled, but the result is concealed from you and you’re told, “Some roll tonight landed double-six.” Your confidence in M goes up but is attenuated by knowing that “some roll” might refer to the roll concealed from you. You are then told, “‘Some roll’ does not refer to the roll you just executed.” Your confidence in M increases even more; notice that it should increase here even when it turns out the roll you just executed landed double-six (though you don’t actually need to learn if it did or not). Rewind the tape again. This time you’re told, “The double-six I referred to is the roll you just executed.” Your confidence in M decreases back to its starting point, lest you commit the Inverse Gambler’s Fallacy.
I take it this gets us closer to what Bostrom describes, as you don’t know whether the roll you’ve seen is the double-six at the start. But I think it not close enough. How about the following?
Rewind the tape. You walk into the hall, and just after entering you see the dice rolled. You see them land double-six. You’re asked to guess if there’ve been many or not many rolls so far tonight. You ask, “Has some roll tonight landed double-six?” You’re told, “Yes.” You think that, most likely, the “yes” refers to the double-six you just saw. But it might not, so your confidence in M increases. You qualify the question: “Did some roll land double-six before I arrived?” And so on.
This all seems perfectly fine, and doesn’t conflict with White’s starting intuition. They don’t perfectly align with the cosmological translation, but strike me as close enough. The key is that we have plenty of reason to think observing ourselves in this specific universe makes so that, “some universe” tells us nothing new when it happens to be our own. And this seems obviously true in the absence of seeing a die rolled: if you learn E’ without having seen any dice rolled, you de facto learn that “some” isn’t the one you just saw. So, again I claim that the special thing about α is that we’ve already seen it (this is also why I don’t include zero as an option for ¬M).
Finally, this calls for a closer look at the question of which universe is meant to be picked out by the statement “some universe.” Which leads in turn to the question of how one learns some universe is life-permitting. This too seems to be missing from Bostrom’s analysis, and might also help answer his question about what’s special about our universe that it doesn’t increase our confidence in M. This is especially true when the way we’ve learned it’s life-permitting is simply by our being alive and being in it. At any rate, White seems to hint at the importance of how we learn E’ when he mentions an example in which “astronomers discover that one or more other big bangs have occurred, and ask us to guess if there have been one or many” (White 265).
That example is more important, however, for the fact that White means us to compare it to our own case, or, by analogy, that of the gambler who has only seen one dice roll.
How does this hold up to Bostrom’s formal analysis?
Let D amount to (where R = “roll”): {DD*, DR1, DR2, … ,DR(n–1)}. So, if M is true and doesn’t include the roll the gambler sees, n will equal 101; if M is false, n will equal two. But wait, I already see a problem. Recall that I excluded D* from D. If we include “the roll you just saw” in P(D), and that roll landed double-six, then you know D is true with probability 1. But what you don’t know is whether the statement “some roll landed double-six” refers to the roll you just saw. So, in order to accommodate Bostrom’s analysis, I’ll retool D to mean “the statement ‘some roll today landed double-six’ refers to the roll you just made or to some other roll made before you arrived.'” This is getting complicated. But I’ll stick with it, keeping in mind that I could turn the knobs of the story and math in other ways; hopefully this gets across the general idea.
We now consider how to evaluate M given various conditions under which D is true.
Assume D is true, conditional on D*; then condition M on DD*. That’s P(M|DD). It seems to me that the proper way to think of this is as “the probability that there have been many rolls tonight given that the statement ‘some roll landed double-six’ refers to the roll you just executed.” It seems reasonable to me that this should be equal to P(M). Recall above that I put P(M) at 1/2 and P(M|D) (which is the same as P(M|DD*) at 1/2. So that works.
If we conditionalize on {DR1, DR2, … ,DR(n–1)}, this simply means, “not the roll you executed,” or: P(M|D¬D*) This should indeed increase your confidence in M. This amounts to the first formulation I presented as P(M|D), which was roughly 97.1%.
This is all well and good, but of course isn’t how Bostrom treats the analogous “some other…” set. That is, he doesn’t stope at conditioning on E¬α, but goes as far as conditioning on some specific β-universe. Does this work? First, we obviously can’t be told “the 100th roll before you got here landed double-six,” as this is question-begging (a point [Hacking 335] makes as well). And if you learn that the roll just before you got here landed double-six, you again commit the Inverse Gambler’s Fallacy if you infer M from that. Same goes for learning that the roll two or three rolls back landed double-six. At some point, the number of rolls back will increase confidence in M simply by virtue of their being bigger numbers.
While it’d be fun to consider the possible range of scenarios, this clearly isn’t what Hacking or White are going for. Though I can see why one might wonder whether White has chosen a good example of learning “some universe is life-permitting” by way of learning that astronomers have discovered “one or more other big bangs” (White 265) and that one of them produced life. My claim is that this is not like learning of a specific roll, e.g., that there was a roll just before you got here that landed double-six, as the “or more” could refer to infinitely many universes.
So, how might you learn that some specific DRj landed double-six in such a way that doesn’t threaten fallacy? The only other sort of option I can think of is in the case I mentioned above, in which the hall gives each day’s roll a name and you learn that, at some point, a roll named Orff landed double-six. Though it carries a kind of specificity, the statement’s vagueness makes it no different than simply learning D—in it’s basic form: “some (specific) roll landed double-six”—and thus should indeed increase your confidence in M. In other words, “Orff” is not “the most specific version of what is known.” See again something I quoted earlier from Hacking:
We must insist that F, the fact to be explained, is the most specific version of what is known and not a disjunctive consequence of what is known. In the case of double-six the most specific version is that double-six has just occurred. That is the fact to be explained, and a long run of previous trials is of no value in explaining that fact. (Hacking 335–336)
All told, and given the admittedly short amount of time I’ve spent thinking about the problem, it seems to me that the Hacking/White intuition could be formally translated into a cosmological case that more clearly survives Bostrom’s formal challenge. I won’t attempt to develop that here (for one thing, I think the verbal qualifiers mock value assignments are enough, given that we’re dealing with subjective probabilities); but I will say that it seems it will need to reflect the critical the distinction between this life-permitting universe and some life-permitting universe, as illustrated in the gambler example. In a nutshell: “The mistake is in supposing that the existence of many other universes makes it more likely that this one—the only one that we have observed—will be life-permitting” (White 5).
The discussion up to this point rests on some important assumptions, however, which brings us to the rest of Bostrom’s critique, which is more about White’s factual claims than about his appeals to probability. I don’t have much to say about factual claims, except that if new information challenges White’s underlying assumptions, this could of course change the math. One of these is that universes are causally independent:
Wheeler universes, like dice, “have no memories,” the individual oscillations are stochastically independent. Previous big bangs in the sequence have no effect on the outcome of any other big bang, so they cannot render it more likely to produce a life-permitting universe. Although Hacking does not mention them, similar points apply to models of coexisting universes. These universes are usually taken to be causally isolated, or if there is any causal relation between them, it is not of a type that could increase the probability of this universe being life-permitting. (White 263)
Bostrom characterizes the memoryless-ness assumption as “highly problematic” on the grounds that “there’s no empirical warrant for it” (Bostrom 21). He notes that there are models—even if highly speculative ones—that rely on causal dependence, even if indirectly, as when universes carry information about one another due to having a “partial cause in common… [as] is the case in the multiverse models associated with inflation theory (arguably the best current candidates for multiverse cosmology)” (Bostrom 21). In fact, “the majority of multiverse models that have actually been proposed, including arguably the most plausible one, directly negate White’s claim” (Bostrom 22). (For a more recent comment on these ideas from physicist Lawrence Krauss, see this footnote.7)
Bostrom goes a step further, claiming that P(M|E) = E doesn’t always hold even when universes are causally unrelated, and even when setting aside concerns about anthropic reasoning. Namely, if we are concerned with epistemic, or subjective, probability rather than objective chance, then a lack of correlation in physical outcomes does not necessarily mean “the outcomes of those events are uncorrelated in one’s rational epistemic probability assignment” (Bostrom 22). He then provides a toy example (Bostrom 22), which I’ll summarize.
Suppose you have knowledge K that justifies some degree of confidence in the existence only three possible worlds:
W1, in which there’s one big universe a and one small universe d;
W2, in which there’s one big universe b and one small universe e;
W3, in which there’s one big universe c and one small universe e.
You get some new information: you live in universe e. You then conditionalize on this information, which is to say you remove W1 from the set of possible worlds. In other words, P(“The big universe is b or c”|K&”The little universe is e”) > P(“The big universe is b or c”|K).
Fair enough. One need not give any thought to the causal relations of these universes in order to make deductive inferences about their relative probabilities. I can’t imagine White objecting to this example in itself, though he may object on the same grounds that make me suspicious: do we have any reason to think our actual situation would permit such an assignment? Do we have some analogous K? If not, and we assign probabilities as though we do, are we not engaging in precisely the sort of capricious or arbitrary guesswork critics point to as the worst place Bayesian/subjective statistics can take us?
Or perhaps this is precisely Bostrom’s point. That is, White is playing loose with the subjectiveness of our situation: he’s willing to make the epistemic claim that P(M|E’) > P(M) on subjective grounds, yet claims it fallacious to do the same for P(M|E) > P(M). He could just as easily claim P(M|E’) = P(M), and that’d at least be consistent. I doubt he’d want to give up P(M|E’) > P(M), nor should he want to give that up (for reasons we’ve well covered here). Writes Bostrom: “All these problems are avoided if we acknowledge that not only P(M|E’) > P(M) but also P(M|E) > P(M)” (Bostrom 23).
Still, these latter arguments from Bostrom actually move me even more towards agnosticism about P(M) and the P(M|E). Which is to say I assign them the same probability, as have White and Hacking (and Dowe, whom I’ve not discussed here), pending some sort of independent information about M, for example from empirical work in physics (more on this in my Concluding Thoughts). As White puts it: “Perhaps there is independent evidence for the existence of many universes. But the fact that our universe is fine-tuned gives us no further reason to suppose that there are universes other than ours” (White 3).
Bostrom then shifts our attention to Surprise, which I’m eager to discuss, and is more up my alley than the present discussion. I’ll save that for Part II of my exploration of this chapter.
∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦∞♦
Concluding thoughts:
Given the truth of White’s underlying assumptions, I agree with him that it seems fallacious to increase our confidence in there being infinitely many universes on the evidence that our universe is fine-tuned. Though knowing that there are infinitely many universes would make it less surprising that at least one them (e.g., ours) turned out fine-tuned. But the power of event B to make event A less surprising isn’t necessarily a good reason to have more confidence that B is true given A. Learning that there was a double-six at some point tonight would be made even less surprising by there having been 100 million rolls than if there had only been one hundred (or 99 million, technically; but I don’t see that it should make any subjective difference, any more than 440.0000001 Hz is any closer to the subjective experience of what we call the note “A-440” than is the physically closer frequency 440.000001 Hz).
But I certainly wouldn’t have any confidence in there having been 100 million rolls—I’d assign zero probability to that. Same goes for 99 million and 98 million rolls. But where to draw the line here is unclear. I conveniently used 100 rolls for M above, but it could have been 101 or 300. At some point, one simply feels that, “Ok, there’s no way there were that many rolls tonight.” Still, even with technically non-zero probabilities, at some point the explanation that makes an improbable event less surprising will seem to dominate. If I’m surprised to learn my friend won the one-in-ten-million lottery, I’ll accept the explanation that he won rather than a less-surprising-making scenario in which he cheated.
Compare this to: If I see a seemingly normal quarter lands heads eight times in a row, I’ll be a little surprised but will accept it. If I see it land 23 times in a row (about the same probability as my friend winning the lottery), I’ll be really surprised, but will accept it, particularly if it then goes back to behaving as expected and I’m the one flipping. If I see it land 100 heads in a row, I’ll assume something’s rigged no matter how normal it looks and even if I’m the one flipping it and even if it goes back to behaving as expected, or if it only lands heads when I’m the one flipping it, and so on. More on this sort of example, and surprise in general, in the second installment on this chapter.
Perhaps this gets at an important disconnect with Hacking’s analogy. It’s hard to know what counts as a reasonable assessment of the number of universes.
I don’t claim to know anything about which multiverse theory is our best bet. And so I’m agnostic, if I’m being honest (though some versions strike me as incredibly ludicrous, and I can never imagine literally believing them). How might I try to come to some conclusion on that count? Suppose I offered a prize to the expert who could convince me that their favorite theory is the best one. Supposing one actually convinces me (rather than merely giving the best unconvincing argument), should I trust this new-found confidence simply on the grounds that one person was, for whatever reason (e.g., charm, cleverness, a smart-sounding Dutch accent)?
Or maybe I should drop everything and shift my attention to physics and learn all the relevant equations and theories and so on in order to see which one clicks. But then why should I prefer the model I prefer to those preferred by others? And aren’t so many steps along the way—which classes I take, books I read, teachers I study with, prerequisite courses I’m drawn to or just happen to take because I need another class and the one I wanted is booked up, folks I meet with for coffee—a gradual assembling of a predilection for this or that sort of theory? And even if my inherent dispositions destined me to accept some particular theory, if only I understood it, it’s not as though I chose my dispositions or couldn’t have ended up with different ones.
All that said, my gut intuition—something that, in the context of probability, one is constantly trying to tame and correct and cover up like an infected, brain-sized pimple—tells me it’s at least a fair bet that there’s more than one universe out there. That same gut says maybe we just shouldn’t think it’s such a rare thing for a universe—or even some particular region of a universe—to be life-permitting. I don’t have anything to back this up except that it’s the general impression I’ve developed after years of hearing experts weigh in on and debate such things. As is often pointed out by the experts, though: today’s candidate theories are models of the world that reflect our best stabs at interpreting the math we’ve developed or discovered so far. Fair enough.
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Footnotes:
- Page 337 of Hacking’s article: “The Inverse Gambler’s Fallacy: The Argument from Design. The Anthropic Principle Applied to Wheeler Universes,” Mind, New Series, Vol. 96, No. 383 (Jul., 1987), pp. 331–340. At JSTOR.
- Hacking 333.
- The referenced papers: “Fine-Tuning and Multiple Universes,” Roger White, Noûs, Vol. 34, No. 2 (Jun., 2000), pp. 260-276 (at JSTOR). “Multiple Universes, Fine-Tuning and the Inverse Gambler’s Fallacy,” Phil Dowe (1998), unpublished manuscript (I couldn’t find a copy).
- I’ve quoted a little more than (Bostrom 18–19) does.
- We can consider this the same as (where “∧” means “and”): ∃x[P(x) ∧ ¬∃y(P(y) ∧ y ≠ x)].
- Appears in (White 262–263)
- Bostrom’s book came out in 2002. Given the developments in particle physics in the intervening years, it’s interesting to note the following comments on inflation from Lawrence Krauss’s 2017 book, The Greatest Story Ever Told—So Far:
If the successors to LIGO, or BICEP, in this or the next century are able to measure directly the signature of gravitational waves from inflation, it will give us a direct window on the physics of the universe when it was less than a billionth of a billionth of billionth of a billionth of a second old. It will allow us to directly test our ideas of inflation, and even Grand Unification, and perhaps even shed light on the possible existence of other universes—turning what is now metaphysics into physics.
For the moment, however, inflation is merely a well-motivated proposal that seems to naturally resolve most of the major puzzles in cosmology. But while inflation remains the only first-principles theoretical-candidate explanation for the major observational features of our universe, it relies on the existence of a new and completely ad hoc scalar field—invented solely to help produce inflation and fine-tuned to initiate it as the early universe first began to cool down after the Big Bang.
Before the [2012] discovery of Higgs particle, this speculation was plausible at best. … (Krauss 293–294)