Quantum Immortality and Non-Classical Logic
aa r X i v : . [ phy s i c s . h i s t - ph ] J u l Quantum Immortality and Non-Classical Logic
Phillip L. Wilson
School of Mathematics & Statistics, University of Canterbury,New Zealand. [email protected]. Te P¯unaha Matatini, New Zealand.July 29, 2020
Abstract
The
Everett Box is a device in which an observer and a lethal quan-tum apparatus are isolated from the rest of the universe. On a regularbasis, successive trials occur, in each of which an automatic measurementof a quantum superposition inside the apparatus either causes instantdeath or does nothing to the observer. From the observer’s perspective,the chances of surviving m trials monotonically decreases with increasing m . As a result, if the observer is still alive for sufficiently large m shemust reject any interpretation of quantum mechanics which is not themany-worlds interpretation (MWI), since surviving m trials becomes van-ishingly unlikely in a single world, whereas a version of the observer willnecessarily survive in the branching MWI universe. Here we ask whetherthis conclusion still holds if rather than a classical understanding of limitsbuilt on classical logic we instead require our physics to satisfy a com-putability requirement by investigating the Everett Box in a model of acomputational universe using a variety of constructive logic, RecursiveConstructive Mathematics. We show that although the standard Everettargument rejecting non-MWI interpretations is no longer valid, we never-theless can argue that Everett’s conclusion still holds within a computableuniverse. Thus we claim that since Everett’s argument holds not only inclassical logic (with embedded notions of continuity and infinity) but alsoin a computable logic, any counter-argument claiming to refute it mustbe strengthened. The several interpretations of quantum mechanics can be divided into twoclasses. One class contains a single interpretation, the many-worlds interpre-tation (MWI) of Hugh Everett [Eve57b, Eve57a], so named by Bryce DeWitt[DeW70, DeW72]. In the MWI, the wavefunction never collapses and essen-tially the universe can be taken to be governed by a single, objectively real,universal wave function. Although when he introduced this idea in his thesis of1957, Everett never spoke of a branching universe in which the universe splitsinto n branches (or worlds ) whenever a quantum measurement has n possibleoutcomes, this is nevertheless the common way in which MWI is discussed, and1e will to some extent use that language here. The other class contains all otherinterpretations of quantum mechanics. In these non-MWI interpretations, thereis only ever one world, and the wavefunction collapses whenever a measurementis taken: if a measurement has n possible outcomes, only one is ever realised.The famously accurate predictions obtained by calculating solutions to theSchrödinger equation do not depend upon the interpretation of quantum me-chanics. However it is desirable from a realist perspective to distinguish betweenthese interpretations. Having divided them into two classes, it is natural to startby asking whether we can distinguish between the classes, either in real-world orthought experiments. In particular, if we can show that the class of non-MWIinterpretations can be rejected, then we are left simply with the many-worldsinterpretation.One of the better-known thought experiments which claims to distinguishbetween these classes is a variant of the classic Schrödinger’s cat experiment inwhich the cat, or rather a human in place of the cat, is the observer [Teg14,Teg98, Lew00]. The life of the observer depends upon the outcome of an au-tomatic quantum measurement called a trial . Many trials occur, one after theother on a regular basis. As described in more detail in §2, this Everett Box isable to distinguish between MWI and non-MWI interpretations. The argumentis a probabilistic one: though the observer might get lucky and survive a fewtrials, continued survival in a non-MWI universe is extremely unlikely. On theother hand, a version of the observer is guaranteed to survive with probability1 in one branch of a MWI universe, guaranteeing the so-called
Quantum Im-mortality of that observer. The former non-MWI option is rejected as being toounlikely. Thus only MWI is a valid interpretation.We call the line of argument summarised above and presented in §2
Everett’sargument and to its conclusion as
Everett’s conclusion , more in homage to thegenesis of these ideas in Everett’s seminal thesis [Eve57b] than because Everetthimself clearly formulated them. Indeed, Everett never formally defined thisexperiment, but variants of it have been given independently by several authors[Squ86, Teg98]. It is not universally accepted. It has been attacked from variousdirections, not least in terms of its real-world applicability, for instance aroundthe definition of death (or at least of a discrete binary distinction between“alive” and “dead”) — see [Teg14, Teg98]. From the philosophical perspective wesee critiques based on the classical philosophical problem of individual identityand its persistence, what it means to “expect” a subjective outcome like one’sown death as opposed to predicting an objective event, the distinction betweenactual and probable events, and the meaning of probabilistic thinking in theMWI context; see [Lew00, Pap04, Ara12, Seb15, Vai18], and references therein.Everett himself anticipated some of these objections in [Eve57b].There are two other ways in which Everett’s argument is critiqued, bothof which are much more general in their scope. They concern (1) the role ofinfinity and the infinitesimal in physics, and (2) the role of the computable. Amotivation for the first of these is that if we live in a finite universe which hasexisted for finite time, and if the fields, matter, time, and space of the universeare all discrete at small enough scales discrete, then we should reject all objectsand arguments which employ the infinite and the infinitesimal. Such strictly finitist theories include digital physics [Whe90], cellular automata [Wol02], loopquantum gravity [RS88], and more besides — see [Sch97] and references therein.The questionable role of infinity in Everett’s argument has been highlighted by2Teg14] amongst others.The second critique, namely that our current theories of physics are non-computable, is the focus of the present work. Requiring a computable theory ofphysics is essentially the same as requiring all knowledge to be obtained throughan algorithmic process in finite time. It is not the same as requiring only finiteobjects, but it does necessitate working within so-called non-classical logics,as we outline in detail below. The desirable quality of computability in thefoundations of physics is not obtained by classical logic.Thus while probabilities and probabilistic thinking have been highlightedas potential concerns with Everett’s argument [Lew00, Pap04], and while therole of the infinite and the infinitesimal in physics [Teg14] have also been calledinto question in this context, to our knowledge no-one has examined the ar-gument from a computable perspective before, and in particular from withinnon-classical logic.Here we show that Everett’s argument that we must reject all non-MWIinterpretations of quantum mechanics is based on a classical understanding oflimiting behaviours of functions which need not hold in other, non-classical log-ics. In particular, we show that the argument fails in a constructive logic calledRecursive Constructive Mathematics, commonly referred to as RUSS, in whichall results are computable. Within RUSS, we show that the existence of so-called pathological probability distributions mean that we must reject Everett’sargument that all non-MWI interpretations are wrong. However, we are able toshow through a new argument that Everett’s conclusion holds even in a universe(or universes) governed by such non-classical logics, thus strengthening the ar-gument in favour of the MWI and requiring any counter-arguments to also bevalid in these non-classical logics.In §2 we give a brief overview of Everett’s argument for how the Everett Boximplies a rejection of all non-MWI interpretations of quantum mechanics. Next,in §3 we define computability and outline the arguments in favour of requiringcomputability in our theories of physics, before giving a summary of the mainresult from [MJW19] on which we base the principal argument in this paper.With this background we prove in §4 the Pathological Immortality Theorem,which shows that Everett’s argument does not work in RUSS. However, in §5 wepresent a constructive, computable proof that Everett’s conclusion neverthelessholds in a universe whose logic is that of RUSS. Finally, we summarise anddiscuss our results in §6.
A conscious observer is placed in a box with a lethal quantum apparatus. Thecontents of the box are completely isolated from the rest of the universe. Al-though the thought experiment does not depend on the details of the lethalapparatus, a particularly clear example is given by [Teg98] and called the “quan-tum gun”. The quantum gun consists of a gun coupled to a quantum system ofa particle in a superposition of two states. At regular time intervals, a measure-ment of this system is made automatically, and if it is found to be in one statethe gun fires a bullet, while if it is in the other nothing happens. After eitherfiring or not firing, the quantum gun resets: a new superposition is set up and3he memoryless process repeats . Each independent occurrence of this processwe call a trial . We take this or a similar lethal setup to be indefinitely repeatableand to occur every second . The apparatus and the observer are isolated fromthe rest of the universe, and this setup constitutes the Everett Box.What is the experience of the observer? It is rather starkly illustrated byTegmark’s gun if we contrast the Everett Box with a similar experiment inwhich instead of being aimed at the observer the gun merely fires or does notfire depending on the measurement. In this case, the observer can expect to heara random string of bangs and clicks: the bangs correspond to the gun firing,the clicks to it not firing and the equipment resetting. Over time the relativeproportion of bangs and clicks will tend towards the relative likelihoods of thosetwo outcomes. In the standard formulation, both outcomes occur with equalprobability and thus the observer expects over time that 50% of the sounds willbe bangs, and 50% clicks. Everett’s argument is actually independent of theselikelihoods, which need not be either equal or constant [Eve57b, Eve57a, Teg14].It is such a general case we consider in this paper.The preceding description is not that of the Everett Box, because the lifeand hence consciousness of the observer does not depend upon the outcome ofthe measurement. In the Everett Box, the gun is aimed at the observer in sucha way that should it fire then death is certain and swift . In this case, whatshould the observer expect?The answer depends upon which class of interpretation of quantum mechan-ics holds in our universe. If there is only one world, then each trial involves thecollapse of the wavefunction and a single outcome occurs for the observer: ei-ther they hear a “click” or they are instantly killed (and so hear nothing). Theymight get lucky once, they might get lucky twice, but as time goes on and thenumber of trials increases, the odds of them surviving decreases exponentially.If, however, there are many worlds, the totality of which contain all possibleoutcomes and histories, then by necessity there is always an observer alive afterany number of trials. For example, after one trial there are two versions of theobserver, the universe having branched into two worlds at the moment of thequantum measurement. In one world the observer heard “click” while in theother they died. After two seconds there are three worlds. In one of them, theobserver’s history shows that they heard “click-click”. In another world, theyheard “click” and then died on trial 2. In the third they died on trial 1. Afterthree trials there is an observer whose history is “click-click-click”, and after anynumber, m , of trials there will always be one world in which the observer hasheard m clicks. After a large number of trials there are many worlds, in allbut one of which the observer is dead, but crucially there remains one livingobserver. Thus the subjective probability of surviving m trials is 1 for any m ,because there is a world in which the observer is still alive after any number oftrials.Thus from the observer’s perspective this experiment has the potential to dis-tinguish between the two classes of interpretation, though the stakes are high.The argument runs as follows. The chances of remaining alive after a large This slight variant of Tegmark’s quantum gun of [Teg98] was given in [Teg14]. The time interval is not important to the subsequent argument, other than to allow formany repetitions within a human lifetime. Consciousness surviving death is not a part of the thought experiment. Both conditions are necessary as outlined in [Teg98]. ǫ ≪ couldbe set to the traditional σ -level, or indeed to a level of any stringency due tothe monotonicity of the probability of remaining alive. Furthermore, with eachsubsequent survived trial the confidence in rejecting non-MWI interpretationsincreases. Of course, if we do not live in an MWI universe then the experimentsimply kills the observer within a short time. They do not know that they donot live in a MWI universe, but neither do they know anything ever again.In more rigorous terms, in non-MWI interpretations the probability of be-ing dead after m trials, P ( m ) is, in the standard presentation in which theprobability of death at each trial is , simply P ( m ) = 1 − (cid:18) (cid:19) m which tends to unity as m → ∞ . The same conclusion holds regardless of theprobability of staying alive on trial k , which we denote p k . In this case, because p k < for all k we still have P ( m ) = 1 − m Y k =1 p k → as m → ∞ . (1)While (1) will remain true throughout this paper, we will see that in the com-putable logic RUSS we can no longer use it to conclude that the observer neces-sarily must expect to be dead after any finite number of trials. First, we mustreview what it means to be computable. A problem is said to be computable if it can be solved in an effective man-ner, which can be more formally defined in a number of models of computa-tion [Coo04, CPS13, Bri94]. Loosely speaking, computable problems are thosewhich can be solved algorithmically in finite time. The major milestone incomputability theory is the Turing-Church thesis identifying computable func-tions on the natural numbers with functions computable on a Turing machine[Bri94, BP18, Dea20, CPS13].It is not simply the rise in computer simulations, nor the “shut-up-and-calculate” instrumentalist approach to physics [Mer04], which have led someauthors to suggest that computability should be a requirement for our theoriesof physics [Zus69, RS88, Sch97, tH99, Fre03, Llo05, Wol02]. It is instead thenotion of the effective method embedded in computability that is important.A method is called effective for a class of problems when it comprises a finite5et of instructions which can be followed by a mechanical device , that theseinstructions produce a correct answer, and that they finish after a finite numberof steps [CPS13].The desirability for computability in physics is therefore a product of a desireto know, and a belief that the universe is ultimately comprehensible to us. Thereasoning in the syllogism goes that if we accept the two premises that (1) theuniverse is entirely comprehensible to the human mind, and (2) there is nothingextra-computational happening in the human mind, then we must accept theconclusion that physics is necessarily computable.However, our current theories of physics are not computable, built as they areon classical mathematical ideas which in turn rely on classical, non-computable,logic [BP18]. There are two issues here. The first concerns the notion of infinityand the related notion of continuity. Infinities abound in our physical theories,whether they are in limiting behaviours (as examined in non-classical logicsin the present paper) or in the related idea of continuous matter or continuousfields. In the latter case, even though we know that neither matter nor fields arecontinuous in our universe, we treat the “gap” between our continuous theoriesand discrete nature as being essentially a rounding error: the high accuracyof predictions made with the (presumptively Platonic) continuous theories isbecause our universe is approximately continuous. It is, after all, perhaps onlydiscrete below the Planck length, or on time scales shorter than the Planck time.The second issue, and the one that concerns us in this paper, is the no-tion of the underlying logic of the universe. Classical logic is not computable,relying as it does on non-computable notions such as the Law of Excluded Mid-dle (LEM) and omniscience principles [BR87, BP18]. Why should we workwith a logic that does not allow for computability if we wish our physics to becomputable? One answer is similar to the response to continuity and infinity:because this logic works, to an astonishing degree [Wil18]. A second responseis simply to reject the second premise given above. Perhaps there is somethingextra-computational happening within the human mind . This is consistentwith a robustly Platonic vision of the universe. If mathematical objects existin a Platonic realm of forms to which our minds (somewhat mysteriously) haveaccess, then the necessity of computability can be rejected. This is also consis-tent with the view above that our physical universe is only an (albeit excellent)approximation to a Platonic form.If however we insist with the authors above that our logic must be com-putable, then we necessarily have to work with non-classical logics which arecomputable. In particular, we should work within so-called constructive inter-pretations of logic [BP18], in which the classical interpretations of disjunctionand existence are rejected in favour of constructive ones. For example, the quan-tifier “there exists” becomes “we can construct (that is, give an effective methodfor defining) an object for which the given statement is true”. There are severalvarieties of constructive mathematics [BR87]. It should be noted that not allvarieties reject notions of infinity. Bishop’s Constructive Mathematics (referredto as BISH) [BB85], for example, admits many classical mathematical objectswhich rely on infinities and continuity, but insists that proofs using these objectsmust proceed constructively (and are therefore computable). This illustrates the The idea here is not that they must be followed by such a device, but that even a humanfollowing them needs no ingenuity in order to derive a correct answer. This perspective overlaps somewhat with the notion of hypercomputation [Cop02, Cop04]. epistemological constructivism of BISH whichremains agnostic on the ontology of mathematical objects, and the ontologicalconstructivism of other varieties of constructive mathematics which insist thatboth objects and proofs (procedures) must be computable [BP18, BR87]. Ithas been said that computable mathematics is simply mathematics done withintuitionistic logic [BP18].In order to subject Everett’s argument to a strong scrutiny in a non-classicallogic, we here choose an ontologically constructive variety of constructive logic,Recursive Constructive Mathematics, RUSS [BP18]. RUSS is a constructiveversion of recursive function theory, in which functions on the natural numbersare defined recursively. Essentially, RUSS takes the classical recursive analysisin the tradition of Turing and Church but uses only intuitionistic logic. Inthe following subsection, we briefly outline the theorems of a recent work incomputable probability based on RUSS which will be central to the argumentof this paper.
Working in RUSS, [MJW19] proved a seemingly counter-intuitive theorem, whichwe call here the
Infinite Monkey Theorem (IMT) . To state the IMT we first needsome notation. The IMT was written in the playful language of the famous apho-rism that a large enough group of monkeys with typewriters will reproduce thecomplete works of Shakespeare, but as is made clear in [MJW19], the IMT isreally about computable probability distributions, as indeed is our focus in thepresent paper.Retaining the metaphor of [MJW19], we work in an alphabet A (of size | A | ,including punctuation) and call a w -string any string of characters of length w ∈ N . For example, “ banana ” is a 6-string over the alphabet { a, b, n } . Eachmonkey works on a computer keyboard with | A | unique keys and each monkeytypes a w -string in finite time. We define M to be an infinite, enumerable setof monkeys (the monkeyverse ), and for any m ∈ N the m -troop of monkeys tobe the first m monkeys in M . We then have Theorem 1 (Infinite Monkey Theorem) . Given a finite target w -string T w anda positive real number ǫ , there exists a computable probability distribution on M of producing w -strings such that:(i) the classical probability that no monkey in M produces T w is ; and(ii) the probability of a monkey in any m -troop producing T w is less than ǫ . [MJW19] established an even stronger, target-free version of this theorem,which requires only a knowledge of w , not of T w .The theorem and its proof are computable. That is, while it is classically truethat it is impossible that no monkey reproduces the works of Shakespeare (part(i)), it is possible to construct a so-called pathological probability distributionon the monkeyverse such that the chances of actually finding the monkey thatdoes so can be made arbitrarily small (part (ii)). The key point in part (ii) isthat this is true for any finite m -troop of monkeys; the pathological distributiondoes not require knowledge of the size of the m -troop, it is simply pathologicalfor all finite sets. 7he monkeys correspond to any finite back-box process occurring in finitetime. The general conclusion drawn in [MJW19] is that in a computable universethe space of all possible probability distributions on enumerable sets containsa non-empty set of pathological distributions for which the IMT holds. This isin contradistinction to a universe governed by classical logic in which the IMTdoes not hold. It is this distinction that we exploit in the remainder of thepaper, by examining the impact of the existence of pathological distributionson the enumerable set of trials in the Everett Box. With the notation from §2 we can state that the probability P ( m ) of dyingwithin m trials is given by P ( m ) = 1 − m Y k =1 p k (2)for any m ∈ N , where p k is the probability of not dying on trial k . The key thinghere, in contrast to the manner in which the Everett Box is normally described,but in keeping with the more general case which Everett himself allowed for in[Eve57b], we consider a quantum apparatus which gives varying probabilities ateach trial. We note that it is not that the probability of death at each trial, p k ,cannot be known in advance; after all, in the standard formulation, p k = 0 . forall k . The restriction is that the observer cannot predict in advance whether shelives or dies on trial k , and that her fate is determined purely by the unknowablequantum state of the apparatus. We can now state the following theorem. Theorem 2 (Pathological Immortality Theorem) . While classically it is im-possible that an observer in an Everett box remains alive as the number of trialstends to infinity, there is a computable probability distribution on the trials suchthat the probability that the observer is alive after any finite number of trials isarbitrarily close to 1.Proof.
The result follows directly from the proof of the IMT in [MJW19]. Inparticular, we place the objects of the IMT and the objects of the PathologicalImmortality Theorem (PIT) in one-to-one correspondence as outlined in thefollowing table.IMT PIT p k probability that k th monkey failsto reproduce Shakespeare probability of not dying on k th trial P ( m ) probability that m -troop does re-produce Shakespeare probability of dying within m tri-alsThe PIT therefore gives us the apparently counterintuitive result that whileclassically it remains true that the observer’s probability of being alive after m trials tends to 0 as m tends to infinity, the classical interpretation of that8esult as being that after a certain finite number of trials the probability ofthe observer being alive should be so small that she should be surprised atremaining alive is not true in a computational sense, in which that probabilitycan remain arbitrarily close to 1 for any finite number of trials. As outlinedin [MJW19], it is important to note that the apparent contradiction here isonly between the classical notion of the limit and the existence of computablepathological distributions within RUSS; we are deliberately comparing resultsfrom non-commensurate logical systems in order to show that classical logic maylead us astray in a computable universe.At first blush the PIT would seem to suggest that in a universe run on com-putable logic Everett’s conclusion is no longer valid. If the quantum apparatushappens to produce a pathological distribution then it is no longer unlikely thatthe observer remains alive after any finite number of trials, since that likelihoodcan remain arbitrarily close to 1. As a result, since the observer can never knowfor sure that she is not in a pathological distribution, she cannot surely statethat remaining alive after any finite number of trials is unlikely and so she cannever reject non-MWI interpretations of quantum mechanics. However, in thenext section we argue that Everett’s conclusion should still be taken to holdeven in a computable universe and even with the existence of the PIT.
We state our main result as a theorem.
Theorem 3 (Computable Everett Box) . The Everett Box implies the rejec-tion of non-MWI interpretations of quantum mechanics even in a computableuniverse modelled by RUSS.Proof.
Suppose an observer in an Everett Box in a RUSS-universe has survivedmany trials. There are two situations to must consider depending on whetherthe probability distribution on the trials is pathological or not. To reiterate, theobserver does not know and has no way of knowing which situation holds.First, if the probability distribution is not pathological then the standard,classical-logic Everett argument holds, and the observer concludes that she mustreject all non-MWI interpretations of quantum mechanics.On the other hand, suppose that the distribution is pathological. SinceTheorem 5 of [MJW19] showed that such distributions are vanishingly rare thenthe observer must reject all non-MWI interpretations since in a single world theodds of being in such a distribution are vanishingly small. However, in theMWI there will always be an alive variation of the observer in a pathologicaldistribution and hence being one such consciousness is not surprising.To state the proof in other terms, we note that in a classical universe, soEverett’s original argument goes, the observer rejects non-MWI interpretationsbecause the odds of surviving repeated trials are so low, whereas in a RUSScomputable universe she reject non-MWI interpretations for the same reasonif she happens to be in a non-pathological distribution, or because the oddsof being in a pathological situation where the PIT holds in a single world are We take this to be equivalent to the requirement for computability in the logic we use inour theories of physics.
We have argued that Everett’s thought experiment implies the rejection of allnon-MWI interpretations of quantum mechanics even when the computabilityrequirement is added to physics through employing RUSS, a constructive, com-putable logic. Our main point is therefore that those whose rejection of Everett’sconclusions depends on some future recasting of physics in a computable formdo not have that option if RUSS is the correct logic on which to base physics.We have shown, in fact, that Everett’s argument still holds in at least one com-putable logic. The case to be made against Everett’s conclusion therefore mustbe stronger than has previously been appreciated.This naturally raises the question as to whether Everett’s conclusion holdsin other computable logics. For example, can we reproduce the argument inBISH, a computable logic which preserves most classical mathematical objects,including some of those which involve either infinity or continuity? What aboutin other logics which do not allow for such objects? And of course, what happensin a completely finitist universe?We have three final points to make. The first is to point out that although theargument here is given in favour of Everett’s conclusion, the argument behindthe proof of PIT works in any situation, quantum or otherwise, in which theprobability of an event occurring tending to 1 in the limit of an infinite sequenceof trials is taken to mean that the probability of that event not having happenedin any finite sequence of trials necessarily tends to 0. In RUSS, this is not true.The second point is to contrast the number of ways in which quantum im-mortality is possible in the classical world and in RUSS. In the former, thereis always a single branch of the universe which contains a living consciousnessregardless of the number of trials. But in RUSS, there are many such worlds:both the classical survivor and any observer who finds herself in a pathologicaldistribution. There are in fact countably infinitely many of these even thoughthey are vanishingly rare, since the requirement for a distribution to be patho-logical is simply that p k > − ǫ for all k [MJW19]. This is similar to the naturalnumbers having measure zero in the reals.The final point is to observe that one can consistently reject Everett’s ar-gument without rejecting the MWI. There are compelling reasons for adoptingthe MWI as a formal mathematical approach to quantum mechanics, separatefrom any metaphysical issues of what “worlds” mean, how probabilities are tobe interpreted when all possibilities are actually realised in the set of all worlds,and how that affects one’s decisions — see [Deu99, Vai18] and references therein.Nevertheless our central idea remains that if one accepts Everett’s argument ina theory built on classical logic, then one must accept it even in a computabletheory built on RUSS, and that arguments against it must also work or havetheir counterparts in a computable theory.10 eferences [Ara12] I. Aranyosi. Should we fear quantum torment. Ratio , 25(3):249–259,2012.[BB85] E. Bishop and D. S. Bridges.
Constructive Analysis . A Series ofComprehensive Studies in Mathematics. Springer-Verlag, New York,1985.[BP18] Douglas Bridges and Erik Palmgren. Constructive mathematics. InEdward N. Zalta, editor,
The Stanford Encyclopedia of Philosophy .Metaphysics Research Lab, Stanford University, summer 2018 edition,2018.[BR87] D. S. Bridges and F. Richman.
Varieties of Constructive Mathematics .LMS Lecture Notes Series. Cambridge University Press, Cambridge,1987.[Bri94] D. S. Bridges.
Computability: a Mathematical Sketchbook . Springer-Verlag, 1994.[Coo04] S. B. Cooper.
Computability Theory . Chapman & Hall, 2004.[Cop02] B. J. Copeland. Hypercomputation.
Minds and Machines , 12:461–502,2002.[Cop04] B. J. Copeland. Hypercomputation: philosophical issues.
TheoreticalComputer Science , 317:251–267, 2004.[CPS13] B. J. Copeland, C. Posy, and O. Shagrir.
Computability: Turing,Godel, Church, and Beyond . MIT Press, 2013.[Dea20] Walter Dean. Recursive functions. In Edward N. Zalta, editor,
The Stanford Encyclopedia of Philosophy . Metaphysics Research Lab,Stanford University, summer 2020 edition, 2020.[Deu99] D. Deutsch. Quantum theory of probability and decisions.
Proceedingsof the Royal Society of London A , 455:3129–3137, 1999.[DeW70] B. DeWitt. Quantum mechanics and reality.
Physics Today , 23:30–40,1970.[DeW72] B. DeWitt. The many-universe interpretation of quantum mechan-ics. In B. d’Espangat, editor,
Foundations of Quantum Mechanics .Academic Press, New York, 1972.[Eve57a] H. Everett, III. “Relative state” formulation of quantum mechanics.
Reviews of Modern Physics , 29(3):454–462, 1957.[Eve57b] H. Everett, III.
Theory of the universal wavefunction . PhD thesis,Princeton University, Princeton, NJ, 1957.[Fre03] E. Fredkin. An introduction to digital philosophy.
International Jour-nal of Theoretical Physics , 42(2):189–247, 2003.11Lew00] P. J. Lewis. What is it like to be Schrodinger’s cat?
Analysis ,60(1):22–29, 2000.[Llo05] S. Lloyd. A theory of quantum gravity based on quantum computa-tion. arXiv:quant-ph/0501135 , 2005.[Mer04] N. D. Mermin. Could Feynman have said this?
Physics Today ,57(5):10–11, 2004.[MJW19] M. McKubre-Jordens and P. L. Wilson. Infinity in computable prob-ability.
Journal of Applied Logics — IfCoLog Journal of Logics andtheir Applications , 6(7):1253–1261, 2019.[Pap04] D. Papineau. David Lewis and Schrodinger’s cat.
Australian Journalof Philosophy , 82:153–169, 2004.[RS88] C. Rovelli and L. Smolin. Knot theory and quantum gravity.
PhysicalReview Letters , 61(10):1155–1158, 1988.[Sch97] J. Schmidhuber. A computer scientist’s view of life, the universe, andeverything. In C. Freska, editor,
Foundations of Computer Science .Springer, 1997.[Seb15] C. T. Sebens. Killer collapse: empirically probing the philosophicallyunsatisfactory region of grw.
Synthese , 192:2599–2615, 2015.[Squ86] E. J. Squires.
The Mystery of the Quantum World . Hilger, 1986.[Teg98] M. Tegmark. The interpretation of quantum mechanics: Many worldsor many words?
Fortsch.Phys. , 46:855–862, 1998.[Teg14] M. Tegmark.
Our Mathematical Universe . Vintage Books, New York,2014.[tH99] G. ’t Hooft. Quantum gravity as a dissipative deterministic system.
Classical and Quantum Gravity , 16(10):3263–3279, 1999.[Vai18] Lev Vaidman. Many-worlds interpretation of quantum mechanics.In Edward N. Zalta, editor,
The Stanford Encyclopedia of Philoso-phy . Metaphysics Research Lab, Stanford University, fall 2018 edition,2018.[Whe90] J. A. Wheeler. Information, physics, quantum: the search for links.In W. H. Zurek, editor,
Complexity, Entropy, and the Physics of In-formation . Addison-Wesley, 1990.[Wil18] P. L. Wilson. What the applicability of mathematics says about itsphilosophy. In S. O. Hansson, editor,
Technology and Mathematics .Springer, 2018.[Wol02] S. Wolfram.
A New Kind of Science . Wolfram Media, 2002.[Zus69] K. Zuse.