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Time travel possibilities within a Gödelian space-time structure Betton, Carla D. 1994

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TIME TRAVEL POSSIBILITIES WITHIN A GODELIAN SPACE-TIME STRUCTURE by CARLA D. BETTON B.A., The University of Alberta, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Department of Philosophy) We accept this thesis as conforming to the required standard _ THE UNIVERSITY OF BRITISH COLUMBIA December 1994 ©Carla D. Betton, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Department DE-6 (2/88) A B S T R A C T This thesis examines the question as to whether Paul' Horwich has preserved the possibility of time travel into the past, within a Godelian space-time structure. In my thesis I argue that Horwich is not successful in his attempt to defend time travel. Horwich himself establishes conditions under which he determines that time travel to the local past would result in "bilking attempts" being manifested. Horwich sees this as unacceptable, as it would result in the past being changed, which gives rise to a logical contradiction. Horwich avoids having to posit a non-G6delian space-time structure however, by attributing the non-existence of time travel within bilking range to excessive fuel requirements. However, fuel requirements to the spatially distant past are not technologically inaccessible, therefore Horwich claims time travel is still possible, as the spatiality distant past is outside of bilking range I argue that trips to the spatially distant past are subject to the same bilking conditions Horwich placed on the local past. Consequently time travel is not possible to any past, as it would result in either a logical contradiction , or multiple inexplicable coincidences. In Chapter I, I examine some of the work of Albert Einstein and Kurt Gbdel that was relevant in sparking much of the modern literature on time travel. Kurt Gbdel proposed a theory of a curved space-time based on field equations yielded by Einstein's General Theory of Relativity, that would mathematically allow for the existence of closed causal chains. This opened up the possibility of time travel into the past. Chapter II is an overview of the relevant literature on time travel. I have attempted to give a sampling of a number of theories in this area, on both sides of the debate. Chapter III is an in-depth look at Horwich's theories on time travel and backward causation, taken from Chapters 6 and 7 of his book Asymmetries in Time - Problems in the Philosophy of Science. Here he defends backward causation against the bilking argument, and attempts to defend time travel against four different paradoxes. In Chapter IV, I raise four concerns I have with Horwich's theories: 1) that he has avoided the issue of temporal parts, thereby overlooking an oddity that results from time travel. 2) that he uses backward causation to refute one of the paradoxes, yet by his own admission, time travel is a form of backward causation that requires bilking attempts of the form that violate the V-correlation, and therefore must be rejected. 3) that the spatially distant past is not out of bilking range as Horwich argues; and is therefore just as inaccessible as the local past and 4) that the spatially distant past as Horwich defines it is ultimately inaccessible as a journey back to the distant past would take longer than the proper duration of the lifetime of the system that was sent back.. I conclude that while Horwich may be successful in defending Gbdelian space-time structure, he does not preserve the possibility of time travel within a i v xirved space-time structure, It is my argument that time travel of any sort is mpossible, as logical contradictions wilkbe the end result. I T A B L E O F C O N T E N T S Abstract Table of Contents Acknowledgement vi Dedication . vii Introduction ' 1 Chapter I - Einstein & Gbdel - Setting the Stage 4 Einstein: Special Relativity 4 Einstein: General Relativity 6 Gbdel: Closed Causal Chains 8 Chapter II - Theories of Time Travel 12 Time Travelling Paradoxes 12 Backward Causation 17 Where Are All The Time Travellers 20 Causal Loops 21 Chapter III - Horwich's Arguments 25 Oxymoron Paradox 25 Leibniz' Law 26 Changing vs. Influencing the Past 27 Autofanticide Paradox 28 Backward Causation Again 30 Chapter IV - Concerns About Time Travel 36 Summary 36 Questions to be Raised 39 Question #1 39 Question #2 42 Question #3 44 Question #4 47 Conclusion 50 Footnotes 5 1 Bibliography 5 5 v i ACKNOWLEDGEMENT Firstly, to my parents, thank you for your love and your belief in me, and especially to my Dad, whose footsteps broke my path. Thank you to my sister and brother and my husband's family for all their caring and support. And thank you also to my supervisor Jack Stewart for his help, patience and guidance, and to Steve Savitt for his valuable comments during the writing of this thesis. DEDICATION To Dana "I've been around enough to know that you're the one I want to go through time with." I N T R O D U C T I O N Travelling through time. Is this the stuff that science fiction is made of, or does it have a place in the academic world of the philosopher, the scientist and the mathematician? While many fiction writers delight in the creative license such a topic gives them, there are many scholars who also find this to be a fascinating subject, one worthy of much thought, discussion and debate. The subject of time itself is a mystery as old as man. Throughout the ages it has been pondered; evasive, frustrating, infinitely simple and at the same time frighteningly incomprehensible. The literature that has been devoted to this subject is endless. And yet, until very recently, the idea of actually travelling through time remained a mere fantasy, a notion so farfetched that any serious study of time gave it a wide berth. Up until the end of the nineteenth century, Newtonian thinking was the dominant force among philosophers and scientists. According to Isaac Newton, both space and time were absolute. He believed that time and space existed independent of anything else. Newton first set out these ideas in the Principia in 1687 where he said that" they are autonomous entities existing independently of things". He went on to explain that "absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Time is [also] a reality in its own right. Absolute, true and mathematical time, of itself and from its own nature flows equably without relation to anything external." 2 Along with numerous other discoveries however, the advent of the twentieth century brought Einstein's theories of special and general relativity. In a short period, Einstein was to revolutionize the way people thought about time. Among other things, his theories reinforced a move away from absolute time to a notion of relative time. This was to have a significant impact on many areas of scientific and philosophical study. Most importantly for this paper, it sparked a paper by Kurt Godel in 1949 in which he suggested that space-time could be curved in such a way as to allow for the existence of closed causal chains - and the possibility of backward time travel! And even better, he had equations to substantiate his claims! Once Gbdel's paper had been published, the door was now open for 'respectable academicians' to pursue theories of space-time structure that would allow for travel into the past, as well as explore the theoretical paradoxes that began showing themselves as the field of study expanded. In the past forty years, the work in the area of time travel has been greatly developed. Many new theories have been suggested, and the two sides are becoming more clearly established in the debate over the possibility of actually travelling in to the past. The purpose of my thesis is to examine the work of one philosopher who has firmly declared himself to be on the affirmative side of the camp. Paul Horwich, a philosopher from M.l.T. has taken a decided stance in favour of the possibility of travelling through time into the past. In his book Asymmetries in Time - Problems in the Philosophy of Science (1987), Horwich devotes all of Chapter 7 to defending the possibility of time travel against four of the most common arguments used against the idea. 3 In my thesis I will argue that Horwich falls prey to his own arguments. I will show how he effectively eliminates the possibility of time travel into the local past on the basis of 'bilking attempts', and then go on to argue that the same argument can be used against time travel into the spatially distant past - a concept that he endorses as a real possibility. I will further show that Horwich must necessarily invoke backward causation in order to substantiate his refutation of one of the paradoxes (changing vs. influencing the past) and that by his own admission this is a contradiction, as backward causation in time travel requires bilking attempts that violate the principle of V-correlation. I will also argue that Horwich has overlooked the important issue of temporal parts, and that he ultimately provides us with a definition of the local past that leaves the spatially distant past inaccessible to any system. My thesis is divided into four chapters. In Chapter I, I examine the role that Albert Einstein and Kurt Gbdel played in opening the door for modern theories of time travel. Chapter II is a survey of much of the relevant literature in the area of time travel. I have included in this chapter work from the areas of philosophy, physics, theology and science fiction. This chapter is meant to give the reader an overview of the different theories of time travel, and the wide variety of areas in which such work can be found. Chapter III is devoted solely to the work of Paul Horwich in this area, specifically Chapters 6 and 7 of his book Asymmetries in Time, which are on backward causation and time travel respectively. Finally in Chapter IV, I present a summary of relevant points, and raise four concerns I have with Horwich's arguments. I conclude that Horwich has not successfully preserved the possibility of time travel within a Gbdelian space-time structure. 4 C H A P T E R I E I N S T E I N A N D G O D E L - S E T T I N G T H E S T A G E Einstein: Special Relativity In 1905, Albert Einstein (1879 - 1955) presented a paper in which he introduced the idea of abandoning Newton's notion of absolute time, as well as the popularly accepted concept of a medium through which light waves were thought to travel. Working as a clerk in a Swiss patent office, this young German was virtually unknown to the scientific world. His revolutionary paper Zur Elektrodynamik bewegter Korper suggested that the laws of science should be the same for all free moving observers, regardless of their speed. This would not only encompass Newton's laws of motion, but also the speed of light. This postulate formed the basis of the Special Theory of Relativity. . One of the most important consequences of this new theory was that Einstein equated mass with energy, resulting in his famous equation E = MC 2 (E=energy, M=mass and C=the speed of light). The implication of this equation was that nothing could travel faster than the speed of light (186 000 miles per second). Hawking, (1988) gives a very clear explanation of this phenomenom: Because of the equivalence of energy and mass, the energy which an object has due to its motion will add to its mass. In other words, it will make it harder to increase its speed...As an object approaches the speed of light, its mass rises ever more quickly, so it takes more and more energy to speed it up further. It can in fact never reach the speed of light, because by then its mass would have become infinite, and by the equivalence of mass and energy, it would have taken an infinite amount of energy to get there. For this reason, any normal object is forever confined by relativity to move at 5 speeds slower than the speed of light. Only light, or other waves that have no intrinsic mass, can move at the speed of light. 1 This theory also revolutionized the way we look at time and space. In the past, according to Newton, observers would have had to agree on the amount of time something took to move from point to point, as time was thought to be absolute. However, if it was light that the observers were measuring, the result would be varying speeds of light, since the speed of anything is measured by dividing the distance something travelled by the time it took to get there. Since Einstein's theory postulated a constant speed of light, the implications were startling. The distance the light travelled and the time it took the light to get there would both be relative for the observer. Consequently, observers could assign different times and positions to the same event, provided they were moving relative to one another. The constancy of the speed of light was to become a backbone of measurement in scientific circles. It allowed for extremely accurate measurement of distance, (for example, a metre is the distance travelled by light in 0.000000003335640952 seconds) producing such terms as light second and light year. The theory of special relativity also demonstrates the interconnectedness between time and space. An event has three co-ordinates of space and one co-ordinate of time, which has come to be known in relativity as a four dimensional space called space/time. There was however, one major problem with the special theory of relativity. It did not account for gravitational effects as set out by Newton's 6 accepted theory of gravity. Einstein had to develop an entirely new theory to deal with that issue. Einstein: General Relativity The inconsistencies that resulted from the special theory of relativity with respect to gravity dealt with the fact that according to Newtonian gravity, gravitational effects would move at infinite velocity - ie. effects happen instantaneously. However, according to special relativity, everything must travel at or below the speed of light. Special relativity does not allow for instantaneous movement. In 1915, Einstein proposed a theory that he felt would be consistent with special relativity. It is what is now known as the General Theory of Relativity, and is fundamentally a geometrical theory. He suggested that gravity is a consequence of space-time being not flat, but curved by the distribution of energy and mass in it. It is not a gravitational force that makes bodies travel in a curved orbit; it is bodies trying to follow a straight line in a curved space. The bodies are travelling in a geodesic - the shortest (or longest) path between two points. For example, it is the mass of the sun that curves space-time around it so that the planets that orbit it appear to be moving in a circular pattern in three dimensional space. In fact, they are following a geodesic line in four dimensional space-time. Einstein's famous field equations were a result of this purely geometrical theory as determined by the distribution of matter and energy in each region of space-time. It was in fact the orbit of the planet Mercury which gave the first confirmation of Einstein's new theory. Newtonian gravity had predicted the orbit 7 of the planets very accurately, but there was a small discrepancy with Mercury which had been noticed by scientists even before 1915. General relativity accounted for this deviation, and later other smaller deviations were accounted for in other planetary orbits. . According to general relativity, light rays must also travel in geodesies, giving the effect of being 'bent' by gravitational fields. This implies for example that a star whose light passes near enough to our sun to be affected by its gravitational field, would have its light bent inward toward the sun's mass. This would give the impression to an observer on Earth that the star is positioned differently from what it actually is. General relativity also predicts that'time will run slower near a massive body, because of the relation between the energy of light and its frequency - the length of time between each wave of light. As light travels toward a body such as the earth, it loses energy, and its frequency decreases. Consequently, everything appears to be taking longer to happen. This is why someone travelling at the speed of light (theoretically) would not age - they would be staying with the same wave the entire journey, so no time would go by. This only seems like an absurd conclusion when one is fixed on the notion of absolute time. Einstein's theory of relativity puts an end to such a notion. Both - time and space become reduced to dynamic qualities that both affect and are affected by everything in the universe. 8 Godel: Closed Causal Chains In 1949, Kurt Gbdel wrote a short paper entitled " A Remark About the Relationship between Relativity Theory and Idealistic Philosophy".2 This paper, one among the innumerable number sparked by Einstein's theories, dealt with a discovery Gddel had made with regard to general relativity. Gbdel determined that Einstein's field equations yielded certain solutions that would allow for the existence of closed causal chains in certain space-time structures. The abandonment of an objective lapse of time, and consequently the advent of the relativity of simultaneity indicative of Einstein's theories, seemed to leave all observers with their own 'set of nows' according to Gbdel. However, he argued that this is only the case in the "abstract space-time scheme of special relativity theory and in certain empty worlds of general relativity theory". Gbdel went on to suggest that as soon as matter is introduced along with the curvature of space it produces (established by general relativity) the equivalence of observers is basically destroyed. Those who follow the mean motion of matter are in fact distinguished from the rest. This could potentially establish a 'true' or objective notion of time, since time would be the same for all such observers in all possible universes. It could be concluded from this, that it is not necessary to abandon the idea of absolute time. Gbdel however, does not embrace this conclusion. He argues that there exists another kind of cosmological solution for universes - distinguished by the fact that "the compass of inertia in them everywhere rotates relative to matter, which in our world would mean that it rotates relative to the totality of galactic systems".4 He calls these "rotating universes". In these rotating universes, observers would not all fit into one objective world time, because "for each 9 possible concept of simultaneity and succession, there exist others which cannot be distinguished from it by any intrinsic properties, but only by reference to individual properties, such as, eg. a particular galactic system".5 Another feature of such worlds, according to Godel, are temporal conditions which could allow for travel into the past. He says: "by making a round trip on a rocket ship in a sufficiently wide curve, it is possible in these worlds to travel into any region of the past, present and future and back again, exactly as it is possible in other worlds to travel to distant parts of space". 6 Paul Horwich (1987) provides a relatively accurate diagram of such a Godelian closed causal chain as produced by these solutions. Horwich explains: For the sake of picturability, there are only two space dimensions. The space-time has two parts. There is an internal cylinder whose light cone (within which all possible world lines must lie) are parallel to the central vertical axis. The rest of the space-time surrounds the cylinder. In that Figure #1 1 0 outer part the time dimension is circular, threading the light cones which are perpendicular to the central axis. Now consider a world line that begins in the central axis, moves upward and outward until it reaches the cylinder boundary, then spirals down outside the cylinder, reenters it, and moves upward and inward to the center again. The endpoint of this line is earlier than its origin, although at every point (neglecting the cylinder boundary) it is oriented toward the future.7 Of course, the obvious question to be raised is - of what consequence is this to our world? Just because certain conditions prevail in possible worlds, need this have any actual bearing on our real world? Gbdel is inclined to think that it does. It is true that our world cannot be represented by the static rotating solutions under discussion. However, there are expanding rotating solutions, according to Gbdel, in which absolute time could be done away with and which could be representative of our world. That is not to say our universe does exhibit this type of curved Gbdelian space-time structure, only that it is not impossible according to Gbdel's arguments that it could. Gbdel's theory opens the door to an actual physical theory of time travel to the past respectably rooted in established scientific notions. Expectedly, scientists and philosophers alike have responded skeptically to such a proposition. Numerous problems with actually travelling back in time have been raised, some suggesting Gbdel's solutions are incompatible with other facts known about the universe, some challenging the physics or math involved, others deeming it conceptually impossible or resulting in anomalous consequences, and still others, including Gbdel himself, rejecting the notion based on the technological impossibility of attaining the velocities necessary to complete such a trip. And there have also been those who have embraced the idea, anxious to forge ahead into new possibilities, and to prove that any 1 1 difficulties raised with this idea can be overcome. The next chapter will explore some of the arguments that have been put forward on both sides of the controversial debate that Godel and Einstein set the stage for. 1 2 C H A P T E R II T H E O R I E S O F T I M E T R A V E L Time Travelling Paradoxes Paul Nahin, in his excellent book Time Machines (1993), gives one of the most comprehensive overviews to date of the existing literature on time travel. Delving into philosophy, physics, theology and science fiction, Nahin presents an extensive account of the ongoing debate on this issue. While he presents both sides of this controversial topic and the majority of his book is dedicated to presenting other writers thoughts and theses on this subject, his own personal support for the possibility of time travel is intricately threaded throughout the pages. Of primary concern to Nahin is the elimination of the position that backward time travel is impossible because of the logical paradoxes and causality violations it seems to entail. He also seems to support the proposition that the principle of self-consistency will safeguard the Universe, should backward time travel occur. Nahin begins his exploration of time travel with examples in literature of paradoxes in which the past is changed. In Ray Bradbury's "A Sound of Thunder"(1980)1 a tour company takes people back to the past to hunt dinosaurs. One such traveller accidentally kills a butterfly, thus beginning a domino effect of changes that spread ripple-like into all the corners of history, resulting in a completely different present world to which the time traveller returns. 1 3 On the opposing side of this position is Fritz Leiber who wrote about how difficult the past was to change in his 1976 story "Try and Change the Past". Leiber wrote: Change one event in the past and you get a brand new future? Erase the conquests of Alexander the Great by nudging a Neolithic pebble?...Brother, that isn't the way it works at all!! The space-time continuum's built of stubborn stuff and change is anything but a chain reaction. Change the past and you start a wave of changes moving futureward, but it damps out mighty fast. Haven't you ever heard of temporal reluctance or the Law of Conservation of Reality?3 He also wrote in his novel The Big Time (1976): Most of us [begin as participants in the Change War] with the false metaphysic that the slightest change in the past - a grain of dust misplaced - will transform the whole future. It is a long time before we accept with our minds as well as our intellects the Law of the Conservation of Reality: that when the past is changed, the future adjusts barely enough to admit the new data. The Change Winds meet maximum resistance always.4 Both of these views allow for what Nahin considers the impossible - changing the past. Philosopher David Malament (1984) summed up the position that Nahin is arguing from when he wrote: "[one] view is that time travel....is simply absurd and leads to logical contradictions. You know how the argument goes. If time travel were possible, one could go backward in time and undue the past. One could bring it about that both the conditions P and not P obtain at some point in space-time. For example, I could go back and kill my earlier infant self making it impossible for that earlier self to grow up to be me. I simply want to remark that arguments of this type have never seemed convincing to me.... The problem with these arguments is that they simply do not establish what they are supposed to. To be sure, if I could go back and kill my infant self, some sort of contradiction would arise. But the • only conclusion to draw from this is that if I tried to go back 1 4 and kill my infant self, then for some reason I would fail. Perhaps I would trip at the last minute. The usual arguments do not establish that time travel is impossible, but only that if it were possible, certain actions could not be performed.5 Nahin does, however, point out that this last point still involves the problem of free will vs. determinism. Still, the gist of what Malament is saying is the vantage point from which Nahin and many other argue for the possibility of time travel. Those who argue against it more often than not invoke some variation of this paradox to prove their point. Kurt Gbdel himself was among those who raised such a paradox. Although practically all the literature concerning time travel over the past 40 years is an offspring of Godel's famous 1949 paper, Godel himself felt that logically such a paradox resulted in a contradiction, and therefore must be avoided. He took this position even though it was he that offered the first theoretical possibility of time travel that was firmly grounded in physics and mathematics. Logically, Godel could not come to terms with his variation of the example that Malament used. That is - Gbdel realized that if it was possible to travel to the past, then a person could travel to a time when their younger self was alive, and could do something to that self that they know did not occur. A stronger version of this argument has become one of the most common arguments against time travel - the Grandfather Paradox. This paradox simply says that if a person could travel to the past, they could go back to a time when their grandfather was alive and kill him before their father had been conceived. The contradiction obviously arising from the fact that if one's father had not been conceived, then the time traveller could not have been conceived, and therefore would not exist to be able to travel back in 1 5 time to commit the deed. This somewhat violent version of the paradox is the most commonly used one among philosophers, although Gbdel felt that even his milder version was strong enough to thwart the logical possibility of time travel. Murray MacBeath, in his paper "Who Was Dr. Who's Father"(1982) wanted to point out that this much used grandfather paradox had a crucial gap in it that needed tightening up. He argues that the example: assumes that your grandfather was not a time traveller. For, if he were, he could have conceived your father in (say) 1920, travelled back to 1915, and thus been there for you to kill him before your father had been conceived What must be claimed is that, if time travel were possible, it would also be possible for you to kill your grandfather at a point earlier in his particular time than the conception of your father.6 There are however, many modern philosophers who simply do not believe that the Grandfather paradox defeats the possibility of time travel, including MacBeath, who only discusses this paradox in order to later refute it. Among others who find this sort of logic flawed are [Lewis, 1976], [Malament, 1984], [Horwich, 1987], and [Nahin, 1993]. In Chapter 4 of his book Time Machines. Nahin discusses time travel paradoxes and some of their explanations. He defines 'paradox' as "something that appears to contain contradictory or incompatible parts, thus reducing the whole to seeming nonsense".7 This is certainly how many people would view the result of a time travel expedition in which the grandfather paradox was . manifested. It is a logical impossibility because of the inexplicable parts. Nahin and others however, disagree with this conclusion. 1 6 Nahin feels the mistake is made in assuming that the past can be changed. He is prepared to admit that this is indeed a logical contradiction and hence an impossibility. You cannot travel back in time and kill your grandfather simply because you did not. Nahin explains: Professor Hospers (1967) unleashes what he thinks is the supreme argument against time travel...'Many centuries B.C., the pyramids were built, and when all of this happened you were not there - you weren't even born. It all happened long before you were born, and it all happened without your assistance or even your observation. This is an unchangeable fact: you can't change the past. That is the crucial point: the past is what happened, and you can't make what has happened not have happened. Not all the king's horses or all the king's men could make what has happened not have happened, for this is a logical impossibility. When you say that it is logically possible for you (literally) to go back to 3000 B.C. and help build the pyramids, you are faced with the question: did you help them build the pyramids or did you not? The first time it happened, you did not: you weren't there, you weren't even born yet, it was all over before you came on the scene. All you could say, then, would be that the second time it happened, you were there -and there was at least a difference between the first time and the second time: the first time you weren't there and the second time you were.' Hospers' puzzle is, of course, the grandfather paradox in different clothes. Hospers' error in his argument against time travel, in particular, is precisely at the point where he states his belief that 3000 B.C. occurs twice. In fact, there is no reason for believing this - 3000 B.C. (or any year!) happens just once. If you will go back to 3000 B.C., then you were there; and if you WEREN'T there then you won't go. You don't remember 3000 B.C. even if you were there (and even though that year is in the past) because your time trip is not in your past but rather in your (personal) future. This all seems odd of course, but it is not illogical.8 1 7 David Lewis, in his paper "The Paradoxes of Time Travel"(1976) concurs with Nahin's position. Using the grandfather paradox as a referent, Lewis writes: Tim cannot kill Grandfather. Grandfather lived, so to kill him would be to change the past. But the events of a past moment are not subdivisable into temporal parts and therefore cannot change. Either the events of 1921 timelessly do include Tim's killing of Grandfather, or they timelessly don't. We may be tempted to speak of the "original" 1921 that lies in Tim's personal past, many years before his birth, in which Grandfather lived; and of the "new" 1921 in which Tim now finds himself waiting in ambush to kill Grandfather. But if we do speak so, we merely confer two names on one thing. The events of 1921 are doubly located in Tim's (extended) personal time, like the trestle on the railway, but the "original" 1921 and the "new" 1921 are one and the same. If Tim did not kill Grandfather in the "original" 1921, then if he does kill Grandfather in the "new" 1921, he must both kill and not kill Grandfather in 1921 - in the one and only 1921 which is both the "new" and the "original;" 1921. It is logically impossible that Tim should change the past by killing Grandfather in 1921. So Tim cannot kill Grandfather.9 This line of reasoning is the most commonly accepted method of refuting the Grandfather Paradox. There is only one past, and it is unchangeable. Everything that has happened are the only things that can happen, including events that lie in one's own personal future. However, allowing events in the past to be part of one's future brings up the next issue - backward causation and affecting the past. Backward Causation The issue of backward causation introduces an entirely new controversial topic - one that in itself continues to be widely debated. Many philosophers will not accept the logical possibility of backward causation, let alone the possibility 1 8 of time travel. Paul Horwich, in his chapter on time travel argues that one can affect but not change the past. This argument is somewhat unclear, and still seems to create a paradox - how could one affect the past without changing it?. A contradiction can be avoided however, if backward causation is admitted. Nahin writes: you could not travel anywhere into the past unless you've already been there, and when you do make the trip, you will do what you have already done. You could not ....change the course of history by revealing twentieth century physics in the eighteenth century. This does not mean you will necessarily be ineffectual during your stay in the past. Not being able to change the past is not equivalent to being unable to influence or affect the past. You cannot prevent either the Black Death in London of 1665 or the Great Fire the following year, but it is logically possible that you - a careless time traveller - could be the cause of either or both. 1 0 J. Friedman, in his paper "Cauchy Problems in Spacetime with Closed Timelike Curves"(1990), referred to the concept of 'what has happened will happen and what hasn't happened won't happen' as the principle of self-consistency. He says: "the only solutions to the laws of physics that can occur locally in the real Universe are those which are globally self-consistent".1 1 In other words, Nahin adds, "strict causality is not invoked but only a logical consistency between events at different times is required". 1 2 The philosophical debate on the issue of backward causation prompted a paper by L. Dwyer called "Time Travel and Some Alleged Logical Asymmetries between Past and Future" (1978) in which he argues in support of time travel and backward causation. 1 9 Time travel, entailing as it does backward causation, does not involve changing the past. The time traveller does not undo what has been done or do what has not been done, since his visit to an earlier time does not change the truth values of any proposition concerning the events of that period. Thus even before the time traveller enters his rocket in 1978 to begin his successful mission to the year 3000 B.C., an accurate catalogue of all the events occurring in Ancient Egypt that year would include an account of his arrival from the sky, as well as an account of his various actions and reactions in that new environment.13 Not all philosophers are in support of this extraordinary concept. L. Spellman in his paper "Causing Yesterday's Effects" (1982) writes: I do not intend to enter the debate over whether time travel is conceivable. What I do want to hold , however, is that there is no need to call Dwyer's example backward causation. Dwyer describes pyramid building as in the [time traveller's] 'causal future', but part of his 'chronological past'. But if time travel is possible, what reason is there for saying that pyramid building precedes the rocket's firing? Rather, from the [time traveller's] point of view, what happens is perfectly ordinary causation - an earlier event (rocket firing) causes a later one (arrival at one's destination) While there is nothing illogical about Spellman's view, Nahin points out that it does not hold for a "non time-travelling observer who watches the entire process linearly from 3000 B. C. to A. D. 1978". 1 5 For him, the process certainly does involve backward causation. One must necessarily admit backward causation if one is to accept the thesis of Nahin's book that the past can be affected. As he explains, there is no contradiction involved "if one accepts the idea that the world lines of the time 2 0 traveller are bent back into the past; and so time travel does not change the past but rather is built into the past". 1 6 Where Are All the Time Travellers? Another objection that is raised against time travel takes a much different approach. Arthur C. Clark, in his 1985 paper "About Time" writes: the most convincing argument against time travel is the remarkable scarcity of time travellers. However unpleasant our age may appear to the future, surely one would expect scholars and students to visit us, if such a thing were possible at all. Though they might try to disguise themselves, accidents would be bound to happen - just as they would if we went back to Imperial Rome with cameras and tape-recorders concealed under our nylon togas. Time travelling could never be kept secret for long. 1 7 Other philosophers and scientists have also used this argument, with G. Fulmer in his paper" Understanding Time Travel" (1980) writing: "actually I know of only one argument against the possibility of time travel that seems to carry any weight at all. This is the fact that it does not appear ever to have happened". 1 8 Even physicist Stephen Hawking in his "Chronology Projection Conjecture" (1992) argues against time travel into the past because "we have not been invaded by hordes of tourists from the future". 1 9 Another answer to the question 'where are they?' given by [del Rey, 1951] and [Podolny, 1970] is that it is only possible to travel through time into the future. And a still more interesting solution is that such time travellers do actually exist, but have somehow kept their presence unknown. A somewhat dramatic description is given of this possibility in the 1993 book Einstein's Dreams by Alan Lightman. He describes a scene in which: 2 1 Now and then, some cosmic disturbance will cause a rivulet of time to turn away from the mainstream, to make connection backstream. When this happens, birds, soil, people caught in the branching tributary find themselves suddenly carried to the past...They wear dark, indistinct clothing and walk on their toes, trying not to make a single sound, trying not to bend a single blade of grass. For they fear that any change they make in the past could have drastic consequences for the future. 2 0 Of course it has already been shown that any travellers to the past could not actually bring about a change. But it is still interesting to speculate on the behavior of such a traveller if he or she believed it was within their power to change the course of history. Causal Loops The final area to be discussed in this chapter is the issue of causal loops. Jonathan Harrison's 1979 paper "Jocasta Crimes" 2 1 centered around a story that illustrates a causal loop in time. In his story, a young lady in 1984 finds a man who has been frozen in a metal chest. She 'thaws' him out, and later they marry and have a son. This son grows up, builds a time machine, and at the age of 16, sets out with his father to travel to the past. The trip takes longer than they had planned and their food rations run short. In desperation, the son kills and eats his father to survive. When the time machine stops, it is 1941. The machine proves to be irreparably damaged, so the son (wanting to be reunited with his mother and unable to obtain the parts to make a new time machine), uses available material to build a freezer that will keep him in a state of suspended animation. In 1945, at the age of 21, he puts himself into the freezer. Thirty-nine years later in 1984, a young lady finds a man who has been frozen in a metal chest. She 'thaws' him out, and later they marry and have a son... 2 2 There are several peculiar implications in this story. 1) that a man can father himself, 2) that a man can eat himself up and 3) that a man can die before he is bom. Any one of these implications would be enough for some philosophers to rule out the possibility of time travel. However, as Murray MacBeath put it in his 1982 paper "Who Was Dr. Who's Father?": "Those who like me believe in the possibility of time travel are faced with a choice between 1) unearthing some other assumptions which, rather than the time travel assumption, can be shown to be responsible for the unacceptable implications 2 2 of Harrison's story and 2) accepting the implications of the story". MacBeath decided to go with option 2), as he argues that there are "no 2 "3 logical obstacles to adopting [this] course of action". However, the implications of the story ultimately cannot be accepted, as MacBeath himself concedes later on, because of the genetic impossibility of a man fathering himself. MacBeath explains: Clearly any genetic inheritance that [the son] receives from his father involves a loop: junior receives the inheritance genetically from senior, while senior receives it from junior simply by aging. It is worthwhile adding that, quite apart from the loopiness of [such] an arrangement...it would require that either [the mother] contribute nothing to junior's genetic inheritance, or that [the mother] and [the father], by a coincidence of the utmost remoteness, have in common one copy of every last one of their genes. If [the son] is not to inherit genetically from [the father], his entire genetic inheritance must come from [his mother]. But if any offspring were to derive its genetic inheritance entirely from its mother, it would not have a father (it would have to be the product of parthenogenesis) and it would not be a son, it would have to be a daughter. 2 4 2 3 Physicists such as [Friedman et al., 1990] reject the possibility of time travel because of concern about multiple trips around such closed loops. Friedman explains his position by arguing: the [fact that] the principle of self-consistency is not tautological becomes clear when one considers the following alternative: the laws of physics might permit closed timelike curves (CTC's); and when CTC's occur, they might trigger new kinds of local physics which we have not previously met. For example, a quantum mechanical system, propagating around CTC's, might return to where it started with values for its wave functions that are inconsistent with the initial values; and it might then continue propagating and return once again with a third set of values, then a fourth, then a fifth... The principle of self-consistency by fiat forbids changing the past . 2 5 Another common issue concerning causal loops is the acquiring of information. Many stories about time travel include a 'loopy' situation in which someone from the present tells a younger self how to build a time machine. The younger self deliberates, studies and experiments, and in some years manages to construct a time machine. That person could then travel back in time, meet up with a younger self and relay the information on how to construct a time machine. The question is of course - where did the information come from in the first place? For those who support the notion of time travel, causal loops seem to be the trickiest obstacle to maneuver around. That they produce paradoxical situations appears to be a given, and this would seem to pose a problem for logical analysis. However, David Lewis, an avid supporter of the logical possibility of time travel, meets this difficult situation with an interesting 24 comeback. In his 1976 paper "The Paradoxes of Time Travel", Lewis sidesteps the problem with this tactic: Where did the information come from in the first place? There is simply no answer. The parts of the loop are explicable, the whole of it is not. Strange! But not impossible, and not too different from inexplicabilities we are already inured to. Almost everyone agrees that God, or the Big Bang, or the entire infinite past of the Universe, or the decay of a tritium atom, is uncaused and inexplicable. Then if these are possible, why not also the inexplicable causal loops that arise in time travel? 2 6 Lewis' quote is perhaps, a fitting final word on the seemingly unsolvable issue of time travel. Neither side is prepared to admit defeat in this on going struggle, however many interesting theories have emerged from the debate. One of these theories is the position taken by philosopher Paul Horwich. Mention has been made of Horwich several times throughout this thesis, and we are now ready to take a closer look at the ideas that are currently under contention. 2 5 C H A P T E R III H O R W I C H ' S A R G U M E N T S Paul Horwich, in his book Asymmetries in Time - Problems in the Philosophy of Science, devotes Chapter 7 to the supporters of backward time travel within a Gbdelian space-time structure. In this section Horwich is arguing against the position that "time travel cannot occur since it would engender anomalous consequences".1 Horwich examines four of the most common paradoxes that emerge during time travel debates. The first three he seems to have little difficulty in disposing of, as they are dealt with quite briefly. The fourth is a tougher 'nut to crack', and requires Horwich to draw on work he presented in an earlier chapter of the book - primarily Chapter 6 which is on Backward Causation. However, Horwich seems to feel confident that by the end of Chapter 7, he has effectively refuted all four of the time travel paradoxes; consequently maintaining the possibility of time travel within Gbdelian space-time. Oxymoron Paradox The first paradox Horwich calls the "oxymoron paradox". The question that is posed here is simply "is time travel an oxymoron?"2 The implication here is that "to travel in time is to transverse some temporal interval in a time that differs from the duration of that interval. Thus we have a straight forward contradiction".3 Horwich explains that this dilemma is quickly solved once we note that "the magnitude of the temporal interval that is traversed and the duration of the journey are measured in different frames of reference".4 Pursuant to the theory of relativity, if time is not absolute, but relative to different 26 frames of reference, one could travel 500 years of Earth time to the past (which would be measured by Earth time clocks), but only take 15 minutes in the time machine to complete the journey (which would be measured by clocks inside the time machine). Thus no contradiction ensues. Leibniz's Law The second paradox that Horwich examines is an apparent contradiction of Leibniz's Law that results from backward time travel. The law essentially states that identical objects have all of the same properties. This appears to be incompatible with time travel in that it seems to allow for two identical objects to have different properties. The example Horwich raises is one of a man 'Charles', who in 1960 is cleanshaven, and by 1970 has grown a beard. The Charles of 1970 then travels back to 1960. The early Charles and the late Charles are the same person, yet at 1960, one is now bearded and the other is not. This would appear to violate Leibniz's Law. The dilemma, however, is easily solved by the introduction of temporal indexes of the properties. Horwich writes: We perform the same sort of maneuver we used earlier when we insisted that the temporal index of the property be made explicit. In this case we must now insist that the Charles-proper-time index of the property be made explicit. This is defined by a clock that Charles carries with him throughout his life, and it applies only to events that take place close to him. Similarly the Earth times 1960, 1970 and so on, apply straight forwardly, in a Godelian universe, only to events that occur near the Earth (although it is possible to extend their application to distant events, in ways that are to some extent conventional). Now, if the property, having a beard, is used proper timelessly to mean having a beard in 1960 at some Charles proper time or other, then it is true of Charles I and of Charles II that they 27 exemplify it. Also if some proper-time index is built into the property, then it is either true or false of both. When it is said that Charles I does not have a beard in 1960 but Charles II does, the proper time index is suppressed. But were we more explicit, we would say that Charles I has no beard in 1960 at some proper time t, whereas Charles II has a beard in 1960 at some different proper-time t'. As before, we have no reason to deny that Charles I is Charles II, and Leibniz's Law remains perfectly satisfied.5 Changing vs. Influencing the Past Having made quick work of these first two apparent paradoxes, Horwich moves on to the issue of changing the past. As seen in Chapter 3, a great deal of attention has been paid to the contradictory idea that one could travel to the past and bring about a change in it. This notion has been the basis of many a rejection of backward time travel. Although his arguments are somewhat lacking in the clarity that was provided by Nahin and others, Horwich refutes this 'paradox' along much the same lines. The argument goes something like this: A person travelling back into the past could bring about a change that we know did not occur, resulting in a contradiction. Therefore we can assume that either backward time travel is impossible, or that certain restrictions of freedom, preventing one from bringing about contradictions, are put on all time travellers going into the past. Horwich claims that this sort of argument is invalid. He bases this on a crucial distinction between changing the past and influencing the past. Horwich explains that while changing the past is logically impossible, there is no contradiction involved with influencing the past. He likens it to the future, in which we can use our influence, however, we cannot bring about an event that will not occur. 28 Horwich goes on to explain that just because someone did not do something while travelling back in time, (eg. Charles attending the Battle of Hastings in 1066) does not mean that they were not free to do it. It is not necessary to have inexplicable constraints on a time traveller's behavior to allow the possibility of time travel to exist. Horwich argues: From the fact that someone did not do something, it does not follow that he was not free to do it. Consequently, from the fact that Charles was not at the battle [of Hastings], it can not be inferred that if he were to travel back in time, he would not be free to attend it. Laws of logic do not involve the kind of limitation on our ability to choose and act freely of which it is appropriate to give causal explanations. In particular, we are not required to explain why it is that if Charles did not attend the battle, then when he travels back in time he does not attend it. This is merely an instance of P - > P - 6 It is interesting to note that the last sentence of the above quote looks like it should read p -> ( q -> p ). However, Horwich is correct in that (q) does drop out of the equation, leaving the conditional as he states it. Horwich also seems to be rather offhand in his treatment of this paradox. He dismisses it quickly, and with not so much as a mention of backward causation (which was the basis on which Nahin et al. allowed for influencing the past). However, the fourth paradox, on which Horwich spends considerable time, is really nothing more than a specific and involved version of paradox three. He calls it the autofanticide paradox, and the treatment of it is neither handled as lightly, nor is the issue of backward causation overlooked. Autofanticide Paradox The autofanticide paradox is just a variation on the grandfather paradox previously discussed. In this version, a time traveller journeys back to the past 29 and kills him or herself as an infant, resulting in a self-defeating causal chain or paradox. Since this is logically impossible, then time travel must not exist. The first question Horwich asks is "would the existence of closed time-like curves imply the possibility of self-defeating causal chains?" 7 While many opponents of time travel would argue yes, Horwich takes a different view. It is not the case, he writes, that "if some causal chain can be located along some time-like line, then it should be locatable along any sort of time-like line - even a closed one if there is such". 8 Firstly, he points out that it is "quite obvious that there are constraints on the time-like curves that may act as loci for particular sorts of causal chains" 9, and secondly, "any causal chain must be consistent with those chains which are located along intersecting time-like l ines". 1 0 Consequently, he feels it is a mistake for one to suppose that any sort of causal chain nay be located within Godelian closed time-like curves. Closed causal chains are merely subject to consistency conditions. He concludes that "the existence of closed time-like curves and the possibility of time travel do not imply the possibility of self-defeating causal chains such as....the autofanticidal maniac. Such systems are excluded by just those restrictions on causal chains and their interrelationships that ordinarily apply in open time". 1 1 Having neatly ruled out the possibility of self-defeating causal chains, Horwich does go on to discuss the problem of attempted autofanticide. Surely it must be a possibility that if time travel exists, some traveller will attempt to bring about a self-defeating causal chain such as autofanticide. Admittedly we know that success is impossible, for as Horwich already stated "my inability to go back in time and kill myself as an infant is just a special case of my inability to go back and kill anyone before their death, and this is impossible for the same reason 30 that I cannot, right now, kill someone before his death. The difficulty has nothing to do with time travel". 1 2 However, there is something disturbing about the repeated failures that must necessarily accompany such attempts. Backward Causation Again At this point, Horwich finally brings backward causation into the picture - a subject he devoted all of Chapter 6 to in his book Asymmetries in Time -Problems in the Philosophy of Science. In Chapter 6, Horwich focused on what be called the "bilking argument". He draws a similarity between this and self-defeating causal chains in time travel. The bilking argument is an objection to backward causation. Versions of this argument are found in [Tolman, 1917], [Flew, 1954], [Black, 1956], [Pears, 1957], [Dummett, 1964], [Earman, 1972] and [Mellor, 1981], among others. Essentially the concept is that backward causation would be defeated by acting to prevent the cause after an effect had happened, or vice versa, producing a cause when no effect had taken place. As Horwich puts it: "If this policy is carried out, then E will often occur in the absence of C, and C will frequently fail to bring about E. So the backward causation hypothesis is false. If, on the other hand, the attempt to carry out the policy fails, this indicates that the agent's ability to produce C depends on the prior presence of E, which in turn means that E is a necessary causal antecedent of C. Thus whatever happens, the hypothesis will be falsif ied. 1 3 Horwich, however, does not believe that the bilking argument refutes backward causation. Instead he believes " its virtue is to underline the fact that backward causation would be associated with the occurrence of inexplicable 31 coincidences (correlated events that are not causally connected)". 1 4 This conclusion could result in an argument looking something like this: P1) backward causation entails inexplicable coincidences P2) inexplicable coincidences are highly improbable C) backward causation must be highly improbable and therefore should never be postulated 1 5 Horwich challenges the soundness of premise P2). He distinguishes between two kinds of inexplicable coincidences. The first kind, which is improbable, he calls 'Humean' because it involves causation of the type characterized by Hume ("a cause is an earlier member of a chain of direct nomological determination".16), and the second kind he calls 'non-Humean', as it differs radically from our normal idea of causation. This second kind, Horwich argues, is not improbable. Most correlated events are "constituents of a V-shaped pattern of correlation".1 7 Either there is a chain of events between them, or there is an earlier common cause that links up the two events. This V-correlation would look like Figure #2: A B C 32 If we combine the principle of V-correlation with Humean causation, we get the principle of causal correlation. If one were to violate the principle of causal correlation, one would either abandon Humean causation, or violate the V-correlation. The latter type of violation would look like Figure #3: This is a highly improbable form of uncaused correlation or 'coincidence'. However, the former type of violation does not violate the V-correlation. This is illustrated in Figure #4: Space Horwich explains this bizarre account of uncaused correlation. "Among these correlated events, causation works toward the future before C and toward the past after C . That is, A causes x1 which causes x2 which causes C. And B causes y1 which causes y2 which causes C. This...would involve an uncaused correlation between A and B, but there would be no violation of V-correlation".1 8 3 3 The point behind this whole procedure is the concoction of a sort of uncaused correlation that does not violate the V-correlation, and consequently need not be labelled empirically improbable. Horwich grants that this may be both "theoretically unacceptable and conceptually unattractive"19, however, such coincidences cannot be described as either infrequent or unlikely. So, if there is no contradiction in admitting to correlations that are not causally connected, then the bilking argument need not refute backward causation simply because it implies the existence of inexplicable coincidences. We simply need to abandon the belief that correlated events must be causally connected. Horwich uses examples of tachyons and positrons to illustrate this concept in Chapter 6. But how does this relate to time travel in Chapter 7? Horwich identifies time travel as a type of backward causation. Consequently, self-defeating causal chains would be similar to the concept of bilking. If Horwich feels he has effectively eliminated the bilking argument as an objection to backward causation, then perhaps similar tactics can be used to save time travel from attempts at instantiating self-defeating causal chains. Unfortunately, Horwich sees time travel as falling under the category of backward causation that violates the V-correlation, resulting in an inverse fork, such as the one in Figure #3. In this pattern "an event is nomologically overdetermined by two distinct preceding chains of events". 2 0 The result is that highly improbable coincidences would be necessary to preserve this correlation from bilking attempts. 3 4 An example of a bilking attempt within Godelian space-time, could be attempting autofanticide, or simply bringing about an event that one knows was not done. In the case of autofanticide, since we know it was not successful, many inexplicable coincidences would have to take place to continually ensure that the death was not brought about (the gun jamming, the killer tripping at the last minute, a doctor being on the scene to remove the bullet, etc.) Since such coincidences are implausible, Horwich concludes the existence of time travel must imply the non-existence (or very rare existence) of bilking attempts. However, we know this would be inconsistent with human nature. If we have the ability to travel back in time, attempts at self-defeating causal chains will be engendered. Consequently, Horwich argues, we must infer that we will not acquire the ability to travel into the recent past where bilking attempts will be made. Horwich, however, is not prepared to conclude from this that the space-time structure of our world is not described by Godel's solutions, or that all forms of time travel are eliminated. The structure of space-time need not be the reason for the non-existence of bilking attempts. Horwich argues that: to perform a bilking experiment, it would be necessary to travel (or send something) into the local past - to some space-time point that is not only earlier than now, but also fairly near here - and it would also be necessary that the (proper) duration of the journey not be longer than the system that is sent back. But these requirements turn out to present insuperable practical difficulties. Specifically, a calculation (eg. given by Malament, 1985b) of the vast amount of fuel required to complete such a trip shows that it would always be technologically impossible to do this... Thus, the nonoccurrence of bilking attempts is explained without having to suppose the non-existence of closed time-like lines, and therefore provides no evidence for that supposition. 2 1 3 5 Horwich points out however, that trips into the spatially distant past would not be ruled out because they require far less fuel. Thus the problem of self-defeating causal chains has been accounted for without disrupting Horwich's overall thesis that based on Gbdelian space-time, time travel into the past is possible. It is Horwich's conclusion that he has successfully defended time travel in Chapter 7 against all four of the paradoxical arguments that were raised against it. He admits that short trips into the local past would allow for attempts at manifesting contradictory self-defeating causal chains, and consequently must be ruled out. However, he feels that because excessive fuel requirements provide an adequate explanation for this rejection, it is not necessary to posit a non-Gbdelian space-time. He is of the position that he has preserved not only the existence of closed causal chains, but also the possibility of time travel within a Godelian world - travel to a distant past that is outside of bilking range. 3 6 C H A P T E R IV C O N C E R N S A B O U T T I M E T R A V E L In the past few chapters, many points on the subject of time travel have been examined. A quick summary of the main issues may be helpful at this point to clarify the line of argument under discussion, and to establish a focal point from which to move forward. Summary 1) Gbdel discovers that the field equations of General Relativity yield solutions that would allow for Closed Causal Chains in certain space-times. 2) Any situations involving logical contradictions that may arise from the existance of Closed Causal Chains are ruled out by Gbdel on the basis of prohibitive fuel requirements. 3) Many people have developed theories to defend time travel against both the argument of logical contradiction and the argument of prohibitive fuel requirements. 4) One point on which everyone agrees - the past cannot be changed. To do so would involve a paradox. 5) The critical flaw in many arguments against time travel is the notion that something happens twice - that is, the year 1066 happens once without Charles in it, and again with him in it. Everything happens only once. Consequently if you will go back to 1066 then you were there; and if you weren't there, then you won't go. 3 7 6) Horwich did not make this point clearly in his argument on changing vs. influencing the past - this omission seemed to leave his argument open to attack, (see Question #2, p. 42 of this chapter). 7) However, allowing events in the past to be part of one's personal future necessitates admitting a form of backward causation. 8) As Nahin and others have argued, active participation in the past can be allowed without logical contradiction if it is viewed as an effect that merely precedes its cause. 9) Backward causation is itself a widely controversial and less than a generally accepted theory. It may not be so easy to use it as an explanatory factor in time travel - it may simply mean using one paradox to explain away another. 10) Horwich uses Chapter 6 of his book to argue in favour of backward causation. 11) Generally two objections to backward causation are raised: i) "the normal time order of cause and effect has been 'built in' to our concept of causal relation - that a cause is by definition, an earlier determining condition."1 ii) what Horwich calls the "bilking condition", that "any backward causation hypothesis would be necessarily refuted by the following experiment: repeatedly wait to observe the presence (or absence) of the alleged effect E, and then try to prevent (or produce) the subsequent, alleged cause C. " 2 • ' . 12) Horwich discusses the first objection based on Quine's arguments (1951) "against the possibility of any 'truth by definition' or 'truth by convention', and because it is possible to imagine evidence that would produce at least some temptation to postulate backward causation".3 38 13) The second objection - the bilking argument, is given considerable attention. Horwich does not believe that the possibility of bilking attempts refutes backward causation. He argues instead that they would be associated with the occurrence of a certain type of inexplicable coincidences that do not violate Humean causation or the V-correlation. Such coincidences, he claims, while being conceptually awkward, are neither unlikely nor infrequent. 14) Horwich concedes that "the reasons that we do not see the sorts of things that are regarded in the philosophical literature as candidates for backward causation are the reasons, whatever they may be, that entropy increases, that choices precede the events they determine and that there are no inverse forks".4 However, he goes on to add that "this by no means implies that there will arise no circumstances in which it will be possible to postulate backward causation".5 15) If affecting the past during time travel is a form of backward causation, then Horwich can apply similar arguments to refute bilking considerations. That is - attempts at bilking need only produce uncaused correlations to explain their lack of success. 16) Such arguments will not preserve the possibility of time travel within bilking range, because time travel falls under the category of an inverse fork structure, which occurs when "an event is nomologically overdetermined by two distinct events". Coincidences of the improbable type would be required to preserve the correlation between the two branches of the fork. 17) Horwich concludes that "we know that the ability to travel backward in time would engender attempts at self defeating causal chains [such as autofanticide]. Consequently we can infer that no such ability will be acquired".7 39 18) The non-existence of such self-defeating causal chains need not imply that the structure of space-time in non-Godelian, nor that all forms of time travel are ruled out. To travel into bilking range, Horwich argues, would require such vast amounts of fuel that such a trip would be technologically impossible: Consequently the non-occurrence of bilking attempts is accounted for without having to posit a non-Godelian space-time structure. 19) The point of Chapter 7 in Horwich's book was to preserve the possibility of time travel within Gbdelian space-time structure, not refute it. Horwich feels he has done this, because he goes on to argue that while the local past may be inaccessible, "there are trips into the 'spatially distant past' for every possible definition of that notion, that require much less fuel, and so would not be ruled out".8 Questions to be Raised While much of what has been presented here on the various theories of time travel is fairly straightforward, and while Horwich has made many of his points clearly, there are certain areas that I feel raise some questions. There are four main concerns that arise out of the work that has been examined, and they shall be discussed accordingly. Question #1 Does the notion of temporal parts play a role in producing a contradiction when a traveller goes back to the past? In Chapter 7 of his book, Horwich seems to gloss over this particularly interesting question. When he deals with the Leibniz's Law paradox - the identity of indiscernibles, he is quickly able to dismiss the paradox by using 40 proper time. That is, if Charles has a beard and travels back to 1960 and meets himself without a beard, we do not have the same Charles with two different properties, because according to Horwich, the two Charles have different temporal indexes, thus distinguishing them from each other and causing no contradiction. By temporizing the predicate, Horwich avoids any problem with this paradox. However, this issue is not so easily dismissed when one considers the notion of 'temporal parts'. Consider for example, Figure #5: Now, in the first segment of the diagram, we have Charles living normally, in world time, from 1960 to 1980. At some point during this period, he occupies the world time 1970. In the second segment of the diagram, we have Charles getting into a time machine and travelling back to 1960. At some point during the trip, he again will occupy the world time 1970. Now, he can not be the same Charles as the one who was in the first segment, because as Horwich already explained, this would violate Leibniz's Law. -So we will call the first Charles, -Charles I, and the second one Charles II. The third segment is everything that happens to Charles after he has reached 1960 for the second time. Whether or 4 1 not he remains there and lives, or flies back to the present, he also will occupy the world time of 1970 at some point. So we will call him Charles III. So far, according to Horwich, there is no problem here, because each Charles has a particular proper time index assigned to him. However, what happens when we begin to speak of the temporal parts of Charles? In Figure #5, the first section, 1960 - 1980, involves Charles I. Now, what if Charles I had a friend Bob, who grew up with him during this time, helped him build his time machine, research the trip back etc. When 1980 rolled around, Bob was on hand to put Charles I into his time machine and see him off. At the point of blast off, their, world time directions would suddenly split. Charles would be headed backward in world time, while Bob would continue to move forward. Having done all of the research with Charles, Bob would know when Charles was scheduled to arrive at his destination. And on that day, Bob would announce that today, Charles has arrived in 1960. The only problem is, the Charles Bob knew was Charles I, and based on the diagram, and Horwich's own arguments, Charles I does not exist past the world time 1980. At the point he left in his rocket ship, he became Charles II, thus effecting no contradiction by being in the same world time twice. But, if no temporal part of Charles I gets past 1980, what can Bob say of Charles after 1980? He wants to say that Charles is travelling back in time, but there needs to be some temporal piece of him that is being referred to. And in the year 2000 for instance, Bob wants to say that Charles has arrived in 1960. But in order to be able to do that, he is in effect saying that there is a world time 2000 temporal part of Charles which is at the world time 1960. This seems to result in quite an 4 2 odd situation, one that perhaps should have played a part in Horwich's chapter on time travel. Question #2 In Horwich's third paradox (changing vs. Influencing the past), backward causation must necessarily be invoked as explained in Nahin's writing on this subject, in order to allow the future to 'bend' into the past and prevent a contradiction from taking place - namely changing the past. However, do not Horwich's own arguments prevent backward causation from being a viable solution to influencing the past during time travel? In Nahin's book, he quite clearly, states that there is no difficulty in accepting that the past can be influenced provided one is willing to accept that this entails backward causation. In effect, he is arguing that there is no problem with bending back the lines of time into the past. An example is used of a time traveller who could not go back in time and prevent the Black Death of London, but who, during his trip back could have been the cause of it. Now, Horwich, in his discussion of paradox 3, suggests a similar scenario where Charles goes back to the year 1066, and is able to influence events of that time during his trip. In order for no logical contradiction to manifest itself (changing the past), we must necessarily invoke backward causation. That is, the effects of Charles' trip in world time would precede the cause. The problem arises however, in the conclusion Horwich reaches concerning time travel, bilking attempts and backward causation. Horwich writes: 43 In the case of certain backward-causation hypotheses...the upshot is merely to underline a certain oddity in causal descriptions of the phenomena; namely, the need to acknowledge uncaused correlations. But in these cases the bilking objection has no tendency to suggest any empirical improbability of the underlying nomological structures. In the second place, however, there are conceivable backward-causation hypotheses whose underlying structure are so-called 'inverse forks' - patterns of phenomena in which an event is nomologically overdetermined by two distinct preceding chains of events. In these cases the bilking argument helps to show that improbable coincidences are required for the correlation between the two branches of the fork to be preserved. We shall see that the case of time travel falls into the second category. Both inverse forks and closed causal chains require a certain lack of randomness in the initial conditions of the universe. In both cases the need for such a constraint is highlighted by bilking considerations, and one can question, on empirical grounds, the likelihood that this constraint is actually satisfied. 9 (italics added) Time travel, then, involves the type of bilking considerations that violate the principle of V-correlation according to Horwich. If time travel had fallen under the other category of a backward causation hypothesis with uncaused correlations of a non-Humean structure, then all might be well. As it is, Horwich has proven it is empirically improbable that inexplicable coincidences will occur every time a bilking attempt is made, consequently he has to conclude that we will not acquire the ability to engender such self-defeating causal chains. Consequently, it does not seem possible that a time traveller could influence but not change the past. Bilking considerations would result in logical contradictions taking place under such circumstances. Horwich has in effect refuted himself out of the changing vs. influencing the past paradox. Initially it appeared as though backward causation would allow time to be "bent", which would preserve the argument from the logical 44 contradiction that would result from changing the past. However,under such circumstances, one would still be well within bilking range. By his own arguments, that is not a viable possibility, therefore a time traveller would not be able to influence the past. Question #3 Is there a spatially distant past that is in fact outside of 'bilking range'? An underlying current running throughout Horwich's chapters on backward causation and time travel is the acknowledgment that attempts at self-defeating causal chains within Humean structures of causation result in unacceptable contradictions occurring. More simply put, if an effect has taken place in the past, a paradox arises if it is within my power to prevent the cause from occurring. Either I would do so, and a contradiction would result, or I would attempt to do so, and with every attempt, some circumstances would arise to prevent me. The latter occurrence Horwich sees as just as unacceptable and improbable as the former. This being the case, is there any sort of time travel possible which would take a person outside of such bilking ranges? Horwich is somewhat vague when he refers to the conditions that constitute being within bilking range - that is, the local past: i) it would be a space-time point that is not only earlier than now, but is also fairly near here. ii) it would also be necessary that the (proper) duration of the journey not be longer than the lifetime of the system that is sent back. 1 0 45 It is the phrase 'fairly near here' that leaves the lines between the local past and the spatially distant past somewhat ambiguous. How far away from 'here' does one have to get to be considered outside of bilking range? Horwich goes on to say that "there are trips into the 'spatially distant past', for every possible definition of that notion, that require much less fuel and so would not be ruled out" . 1 1 ' Horwich explains that this: i) preserves the possibility of closed time-like lines within Godelian space-time, ii) removes the threat of bilking attempts and iii) is technologically accessible. While I do not have an issue with points i) and iii), I would question point ii). Any one of the "every possible definitions" would be helpful in nailing down exactly what is meant by the term "spatially distant past" in the context of this bilking argument. It seems to imply that either one could go far enough back into the past that one would be outside of bilking range, or else that one could go back into the past at a point in space that is far enough away from our world, and then be outside bilking range, or a combination of the two. But how does travelling to the spatially distant past remove the threat of bilking attempts? If any action is carried out during the time the traveller spends in the past, then the cause of that action (the trip back in time) would be within that travellers power to prevent, resulting in a contradiction. In other words, since we are not arguing for a pre-determined future, if I was going to board a time travelling vessel tomorrow to go to the spatially distant past, I could change my mind. What if the effect (an action of mine) was already in place in the spatially distant past, waiting for its cause to occur? Just because 46 the trip through time is in the past according to the world time-lines, does not change the fact that it is in my personal future time-lines. Therefore it is still within my power to change. This is not to say that I can engender a contradiction by changing the past, for admittedly that is impossible. It is only to point out that the opportunity to attempt to bring about such a self-defeating causal chain is still very much intact. Travelling to the spatially distant past does not protect against such a bilking set-up, simply because alltrips to anywhere in time must begin now - in the present. Consider Figure #6: If this is a diagram of a time traveller's personal time line, then it is always future oriented, even though part of it is going backward in world time. Let us say that the traveller is born at point A, leaves on his time trip at point B, arrives at his destination in the past at point C and terminates his presence in the past (either by flying back, or by dying) at point D. Regardless of how far back into the past point C is, point B is always in the traveller's present at some time. Consequently, it is always subject to bilking attempts, because the traveller can always try to decide at point B minus 1 day (hour/minute) that he is not going to take the trip. If the trip has been taken (because the effects of it - arrival at point Time 47 C - are already 'in place' in world time), then it must happen. But improbable coincidences must occur to prevent the traveller from successfully bilking this trip back in time. In fact, since the action of not taking a trip is a relatively easy one, coincidences would probably have to occur repeatedly to prevent the traveller from exercising this seemingly simple action. This invariably forces the spatially distant past to be subject to the same conditions that Horwich imposed on the local past, leaving it technologically accessible but logically inaccessible. QUESTION #4 Does the definition Horwich gives for the local past imply that the spatially distant past is inaccessible to all systems? With respect to the arguments that Horwich has presented in his book, this is one of the more problematic questions that come out of his theories on time travel: As stated in the previous question, Horwich says that there are trips to the spatially distant past that need not be ruled out. He explains that such trips will fulfill three important conditions: i) they preserve the possibility of closed time-like lines within Gddelian space-time, ii) they remove the threat of bilking attempts, and iii) they are technologically accessible. I have already argued that condition ii) is not met by limiting travel to the spatially distant past. The next question is whether or not the spatially distant is in fact even accessible. The issue here is not one of technology in the sense that Horwich means (fuel requirements), but rather the length of time it would take a system to reach the distant past. When Horwich is defining the 'local past; he uses a two part definition that has been stated in the previous question. The interest here lies 48 with the second half of that definition. It reads: " it would also be necessary that the (proper) duration of the journey not be longer than the lifetime of the system that is sent back". 1 2 If any point in space-time that can be reached before the end of a lifetime of a system is defined as the local past, then the spatially distant past can never be reached. It is only logical that the spatially distant past is inaccessible if a system ceases to exist before it arrives. As well, Horwich does not seem to be restricting 'system' to a human one. One can assume from the breadth of the definition that it would apply to any system; humans, animals, robots, rocket ships etc. And although this is perhaps a more vague definition than one would like, it is as specific as he gets on this issue, as the first part of the definition of the 'local past' refers merely to a space-time point that is not only earlier than now, but also fairly near here. Ambiguous to say the least. Consequently, even if it is the case that fuel requirements are such that the spatially distant past is technologically accessible, Horwich's own definition implies that no system would be able to reach such a point in space-time before it expired. This would appear to be inconsistent with his conclusion that time travel is possible to the spatially distant past. This does not necessarily imply that space-time structure is non-Gbdelian. Just because the local past is not accessible due to excessive fuel requirements, and the distant past is inaccessible due to system duration restrictions, does not mean that Gbdel's theory of a curved space-time is incorrect. Although, one might question what sort of physical theory of curved space-time this is, if there is no form of matter that can actually travel it into the past. It does, however, mean that contrary to Horwich's arguments, the 49 possibility of time travel within such a space-time structure has not been preserved. Horwich has, in effect provided arguments that are to his own undoing. He has implied that backward causation is necessary to have influence over the past, yet he has shown that the bilking considerations that arise from such backward causation violate the principle of V-correlation. He has argued against bilking attempts being permissible in the local past, however, I have shown that trips to the spatially distant past are subject to the same anomalous conditions, and he has provided us with a definition of the local past that implies no system can access the spatially distant past within the proper duration of its lifetime. It is my position that although the possibility that space-time is Gbdelian remains intact, Horwich has not adequately preserved the possibility of time travel through that space-time into the past. 5 0 C O N C L U S I O N Horwich has not preserved the possibility of time travel into the past within Gbdelian space-time structure. This thesis has examined Horwich's theories about backward time travel, and discussed the four paradoxes that he set out to defend time travel against. Horwich introduced the bilking argument, and showed how it was connected with backward causation. By his own admission, time travel involves the type of bilking considerations that violate the V-correlation, therefore the bilking argument could defeat the notion of travel into the past. Horwich strives to preserve the possibility of space-time being Gbdelian by embracing the argument that trips to the local past are technologically impossible based on the vast amount of fuel that would be needed. However, he argues that trips to the spatially distant past remain possible. I have argued that Horwich has overlooked an important issue - that of temporal parts, which when considered within the concept of backward time travel, seems to produce odd conditions. Further, I argue that trips to the . spatially distant past are subject to the same bilking conditions as trips to the local past, which would again result in a contradiction. I have also argued that by Horwich's own definition, the spatially distant past is not accessible to systems travelling back in time, as any space-time point they reached during their lifetime would by definition be the local past. Consequently it is my conclusion that while space-time may by of a Gbdelian nature, time travel within it is impossible. 5 1 FOOTNOTES CHAPTER 1 1 Hawking, S. W. (1988). A Brief History of Time. New York: Warner Press, pp.20 -21. 2 Godel, K. (1949). "A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy." In Albert Einstein: Philosopher-Scientist. Vol. 7 of the Library of Living Philosophers. Edited by P. A. Schilpp. Enanston, IL: Open Court, p.559. 3 Godel, K: (1949). p.559. 4 Godel, K. (1949). p.560. 5 Godel, K. (1949). p.560. 6 Godel, K, (1949). p.560. 7 Horwich, P. (1987). Asymmetries in Time. Cambridge, MA: The MIT Press. p 113-114. CHAPTER 2 1 Bradbury, R. (1988). 'The Toynbee Convector." In The Tovnbee Convector. New York: Ballantine. 2 Leiber, F. (1976). The Big Time. Boston, MA: Gregg Press 3 Leiber, F. (1976) 4 Leiber, F. (1976) 5 Malament, D. B. (1984). " Time Travel" in the Godel Universe." Proceedings of the Philosophy of Science Association 2:91 -100. 6 MacBeath, M. (1982). "Who Was Dr. Who's Father?" Synthese 51 (June):397-430. 52 7 Nahin, P. J. (1993). Time Machines: Time Travel in Phvsics. Metaphysics, and Science Fiction. New York: American Institute of Physics, p. 168 8 Nahin, P. J.(1993). p. 176 9 Lewis, D. (1976) 'The Paradoxes of Time Travel." American Philosophical Quarterly 13 (April): 145-152. p. 150. 1 0 Nahin, P. J. (1993). p.183. 1 1 Friedman, J . , et al. (1990). "Cauchy Problem in Spacetimes with Closed Timelike Curves." Physical Review D 42 (15 September): 1915-1930. 1 2 Nahin, P. J. (1993). p. 184. 1 3 Dwyer, L. (1978). "Time Travel and Some Alleged Logical Asymmetries Between Past and Future." Canadian Journal of Philosophy 8 (March): 15-38. 1 4 Spellman, L. (1982). "Causing Yesterday's Effects." Canadian Journal of Philosophy 12 (March): 145-161. 1 5 Nahin, P. J. (1993). p. 191. 1 6 Nahin, P. J . (1993). p. 191. 1 7 Clarke, A C . (1985). "About Time." in Profiles of the Future. New York: Warner. 1 8 Fulmer, G. (1980). "Understanding Time Travel." Southwestern Journal of Philosophy 11 (Spring): 151-156. 1 9 Hawking, S W. (1988). A Brief History of Time. New York: Bantam. 2 0 Lightman, A. (1993) Einstein's Dreams. New York: Warner Press, p.20. 2 1 Harrison, J. (1979). "Jocasta's Crime." Analysis 39 (March): 65. 2 2 MacBeath, M. (1982). "Who Was Dr. Who's Father?" Synthese 51 (June):397-430. 2 3 MacBeath, M. (1982). p.397. 2 4 MacBeath, M. (1982) pp.427-428. 53 2 5 Friedman, J., et al. (1990). "Cauchy Problem in Spacetimes with Closed Timelike Curves." Physical Review D 42 (15 September): 1915-1930. p.209. 2 6 Lewis, D. (1976) "The Paradoxes of Time Travel." American Philosophical Quarterly 13 (April): 145-152. p. 149. CHAPTER 3 1 Horwich, P. (1987). Asymmetries in Time. Cambridge, MA: The MIT Press. p. 111. 2 Horwich, P. (1987). p. 114. 3 Horwich, P. (1987). p. 114. 4 Horwich, P. (1987). p.114. 5 Horwich, P. (1987), p. 115. 6 Horwich, P. (1987). p. 116. 7 Horwich, P. (1987). p.118. 8 Horwich, P. (1987). p.118. 9 Horwich, P. (1987). p. 118. 1 0 Horwich, P. (1987). p. 118-119. 1 1 Horwich, P. (1987). p. 119. 1 2 Horwich, P. (1987). p. 119. 1 3 Horwich, P. (1987). p.92. 1 4 Horwich, P. (1987). p.93. 1 5 Horwich, P. (1987). p.97. 1 6 Horwich, P. (1987). p.98. 1 7 Horwich, P. (1987). p.98. Horwich, P. (1987). p.98-99 Horwich, P. (1987). p.99. Horwich, P. (1987). p. 120. Horwich, P. (1987). p. 122. CHAPTER 4 1 Horwich, P. (1987). p.91. 2 Horwich, P. (1987). p.92. 3 Horwich, P. (1987). p.91. 4 Horwich, P. (1987). p. 108. 5 Horwich, P. (1987). p. 108. 6 Horwich, P. (1987). p. 120. 7 Horwich, P. (1987). p. 121. 8 Horwich, P. (1987). p. 122. 9 Horwich, P. (1987). p. 120. 1 0 Horwich, P. (1987). p.122. 1 1 Horwich, P. (1987). p. 112. 1 2 •: Horwich, P. (1987). p. 122. 1 8 1 9 2 0 2 1 B I B L I O G R A P H Y Bradbury, R. (1988). "The Toynbee Convector." In The Tovnbee Convector. New York: Ballantine. Clarke, A. C. (1985). "About Time." In Profiles of the Future. New York: Warner. Davies, P. C. W. (1977). The Physics of Time Asymmetry. Los Angeles: University of California Press. Deutsch, D. (1991). "Quantum Mechanics Near Closed Timelike Lines." Physical Review D 44 (15 November): 3197-3217. Dwyer, L. (1978). "Time Travel and Some Alleged Logical Asymmetries Between Past and Future." Canadian Journal of Philosophy 8 (March): 15-38. Earman, J . (1989). World Enough and Time and Space. Cambridge: MIT Press. Einstein, A. (1905). "Annalen der Physik" 17. Friedman, M. (1983). Foundations of Space-Time Theories. Princeton, NJ: Princeton University Press. Friedman, J. , et al. (1990). "Cauchy Problem in Spacetimes with Closed Timelike Curves." Physical Review D 42 (15 September): 1915-1930. Fulmer, G. (1980). "Understanding Time Travel." Southwestern Journal of Philosophy 11 (Spring): 151-156. Gbdel, K. (1949). "A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy." In Albert Einstein: Philosopher-Scientist. Vol. 7 of the Library of Living Philosophers. Edited by P. A. Schilpp. Enanston, IL: Open Court. Harrison, J. (1979). "Jocasta's Crime." Analysis 39 (March): 65. Harrison, J. (1971). "Dr. Who and the Philosophers, Or Time Travel for Beginners." Aristotelian Society Supplement 45:1-24. Hawking, S. W. (1988). A Brief History of Time. New York: Bantam. 56 Horwich, P. (1987). Asymmetries in Time: Cambridge, MA: The MIT Press. Horwich, P. (1975). "On Some Alleged Paradoxes of Time Travel." Journal of Philosophy 72 (14 August): 432-444. Leiber, F. (1976). The Big Time. Boston, MA: Gregg Press. Lewis, D. (1976) "The Paradoxes of Time Travel." American Philosophical Quarterly 13 (April): 145-152. Lightman, A. (1993) Einstein's Dreams. New York: Warner Press. Malament, D. B. (1984). " Time Travel' in the Gbdel Universe." Proceedings of the Philosophy of Science Association 2:91-100. MacBeath, M. (1982). "Who Was Dr. Who's Father?" Synthese 51 (June):397-430. Nahin, P. J. (1993). Time Machines: Time Travel in Physics. Metaphysics, and Science Fiction. New York: American Institute of Physics. Reichenbach, H. (1958). The Philosophy of Space and Time. New York: Dover. Sklar, L. (1974). Space. Time and Soacetime. Los Angeles: University of California Press. Smart, J . J . C. (1964). Problems of Space and Time. New York: MacMillian Publishing Co., Inc. Spellman, L. (1982). "Causing Yesterday's Effects." Canadian Journal of Philosophy 12 (March): 145-161. 

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