1 THE DOOMSDAY ARGUMENT AND THE S ELF-INDICATION ASSUMPTION: REPLY TO OLUM Ć irkovi and Milan M. By Nick Bostrom ć In a recent paper in this journal, Ken Olum attempts to refute the Doomsday argument by appealing to the self-indication assumption (SIA) , the idea that your very existence gives you reason to think that there are many observers. In contrast to earlier refutation attempts that use this strategy, Olum confronts and try to counter some of the objections that have been made against SIA. We argue that his defense of SIA is unsuccessful. This does not, however, mean that one has to accept the Doomsday argument (or the other counterintuitive results that flow from related thought experiments) . A developed theory of observation selection effects shows why the Doomsday argument is inconclusive and how one can consistently reject both it and SIA. I. THE RELATION BETW EEN THE DOOMSDAY ARGMENT AND SIA sday argum atically underestim ated ent purports to show that we have system The Doom the probability that hum ankind will becom e extinct relatively soon. Originated by 1 , the Doom sday argum ent argues Brandon Carter and developed at length by John Leslie ation residing in the that we have neglected to fully take into account the indexical inform fact about when in the history of an species we exist. In a nutshell: Your birth the hum 1 Orig inally in J. Leslie, “Risk ing th e W orld 's En d,” Bul letin of the C anadi an N uclear Soci ety , M ay: 10-15. (1989). He has a m ore com ve treat ment in The End of the Worl d: The Sci ence and Et hics of prehensi dn’t ma ctio Hu (London: Rout ledge, 1996). C arter di n Extin publ ish on t he Doom sday argum ent . A n different versi on of t he Doom sday argum ent , whi ch we shal l not di scuss here, was i ndependent ly ure prospect discovered by t, “Im plicat ions of t he C operni can pri nciple for our fut Richard Got s,” Na ture 363 (27 M ay): 315-9 (1993). 1

2 rank (i.e. your position in the sequence of all hum ans) is roughly 70 billion. That you ore probable, if the total should have such a low birth rank is less surprising, and m ans that will ever have existed is, say, 200 billion rather than, say, 200 num ber of hum trillion. (By ‘probability’ we here m ean rational subjective credence.) Given these unequal conditional probabilities, one can derive f rom that the Bayes’ theorem pending doom probability of im goes up after conditionalizing on your birth rank. That is, after realizing the full evidential im port of you having a relatively low birth rank, such as 70 billion, you should increase your probability estim ate of hypotheses according to ans (such as 200 billion in total) at the which there will be relatively few extra hum ore hum ans (such as expense of hypotheses according to which there will be very m any m a total of 200 trillion). ption in the Doom sday argum ent is that you should reason as if you A key assum e sense a random sam ple from the class of all observers that will ever have were in som 2 : -sam existed. This is encapsulated in the self ption pling assum (SSA) One should reason as if one were a random the set of all sam ple from observers in one’s reference class. We assum oment that one’s reference class consists of all observers, although, e for the m as it turns out, this supposition is far from innocent. ethodological pr escription specifying certain types of SSA should be read as a m P(I am conditional credences of the form such and such an observer | The non-indexical sam ple” properties of the world are such and such). The phrase “as if one were a random is sim mendations. There is no intim ation of any physical ply shorthand for these recom 2 Orig inally in N. Bo stro m, “In vestig atio ns in to th e Do omsday arg ument,” Unpubl ished manuscri pt . (1997); http://www.ant /prepri nts/inv/ invest igations.ht ml nciples.com see al so “The Doom sday c-pri hropi ent is Al ive and Ki cki ng,” Mind 108(431): 539-50 (1999); “Observer-rel ative chances i n ant hropi c Argum ++ , and Quant um Erkennt nis 52: 93-108 (2000); “The Doom sday argum ent , Adam & Eve, UN reasoni ng? ” ect Joe,” hese 127(3): 359-87 (2001); and especi ally Ant hropi c Bi as: Observat ion Sel Synt ion Ef fect s in Sci ence and Phi losophy (New York: Rout ledge, 2002). 2

3 random ization m – som e kind of stochastic tim e-traveling stork? – responsible echanism for distributing observers in the world. SSA is what generates the conditional probabilities that, when f ed into the ent and com pirical inform ation about our birth ranks, Doom bined with em sday argum lead to the probability shift in favor of doom soon. It would take us too far afield to ents for why it (or exam ine what grounds there are for accepting SSA. Detailed argum uch like it) should be applied in a wide range of cases can be found in the ething m som 3 literature. 4 accepts SSA. How then does he hope to avoid the conclusion of the Olum sday argum Doom By appealing to another principle, the self- indication assum ption ent? (SIA) that says that the sheer fact that you exist should lead you to adjust your prior probability assignm ent in favor of hypotheses according to which there are lots of 5 observers: (SIA) Given the fact that you exist, you should (other things equal) favor any observers exist over hypotheses on which hypotheses according to which m few observers exist. If we explicate SIA as saying that the probability boost that a hypothesis gets is proportional to how m any observers it says exists, i.e. (, | ) ( ) = ∝ = ⋅ N n I exist P N n n P tot tot ” is the proposition that there are n observers, and is the = = P N n ( ) (where “ N n tot tot prior probability of this proposition), then it is easy to show that we get an exact obability shift in favor of hypotheses that ent-like pr sday argum cancellation of the Doom es from postulate few observers that com ove was appl ying SSA to our birth ranks. This m 3 E. g. B ostrom (2002). 4 K. Ol um , “The Doom sday Argum ent and t he Num ber of Possi ble Observers,” Phi losophi cal Quart erly 52(207): l page refe rences refer t o this paper. 164-84 (2002). Al 5 Bostrom (2002). 3

4 6 The exact cancellation was first m ade, albeit not very transparently, by Dennis Dieks. s to have been independently discovered by Paul first shown by Kopf et. al and seem 7 W hile these authors where satisf ied with noting that SIA Bartha and Chris Hitchcock. m sday-effect of SSA, Olum cancels the apparent Doom oves the debate a step forward by trying to address som e of the objections against SIA that defenders of the Doom sday argum ent have presented. II. OLUM’S DEFENSE OF SIA SIA m ethodological prescription or a purported principle of ay seem quite dubious as a m hy should reflecting on the fact that you exist rationally com rationality. W pel you to redistribute your credence in favor of hypotheses that say that there are m any observers at that there are few? the expense of those that claim This probability shift, it should be stressed, is m pirical considerations such as eant to be a priori; it is not attributed to em ust have taken m plex life form like that it m any generations for evolution to lead to a com yourself, or that we’ve discovered that the cosm os is very big and probably contains vast num bers of Earth-like planets. The support for “fecund” hypotheses (the support being ber of observers postulated) com es from the sole proportional to the degree of the num fact that you exist. , SIA m ay seem im plausible, and our view is that SIA is no less Prima facie plausible ultimo facie . Probably the m ost positive thing that can be said on its behalf is im that it is one way of getting rid of the count erintuitive effects of the Doom sday argum ent 8 to and related thought experim ents of Bostrom ents, including the experim Adam & Eve 6 D. Dieks, “Doom sday - Or: the Dangers of Statistics,” Phi losophi cal Quart erly 42(166): 78-84 (1992); sday Argum see also “The Doom ” in preparation (available at ent, ://p hilsci-arch http u/documents/d isk 0/00/00/02/47/in dex.html ). ive.pitt.ed 7 http://xxx.l T. Kopf et oom ,” Preprint , gl anl.gov/ abs/ gr-qc/ 9407002 (1994); P. al., “Too soon for doom Bartha and C . Hi tchcock, “No One Knows t he Dat e or t he Hour: An Unort hodox Appl icat ion of R ev. Bayes’s Theorem ,” Philosophy of Science (Proceedings) 66: S329-S53 (1999). 8 Bostrom (2001). 4

5 which Olum alludes in his paper. Yet this palliative relief es at a cost. Consider the com 9 Presumptuous Philosopher gedanken: It is the year 2100 and physicists have narrowed down the search for a theory of and T (using everything to only two rem T aining plausible candidate theories, 2 1 the world is very, metry). According to T super-duper sym considerations from 1 inite and there are a total of a trillion trillion observers in the very big but f , the world is very, very, very big but finite and there are os. According to T cosm 2 a trillion trillion trillion observers. The super-duper sym metry considerations are indifferent as between these two theories. Physicists are preparing a sim ple ent that will falsify one of the theories. Enter the presum ptuous experim philosopher: “Hey guys, it is com pletely unnecessary for you to do the is about a trillion tim es experim ent, because I can already show to you that T 2 ent !” (whereupon the philosopher runs the argum T more likely to be true than 1 that appeals to SIA). Whereas we regard this as som , Olum bites the bullet and ething close to a reductio to the view that this philosopher would be mits him acknowledges that his position com right, provided that the prior probabilities for the two theories (based on the super-duper metry) were roughly equal. sym Olum tries to take the sting out of this consequence by offering us an analogy, meant to dem onstrate that we can justifiably reject the thought experim ent’s assum ption would have roughly equal priors. Suppose that a stranger com and T es up to you T that 1 2 in the street with the claim that if you give him one dollar, he will give you ten dollars in orrow. Presum ably you will decline his offer, return tom axim izing your expectations) that you think the chance which shows (if you are m he will com e through as he says is less than 10%. On the other hand, it would be strange to say that the chance that he will com e through is less than one in a million, since som es people m aking statem ents like this are honest. ... etim 9 Bostrom (2002). 5

6 Nevertheless if the payoff is raised to ten m illion dollars, you still will not give the dollar, which shows that now you think the chance for a payback is in fact less than 1 in 10 m illion. In order not to have a proposed payback large enough to ust think that deprive you of m the likelihood of getting paid decreases oney, you m at least inversely with the proposed payback. Applying this to cosm ology, it is possible that one should think that a theory involving a very large universe is unlikely in proportion to the size of the universe it proposes. (p. 183) s tenuous; we doubt that m any will find m The analogy seem fort in it. At any uch com rate, what Olum s to be suggesting is that there are independent reasons (although he seem ay be) for thinking that the universe is not very big, and that doesn’t specify what those m these reasons would balance out the SIA-induced probability shift that the presum ptuous philosopher is advocating. These alleged reasons, whatever they are, couldn’t stem from the Doom sday argum ent, since the presum ptuous philosopher is lacking crucial em pirical inform ation on which the Doom sday argum ent relies, viz. what his own sequential position is in the population whose size he it trying to predict. For we can suppose that the philosopher and his physicist interlocutors are clueless about their birth ranks in the class of all observers os. In this respect, they would be in the sam e situation as you and I, who are, in the cosm any obs ervers cam of course, likewise ignorant about how m e into existence before we did som ewhere in the cosm os. (W e could additionally m ake the assum ption, without destroying the coherence of the thought experim ent, that the philosopher and the physicists don’t know their sequential position in the hum an species. The reason why this assum ption is not m ade in the original thought experim ent is that it is not needed. Inform ation about one’s hum an birth rank would not suffice, even if the basic form of Doom sday argum ent were correct, for deriving a prediction about the total num ber of 6

7 observers in the cosm os; for in order to do that, one would need to know one’s cosmic 10 ) ong all an ones. birth rank, one’s sequential position am observers, not just the hum It is hard to see what other kind of justification there could be for thinking a theory a priori unlikely in proportion to the size of the universe it proposes. Maybe one could attem e variant of Occam ’s razor. W e could form ulate a pt to base a case on som ust not postulate m ore space or m new sim plicity principle stating that one m atter than is ally necessary to account for known phenom ena, and appeal to this as the reason minim for Olum ’s new prior probability function (which we can dub the “OLUM-prior”). This proposal is problem atic for several reasons. First, Occam ’s razor and various plicity principles used in science are not so m uch injunctions against postulating sim atter (or space) as against excess theory. Sim excess m ony usually plicity and parsim y of independent assum ptions means econom , avoiding proliferation of unrelated s, keeping the num ber of free param echanism all as possible, not explanatory m eters as sm introducing too m any ad hoc clauses to account for recalcitrant findings, keeping the , and things of this sort. It is not generally taken to im basic ontology neat and slim ply an aversion to thinking that the world is very big bers of electrons, or that there are vast num 11 photons etc. out there. ological theory, which is the scientific field Second, if we look at the case of cosm that the presum ptuous philosopher hopes to influence, we see that spatial or m aterial parsim ony is em phatically not part of established best practice in this discipline. All cosm ologists agree that the cosm ously large; a m ajority probably thinks it is os is enorm ble of physically real universes infinite and that our universe is just one in a vast ensem ong those best placed to (in particular in the currently popular inflationary scenarios). Am aller universes and one judge these things, there is no sign of any preference for sm detects no epistem punction about hypothesizing that the world is very, very big. ic com 10 Bradl ey Mont on, in a paper fort hcom ing i n this journal , argues t hat the Doom sday argum ent is appl icabl e in the absence of knowledge about e di sagree but lack the space here for a detailed one’s birth rank. W analysis. 11 For i nstance, t his is general ly not used as an argum ent agai nst m any-worl ds versi ons of quant um mech anics, to wh ich we sh all retu rn sh ortly. 7

8 Third, the reference to “super-duper sym metry” in the thought experim ent was meant to suggest a situation where relatively clear-cut theoretical considerations existed and T roughly sim ilar prior probabilities. Unless the coherence of this for assigning T 1 2 possibility is disputed, we can use the hypothetical case presented in the gedanken as a SIA, whatever our actual circum stances m ay be in regard to test bed f or the plausibility of the OLUM-prior. But suppose, all this notwithstanding, that our a priori credence distributions ought to be skewed against hypotheses that say that the world is very big, and that this relation is one of proportionality: a universe twice as big gets, other things equal, half as much prior probability. Now, if we adopt this OLUM-prior, we can counter the ptuous philosopher’s argum ent, not by negating SIA, but by pointing out that the presum erely serves (when com SIA m pensate for the bined with the fact that you exist) to com and T end up with equal probabilities, as aller worlds. OLUM-prior’s favoring of sm T 1 2 ologists in the gedanken thought all along. (First we were asked to buy SIA as the cosm sday argum ent; now we are asked to buy the OLUM-prior an insurance against the Doom to insure us against SIA.) It m ay look as though the only effect of the OLUM-prior is to cancel out the effect of SIA. This is not quite the case. The reason is that SIA speaks about population size while the OLUM-prior speaks about spatial size . In Presumptuous Philosopher , this distinction m ade no difference because there we supposed that we had background e knowledge to the effect that the density of the observer-population was the sam said that the cosm os was both bigger and had m ore according to both theories; T 2 e can drive a wedge between SIA and the OLUM-prior, . W T observers than did 1 however, by considering a variation of the thought experim ent where the two theories disagree about the population density. Presumptuous Philosopher II : To this end, consider The year is now 2200, and m ore careful study in the intervening century has metry considerations actually do not favor T revealed that the super-duper sym 1 , but instead T and T , and they are neutral as between these two new T and 3 4 2 it with and T plicity, f instantiate theoretical virtues such as sim T theories. 4 3 8

9 evidence, etc. to sim ilar degrees, and contem ologists therefore assign porary cosm and T is that the roughly equal credences. One difference between them T 3 4 T is true than if is true. This es greater if observers is a trillion tim T density of 3 4 difference, however, is not directly observable, for both theories agree that the density of civilizations is so low that the probability that there should be another within our horizon is negligible. The theories also agree about the size of the os, and, m oreover, that the size of the cosm cosm os is so exceedingly vast that whichever of the theories is true there was virtually no probability that it should any quintillions of ore contain less than m civilizations. There is also another m subtle difference between the predictions that the theories m ake, a difference that is testable in an easy experim ent that cosm ologists are just about to perform . ptuous philosopher, who thanks to a breakthrough in life- Again the presum akes an appearance and argues, on extension research is still alive and well, m being correct is one in a trillion; grounds of SIA, that the probability of T 3 wherefore it is com pletely unnecessary to do the experim ent. Luckily, the physicists do not abort the experim ent but instead offer the philosopher a bet on the outcom e, agreeing to pay him one thousand dollars if the test in return for ten thousand dollars if it favors . The philosopher T T com es out in favor of 4 3 gladly accepts. As it happens, the physicists win the bet and get ten thousand dollars. As illion chance that the experim ent has yielded a m isleading result, a there is a one-in-a-m ent is proposed to verify the first. Despite the setback, the philosopher’s second experim erely m is hardly perturbed; he still assigns a probability of SIA-based confidence in T 4 , so he accepts a repeat bet with he physicists. The presum ptuous one in a m illion to T 3 philosopher is m self. aking a fool of him and The OLUM-prior doesn’t forestall this unpalatable consequence because T 3 ber hat they centrally disagree about is the num agree about the size of the cosm os. W T 4 of observers, and it is this difference that, when conjoined with SIA, gives rise to the im plausible conclusion that the philosopher is quixotically sticking by. One way out of this predicam ent would be by m odifying the OLUM-prior to one (call it the OLUM*-prior) that levies on hypotheses a probability tax that is proportional 9

10 to the num ber of observers they entail, rather than to the im plied size of the universe. Starting with the OLUM*-prior, and then taking account the fact that you exist as SIA says you should, the two effects – the first arguing for few observers and the second for ination from the many observers – cancel each other out. In light of the com bined illum ilar degrees T would assign them sim and T OLUM*-prior and SIA, people considering 3 4 line that seem s reasonable. of credence, a bottom The problem with this solution (apart from the fact that it appeals to an assum s as lacking in justification as the original ption, the OLUM*-prior, that seem uch. The OLUM*-prior “tam OLUM-prior) is that it proves too m es” the unruly SIA- induced probability shif t sim ply by killing it. The probability distribution we get af ter e as we had before hearing taking both the OLUM*-prior and SIA into account is the sam e have been offered two unsupported new principles whose net effect is nil. of either. W This brings us right back to where we started, for with no net epistem ic subsidy of hypotheses that favor m any observers, there is nothing to counterbalance the pessim istic probability shif t in f avor of few observers argued f or by the Doom sday argum ent and in related paradoxes, which are therefore left standing tall. III. OBSERVATION SELECTION THEORY AND QUANTUM MECHANICS ’s paper could be forgiven for com ing away with the im pression that we Readers of Olum 12 This is not the case. accept all the strange consequences of an unrestricted use of SSA. We agree with Olum that these consequences are very hard to accept. But rather than taking this as a justification for SIA, which, as we’ve argued above, com es with its own set of equally unacceptable consequences, we see it as a reason for replacing SSA with som e other principle that does a better job at guiding the use of indexical inform ation. Such a principle needs to be part of a general theory of observation selection effects, a theory that explains how to reason when indexical inform ation is linked to non- indexical hypotheses (and vice versa) and how to avoid anthropic biases in the course of such reasoning. The theory m ust to cater to legitim ate scientific needs as well as being paradox-free in philosophical thought experim ents. One of us has recently presented such 12 One of us has el aborat ed on t hese el sewhere (B ost rom 2001). 10

11 13 It shows that by taking more indexical a theory and placed it in a Bayesian fram ework. ation into account than SSA does (SSA considers only inform ation about which inform which temporal part of this observer you are, but you also have inform ation about e.g. you are at the current m oment) it is possible to relativize your reference class so observer different tim es, depending partly on your that it m ay contain different observers at epistem ic situation on the occasion. SSA, therefore, describes the correct way of assigning probabilities only in certain special cases. This enables us to avoid the original paradoxes without invoking SIA. In particular, the Doom sday argum ent is shown to be inconclusive in that it depends on particular assum ptions about the part of one’s orm ation – subjective prior probability distribution that has to do with indexical inf ptions that one is free to reject, and indeed, arguably, ought to reject in light of assum their strongly counterintuitive consequences. This theory of observation selection effects has applications for both philosophy and several scientific areas including cosm ology, evolution theory, therm odynam ics, e theory problem s involving im perfect recall, and quantum physics. traffic analysis, gam As a brief illustration, we can consider the application to quantum physics that Olum 14 ), the refers to in his paper. In this application (for which we’re indebted to Don N. Page theory concurs with what we would get if we sim ply used SSA with the universal reference class (the one containing all observers), so Olum ’s rem arks on this point are pertinent to the view we actually espouse. The idea is that we can in principle drive an em pirical wedge between single-history versi ons and m any-worlds versions of quantum theory by considering cases where different num bers of observers com e to exist e of som m easurem ent-like interaction (perhaps depending on the outcom e early quantum e random factor in the sym metry-breaking that determ som ined the strengths of the force constants). 13 Bostrom (2002). 14 D. N. Page, “Can Quantum Cosm ology Give Obse rvat ional Consequences of M any-W orl ds Quant um real Theory? General Rel ativity and Rel ativistic Ast rophysi cs, Ei ght h C anadi an C onf erence, Mont ” in , Quebeck ville, New York: Am erican Institute of Physics, , eds. C. P. Burgess and R. C. Myers, 225-32 (Mel 1999); also personal com muni cat ions. 11

12 The point can be m ost sim ply by considering a quantum cosm ological toy ade m model: -30 World 1: Observers; m easure or probability 10 -30 easure or probability 1-10 World 2: No observers; m Given these m easures, the single-history version predicts with overwhelm ing probability -30 orld 2 would be the (only) r ealized world. (How we could figure out ) that W (P = 1 - 10 easure of “worlds”, i.e. of whole bran ches of the universal wave function, is a the m separate problem that need not concern us here; we can think of the proposed test as physics itsel f but of versions of quantum being not of versions of quantum physics taken in conjunction with som cosm ological theory that specifies these e particular quantum orld 1 has been realized, this gives us strong measures). If we exist, and consequently W ry, given this particular toy m odel. By reasons for rejecting the single-history theo any-worlds version, both W orld 1 and W orld 2 exist, and since W orld 2 contrast, on the m has no observers, what is predicted (by SSA) is that we should observe W orld 1, easure. In this exam ple, if the choice is between the notwithstanding its very low m single-history and the m any-worlds versions, we should therefore accept the latter. Here’s another toy m odel: -30 10 observers; m easure or probability 1-10 World A: 10 50 -30 easure or probability 10 observers; m World B: 10 odel, finding that we are in W orld B does not logically refute the single-history In this m ake it extrem ely im version, but it does m probable. For the single-history version gives a -30 to us observing W orld B. The m any-worlds version, on conditional probability of 10 the other hand, gives a conditional probability of approxim orld ately 1 to us observing W 15 Provided, then, that our subjective prior probabilities f or the single-history and the B. 30 − 50 ⋅ 10 10 15 ≈ P 1 = 50 30 − − 30 10 − ⋅ + ⋅ ) ( 1 10 10 10 10 12

13 many-worlds versions are in the sam e (very big) ballpark, we should in this case again accept the latter. (The opposite would hold, of course, if we found that we are living in World A.) charges that “the fact that [thi s] treatm ent yields am biguous predictions Olum ology” (p. 183). In our cosm argues against its use in evaluating theories of quantum theory many-worlds version of quantum view, however, keeping single-history and pirically inseparable is not an adequacy condition on theories of observation selection em ethodology effects. On the contrary, it would be quite interesting if we could have a m that m akes the above kind of consideration intelligible and that m ay enable us to us to tease these very different conceptions of the world apart observationally. (There are 16 ake such a discrim ination, which for how one could m several other recent suggestions plem are neatly com ented by the present considerations.) up: Olum and we agree that an unrestricted use of SSA (with the universal To sum ber of counterintuitive reference class and in the absence of SIA) has a num consequences. SIA cancels these consequences. SIA, however, as the Presumptuous Philosopher gedanken reveals, has counterintuitive consequences of its own that seem just as bad as the ones it was m eant to defeat. Olum suggests that we can fix this problem by adopting what we have dubbed the OLUM-prior. W e have argued that the OLUM- anyway (as revealed by prior is unjustified and that it doesn’t fix the problem II ). W hat we called the OLUM*-prior would fix the problem , Presumptuous Philosopher but it is equally unjustified, and, crucially, adopting it has the effect of canceling SIA, bringing back us back to square one. SIA is a blind alley. ay Threatened by paradoxes on both sides, whether we accept or reject SIA, it m appear as if we have a crisis. But in fact, what we have is a philosophical opportunity – we have som e powerful theory-constraints that can be used to evaluate reasoning about inform ation that has an indexical com ponent. If we develop a theory of observation selection effects that heed these constraints along with m any other desiderata and criteria derived from both philosophical and scientific applications of anthropic reasoning, we 16 R. Pl aga, “Proposal for an experi mental test of t he m any-worl ds interpret ation of quant um m echani cs,” Found. Phys. 27, 559 (1997). 13

14 can hope to get serious theoretical leverage – m ethodological techniques and results that can turn over boulders of ignorance and reveal som ething new about the world. 14

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