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The passage of time and the evolution of the universe-tree Reiffer, Randy 1991

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The Passage of Time and the Evolution of the Universe-Tree by Randy Reif f e r . A., The U n i v e r s i t y o f B r i t i s h Columbia, 1988 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES Department of Philosophy We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 1 May 1991 C Randy R e i f f e r 1 991 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. Department of M / USb CtT*7 The University of British Columbia Vancouver, Canada Date Z% "#7 f 99/ DE-6 (2/88) 1.0 Abstract McTaggart's paradox leads to the conclusion that time i s unreal. The paradox has two key premises: the premise that time flow i s a necessary component of time, and the premise that time flow e n t a i l s a contradiction. The paradox therefore derives i t s strength from the apparent contradiction entailed by the passage of time. The conclusion of McTaggart's paradox, that time i s unreal, could be otherwise stated as the claim that there are no l o g i c a l l y possible models of time with both the property of the e a r l i e r / l a t e r r e l a t i o n and the property of time flow. Given t h i s analysis of the paradox, I claim that McTaggart's conclusion can be rejected. Bigelow (1989) has argued that the l o g i c of McTaggart's paradox suggests a modal solution. One possible modal solution could be along the l i n e s that McCall (1966, 1976, 1984, forthcoming) i s developing. I show that McCall's model i s a l o g i c a l l y and p h y s i c a l l y possible model of time flow. McCall's model explains the passage of time through the metaphysical mechanism of the evolution of the universe-tree. The universe-tree i s a changing set of connections between the p h y s i c a l l y possible worlds. In the introduction, section 2, I present an overview of the two main groups of philosophical views on the problem of the passage of time. The debate between the two groups, that of tensers and that of non-tensers, serves to i l l u s t r a t e the d i f f i c u l t i e s that giving an account of the passage of time that must be confronted. The c r i t i c a l d i f f i c u l t y is shown to be to give an account of the changing temporal r e l a t i o n of novness. In section 3, I c r i t i c a l l y evaluate McTaggart's paradox. In so doing, I c l a r i f y the problem of giving an account of the changing temporal r e l a t i o n of nowness. I accept Bigelow's (1989) argument that the l o g i c a l structure of the paradox, together with the elegance of tense l o g i c , suggests a modal solution to the problem of temporal passage. In the remainder of t h i s section, I show that McTaggart's paradox is l o g i c a l l y v a l i d , and that while some of McTaggart's claims are unsound, o v e r a l l the paradox highlights a c r u c i a l philosophical puzzle: to give an account of the ever-changing moving now. In section 4, I show that McCall's model i s a l o g i c a l l y and p h y s i c a l l y possible model of time flow. McCall's model objectively distinguishes the present from past and future events through the dynamic evolution of the universe-tree. The tree consists of a branching connectedness r e l a t i o n between the p h y s i c a l l y possible worlds. The branch-structure is not fixed; the change i s asymmetric, and i t i s t h i s asymmetry that i s the ontological extension of the moving now. The asymmetric evolution \ \ \ of the tree makes the branch-structure to be future-directed. The now moves up the tree of connections, a c t u a l i z i n g one track as the non-selected branches vanish. I show that McCall's model i s l o g i c a l l y possible by proving that i t i s mathematically determinate in i t s ontological extension. Furthermore, the model is p h y s i c a l l y possible, since i t contains enough topological resources to derive l o c a l Euclidean computability. In the f i n a l analysis, section 5, I come to the conclusion that McCall's model i s a good counterexample to the conclusion of McTaggart's paradox. McCall's model i s a l o g i c a l l y and ph y s i c a l l y possible model of time flow. The passage of time is explained by the evolution of the branch structure of the universe-tree. The universe-tree i s the set of connections between the selected and nonselected p h y s i c a l l y possible worlds. The motion of the now in our topological space i s the resu l t of the metaphysics of the l o g i c a l space within which our topological space i s embedded. Thus the passage of time in the actual world i s a function of the changing r e l a t i o n between the actual world and the other possible worlds. IV Table of Contents 1.0 Abstract i i L i s t of Figures v Acknowledgments v i 2.0 Introduction 1 3.0 The Passage of Time 18 3.1 McTaggart's paradox 18 3.1.1 The I n f i n i t e Regress Problem 25 3.1.2 The Possible Worlds Metaphysics Solution . . 34 3.2 The logic of McTaggart's paradox 43 3.3 The soundness of McTaggart's paradox 46 3.4 The conclusion of McTaggart's paradox 55 4.0 The Evolution of the Universe-Tree . 56 4.1 McCall's model 56 4.1.1 The Transfinite Plus Problem 62 4.1.2 The Possible Worlds Metaphysics Solution . . 68 4.2 The l o g i c of McCall's model 74 4.3 The soundness of McCall's model 85 4.4 The conclusion of McCall's model . . 90 5.0 Conclusion . 92 6.0 References 93 V L j s t of. Figures Figure 1: A-series and B-series 3 Figure 2: A Light Cone 6 Figure 3: The universe-tree 59 Figure 4: The now selects a branch 60 Figure 5: The universe-tree evolves 61 Figure 6: Bifurcation of space-time 65 Figure 7: Minkowski space 72 Figure 8: Modal-Minkowski space 73 Figure 9: Light Cones in a Black Hole 88 Acknovledgeroents I would l i k e to thank my parents f o r support and my g i r l f r i e n d B r i d g e t Minlshka f o r encouragement. I would a l s o l i k e to thank Greg Hagen and Dr. Steven S a v i t t f o r h e l p f u l comments on e a r l i e r d r a f t s of t h i s t h e s i s . I would l i k e to thank Mr. Jack Stewart f o r h i s guidance and i n s t r u c t i o n . F i n a l l y , I thank Ian T. Sadler f o r h i s computer e x p e r t i s e and the use of h i s l a p t o p computer. 2.0 Introduction The passage of time seems r e a l . Events seem to change from being future, to being present, and to then being past. Yet the passage of time also seems paradoxical. Real events are always an unchanging, fixed Interval apart, described by either one of the e a r l i e r than or later than or simultaneous with r e l a t i o n s . How can events have both changing and unchanging properties? Perhaps the central problem in twentieth century studies in the philosophy of time i s to explicate the nature of the passage of time. A large spectrum of positions have been claimed in response to McTaggart's 1908 claim, and i t would be a monumental task beyond the scope of t h i s paper to completely evaluate a l l of the plausible candidates. Most philosophers can be placed in one of two main camps of the plausible positions on the problem of the passage of time. One position i s that of the non-tensers, who hold that time consists only of the uncontroversial unchanging r e l a t i o n s , one of the set containing the e a r l i e r than, l a t e r than, and simultanous with r e l a t i o n s . Non-tensers appeal to the timeless sense of ' i s ' as demonstrated in the sentence: "Two plus two i s four." The NT-position thus analyzes the temporal copula ' i s ' as one of only three p o s s i b i l e r e l a t i o n s : e a r l i e r than, l a t e r than,, and simultanous with. The NT-position has a long t r a d i t i o n of d e f e n d e r s and has been d e v e l o p e d by R u s s e l l (1903), W i l l i a m s (1951), Smart (1955), and Grunbaum (1963). The o t h e r main camp of p h i l o s o p h e r s may be c a l l e d t h a t of the t e n s e r s . The t e n s e r s ' camp can be f u r t h e r s u b d i v i d e d - i n t o two e x c l u s i v e groups: one, tho s e who h o l d t h a t time c o n s i s t s of both c h a n g i n g and unchanging r e l a t i o n s ; and two, t h o s e who c l a i m t h a t time i n t r i n s i c a l l y c o n s i s t s of a ch a n g i n g r e l a t i o n , and may t h e r e f o r e have unchanging e x t r i n s i c r e l a t i o n s . E s s e n t i a l t o the view of the t e n s e r s i s t h a t no a n a l y s i s can be g i v e n of the meaning of ' i s ' as i n such s e n t e n c e s a s : "The c a t i s on the mat" which would reduce the ' i s ' t o one of the e a r l i e r than / l a t e r t h a n / s i m u l t a n e o u s w i t h r e l a t i o n s . F o r a n o n - t e n s e r , the ' i s ' i s t o be t i m e l e s s ; the sentence i s a n a l y z e d t o c a r r y the i n f o r m a t i o n t h a t the c a t i s on the mat a t some time t . A c c o r d i n g t o t e n s e r s , r e f e r r e d t o as the T - p o s i t l o n , the ' i s ' i n such s e n t e n c e s i s not t i m e l e s s , but d i s t i n c t i v e l y t e m p o r a l ; the sentence c a r r i e s t he i n f o r m a t i o n t h a t the c a t i s on the mat now, a t t h i s p r e s e n t moment. The ' i s ' i n such c l a i m s about p h y s i c a l e v e n t s i s , a c c o r d i n g t o the T - p o s l t i o n , n e c e s s a r i l y t e n s e d . For a t e n s e r , t h e e x p r e s s i o n " a t the p r e s e n t moment" i s not f u r t h e r a n a l y z a b l e i n t o " s i m u l t a n e o u s w i t h some time t . " S i m i l a r l y , the sent e n c e "Two p l u s two i s e q u a l t o f o u r " i s a n a l y z e d by a t e n s e r t o have a d i s t i n c t i v e l y t e m p o r a l c o p u l a . T - p o s i t i o n s have had t h e i r a d v o c a t e s , though each d e f e n d e r has t y p i c a l l y pursued a l i n e i n c o n s i s t e n t w i t h the o t h e r non-standard l i n e s . Recent d e f e n d e r s 3 of time flow include Broad (1938), Gale (1969), and Schliesinger (1980) in addition to McCall (1966, 1976, 1984, forthcoming). The set of temporal re l a t i o n s — > thought to be past present future o n t o l o g i c a l l y prior by • > A series the NT-position, the set of e a r l i e r than, <— • — ) B-serles earlier simultaneous later l a t e r than, and before with time t after simultaneous with, i s ( also known in the l i t e r a t u r e since figure 1: A-•series and B-series McTaggart as the B-s e r i e s , while the set of temporal re l a t i o n s thought to be on t o l o g i c a l l y prior by the T-position is known as the A-series, and consists of the rel a t i o n s of past, present, and future. The two temporal s e r i e s , A and B, seem to be related to one another in a ce r t a i n respect. CD. Broad (1938) notes that, "At any moment any term in a B-series w i l l be characterized by one and only one term in t h i s series of A - c h a r a c t e r i s t i c s " 1 and that the converse r e l a t i o n i s also true. While i t i s usually thought that the A-series i s moving and the B-series is s t a t i c , motion i s r e l a t i v e , and either s e r i e s could be thought to be moving with "Ostensible Temporality", p. 290. • V respect to the other (see figure 1). The problem of temporal passage i s to explain the phenomenon that an event Is f i r s t future, then becomes present for an instantaneous moment, then vanishes into the past. What i s t h i s phenomenon? The passage of time i s captured by such sentences as: (1) The present was just future and w i l l soon be past. This f l e e t i n g phenomenon can be described in two other d i f f e r e n t but equivalent ways: (2) The future w i l l be present and then i t w i l l be past. (3) The past was once present at a time when what is present was in the future. The analysis w i l l focus on (1). The two sets of properties, A-series and B-series, also appear to be incompatible in ce r t a i n respects. Analysis of sentences of A-series copulas shows that their truth-values changes in time, while the truth-value of sentences with B-series copulas does not change in time. Thus, the A and B series may be thought to be incompatible, since one set of properties changes i t s truth-value i n time; the other does not. The incompatibility is thought by some non-tensers to be due to an incoherency in the A-series. This NT-view i s supported by the f a i l u r e of the following three attempts to explicate the A-series. I w i l l caution that while these attempts f a i l , they f a i l in interesting ways, and that the upshot of this analysis is that the now i s not i i 5 what the NT-position has thought i t to be, not that the A-serles term now cannot be explicated at a l l . Tensers might want to take the phenomenon of time flow as basic or primitive, and thus hold that no e x p l i c a t i o n of i t s nature i s possible. However, a refusal to submit one's view to interrogation by one's peers i s seems quite suspicious in i t s e l f . Why should time flow be a primitive, unexplainable feature of our world? Why should anyone assume that time flow i s unexplainable when there are some explanations available? Before tensers can prove statements about the passage of time, they must f i r s t explicate the nature of temporal passage. They must provide a non-circular and non-question-begging explanation of time flow. Tensers must avoid the threat of the i n f i n i t e regress of explanation of the past, present, and future. So tensers must answer the question: How are the A-series terms related to one another? In response to the question of how the A-series events are related to one another, one might hold that such events can be simultaneously past, present, and future. This attempt to explicate the A-series with the help of the B-series i s of no use. Events cannot be simultaneously past, present and future. The simultaneous l i n e f a i l s because simultaneity i s defined in terms of being present for an instant. Thus, explicating the A-series in terms of the B-series, where the B-series is explained in terms of the A-series, r e s u l t s in a c i r c u l a r d e f i n i t i o n . The simultaneous view e n t a i l s the f a l s i t y of the following necessarily true proposition: (4) What is past i s not future. If the simultaneous view were true, then what i s past would simultaneously be future. Therefore the simultaneous l i n e i s f a l s e . timelike figure 2: A Light cone At t h i s point i t i s informative to consider whether or not the r e l a t i v i t y of simultaneity in Einstein's theories may be of some help to our problem of expl i c a t i n g temporal passage with regard to the A-series r e l a t i o n s (see figure 2 ) . C e r t a i n l y Earman (1974) i s correct in "An Attempt to Add a L i t t l e Direction to the Problem of the Direction of Time" when he suggests that by formulating the questions in the philosophy of time i n a precise and formal manner we can achieve more useful r e s u l t s than otherwise. However, the mathematical space-time formalism does not decide the question of the nature of the passage of time. Indeed, defenders of the NT-position have suggested that the present lack of a mathematical space-time formalism which would represent the passage of time, together with the fact that the present mathematical space-time formalism does explain many other features of our world, e n t a i l s that the passage of time i s unreal. Defenders of the T-position have three possible responses to the NT-argument above: one, that the passage of time is not the sort of phenomenon which could be captured by a formal theory; two, that the formal space-time theories may represent temporal passage, but only in a metaphysically neutral way; and three, that the present formalism i s simply f a l s e , and w i l l have to be replaced with another formalism wherein passage i s accounted for. Philosophers may well be suspicious of the f i r s t possible response, that temporal passage i s not the sort of phenomenon which could be captured by a formal theory, since the purpose of seeking a formal space-time theory i s to describe a l l and only the e n t i t i e s of space and time and their r elations to one another. But there are l i m i t s to what a formal theory, derived from eropircal observation and t e s t i n g , can inform about the reasons why i t , as a formal theory, happens to be true of our world. E x p l i c a t i o n of passage requires a metaphysical theory. Philosophers, I argue, can accept the second possible response, that formal space-time theories may represent passage, but only in a metaphysically neutral manner. If so, then the support the NT-position derived from the f a i l u r e of the present formal space time theory to represent passage i s defused, since even i f the formal theory did represent passage, that representation would not be s u f f i c i e n t to decide the metaphysical problem. The formal space-time theories describe the ontological realm of physics, while e x p l i c a t i o n of temporal passage cannot be decided through experiment and observation per se, but rather through metaphysical analysis. Therefore, the f a i l u r e of the formal space-time theories to represent passage i s not a f a i l u r e at a l l since the task i s supererogatory. Hence, philosophers may remain sk e p t i c a l of the t h i r d possible response, that the present formalism i s simply f a l s e , and that passage needs to be represented therein. Rather, passage needs to be incorporated into a metaphysical account which accords well with the physical account given by the formal space-time theory. How could the formalism introduce temporal passage in a metaphysically neutral manner, i f in fact that formalism does not, at present, represent passage? Light-cones are assigned a 1 d i r e c t i o n of temporal orientation (see figure 2); they are i n t r i n s i c a l l y symmetric about the point of o r i g i n . The description of temporal passage in space-time theories i s given in symbols as <<M,g,V>,0> where M i s a manifold, g i s a metric on M, and V i s the linear symmetric connective compatible with g which defines the a f f i n e structure. 0 i s the temporal orientation at each point p « M. A l l a f f i n e geodesies are either timelike <g(T.,T a)>0), spacelike (g(T a,T c)<0), or n u l l (g(T 0,T a)=0). 2 But i f the manifold M is temporally orientable, then i t is distinguishable from another manifold, Mj_, M^M, such that <<MJL, g, V>,0T>, where 0 T i s the opposite temporal orientation from O. But, and t h i s point i s the key to the argument, the af f i n e structure of i s the same as in M, hence the a f f i n e structure alone is not s u f f i c i e n t to di s t i n g u i s h 0 from 0 T, hence the formalism i s not s u f f i c i e n t to di s t i n g u i s h the d i r e c t i o n of temporal orientation T . Therefore, while Earman i s correct that useful r e s u l t s can be obtained from studying the mathematics of space-time theories, in thi s case, the problem of the passage of time i s metaphysical, and hence cannot be solved through mathematical-physical a p r i o r i theorizing alone. The problem of temporal passage i s more complex, hence the formalism i s neutral on the problem. Another way to put the point i s that the formalism does not contain enough resources to di s t i n g u i s h the d i r e c t i o n of temporal orientation T , hence does not contain For more d e t a i l s see: Friedman, M. (1983), Foundations of  Space-Time Theories, pp.43, 353-358. enough explanatory resources to explain the passage of the moving now. The reason why t h i s r e s u l t should be derived can be seen by a n t i c i p a t i n g the discussion of McCall's dynamic tree model. The phenomenon of temporal passage i s the r e s u l t not only of causal interactions within our own actual world a , but also the r e s u l t of the connectivity r e l a t i o n between a now and the changing set of p h y s i c a l l y possible future worlds. In other words, the formalism of physics only describes causal interaction within a, while temporal passage is the re s u l t of interaction both within a and also between a and the set of p h y s i c a l l y possible future worlds picked out by the connectivity r e l a t i o n . So the l i n e to explicate the A-series in terms of simultaneity or the r e l a t i v i t y of simultaneity has f a i l e d . Perhaps events are timelessly past, present, and future? Unfortunately, t h i s l i n e f a i l s also. Events cannot be timelessly past, present and future. The timeless l i n e f a i l s because 'timelessness' i s non-informative as an explication of the d i s t i n c t l y temporal r e l a t i o n s of pastness, presentness, and f u t u r i t y . In what sense could time be timeless? 'Timelessness' i s either a metaphor, in which case i t i s ambiguous; or 'timelessness' should be interpreted l i t e r a l l y , in which case the analysis reduces to claiming that there i s no temporal passage and hence no time, which is c l e a r l y wrong. w F i n a l l y , one might suppose that events are successively past, present and future. The successive l i n e seems to f a i l because i t seems that i t e n t a i l s either a vicious c i r c u l a r i t y or a vicious i n f i n i t e regress. However, i t i s the case that only most construals of the successive l i n e w i l l end in f a i l u r e . Finding the correct analysis of a successive l i n e of analysis of past, present, and future in order to provide an informative explanation without the vicious i n f i n i t e regress w i l l occupy the central l o g i c a l lock to be picked in th i s paper. The reasons why the successive l i n e generates an i n f i n t e regress of properties, and the proof that the attempts to explicate the i n f i n i t e regress show i t to be vi c i o u s , are covered in a later section (2.1). The solution I adopt i s that events are successively past, present, and future, but that t r a n s f i n i t e l y many worlds are required. To ant i c i p a t e , in McCall's model, the now moves up the universe-tree, successively a c t u a l i z i n g one of a set of ph y s i c a l l y possible (future) worlds; in t o t a l , there are t r a n s f i n i t e l y many worlds, so that an i n f i n i t e amount of branch-attrition can occur. 3 Radical skeptics might claim that the future and the past don't e x i s t , and that there are no contrary A-series properties for an event to possess. Events only seemingly possess these temporal properties of f u t u r i t y and pastness. This skeptical such as Hobbes, perhaps, position contradicts analysis of Indo-Eurpoean grammer which suggests that our temporal copulas of past, present and future are r e f e r r i n g terms which we take to be r e a l . But i f one Is a r e a l i s t about the future, present, and past, since after a l l temporal phenomenon of duration is a physical phenomena which can be o b j e c t i v e l y measured as the proper time along a trajectory; then such r e a l i s t s have adopted a burden of proof to explain the having of the apparently contrary temporal properties also entailed by the r e l a t i o n s of past, present and future; and that burden of proof has proven quite d i f f i c u l t to dislodge. One c r u c i a l point which helps to support the burden of proof that has been accepted by the r e a l i s t of passage i s that i f passage i s r e a l , i t i s a pervasive feature of the actual world, and the more pervasive the asymmetry, the better candidate i t i s to be an i n t r i n s i c property of time. A necessary property of time i s i n t r i n s i c to time, and i f necessary, i s true throughout time and space, and hence pervasive. Reasoning backwards from pervasiveness to necessity and thus to i n t r i n s i c properties i s a compelling form of metaphysical argument, but i t is not l o g i c a l l y necessarily compelling. We c e r t a i n l y cannot v e r i f y whether or not a proposed causal asymmetry i s pervasive by waiting and looking, since waiting and looking w i l l generate only a f i n i t e amount of data and the metaphysical conclusion concerns an i n f i n i t e number of events. Hence we cannot decide the matter of the status of temporal passage in a purely empirical, v e r i f i c a t i o n i s t , or instrumentalist manner. Hence, i t should be no surprise that physical theory cannot analyze temporal passage, since temporal passage e n t a i l s a larger ontology than that entailed by physical theory. Yet i t appears that time passes. A duration occured between reading t h i s sentence and the previous one. Is duration merely a B-series r e l a t i o n , or i s there a d i s t i n c t l y A-series component? I assume hereafter that there i s some re a l phenomena which i s denoted by the phrase 'temporal passage'. That phenomena needs more ex p l i c a t i o n . To provide that e x p l i c a t i o n , metaphysicians are led to believe that temporal passage i s the i n s t a n t i a t i o n of some property or r e l a t i o n ( s ) and i t i s the nature of th i s property or r e l a t i o n ( s ) that a r a t i o n a l philosopher of metaphysics must explicate. By postulating the r e a l i t y of time and temporal passage I am inconsistent with the conclusion of McTaggart's profound paradox. I r e j e c t McTaggart's premise that temporal passage (in the form of the A-series) i s or e n t a i l s a contradiction. McTaggart must be credited with c l a r i f i n g the d i s t i n c t i o n between the co-ordinate system of the A-series, 'past, present, and future', and the B-se r i e s co-ordinates of ' e a r l i e r than' and 'later than' and 'simultaneous with'. But McTaggart cannot be credited with proving the u n r e a l i t y of time; rather, the conclusion forces us to reconsider the b e l i e f s which led to the unpalatable conclusion. I claim that the A-series and the B-series are compatible on a s p e c i f i c interpretation of the nature of time and temporal properties. Further, i t i s i n s u f f i c i e n t explanation to analyze temporal passage t r i v i a l l y as the NT-position (also known as a 'block universe 1 view) requires. I show that the passage of time can be given a non-question-begging and consistent modal inter p r e t a t i o n . This interpretation adopts the possible worlds metaphysics as an explanatory framework for solving the problem, and postulates a complex physical a c c e s s i b i l i t y r e l a t i o n between possible worlds. McCall (1966, 1976, 1984, forthcoming) has been developing and continues to develop a modal model of objective time flow. Time ob j e c t i v e l y flows in the model, as the branch structure of the universe-tree evolves. The universe-tree consists of sets of p h y s i c a l l y possible worlds at a time. One of these worlds i s selected by the now, or the property of a c t u a l i z a t i o n , which conjoins the property of being present. Once the s e l e c t i o n process has occurred, then the non-selected worlds vanish from the tree, and the now moves to a new selection-set. In t h i s manner each present i s characterized by a unique state in the evolution of the universe-tree. Furthermore, the choices we make, and the random quantum events that occur, a l t e r the p r o b a b i l i t i e s associated with future outcomes. Thus McCall*s model provides a coherent metaphysics for the epistemology of time. The directed nature of the branch structure and branch-attrition constitutes the underlying causal process which Reichenbach sought for in the c l a s s i c text The Direction of Time. Is McCall's model a legitimate contender as a solution to McTaggart's paradox? I determine that McCall's model i s a good one by answering the following questions: (i) What i s the lo g i c of the model? ( i i ) What i s the metaphysical committment of the model? These two questions determine whether or not McCall's model i s worthy of deeper consideration. Then I ask: ( i i i ) Is the model r e a l l y a model of time flow? I answer t h i s question with reference to the tree-structure and branch a t t r i t i o n . F i n a l l y I ask: (iv) Is the model not s t i l l a block universe l i k e the other models i f i t i s set in a five-dimensional space? These l a s t two questions s t r i k e at the heart of the matter. If McCall's model i s r e a l l y a model of time flow, then i t must uniquely and coherently model time flow, and meet a l l the other requirements for a model of time such as consistency. It might be thought that McCall's model is just another s t a t i c , block universe. What i f McCall's model i s viewed as i f i t were five-dimensional? Then i t would appear as a s t a t i c , block universe. A universe i s c a l l e d a 'block universe' i f the past, present, and future of the world can be a l l arrayed in a single unchanging structure without inconsistency. The term 'block universe' was o r i g i n a l l y meant to describe the Minkowski manifold space, a four-dimensional e n t i t y . The four dimensions are composed of three dimensions of space and one time dimension. So McCall's model, i f viewed within a five-dimensional s e t t i n g , would appear s t a t i c , just as the world-line of a p a r t i c l e i s s t a t i c in four-dimensional Minkowski space-time. But the model s p e c i f i e s only four-dimensions, and within those four-dimensions, time passes. Within those four-dimensions, i t i s not possible to give a complete set of the pictures of the universe from every time. The structure of the universe changes over time, so that i t looks d i f f e r e n t to d i f f e r e n t observers. The universe-tree evolves, and d i f f e r e n t sets of p h y s i c a l l y possible worlds are future-directed at d i f f e r e n t times. To suppose that McCall's model should be judged by five-dimensional standards i s incorrect. The four-dimensionality of McCall's model is important; i t puts the model on a par with the current models which are four dimensional but do not incorporate time flow. Storrs McCall has further suggested that i f his model can do a l l that the non-time-flow models can do, plus include time flow, then that inclusion i s i t s e l f a very strong metaphysical consideration which should favour acceptance of the truth of McCall's model over the NT-Minkowskian models. It is one step to suggest that the dynamic universe-tree model can do a l l the explanatory work that any non-time-flow model can do, plus include time flow, but i t i s a giant leap (I believe) to prove that the dynamic universe-tree model can do a l l the explanatory work that any non-time-flow model can do. I prove that McCall's model w i l l do the required explanatory work. Thus, the evidence to date suggests the truth of a modal metaphysical solution to the problem of temporal passage. at the P a c i f i c D i v i s i o n meeting of the American Philosophical Association in L.A., C a l i f . , U.S.A., March 1990. 3.0 The Passage of Time 3.1 McTaggart's paradox In t h i s section I s h a l l discuss McTaggart's paradox. McTaggart's paradox i s that since temporal passage i s inconsistent and since temporal passage i s also necessary for time, then time i s therefore unreal. McTaggart must be credited for recognizing a useful d i s t i n c t i o n between two types of temporal r e l a t i o n s . Temporal re l a t i o n s between events are s p e c i f i e d in two d i s t i n c t systems of coordinates, termed the A-series and the B-series. The A-series co-ordinates locate an event as either past, present, or future r e l a t i v e to the 'now'. The B-series co-ordinates locate an event as occurring at a clock time t, such as: "This sentence was typed on the f i r s t day of A p r i l . " The truth-value of sentences with A-series temporal re l a t i o n s changes over time as the now moves into the future; the truth-value of sentences with B-series r e l a t i o n s are fixed for a l l times. For instance, at one time, such as during Plato's l i f e t i m e , Newton's b i r t h was in the future, while at another time, for instance, today, Newton's b i r t h is in the past. Newton's b i r t h has changed from being future to being past, and was now for only one instant. The A-series r e l a t i o n i s dynamic. and the B-serles r e l a t i o n i s s t a t i c . McTaggart thought the A-series r e l a t i o n was somehow more fundamental because i t r e f l e c t s change in time. The B-series r e l a t i o n i s unchanging, and free from the complications of tense in language. McTaggart's c l a s s i c argument for the unreality of time depends upon the d i s t i n c t i o n between the A-series and the B-series and upon his concept of genuine change. McTaggart argues that the A-series i s necessary for genuine change, but that the A-series i s self-contradictory; therefore time, which depends upon genuine change, cannot e x i s t . McTaggart's argument i s e a s i l y rejected, as his theory of genuine change i s implausible. However, defenders of the T-position must explicate the nature of the A-series r e l a t i o n , and hence must overcome a l l e g a t i o n that the A-series i s a contradictory notion, or leads to contradictions. F i r s t , a b r i e f explanation of why McTaggart's theory of genuine change is implausible. A poker's being hot at and cool at t2 does not, according to McTaggart, s u f f i c e as proof of the existence of genuine change because these facts are true for a l l time. Genuine change requires that the sum t o t a l of facts at one time be not the same as the sum t o t a l of facts at another time. Thus genuine change occurs as the now moves into the future, because an event which was future becomes present and then past, and so the fact of i t s being future and i t s being past changes, and hence the sum-total of facts of the world also change over time. The changing truth-value of A-series propositions marks the only type of genuine change possible, according to McTaggart. The contradiction within the A-series i s expressed by saying that events which are future r e l a t i v e to one 'now' are past r e l a t i v e to another 'now1. Hence, explication of the sentence "The poker is hot now" when said at and when said at t2 results in d i f f e r e n t truth-values, and hence there i s no time-neutral body of facts of now-expressions. McTaggart's argument regarding genuine change i s an i n t e l l e c t u a l l y challenging reductio ad absurdum. If the truth-value of sentences regarding temporal facts, such as the fact that the poker i s hot now when now i s and not t^i could change over time, then one could legitimately ask whether or not the second-order fact that f i r s t - o r d e r facts can change the i r t r u t h -value over time, could change i t s truth-value over time? If sentences regarding second-order facts could change the i r t ruth-value, then i t could be true that sentences regarding f i r s t - o r d e r facts don't change the i r truth-value, which i s a contradiction with the changing truth-value of sentences regrading f i r s t - o r d e r f a c t s . But i f a sentence regarding a second-order temporal fact could not change i t s truth-value, then there Is no r e a l l y genuine change after a l l at the second-order l e v e l , contradicting the assumption that genuine change was possible. Another way to put the absurdity of McTaggart's idea of genuine change i s as follows. How could a fact about the past, such as the fact that McTaggart wrote a paper on the nature of time in 1908, change i t s truth-value over time, so that at some future time i t won't be true that McTaggart wrote a paper in 1908? A committment to a m u l t i p l i c i t y of pasts i s quite counterintuitive and hence an unwanted r e s u l t . If McTaggart wrote a paper in 1908, then that fact cannot change i t s truth-value; a presumption of r a t i o n a l analysis i s that facts cannot change thei r truth-value. Logical reasoning presupposes that sentences containing facts are, by their very nature, true for a l l time; to allow facts to change the i r truth-value over time would permit inconsistencies to occur over time. On t h i s point, Horwich (1988) states that "...we reserve the term 'fact' for those aspects of r e a l i t y whose e x p l i c i t descriptions are sentences that are true s i m p l i c i t e r - and not merely true r e l a t i v e to some context or 5 point of view, and false r e l a t i v e to others." Thus McTaggart's paradox c e r t a i n l y must be resisted by re j e c t i n g his theory of genuine change. Change occurs when the temperature of a poker i s hot at t l and cool at t2. The d i f f i c u l t question i s whether or not t h i s observed physical change requires temporal passage as the T-position would have i t , or whether i t can be explained as the NT-position would have i t , without 5 Horwich, Paul. (1988) p. 23. recourse to passage. As Broad (1938) points out, following Russell, the existence of the s t a t i c B-series does not by i t s e l f rule out change in the world. The A-series i s not necessary for a theory of causal change. Thus, the evidence so far would seem to support the NT-position, since the A-series appears to be a superflous concept. If the A-series r e l a t i o n i s a s i g n i f i c a n t temporal phenomenon, then some underlying metaphysical mechanism must be found wherein the A-series r e l a t i o n plays a s i g n i f i c a n t r o l e . So while McTaggart's paradox can be e a s i l y rejected by re j e c t i n g his theory of genuine chanage, there is a gem remaining buried in the rubble. McTaggart's argument shows that the 'moving now' engenders contradictions i f i t i s assumed to mark a single ontological property that i s both timeless as the B-series would suggest and yet also changing in time as the A-series would have i t . A single ontological property cannot to the explanatory work required by both the A-series r e l a t i o n and the B-series r e l a t i o n . The A-series engenders a contradiction only i f some consistent point of reference cannot be found. Without a point of reference, some explanatory map which defines the A-series r e l a t i o n , the A-series r e l a t i o n w i l l engender a contradiction which could be t r i v i a l l y transformed into a vicious i n f i n i t e regress. The vicious i n f i n i t e regress occurs when second-order facts about temporal r e l a t i o n s , or second-order time dimensions, are required, necessitating third-order temporal r e l a t i o n s , and so on, i n a never-ending attempt to plug the explanatory leak. If the A-series r e l a t i o n i s to do r e a l metaphysical work, then some other point of reference than the B-series r e l a t i o n needs to be found. McTaggart has not shown that no possible point of reference exists to do the required ontological work. He has only shown that the timeless truth of B-series propositions cannot do the d i s t i n c t i v e l y temporal work of the A-series. In other words, while the A-series and the B-series are related, they are not one and the same r e l a t i o n - some t h i r d e n t i t y must be found to relate them together as both sets of properties of one and the same t h i r d e n t i t y . Philosophers of the NT-persuasion have t r i e d to reduce the notion of the A-series to the B-series, while philosophers of the T-position have t r i e d to run the reduction the other way round. Both l i n e s of inquiry have proven to be plausible, and i t is quite d i f f i c u l t to determine which way the reduction should run, assuming a directed reductive account i s needed. It turns out that each r e l a t i o n , A-series and B-series, can be defined in terms of the other. Hence i t i s quite unclear which r e l a t i o n would better serve as the underlying mechanism. Hence I believe that no directed reductive account should be sought for. That i s , neither the A-series r e l a t i o n nor the B-series r e l a t i o n i s more bale than the other. Rather, they are both necessary for time, and a t h i r d e n t i t y must be found which can relate them together. So there i s a t h i r d p o s s i b i l i t y : that neither r e l a t i o n i s more basic than the other, but that both the A-series r e l a t i o n and the B-series r e l a t i o n are aspects of time. This view necessitates that the apparent contradiction between the A-series and the B-series be resolved. The changing truth-value of sentences using A-series r e l a t i o n s must be reconciled with the nonchanging truth-value of sentences using B-series r e l a t i o n s , in that both the A-series r e l a t i o n s and the B-series relations are aspects of time. The r e c o n c i l i a t i o n can be achieved by finding another metaphysical e n t i t y to do the work. Bigelow (1989) i n s i g h t f u l l y remarks that possible worlds w i l l keep contrary properties out of each other's hair. 3.1.1 The I n f i n i t e Regress Problem An immediate problem upon encountering possible worlds i s that there are, possibly, very many of them, so many in fact that there are possibly more than i n f i n i t e l y many. Before I claim that in fact t r a n s f i n i t e l y many possible worlds are required in order to explicate temporal passage, I can show that an i n f i n i t e number of e n t i t i e s won't s u f f i c e to explain temporal passage. The following l i n e of argument w i l l show that the i n f i n i t e regress of temporal predication, suggested by McTaggart's paradox, is necessarily v i c i o u s , and hence t h i s l i n e of analysis must be abandoned. Quentin Smith (1986) proposes that the regress i s i n f i n i t e but not v i c i o u s ; t h i s l i n e w i l l be shown to be based on a misunderstanding of the problem. However, analyzing just where Smith's argument goes wrong i s a fascinating problem tackled by Oaklander (1987). I a r b i t r a t e t h i s debate because I believe i t w i l l y i e l d the necessary insight that indeed the solution of McTaggart's paradox when the A-series i s construed as an i n f i n i t e regress i s vicious and therefore unhelpful. If the regress of temporal a t t r i b u t i o n i s both i n f i n i t e and v i c i o u s , and i f no single e n t i t y can be found to do the required metaphysical work, then i f there i s to be an explanation of temporal a t t r i b u t i o n , some other e n t i t y or e n t i t i e s must be found to do the work. The only other candidate is a possible worlds metaphysics s o l u t i o n . I maintain that some explanation of the apparent motion of the 'now* i s required. If the motion of the •now' i s a necessary component of the nature of time, then time is i n t r i n s i c a l l y asymmetric. So I grant one premise of McTaggart's paradox, the premise that time flow i s a necessary property of time. However, I examine c l o s e l y the question of whether or not time flow e n t a i l s a contradiction or an i n f i n i t e regress in orderr to show that time flow can avoid both dangers by postulating t r a n s f i n i t e l y many e n t i t i e s (e.g., worlds). If the analysis of temporal a t t r i b u t i o n y i e l d s an i n f i n i t e regress, then i t must be determined whether or not the regress vicious or nonvicious. If the i n f i n i t e regress i s nonvicious, then there i s no motivation to seek a possible worlds solution which requires t r a n s f i n i t e l y many e n t i t i e s . But the analysis w i l l show that the i n f i n i t e regress is indeed v i c i o u s . Hence a possible worlds solut i o n must be sought f o r . I thus return to consider Quentin Smith's argument that the regress e x i s t s but that i t is nonvicious. I claimed above that Smith's argument i s based on a misunderstanding of the strength of the problem. Below I argue that the regress i s vicious by examining Smith's argument to the contrary. F i r s t Smith outlines the regress as a modern McTaggartean philosopher might express i t . The f i r s t l e v e l of temporal predication applies to events: every event i s present, past, and future. Smith notes that since presentness, pastness, and f u t u r i t y are incompatible predicates, they cannot belong to the same event simultaneously, but must hold successively. In tensed language, t h i s expression means that an event i s present, w i l l be past, and has been future; or that i t i s past, has been future and has been present; or that i t i s future, and w i l l be present and then w i l l be past. In tenseless language, t h i s expression means that an event i s present at a moment of present time, i s past at some moment of future time, and i s future at some moment of past time. The ' i s ' in the tenseless use of language i s the non-temporal copula much l i k e the use of ' i s ' in "Two plus two i s four." The second l e v e l of temporal predication i s spe c i f i e d in terms of the f i r s t l e v e l predicates. For example, in the statement, 'the event i s present at a moment of present time', the moment of time referred to i s not only a present moment, but i s also a past and a future moment. Since i t i s contradictory to assert that the moment i s present, past and future simultaneously, because these are incompatible predicates, i t must be asserted instead, according to Smith, that the moment i s present at some higher l e v e l present moment, and s i m i l a r l y for the past and future. The t h i r d l e v e l of the regress i s introduced by e x p l i c a t i n g the nature of the second l e v e l temporal predicates. The fourth l e v e l i s introduced s i m i l i a r l y , and so on to i n f i n i t y . McTaggart argued that t h i s i n f i n i t e regress was vicious because: The a t t r i b u t i o n of the c h a r a c t e r i s t i c s past, present, and future to the terms of any series leads to a contradiction, unless i t i s s p e c i f i e d that they have them successively. This means, as we have seen, that they have them in r e l a t i o n to terms s p e c i f i e d as past, present, and future. These again, to avoid a l i k e contradiction, must in turn be s p e c i f i e d as past, present and future. And, since t h i s continues i n f i n i t e l y , the f i r s t , s e t of terms never escapes from contradiction at a l l . Smith betrays his lack of understanding of McTaggart's clear rendition of the viciousness of the i n f i n i t e regress by c a l l i n g McTaggart's remarks 'paradoxical.' Smith agrees that: "McTaggart i s indeed correct that the a t t r i b u t i o n of presentness, pastness and f u t u r i t y leads to a contradiction unless they are attributed 7 successively." But then Smith goes on as i f the contradiction can be avoided simply by a t t r i b u t i n g the temporal predicates successively. The problem i s that as soon as the attributes are attributed successively, the analysis is driven onto a higher l e v e l of predication. 6 P a r t s of McTaggart's argument are reprinted i n Gale (1967). This part is reprinted on p.96. 7Sraith (1986), p.384-385. Oaklander, elsewhere a defender of a Russelllan theory of time, argues persausively against Smith's analysis of McTaggart's paradox. Oaklander notes that "...McTaggart's point i s that an appeal to succession does not s u f f i c e to avoid the contradiction. Smith does not deal d i r e c t l y with McTaggart's argument." Smith states what he thinks the paradox i s in McTaggart's argument, and argues that the paradox is based on an elementary l o g i c a l blunder. But Oaklander points out, and I agree, that i f McTaggart's argument were based on that blunder, then the argument would be i n v a l i d . But McTaggart did not make the elementary blunder of which he i s accused by Smith (1986). Smith believes that McTaggart's argument i s paradoxical because the a t t r i b u t i o n of temporal predicates i s nonproblematic; Smith says that: . . . i n each case i t i s s p e c i f i e d that the terms to which they are attributed have them successively... How could (the i n f i n i t e l e v e l s of predication] never escape from contradiction i f i t never was contradictory? The f i r s t set of terms, the events, are contradictory only i f i t is not s p e c i f i e d that these terms have presentness, pastness, and f u t u r i t y successively. But i t - i s s p e c i f i e d that they have them successivelyI Oaklander (1987, p.425) points out that Smith simply asserts that succession avoids the contradiction. I hold that Smith is correct Oaklander (1987), p.425. 9 Smith (1986) "The I n f i n i t e Regress of Temporal At t r i b u t i o n s " The Southern Journal of Philosophy, v o l . XXIV No. 3.p.385. on the point that succession avoids the contradiction, but for d i f f e r e n t reasons (namely, that the succession i s t r a n s f i n i t e ) . Smith is not correct that McTaggart's argument i s paradoxical. Oaklander notes that Smith does not deal d i r e c t l y with McTaggart's argument, but instead gives an analysis of some other paradoxical claims. oaklander shows that Smith's argument goes off the r a i l s when Smith claims that McTaggart believes that the f i r s t l e v e l of temporal a t t r i b u t i o n using the A-series r e l a t i o n i s contradictory. Smith infers from (5) The a t t r i b u t i o n of the c h a r a c t e r i s t i c s past, present, and future to the terms of any series leads to a contradiction, unless i t i s s p e c i f i e d that they have them successively to (6) The a t t r i b u t i o n of the c h a r a c t e r i s t i c s past, present and future to the terms of any series leads to a contradiction, which i s subsequently resolved 0by specifying that they have them successively. If McTaggart's argument were based on the i n v a l i d inference that Smith claims i t i s based on, then McTaggart's argument would indeed be paradoxical. But McTaggart's paradox i s shown to be a v v a l i d argument in a later section of t h i s thesis (3.2). Smith's i n v a l i d inference i s i n v a l i d because a statement of the form 'A unless B' does not e n t a i l a statement of the form 'A and B'. Oaklander, L. Nathan. (1987) "McTaggart's Paradox and the I n f i n i t e Regress of Temporal At t r i b u t i o n s : A Reply to Smith" The  Southern Journal of Philosophy. Vol. XXV, No. 3., p.426. 31 McTaggart i s not g u i l t y of t h i s s i n . Smith claims that the appeal to succession demonstrates that there i s no problem in temporal a t t r i b u t i o n s . But McTaggart is arguing that there i s a problem in temporal a t t r i b u t i o n and that succession w i l l not solve the problem. Oaklander remarks: "For detensers l i k e myself the d i r e c t i o n of time i s based on the unanalyzable temporal r e l a t i o n of succession, but for McTaggart temporal r e l a t i o n s are analyzable in terms of the flow of time or the moving now."11 Oaklander sympathizes with the NT-position. On the point of the unanalyzable succession r e l a t i o n , McTaggart remarks: " I t i s only when the A se r i e s , which gives change and d i r e c t i o n , is combined with the C ser i e s , which gives permanence, 12 that the B series can a r i s e . " For McTaggart, the B-series r e l a t i o n was defined in terms of the A-series r e l a t i o n . Therefore, McTaggart believed that an appeal to succession would f a i l because the B-series r e l a t i o n would in turn be analyzed using the corresponding A-series term, which would involve a vici o u s c i r c l e because i t assumes the existence of time in order to account for the way in which moments are past, present, and future. Time then must be pre-supposed to account for the A s e r i e s . But we have already seen that the A series has to be assumed in order to account for time. Accordingly, the A series has to be pre-supposed in Oaklander (1987), p.426. 1 2McTaggart (1934) "The Unreality of Time." i n : Philosophical Studies, p.118. 32. order to account for the A s e r i e s . And t h i s i s c l e a r l y a vicious c i r c l e . Consider the truth-value of a sentence such as "The poker i s hot now." The word 'now' requires a determinate analysis of an A-series temporal r e l a t i o n i f the sentence has a coherent meaning. The problem that the truth-value of such sentences with A-series r e l a t i o n s changes from to t2 suggests that the analysis of A-serie s r e l a t i o n s should allow th e i r truth-value to s h i f t in time. McTaggart thought of the problem in terms of how to account for change in time, say the poker being hot at one time and cold at another. However, the problem of having a theory of change is a symptom of the deeper problem of finding an account of changing A-series truth-values. Changes in the truth-value of sentences containing A-series r e l a t i o n s can have meaning, then, only i f events acquire and shed the A-properties successively. Given the fact that the poker f i r s t i s hot (at tj_), and then i s cold (at t2>, however, the temporal r e l a t i o n of ' e a r l i e r than' is presupposed by the facts, and that temporal r e l a t i o n can only be explicated by previous appeal to the A-series, since the B-series i s defined in terms of the A-series and the C-series. So far the paradox has been expressed in the l o g i c a l form of a c i r c l e . But a c i r c u l a r argument can be transformed into a vicious i n f i n i t e regress, and vice versa. In t h i s case, i t may be i n s t r u c t i v e to do so. If we avoid the o r i g i n a l contradiction by McTaggart "The Unreality of Time.", p. 124. 33 claiming that the terms never have the incompatible properties simultaneously but rather only at d i f f e r e n t times, that i s , E i s future at t ^ , present and and past at t3, the problem i s avoided only i f : t ^ f t2 / t$. For i f events do not have their A-properties at d i f f e r e n t times, then they are either timelessly or simultaneously past, present, and future, and the paradox i s unavoidable. What, then, i s the basis f o r : fcl ? ^2 * ^3' that i s , for, say, t ^ being e a r l i e r than t2 and t2 e a r l i e r than t3 (assuming, with common sense, that time has a complete linear ordering)? The resolution of t h i s dilemma cannot be found by postulating another time dimension, for the new seri e s i s only a genuine temporal ser i e s i f i t i s free of contradiction. The resolution of t h i s dilemma can only be found by seeking greater explanatory resources. The surprising fact i s not that those resources are a v a i l a b l e , though the explanatory framework of the possible worlds metaphysics, but that temporal passage should turn out to e n t a i l a metaphysical connection between the actual world and a set of possible worlds. •3f 3.1.2 The P o s s i b l e Worlds M e t a p h y s i c s S o l u t i o n John B i g e l o v p r o v i d e s a tremendous i n s i g h t i n t o the n a t u r e of McTaggart's paradox i n h i s (1989) paper MA Modal Theory of Time." B i g e l o v ' s key i n s i g h t i s t h a t a modal s o l u t i o n t o t h e paradox i s su g g e s t e d by the modal s t r u c t u r e t h a t u n d e r l i e s McTaggart's own argument. B i g e l o w a c h i e v e d t h i s i n s i g h t by n o t i n g t h a t McTaggart's argument reduces t o two c r u c i a l p r e m i s e s . The f i r s t i s t h a t an e s s e n t i a l , or n e c e s s a r y p r o p e r t y , of time i s t h a t i t p a s s e s . The second premise i s t h a t the passage of time i s i n c o h e r e n t or i n c o n s i s t e n t , and i n v o l v e s e i t h e r a v i c i o u s i n f i n i t e r e g r e s s or a v i c i o u s c i r c u l a r i t y . Together t h e s e p r e m i s e s e n t a i l t h a t time i s u n r e a l , t h e r e i s no t i m e , and t h a t n o t h i n g ever r e a l l y o c c u r s b e f o r e or a f t e r a n y t h i n g e l s e . What i s t h e passage of t i m e ? On B i g e l o w ' s v i e w , the passage of time i s c a p t u r e d by such s e n t e n c e s a s : (5) An event was once f u t u r e , became p r e s e n t , and has now become p a s t . (6) An event i s p a s t , had become p r e s e n t , and was f u t u r e . (7) The p r e s e n t event was j u s t f u t u r e , and w i l l soon be p a s t . Sentences ( 5 ) - ( 7 ) a r e a l l e q u i v a l e n t , so a n a l y z i n g (7) i s enough, and i t i s (7) t h a t B i g e l o w ' s a n a l y s i s f o c u s e s on. The N T - p o s i t i o n a n a l y s e s (7) a s : (8) What i s s i m u l t a n e o u s w i t h t i s l a t e r than some time which i s e a r l i e r t h a n t , and i s e a r l i e r t h a n some time which Is later than t. The NT-position arrives at (8) from (7) by replacing the A-series terms with th e i r B-series counterparts. (8) is t r i v i a l l y s a t s i f i e d , and hence necessarily true. The NT-view i s that since (8) i s a l l there i s to the passage of time, there i s no problem about temporal passage. But i s (8) a l l that the passage of time amounts to? (8) i s vacuous. (8) conveys no information, and hence reduces the passage of time to a merely abstract existence. Bigelow argues that the NT-analysis leading to (8) i s misdirected from the s t a r t . (8) analyzes the passage of time by appealing to the e a r l i e r / l a t e r r e l a t i o n . Bigelow states: But t h i s r e l a t i o n i s presupposed to be a temporal ordering. Not every r e l a t i o n with the l o g i c a l properties of the e a r l i e r / l a t e r r e l a t i o n , not every linear ordering, counts as a temporal ordering. In order for the e a r l i e r / l a t e r than r e l a t i o n to be a temporal r e l a t i o n , there must be some property of t h i s r e l a t i o n that makes i t d i s t i n c t i v e l y temporal. As i t stands, the B-series i s timeless; what makes the B-series a temporal r e l a t i o n i s that what i s e a r l i e r becomes present before what i s la t e r becomes present, and what i s later i s no longer future but becomes present and then too becomes past. On i t s own, the B-series r e l a t i o n is merely a linear ordering. Bigelow states the point: Bigelow (1989), p.3-4. "...the e a r l i e r / l a t e r r e l a t i o n i s a temporal one only i f i t 15 concerns something which passes." This requirement e n t a i l s that a complete analysis of time cannot be given without considering both the B-series and the A-series properties of time. Questions of conceptual or l o g i c a l p r i o r i t y are del i c a t e because, p a r t i c u l a r l y in the problem of temporal passage, there are plausible arguments running the reduction in both d i r e c t i o n s . Running the reduction in the d i r e c t i o n of making the B-series basic, however, f a i l s to be s a t i s f y i n g because of the t r i v i a l i t y of the indexical analysis of time and tense. Running the reduction i n the d i r e c t i o n of the A-series e n t a i l s taking passage as a primitive notion. But i f e a r l i e r / l a t e r can be defined in terms of the A-series in a coherent explanatory framework, then the A-series i s guaranteed to be more than mere metaphor. The e a r l i e r than / l a t e r than r e l a t i o n can only be defined in terms of the A-series i f some other e n t i t y , other than time i t s e l f , can be found as a point of reference for the nature of temporal passage. McTaggart has shown that time i t s e l f cannot be the point of reference required to explicate temporal passage. The possible worlds metaphysics i s nominated to f u l f i l that function. The possible worlds metaphysics (PWM) holds that a p o s s i b i l i t y P exists in our world a i f and only i f there i s an Bigelow (1989), p.4. a c c e s s i b i l i t y r e l a t i o n R between the actual world a and the possible world 0 i n which P. Suppose that possible worlds a and 0 meet the following three temporal conditions: (i) everything past or present i n a i s past in 0; ( i i ) everything present or future in 0 i s future in a; ( i i i ) everything future in a and not future in 8 i s present or past in p . If ( i ) - ( i i i ) are true in both a and 8 then a i s in the past of 0 and 0 i s in the future of a . Thus i f ( i ) - ( i i i ) then there i s an a c c e s s i b i l i t y r e l a t i o n R between a and 0 such that 0 contains a possible future of a . And mutatus mutandis for possible pasts of o. The solution of the problem of the passage of time i s to describe the r e s t r i c t i o n of th i s a c c e s s i b i l i t y r e l a t i o n R between possible worlds. It turns out that R i s i t s e l f a function from a onto a set of possible worlds r . This solution benefits from a conceptual mapping between previously developed tense l o g i c (Prior 1957, 1958, 1959, 1967) and modal l o g i c following from Kripke. Bigelow notes that: "The use of modal terminology enables us to assimilate the formal t r i c k s of tense l o g i c into the... ontological realm of individuals, properties, and r e l a t i o n s , together with modalities. Bigelow (1989), p.10. IS* The modal theory explicates the passage of time through a counterfactual analysis, as possible worlds analyses are given of the truth-conditions of counterfactual statements. Consider the counterfactual claim: (9) What i s present was once future because i t would have been future i f past things were present. This claim, (9), i s true i f and only i f there i s the appropriate a c c e s s i b i l i t y r e l a t i o n R between worlds a and 8 such that what i s present in a was once future because what i s present in a i s mapped onto the future in 8 and consequently what i s past in a i s present in 8. If we assume the existence of the properties of pastness, presentness, and f u t u r i t y , and we assume that what i s a c t u a l l y present could have been past or future, that i s , that what i s ac t u a l l y present i s possibly past or possibly future, then we can give a non-circular d e f i n i t i o n of the d i s t i n c t l y temporal ordering of the e a r l i e r / l a t e r r e l a t i o n and hence avoid McTaggart's paradox. What i s a c t u a l l y present, that i s , present in a , i s possibly past or future i n 8* The present in a i s uniquely defined, defined that is by a world-relative index. The re a l present-in - o has, on Lewisian counterpart theory, other counterparts, such as pa s t - i n - 8 , while on other construals of the id e n t i t y r e l a t i o n across possible worlds, the present-in - a has a transworld doppleganger. The key point for my purposes is that on the possible worlds analysis, e x p l i c a t i o n of "The poker i s hot now" w i l l r e s u l t in the claim "The poker is hot at the present in a" where the ' i s ' in t h i s l a t t e r claim i s nontensed. The ' i s ' being nontensed has the res u l t that every temporal object in a is present in a, but t h i s r e s u l t i s welcome, since i t i s necessarily true. So, on the PWM analysis, a complete sentence involving A-series r e l a t i o n s requires a world index (although, for most purposes, the relevant world is the actual world, a ) . The e a r l i e r / l a t e r r e l a t i o n i s a linear ordering, so i f the e a r l i e r / l a t e r r e l a t i o n i s a temporal ordering then because i t is linear i t has these fiv e properties: (10) i r r e f l e x i v i t y (nothing i s e a r l i e r than nor later than i t s e l f ; (11) asymmetry ( i f x i s e a r l i e r than y then y i s not e a r l i e r than x); (12) t r a n s i t i v i t y ( i f x is e a r l i e r than y and y is e a r l i e r than z then x i s e a r l i e r than z); (13) connectivity ( i f x i s e a r l i e r than y and z i s not e a r l i e r than y then x is e a r l i e r than z); (14) density ( i f x i s e a r l i e r than z then there i s a world y which i s later than x but e a r l i e r than z ) . In Godelian worlds (12) does not hold; in discrete time worlds, (14) is f a l s e . The account so far does not guarantee that i t has s u f f i c i e n t content. In the interpretation of possible worlds semantics developed by Montague and Cresswell, i t i s possible for a thing to have a property in a world without exi s t i n g in that world. A 4Q property of a thing is taken to be a function from each ind i v i d u a l to the set of possible worlds in which the individual (or in Lewisian semantics, the counterpart of the individual depending on the r e l a t i o n of i d e n t i t y across worlds) has that property. There i s no s e t - t h e o r e t i c a l d i f f i c u l t y in t a l k i n g about the set of worlds in which an object does not e x i s t . If the a c c e s s i b i l i t y function i s to sets of worlds in which objects don't e x i s t , however, then t h i s metaphysics re s u l t s in a nonexistent future and past, and only successive instantiations of present events. If t h i s metaphysical picture were true, then things in the actual world a would have the property of being future or being past only in virtue of the existence of the e n t i t y with that respective property in some other world 8. Bigelow notes that "This conception... as truncated to include only the present, requires us to rethink the modal theory i of time in a number of ways... One world w i l l be 'temporally' accessible from another only when they contain the same things and d i f f e r only in a systematic transposition of the properties of pastness, presentness, and f u t u r i t y . " (p.15). This truncated conception e n t a i l s that temporal a c c e s s i b i l i t y r e l a t i o n s depend not only on the systematic transposition of the properties of pastness, presentness, and f u t u r i t y with respect to world a , but also with respect to things which have properties with respect to worlds even though they do not exist in those worlds (say in world(s) 6 ) . 4± But the truncated conception is not a s u f f i c i e n t analysis of the past and future. The truncated conception i s designed to r e f l e c t the position of presentism, the view that only the present e x i s t s . This truncated view of the temporal spectrum e n t a i l s r a d i c a l skepticism about the past and future and is an untenable ontology because i t i s too parsimonious. The truncated conception attempts to remove the complexities of a theory involving the past and future, removing them only by postulating an assumption contrary to our ordinary i n t u i t i o n s about time. McCall's model, by contrast, requires that the a c c e s s i b i l i t y r e l a t i o n be far more complex than in the truncated modal conception wherein the ontology consists only of a set of events of which i t i s successively true to say that the event is present-in - a. In McCall's model, the a c c e s s i b i l i t y r e l a t i o n i s that of physical p o s s i b i l i t y . Hence, being present means that the event was selected from a range of ph y s i c a l l y possible events. Thus, what events become present are ones that are p h y s i c a l l y possible at the present-in - a . On McCall's model, being present i s a world-relative phenomenon. Being r e l a t i v e in t h i s sense does not undermine in any way the o b j e c t i v i t y of the phenomenon. Motion is r e l a t i v e to a reference frame, but motion i s nonetheless an objective phenomenon. So too with the passage of time, on McCall's model. 4 he passage of time, in the actual world, is a function from the ctual world to a changing set of other possible worlds. 3.2 The lo g i c of McTaggart's paradox An outline of McTaggart's paradox i s as follows. 1. Events are located in the B-series (ordered with respect to e a r l i e r than or later than or simultaneous with) only i f time e x i s t s . eB -» T 2. Time e x i s t s , only i f there i s genuine change. T -» gc 3. There i s genuine change in the world only i f events are located in the A-series. gc -» eA Therefore, from 1. to 3., 4. Events are located in the B-series only i f they are located in the A-series. ((eB -» T) & (T -» gc) & (gc -» eA) ) -» (eB -» eA) 5. If events are located in the A-series, then each event acquires the absolute properties of being past, being now, and being future. eA •• bPNF 6. There i s a contradiction in supposing that any event has any two of these three absolute properties. -bPN & -bPF & -bNF Therefore, from 5. and 6., 7. The A-series cannot e x i s t . ((eA -» bPNF) & (-bPN & -bPF & -bNF) ) -• -eA Therefore, from 4. and 7., 8. Events are not ordered with respect to the B-series. -eB Therefore, from 1. and 8., 9. Time does not e x i s t . -T In t h i s section, I am only concerned with the l o g i c a l structure of McTaggart's argument, not the soundness of the 17 premises. I want to make the point that t h i s version of McTaggart's paradox i s a v a l i d argument. If the premises are granted, then the conclusion follows. I can quickly prove that the above argument i s v a l i d , by generating a contradiction with the premises i f the conclusion, (9.), i s denied. I w i l l carry out t h i s proof using symbols keyed as noted above, following the strategy of reductio ad absurdum. F i r s t , suppose the opposite of the conclusion (9.): 10. Time e x i s t s . T From 1. and 10., 11. Events are located in the B-series. eB From 11. and 4., 12. Events are located in the A-series. eA But: 7. The A-series cannot e x i s t . ((eA -* bPNF) & (-bPN & -bPF & -bNF) ) -» -eA So from 12. and 7. we derive a contradiction: 13. Events are located in the A-series and events are not located in the A-series. (eA & -eA) Therefore, McTaggart's paradox i s a v a l i d l o g i c a l argument, since the denial of the conclusion i s inconsistent with the premises. This e x p l i c a t i o n i s given by Horwich (1988, p.18). The l o g i c of McTaggart's paradox is to set up a contradiction within the A-series r e l a t i o n and then conclude that time is unreal since the A-series r e l a t i o n i s necessary for time flow and flow i s a necessary part of time. Hence, i f the contradiction within the A-series r e l a t i o n can be resolved, the remainder of the paradox may be a salvageable l i n e of argument. A possible worlds metaphysical analysis of the apparent contradiction entailed by the A-series r e l a t i o n s w i l l be seen to defuse the problem and hence salvage the remainder of McTaggart's paradox. Hence, the l o g i c a l structure of McTaggart's argument, taken together with the substantive claim that time flow is necessary for a complete account of time, suggests that a modal solution to the problem of the passage of time would also serve to deny McTaggart's conclusion and thus vindicate our common sense b e l i e f in the r e a l i t y of time. 3.3 The soundness of McTaggart's paradox The escape from the conclusion of McTaggart's paradox can come, therefore, only by denying one of the premises, since we cannot escape the conclusion i f we accept a l l the premises, since the conclusion i s a v a l i d inference from those premises. I w i l l examine each premise in turn in order to determine what arguments can be marshalled to support each premise, in order to find i f at least one premise can be rejected. 1. Events are located in the B-series (ordered with respect to e a r l i e r than or la t e r than or simultaneous with) only i f time e x i s t s . eB -» T. Premise 1. i s obviously true, since as Horwich notes, the consequent that time exists must be construed in a very broad manner, such as meaning that the world exhibits temporality in some sense. If the world exhibits temporality, then the temporal dimension of the world may have a B-series ordering. 2. Time e x i s t s , only i f there is genuine change. T -» gc McTaggart has a p a r t i c u l a r l y strange view of genuine change, 18 and i t is thi s premise of his argument that Norwich c r i t i c i z e s . According to McTaggart, change does not occur when a poker, hot at time t 0 , becomes cold at a l a t e r time t^. McTaggart i s correct to note that the truth-value of these propositions regarding the poker are true throughout time, hence unchanging. McTaggart thought the only genuine change could be i f the sum t o t a l of facts at one time i s not the same as the sum t o t a l of facts at another time. McTaggart's theory of genuine change can e a s i l y be re s i s t e d , by denying his concept of genuine change. A genuine change has occurred when the poker i s hot at t 0 and cold at t ^ . Horwich 19 notes c o r r e c t l y that the unwelcome res u l t s that McTaggart derives from i s theory of genuine change show not that time is unreal but rather that the theory of genuine change i s implausible. 20 But Horwich is not correct to suppose that the tree model of r e a l i t y , such as McCall's model, e n t a i l s that the sum t o t a l of facts at one time i s not the same as the sum t o t a l of facts at another time. On some tree models, such as A r i s t o t l e ' s , claims Horwich (1988), p.19. Horwich (1988), p.20. Horwich (1988), p.25. about the past and present are determinately true or f a l s e , while claims about the future do not have a truth-value u n t i l the predicted events either occur or f a i l to occur. But on other tree models, such as McCall's model, the t o t a l i t y of facts can remain constant in time. Horwich's c r i t i c i s m s of the claim that the t o t a l i t y of facts must change over time are bold, penetrating, and deci s i v e . Horwich (1988) and Mellor (1981) have shown that McTaggart does not substantiate his theory of genuine change. Denial of the theory of genuine change i s s u f f i c i e n t to undermine McTaggart's paradox. As Horwich remarks, "...change is always v a r i a t i o n in one thing with respect to another, the t o t a l i t y of absolute facts 21 about those functional relations remaining forever constant." However, while McTaggart's argument can be rejected simply by denying the theory of genuine change, there i s much of value to be salvaged from the wreckage of the argument. McTaggart's paradox i s the s t a r t i n g point for a theory of time based on the d i s t i n c t i o n between the A-series relations and the B-series. 3. There i s genuine change in the world only i f events are located in the A-series. gc -» eA Horwich (1988), p.25. On McTaggart's theory of genuine change, there can be change in the t o t a l i t y of facts only when the now actualizes or f a i l s to actualize a predicted future event. This movement of the now i s described by the A-series r e l a t i o n s . There are two possible construals of the A-series terms of 'past', 'present', and 'future'. McTaggart must deny both p o s s i b i l i t i e s in order to show that the A-series i s contradictory. One p o s s i b i l i t y i s that the A-series terms are r e l a t i o n a l or monadic predicates. The other p o s s i b i l i t y i s that the A-series terms are simply indexicals. I agree with McTaggart that for there to be (genuine) change and a rea l A-series, then the A-series terms cannot be r e l a t i o n a l predicates nor monadic predicates, nor indexicals. I do not agree with Horwich that McTaggart has f a i l e d to show that time requires the existence of a moving now. Indeed, McTaggart's argument shows that time requires the compatible existence of the B-series with the A-series moving now. Horwich i s quite correct that McTaggart's reasoning i s f i n e , that the moving now i s self-contradictory, i f the A-series r e l a t i o n i s explicated in terms of the B-serles r e l a t i o n . But Horwich would not be correct to suppose that a l l theories of the moving now are doomed to f a i l u r e , but rather only those theories which require the B-serles r e l a t i o n to be previously defined. If another point of reference can be found to do the work required, 5o then the moving now would be restored to legitimacy in our ontology. But f i r s t , I w i l l show that the A-series terms can be neither r e l a t i o n a l predicates nor monadic predicates, nor indexicals. If the A-series terms were construed as r e l a t i o n a l predicates, then the sentence: (14) Event E i s now. could be construed as: (15) Event E i s at t^. However, t h i s construal has replaced the d i s t i n c t i v e l y temporal A-series term with a r e l a t i v e property whose exemplification i s determined by the B-series. This attempt to explicate the A-series has resulted in throwing out the baby with the bathwater, replacing the A-series temporal component with the B-series co-ordinate system. For the B-series co-ordinate system i s defined a f t e r an o r i g i n for the time scale i s selected, and that o r i g i n i s defined by the A-series term. Hence, (15) can only be unpacked to give a sentence exactly the same as (14). Interpreting the A-series by analyzing i t to be a monadic property of events only worsens the problem, for t h i s move i s to take nowness as a primitive property of events in the actual world. It would be impossible to give an account of why some events are actualized in a , and others not, i f nowness i s analyzed as a monadic property, since each i n s t a n t i a t i o n of novness i s o n t o l o g i c a l l y d i s t i n c t from each other. An A-series term l i k e 'now' appears to be an indexical expression. One manner in which time i s l i k e space i s that r e f e r r i n g words such as 'here' and 'now' function as indexical r e f e r r i n g expressions. Certainly, each use of 'now' could be replaced with i t s B-series counterpart, as from (14) to (15). The •now' in (14) i s a manner of r e f e r r i n g to the time which i s simultaneous with the utterance. The construal of (15) would not be acceptable to McTaggart, since i t results in the d i s t i n c t i v e l y temporal component of (14) being explicated by the B-series. McTaggart held that for there to be genuine change, temporal statements would have to be explicated with reference to the A-s e r i e s . So while the word 'now' can be used to simply refer to the time of utterance, the now must have more i n t r i n s i c structure than to be simply an i n d e x i c a l . The indexical analysis i s quite agreeable to non-tensers, because then there i s no other component i n A-series terms except t h e i r B-series counterparts. But the indexical analysis must be rejected by a tenser, because then the A-series terms remain not completely analyzable by t h e i r B-series counterpart, and hence there i s re a l work for the A-series to do after a l l , and i t i s not dispensible and replaceable by the B-series alone without loss of information. 5 2 Therefore, from 1. to 3., 4. Events are located in the B-series only i f they are located in the A-serles. ((eB -» T) & (T -• gc) & (gc -» eA) ) -» (eB -* eA) Premise 4. follows from 1. to 3.. As premise 2. has been rejected, premise 4. i s consequently deniable. 5. If events are located in the A-series, then each event acquires the absolute properties of being past, being now, and being future. eA -•. bPNP Premise 5. i s problematic. There are several d i s t i n c t senses of the word 'absolute*, and i t needs to be c l a r i f i e d which sense i s intended in t h i s premise. McTaggart had in mind his theory of genuine change, in which case, the having of the absolute property of being past requires that being past cannot be translated into 'was once future', since to do so would transform 'being past' into a r e l a t i v e property. So, being past can be analyzed as 'was once future' only i f the property 'was once future' applies to a second-order temporal fact about events. And of course, such secord-order a t t r i b u t i o n s are just as problematic as the o r i g i n a l f i r s t - o r d e r a t t r i b u t i o n , and so the analysis i s pushed to postulate third-order temporal a t t r i b u t i o n s to repair the remaining deficiency. For 'being past' and 'was future' are compatible only i f past, present, and future are d i s j o i n t regions of time. But that assumption i s contrary to the requirement that every event has the absolute q u a l i t i e s of being past, being present or being now, and being future. So the apparent contradiction within the A-series cannot be resolved by introducing second-order a t t r i b u t i o n s . McTaggart moved from the f a i l u r e to resolve the contradiction at each l e v e l of the analysis to the conclusion that the concept of the moving now was not possible, and hence that time was unreal. Given that we should reject McTaggart's theory of genuine change, we need not conclude that the concept of the moving now i s contradictory, but only that the A-series cannot be explicated in terms of i t s e l f nor can i t be explicated in terms of the B-series. Some other point of reference must be found to handle apparently contradictory properties i f a coherent analysis i s possible. That point of reference w i l l turn out to be possible worlds. A l l premise 5. r e a l l y informs us i s that the A-series cannot be explained in terms of i t s e l f . So premise 5. i s rejected, since there i s a possible worlds account which w i l l solve the problem. 6. There i s a contradiction in supposing that any event has any two of these three absolute properties. -bPN & -bPP & -bNF Premise 6. is correct i f construed with McTaggart's sense of absolute properties. Therefore, from 5. and 6., 7. The A-series cannot e x i s t . ((eA -» bPNF) & (-bPN & -bPF & -bNF)) -» -eA Line 7. does not follow from 5. and 6. Premise 5. must be rejected, since unless possible worlds are introduced, the exp l i c a t i o n of the A-series i s impossible. Even though premise 6. i s acceptable, l i n e 7. cannot be derived. Therefore, from 4. and 7., 8. Events are not ordered with respect to the B-series. -eB Line 8. does not follow, since 7. does not follow. Therefore, from 1. and 8., 9. Time does not e x i s t . -T Line 9 does not follow, since 8. does not follow. Hence, the conclusion of McTaggart's paradox can be denied, i f the inter p r e t a t i o n suggested in the analysis of premise 5. can be carr i e d out. That i s the task of a later section (4). 3.4 The conclusion of McTaggart's paradox McTaggart introduced the key d i s t i n c t i o n s and set up the problem of the passage of time. But McTaggart's analysis was led astray by an unsubstaniated and unhelpful theory of genuine change. The theory of genuine change, that the sum t o t a l of facts must change over time, confuses the epistemology of time with the metaphysics of time. However, the l o g i c McTaggart's paradox contained the suggestion of a modal solu t i o n . The remaining task i s to flesh out the nature of that modal solution, a task that i s taken up in the next section. 4.0 The Evolution of the Universe-Tree 4.1 McCall's model 22 In "Objective Time Flow" McCall presents a theory of temporal passage which i s objective, in the sense that the 'now' characterizes the universe independent of the existence of consciousness. Thus McCall's view i s in d i r e c t opposition to 23 Grunbaum's mind-dependence theory of becoming. The representation of objective temporal becoming i s accomplished through a model of the universe in which the past i s a single four-dimensional manifold, while the future consists of a branched structure of p h y s i c a l l y possible manifolds. Time flow i s r e l a t i v i s t i c in t h i s model in that time flow i s frame dependent (but not observer dependent). McCall believes that since s p a t i a l length i s frame dependent but nonetheless objective, so too time flow can be frame dependent and objective. For the most part McCall merely presents t h i s view of time without o f f e r i n g evidence that i t best describes our universe; however, i t i s claimed that the view of the future as a branched structure i s in accord with our ordinary language assumptions McCall, Storrs (1976) "Objective Time Flow" Philosophy of  S c i e n c e 43 p.337-362. 2 3Grunbaum, A. (1971) "The Meaning of Time" i n : Basic Issues  in the Philosophy of Time p.195-228. (which is supposed to be a point in i t s favour), and second, that since t h i s view allows the notion of physical law to be defined in terms of physical p o s s i b i l i t y , i t allows for the empirical p o s s i b i l i t y that the physical laws change over time, just as the physical p o s s i b i l i t i e s change over time. Such reasons are not s u f f i c i e n t l y compelling to accept the metaphysical truth of the future-directed branched structure theory. A metaphysical theory in complete agreement with the precepts of ordinary language i s a metaphysical theory infected with the presuppositions inherent i n the tensed structure of language. Hence ordinary language does not provide good independent reasons for accepting the metaphysical truth of the theory. Second, while i t is possible that cert a i n physical constants, assumed to be invariant over time, such as the g r a v i t a t i o n a l constant, do as a matter of fact change, instead of concluding that the notion of a physical law must be amended to admit of change over time, one could merely conclude that the s a t i s f a c t i o n of the g r a v i t a t i o n a l constant i s not a law, and 24 physical laws never change over time. I n f l a t i o n a r y cosmological models have been developed in which the expansion of the universe i s assumed to have been greater in the early stages of the universe than at present, but no amendment in the semantic notion of a law i s entailed by empirical assumptions about changes over time in the cosmological constant. C e r t a i n l y t h e r e appears t o be a deep s t r u c t u r a l d i f f e r e n c e between t h e p a s t and the f u t u r e i n need of some e x p l a n a t i o n ; however, Grunbaum argued t h a t t h i s d i f f e r e n c e i s due o n l y t o the manner i n which our c o n s c i o u s n e s s p e r c e i v e s e v e n t s . That i s , o n l y the B - s e r i e s of e a r l i e r and l a t e r r e l a t i o n s c h a r a c t e r i z e s mind-independent t i m e , w h i l e t h e A - s e r i e s ( i n c l u d i n g t h e 'now') i s mind-dependent. The c l a s s i c s t a t e m e n t of t h i s view i s by Weyl: The o b j e c t i v e w o r l d s i m p l y i s , i t does not happen. Only t o the gaze of my c o n s c i o u s n e s s , c r a w l i n g upward a l o n g t h e l i f e - l i n e of my body, does a s e c t i o n of t h i s w o r l d come t o l i f e as a f l e e t i n g image i n space which c o n t i n u o u s l y changes i n t i m e . The view M c C a l l d i r e c t l y opposes thus h o l d s t h a t f u t u r e e v e n t s a r e as much ' f i x e d * as p a s t ones, i n t h a t t h e r e i s no r e a l s t r u c t u r a l d i f f e r e n c e between p a s t and f u t u r e e v e n t s . W e l l , i t may be asked a t t h i s s t a g e , i n what way does time f l o w i n M c C a l l ' s model? I n the model, the a c t u a l w o r l d a i s embedded i n a modal phase space t o g e t h e r w i t h o t h e r p o s s i b l e w o r l d s , B i , $2r•••• • The a c t u a l w o r l d e v o l v e s t h r o u g h i t s c h a n g i n g c o n n e c t i o n w i t h v a r i o u s o t h e r w o r l d s . The c o n n e c t i o n i s t h a t of p h y s i c a l p o s s i b i l i t y . Each w o r l d i s a s e t of p h y s i c a l l y p o s s i b l e e v e n t s , where t h e n a t u r e o f the embedding r e l a t i o n i s g i v e n by the c o n n e c t i o n s (see f i g u r e 3 ) . The fundamental laws L of a d e t e r m i n e which w o r l d s a r e p h y s i c a l l y p o s s i b l e a t a t i m e . Weyl, H. (1949) P h i l o s o p h y of Mathematics and N a t u r a l S c i e n c e p. 116. See a l s o : E d d l n g t o n , A.S. (1920) Space, Time and G r a v i t a t i o n p. 51 . "51 figure 3: The universe-tree. The now, a property of i n s t a n t i a t i o n or a c t u a l i z a t i o n , selects one of the branches. As the now selects one of the branches, the other non-selected branches drop off and vanish from the tree (see figure 4). This select!on-and-vanishing process within the model i s a unique ontological s t r u c t u r a l d i s t i g u i s h i n g feature the past from the future. The now selects from the p h y s i c a l l y possible worlds at a branch node. The branches may have p r o b a b i l i t i e s associated with them; i f so, the now selects accordingly. Of course the t o t a l of the p r o b a b i l i t i e s at each node w i l l sum to 1. % » » 0 \ * / \ f \ / \/ 2 0 0 0 - > \ / N c y* \ / F U T U R E 1 9 0 0 — > * * < ^ J* < - P R E S E N T 1 8 0 0 > ""T*^  P A S T i f i g u r e 4: The now s e l e c t s a branch. 6 1 F i n a l l y / a new set of p h y s i c a l l y possible worlds are • * * • \ / / t 2000 ->\ y > / \ / F U T U R E 1 / < — P R E S E N T X P A S T 1900 —> 1800 > | ^ figure 5: The universe-tree evolves. determined, and the proceedure repeats (see figure 5) as the now must make a new s e l e c t i o n . At the time of the Big Bang, the entire universe-tree of a l l the ph y s i c a l l y possible futures were l a i d out. Then, as time passes, the tree i s continuously pruned, losing branches. The tree has been s i m p l i f i e d to show only binary branching. In r e a l i t y , there may be more branches, and a varying number of branches from node to node. The tree may be imagined to have ten branches at each node, a move which McCall adopts in order to si m p l i f y calculations of quantum p r o b a b i l i t i e s across the nodes. 6 2 . 4.1.1 The T r a n s f i n i t e Plus Problem Why adopt the possible worlds metaphysical framework to solve the problem of the passage of time? i n other words, how does the PWM account explicate the A-series? The possible worlds metaphysics embeds the actual world a in a continuum of possible worlds, in an analogous manner to the way that matter and energy are embedded in the actual world. Just as matter and energy exhibit temporal passage in a , temporal passage i s explained with reference to the possible worlds with which a i s connected. What is the nature of th i s embedding space and i t s interconnections? Thermodynamic behaviour of a system i s described by imaginging a 'phase space' within which a l l the possible states of the system are embedded, and then determining the p r o b a b i l i t y that any one of those possible states w i l l be r e a l i z e d . S i m i l a r l y , quantum behaviour of a system i s described by the Shrddinger equation with reference to an ensemble of systems, each of which i s a possible state of the system. The explanation of the asymmetric behaviour of the now w i l l also require a meta-space just as the explanation of asymmetric thermodynamic and asymmetric collapse of the wave function require a meta-level explanatory structure. The t r i c k i s to derive such an explanation that i s consistent with the symmetric aspect of time, the B-serie s language which maps so well onto the s t a t i c Minkowski world-view. 42 Possible worlds metaphysicians d i f f e r in their ontological committment. At one modal extreme i n the debate over the collapse of the wave function, one option i s termed the Many Worlds Theory. This theory, proponded by Wheeler and de Witt, holds that each of the i n f i n i t e l y many worlds, ensembles of systems, i s r e a l , de Witt (1970) required that the universe was: ...constantly s p l i t t i n g into a stupendous number of branches, a l l r e s u l t i n g from the measurementlike interactions between i t s myriads of components. Moreover, every quantum t r a n s i t i o n taking place on every star, in every galaxy, in every corner of the universe i s s p l i t t i n g our l o c a l world into myriads of copies of i t s e l f . At one modal extreme in the debate about time would be the view that a l l of the possible sets of events forming the past, present, and future would exi s t in some r e a l world or other, and our world i s one of these worlds. To some, such as van Fraassen (1970), b e l i e f in continuum-many or more r e a l worlds i s b e l i e f in 26 an i n f l a t i o n i s t metaphysics. Certainly explanations which take a smaller ontological loan are to be preferred; but the loan must be enough to do the explantory work, and the more explanatory work that can be done j u s t i f i e s the larger ontological loan. Occam's Razor does not say to choose the theory that requires the least number of e n t i t i e s , but rather, to choose the theory that requires the least number of e n t i t i e s that f i t s a l l the f a c t s . So i f temporal passage i s a fact, then we must accept the a l l the See, e.g., van Fraassen, Bas C. (1970) An Introduction to the PhUospphy of space and Time. e n t i t i e s required by the theory that most adequately explicates temporal passage. Levis (1986) c e r t a i n l y thinks that only absolute modal realism, the view that each possible world i s r e a l , i s the only view that f i t s a l l the f a c t s . The large ontological committment i s the A c h i l l e s ' heel of PWM. The Quinean objection "How many possible men are there i n the doorway?" os answered on the PWM account by: "Possibly one, possibly two..." This v i r t u a l l y l i m i t l e s s committment to r e a l p o s s i b i l i t i e s i s what I c a l l the ' t r a n s f i n i t e plus problem. 1 There are as many possible worlds as sets, namely, t r a n s f i n i t e l y many. But I hold that we need not postulate e n t i t i e s to supply truth-conditions for a l l counterfactual statements. McCall's model places a r e s t r i c t i o n upon the number of possible worlds i t requires. Not every possible world i s in the model; only those possible worlds which are ph y s i c a l l y accessible from the actual world a are r e a l . The worlds which are accessible are those worlds which are possible given the fundamental laws L of a , i . e . , p h y s i c a l l y possible, and therefore connected to a by the a c c e s s i b i l i t y or connectivity r e l a t i o n . Earman i s a substantival r e a l i s t about spacetime, and hence rejects a simple modal interpretation. Earman (1986) rejects the many worlds interpretation (MWI), and claims: What has r a r e l y been explored i s the implication for space-time structure of taking [MWI] se r i o u s l y . To make sure that the d i f f e r e n t branches cannot interact even in p r i n c i p l e they must be made to l i e on sheets of space-time that are t o p o l o g i c a l l y disconnected after measurement, implying a s p l i t t i n g of space-time... I do not balk at giving up the notion, held sacred u n t i l now, that space-time i s a Hausdorff manifold. But I do balk at tr y i n g to invent a causal mechanism by which a measurement of the spin of.an electron causes a global b i f u r c a t i o n of space-time. a l t e r n a t e f u t u r e s o n d i f f e r e n t s h e e t s o f s p a c e — t i m e bifurcation t h e p r e s e n t past global space—time figure 6: Bifurcation of space-time. Figure 6 i s an attempt to diagram the topological s p l i t t i n g of space-time that Earman is tal k i n g about. Such a s p l i t t i n g of space-time requires a weakening of the topological requirements upon the space, since in the usual Hausdorff space such disconections are not possible. But there i s a weaker space described by the T^ axiom within which such topological disconnections are possible. The cost, i f any, of weakening the topological requirements, and the implications of the T^ axiom, found i n Hughes (1989) The Structure and Interpretation of Quantum Mechanics. w i l l be taken up in a later section of t h i s thesis (4.2). Thus while Earman (1974) should be well respected amongst philosophers of time for the i n s i g h t f u l "An Attempt to Add a L i t t l e Direction to the Problem of the Direction of Time", nonetheless Earman i s not on the r i g h t track in his comment reproduced above. C l e a r l y while the truth of MWI seems to some, e.g., David Lewis, to be self-evident, to many other philosophers i t i s absurd. Ho extraordinary causal mechanism i s required of PWM models; as McCall has shown, when a quantum p a r t i c l e determines which of two possible states i t i s i n , then the quantum p a r t i c l e has selected one global space-time manifold from a choice of two possible manifolds or worlds. The quantum 'choice* i s the b i f u r c a t i o n within the space-time model. It c e r t a i n l y can be shown that McCall's model requires giving up the Hausdorff separation axiom for i t s topology (since the a l t e r n a t i v e i s to give up l o c a l Euclidean computability, which i s c l e a r l y untenable). But Earman does not balk at giving up Hausdorff, presumably because the theorems regarding l o c a l Euclidean computability can s t i l l be derived in the weaker T± space. Again, t h i s l i n e of argument w i l l covered in more d e t a i l i n a l a t e r section of t h i s thesis (4.2). Earman should not balk at postulating a global b i f u r c a t i o n of space-time. The modal l o g i c of PWM models i s s u f f i c i e n t l y well developed to handle a description of a b i f u r c a t i o n a l model. Perhaps Barman i s simply an a c t u a l i s t in his i n t u i t i o n s ; i f so, then i t w i l l take a separate argument to show that actualism can solve the problem of the passage of time. A l l I intend to show in t h i s thesis i s that there is a PWM model which solves the problem of the passage of time, namely McCall's model. The existence of one model of time flow i s s u f f i c i e n t to show that McTaggart's paradox i s unsound. 4.1.2 The Possible Worlds Metaphysics Solution The problem of the passage of time was reduced by McTaggart's paradox to the problem of expl i c a t i n g the A-series r e l a t i o n . The A-series temporal r e l a t i o n can be explicated, I argue, with the help of adopting the possible worlds metaphysics (PWM). The PWM has various d i f f e r e n t interpretations as given by various authors in the f i e l d : Lewis, Montague & Cresswell, Adams, Plantinga, Kaplan. But a l l PWM interpretations have in common the Kripkean o r i g i n that a possible world is a set which i s an ordered quadruple <P, W, R, K>. The set <P, W, R, K> i s such that P i s a sentence (or set of sentences) expressing a proposition that i s true in world W, e.g., 'The cat i s on the mat now'; W i s the name of a pa r t i c u l a r possible world, e.g., the actual world o; R i s the a c c e s s i b i l i t y r e l a t i o n between worlds that describes transworld identities/counterparts and K i s the background set of worlds such that WcK, e.g., K might be the set of a l l possible worlds, or the set of a l l p h y s i c a l l y possible worlds. The A-ser i e s r e l a t i o n i s explicated v i a <P, W, R, K> on McCall's model, since on that model, the A-series r e l a t i o n novness i s a world-r e l a t i v e property. Hence, on McCall's model, the A-series r e l a t i o n nowness n = <P, oj, R, K> where R i s an asymmetric r e l a t i o n from a to a set of p h y s i c a l l y possible worlds 8, and where K is the set of ph y s i c a l l y possible worlds r e l a t i v e to a at t where t i s the instant picked out by n. The f u l l Import of n = <P, a , R, K> cannot be appreciated without contrasting the interpretation of R and K required by McCall's model with the R and K selected by other interpretations. I consider the Lewisian or Many Worlds int e r p r e t a t i o n on the one extreme, and the a c t u a l i s t interpretation t y p i f i e d by Adams on the other extreme. The Lewisian R i s symmetric between a l l worlds in K, and K is the set of a l l l o g i c a l l y possible worlds. The Lewisian K i s the largest ontology possible, and i s therefore not parsimonious at a l l . A l l other things equal, Occam's Razor says that i t i s good to be parsimonious. Hence, i f there i s a model that i s parsimonious to any degree at a l l , then that model should be preferred, by Occam's Razor, to an ontology of the Lewisian K. The set of p h y s i c a l l y possible worlds i s smaller than the set of l o g i c a l l y possible worlds. Hence the McCall K i s smaller than the Lewisian K, and should be preferred on those grounds. The a c t u a l i s t R i s asymmetric since p r i o r i t y i s given to a over a l l other worlds and K, on the other hand, i s very parsimonious. Actualism i s the view that only the actual world a i s r e a l . The a c t u a l i s t model within the PWM i s that a l l other worlds are only possible worlds relative to a . C l e a r l y , the a c t u a l i s t K i s smaller than the McCall K, since there are ph y s i c a l l y possible events which are never instantiated in a . Proof that there i s one such p h y s i c a l l y possible event never instantiated in a can be given as follows. Suppose that in a, Caesar ate eggs for breakfast before crossing the Rubicon r i v e r (denote t h i s claim aCe). Then Caesar's not eating eggs for breakfast before crossing the Rubicon r i v e r is a p h y s i c a l l y possible event in a never instantiated in a (-aCe). According to actualisra, since, in a the claim 'oCe' i s true, then i t ' s not possibly r e a l , r e l a t i v e to a, that -oCe i s also true; i n other words, i t ' s not p h y s i c a l l y possible that Caesar didn't eat eggs before crossing the Rubicon. In t h i s respect, actualism can be seen to be another truncated conception of the needed ontology, since McCall's K encompasses both p o s s i b i l i t i e s (oCe & -aCe). Both aCe and -aCe are mapped onto exclusive future branches. Only what a c t u a l l y occurs i s p h y s i c a l l y possible on actualism, whereas McCall's ontology includes events/worlds which are not, in f a c t , instantiated in a at any time. The upshot of the above analysis i s that the PWM framework has allowed a f a i r comparison of d i f f e r e n t models of time. Furthermore, on t h i s PWM analysis, McCall's model explicates the A-series r e l a t i o n through an asymmetric R and where K>the set of p h y s i c a l l y possible worlds at where n » <P, a, R, K>. This PWM analysis thus demonstrates i t s usefulness to solving the problem of the passage of time for three reasons. One reason i s that the l o g i c of McTaggart's paradox suggests a modal solu t i o n . A second reason i s that McCall's model i s obviously a modal model of objective time flow. And a t h i r d reason for adopting the possible 7 i worlds metaphysics i s , as Bigelow notes, that we can adopt the t r i c k s of modal l o g i c to solve problems in the l o g i c of time. The problem of giving a l o g i c a l analysis of the properties of the A-se r i e s r e l a t i o n has been solved by adopting the t r i c k of modal lo g i c that contrary properties (such as (oCe & -aCe)) are relegated to d i f f e r e n t possible worlds (such that (aCe & p=-aCe)). The possible worlds metaphysics (PWM) gives the general structure of a l o g i c a l space within which d i f f e r e n t models of time can be compared. In McCall's model, that structure i s l i k e a tree, so that the connectedness r e l a t i o n describes the set K of p h y s i c a l l y possible events, and the evolution of that structure describes a and hence which propositions P are true in a . The general s e t t i n g of the PWM within which the tree structure i s embedded can be understood as analogous to a 'phase space'. A 'phase space' i s a conceptual e n t i t y used in thermodynamics, a space which represents a l l the possible entropic states for a closed system. S i m i l a r l y , the l o g i c a l space of the PWM given by set K describes the space of p o s s i b i l i t i e s . Exactly what entropic state the system in at a time may be given in the phase space by points on a l i n e . S i m i l a r l y , the exact location of the actual world a (as given by the possible worlds that are accessible from a) may be given by the r e l a t i o n R. To find a more general s e t t i n g for modal models of time, the concept of a Minkowski space i s expanded upon. A Minkowski space M i n k o w s k i s p a c e — t i m e one dimension of time /N world—line of a particle three spatial dimensions (two supressed) figure 7: Minkowski space. i s a four-dimensional manifold which can be resolved into three s p a t i a l dimensions and one dimension of time (known as hypersurfaces). A massive p a r t i c l e traces a four-dimensional path in the Minkowski space during i t s l i f e t i m e , where the path is the space-time h i s t o r y of that p a r t i c l e (see figure 7). To generalize upon Minkowski space, imagine the Minkowski space to be a sub-set of a s t i l l larger e n t i t y . This larger e n t i t y i s c a l l e d a supermanifold, and i t contains many Minkowskian space-time manifolds. Each manifold i s known, loosely, as a world. The worlds are connected in a structured array, so that some worlds are 'closer' to each other than others. The exact manner in which possible worlds are connected or not i s determined by the a c c e s i b i l i t y r e l a t i o n R; a l l and only the worlds i n the supermanlfold are in K. According to McCall's model, a evolves from K through the successive i n s t a n t i a t i o n of a physical p o s s i b i l i t y . Modal space'N possible worlds ordered by accessibility relation Minkowski space manifolds ordered by B—series relation figure 8: Modal-Minkowski space. But a p a r t i c l e within the supermanifold thus traces out a world-line, a history, just as i t did within a single Minkowski manifold. We can trace the path of a p a r t i c l e within the supermanifold, or within the Modal-Minkowski space as Yourgrau re f e r s to i t (see figure 8 ) . 4.2 The l o g i c of McCall's model I f , as holds on McCall's model, the world i s such that at each instant of time two or more alte r n a t i v e future courses of events are open, then how many p h y s i c a l l y possible futures are there? McCall's modal model has the resources to answer t h i s question p r e c i s e l y , to determine the exact t r a n s f i n i t e size of K. The fact that the mathematics of the model are such that i t can provide a determinate proof of the number of worlds required i s a point in favour of the model. MWI PWM models would not have such resources. The answer to t h i s question helps to show that McCall's model i s determinate, and hence l o g i c a l l y possible. Suppose that each possible future may be represented as a branch of a continuously branching tree. Each branch i s composed of a continuum of binary branch-points. McCall argues that for every branch of the tree there corresponds a function from points on the time-axis measuring the height of the tree to the set {1,2} and vice versa. Given any point x on any branch, assign 1 i f the branch goes l e f t , 2 i f r i g h t . Hence every branch uniquely determines a function £. And conversely, every function f uniquely determines a branch structure, in that given any point y on the time-axis, we can determine the corresponding branch point x: For each x on the same l e v e l as y, consider a l l the predecessors of x, each of which is a left-branch point of a right-branch-point, and construct the function gx corresponding to them, x corresponds to y i f and only 1£ f=gx for predecessors of y. Hence the number of branches equals the number of functions. But the number of functions from the time-axis continuum into {1,2} i s 2 . Hence there i s something - the number of possible futures - which has physical s i g n i f i c a n c e and which has a c a r d i n a l i t y higher than that of the continuum. This r e s u l t holds because McCall assumes that the branch structure of the world i s such that e a r l i e r events are proper subtrees of l a t e r events; that the branches " f a l l o f f the tree as the now moves up the time-axis. This view e n t a i l s that the outcome of events i n t h i s world i s not e n t i r e l y fixed by events in t h i s world, but by events across a l l p h y s i c a l l y possible worlds. The exact manner in which the property of the present i s distinguished in McCall's model i s given by i t s unique topology. That topology i s the weaker T± axiom only, rather than the topology of the standard Minkowski models, which are a l l T 2 or more generally known as H for Hausdorff. How does t h i s l i n e go? By d e f i n i t i o n , a topological space X i s a T^-space i f and only i f whenever x and y are d i s t i n c t points in X, there i s a neighbourhood N of each not containing the other. Obviously, every T± space i s also a TQ space. It can also be e a s i l y shown that there i s a distinguishing property of T± spaces not had in T 0 space only. If X i s a space, then each one-point set in X * McCall, Storrs. (1970) "The C a r d i n a l i t y of Possible Futures" [abstract! The Journal of Symbolic Logic 35 p. 363. i s c l o s e d , which i s e q u i v a l e n t w i t h t h e s t atement t h a t each s u b s e t of X i s the i n t e r s e c t i o n o f the open s e t s c o n t a i n i n g i t . The more r e s t r i c t e d H a u s d o r f f s p a c e , a r e f i n e m e n t o£ t h e T 2 s p a c e , has t h e p r o p e r t i e s of T l 7 t h a t o n e - p o i n t s e t s a r e c l o s e d , but a l s o the p r o p e r t y t h a t each net or f i l t e r on X converges t o no more th a n one p o i n t , and no l e s s t h a n one p o i n t e i t h e r . By d e f i n i t i o n , a space X i s a T 2 space i f and o n l y i f whenever x and y a r e d i s t i n c t p o i n t s of X, t h e r e a r e d i s j o i n t open s e t s U and V i n X w i t h x c U and y c V. Another i n t e r e s t i n g f e a t u r e of H a u s d o r f f . s p a c e s (as the d e f i n i t i o n above i s known as the H a u s d o r f f s e p a r a t i o n axiom) i s t h a t t h e d i a g o n a l i s c l o s e d i n a H s p a c e ; i n symbols, the d i a g A « { ( x , x ) | x c X} i s c l o s e d i n X x X. But i n t e r e s t i n g f e a t u r e s a s i d e , i f the p h y s i c a l laws o f the a c t u a l w o r l d a r e i r r e d u c i b l y s t o c h a s t i c , t h e n f o r each here-now i n s p a c e - t i m e , t h e r e i s a s e t of p h y s i c a l p o s s i b i l i t i e s and a s e l e c t i o n from t h a t s e t , and the s e l e c t i o n g i v e n by an a c t u a l i z a t i o n f u n c t i o n t h a t goes from a w o r l d - a t - a - t i m e t o a p h y s i c a l l y p o s s i b l e f u t u r e w o r l d . so the d e c i s i o n on the p a r t o f a s c i e n t i f i c r e a l i s t t o adopt and a c c e p t i n t o h i s or her o n t o l o g y t e m p o r a l passage can be i n t e r p r e t e d as a d e c i s i o n t o weaken the t o p o l o g i c a l r e q u i r e m e n t of t h e model s p a c e , whether or h o t , i n o t h e r words, t h e For a c l e a r and s i m p l e p r o o f , s e e : W i l l a r d . (1970) G e n e r a l TPPPlogy. p. 86. a c t u a l i z a t i o n function is one-one or one-many, whether or not the set of physical p o s s i b i l i t i e s i s the i d e n t i t y class with t h i s world a, or whether there i s a further physical feature of our world that distinguishes being present in a. What i s at stake in th i s decision? McCall's model requires us to weaken the topological requirement from H to T^, to allow b i f u r c a t i o n and thus a branch-structure. The requirement that i s weakened i s a l o c a l requirement on the geometric structure from one point to another 'nearby' point in the manifold. The manifold, the set of points with geometric structure, in turn breaks down into four dimensions, three of space and one of time. The weakening of the topology allows the time dimension to refer to a non-identical set of manifolds-at-a-time, manifolds which have a global time function and manifold i n which time i s linear (non-circular). Why weaken the requirement? McCall advances many reasons for his model, a model which e n t a i l s the weakening of the topology from H to T^, reasons ranging from the decision theoretic to the theory of quantum mechanics to astrophysics and cosmology. It would be a compendious task beyond the scope of t h i s paper to evaluate McCall's model in each of these sectors. Suffice i t to note that i f the metaphysics of McCall's model holds, then i t follows that the trickle-down e f f e c t s of that metaphysics w i l l have profound consequences on the nature of causal interaction, 19 which i n t u r n w i l l have a f f e c t the n a t u r e s of the m a t t e r - e n e r g y e n t i t i e s i n t h e w o r l d such as you and I and how we go about making d e c i s i o n s and what d e c i s i o n s a r e p h y s i c a l l y p o s s i b l e and what d e c i s i o n s a r e i n f a c t a c t u a l i z e d . Another way t o u n d e r s t a n d the t r i c k l e - d o w n e f f e c t of the m e t a p h y s i c s of M c C a l l ' s model i s t h a t the t r i c k l e - d o w n f o l l o w s from the f a c t t h a t the dynamic motion of the now i s asymmetric i n t i m e . I t i s t h i s asymmetry t h a t d i s t i n g u i s h e s the motion of the now i n t i m e . For two r e a s o n s t o p o l o g i s t s have s t u d i e d i n g r e a t d e t a i l t h e theorems p r o v a b l e on the H a u s d o r f f l e v e l , and c o n s e q u e n t l y not s t u d i e d the n a t u r e of theorems p r o v a b l e i n weaker spaces such as the or the TQ space. One r e a s o n i s t h a t H a u s d o r f f spaces have a ' n i c e r ' p r o p e r t y t h a n i n s p a c e s . The u n d e r l y i n g r e a s o n f o r t h i s n i c e n e s s or g r e a t e r a e s t h e t i c v a l u e i s t h e e x i s t e n c e of unique l i m i t s i n H a u s d o r f f s p a c e s , which has p l e a s a n t consequences ( f o r example, c o n t i n u o u s f u n c t i o n s w i t h H a u s d o r f f range a r e d e t e r m i n e d by t h e i r v a l u e s on a dense s e t , or t h a t t h e d i a g o n a l A = { ( x , x ) | x « X} i s c l o s e d i n [ t h e c o n t i n u o u s c l o s e d 30 image] X x X.) . O f c o u r s e , t h e c o n t i n u o u s c l o s e d image of a H space need not be an H s p a c e , but i t may. W i l l a r d . (1970) G e n e r a l Topology, p.87. 7 ? The other reason why topologists have studied H spaces more than T 2 i s that the standard Minkovskian models that p h y s i c i s t s were working on were H spaces. The underlying reason for t h i s neglect i s that workers benefit from looking to the needs of Mother Nature. But the p h y s i c i s t s were working within a too-limited conceptual framework. The Minkowksi model i s forced to deny the existence of a property that distinguishes the uniqueness of the present events, that they are present in a . Some models go so far as to deny the existence of a cosmic time function, contrary to the observed evidence of very-large-scale structures of matter-energy in the universe (e.g., the "Galactic Wall") which suggests that there i s a unique t r a j e c t o r y of the mean motion of matter in space-time, and a proper time along that privledged t r a j e c t o r y . McCall's model, by contrast, takes the proper time along that t r a j e c t o r y to demarkate the evolution of the universe-tree for the cosmos. The universe-tree, in McCall's model, i s a branched-connected structure of manifolds, or worlds. The tree loses branches, where the branches represent a p r o b a b i l i s t i c , but not necessarily deterministic, set of physical p o s s i b i l i t i e s at-a-time. The branches which are non-instantiated vanish, and the now selects the instantiated branch aft e r the node of p o s s i b i l i t i e s at-that-time. McCall's model i s committed to a unique topological feature of the branching structure that distinguishes the branches from the trunk, without that distinguishing feature, the M l n k o v s k l a n proponent can agree with McCall's premises and deny his conclusion by arguing that the i d e n t i t y r e l a t i o n t r i v i a l i z e s the supposed unique determination of a s p e c i f i c set of worlds in the future d i r e c t i o n from the world in the past d i r e c t i o n of the trunk. That di s t i n g u i s h i n g feature of McCall's model i s the fact that i f the branching structure i s upper cut, then the space i s T^ and not Hausdorff. If the branching were lower cut then the space could be Hausdorff and not Tj_ # although i t could also be Tj_. In a lower cut space, the tree and the branches are indistinguishable from branches and branches. In an uppercut space, the union of the trunk and the branch i s the same as a lower cut space, but the union of any two branches i s either disconnected or i t s e l f an uppercut space. The dist i n g u i s h i n g feature i s the existence of doubled paths under Tl-upper cut but not under H-lower cut, and hence not under T^-lower cut. If there are undoubled paths, then the image of the fuction i s also a function, whereas i f there are doubled paths, then the image of the function i s not a function. The f a i l u r e of the £± function to ex i s t when doubled paths ex i s t distinguishes the upper cut space . S I from a lover cut space. This asymmetry distinguishes the trunk 31 from the branches. To the further s k e p t i c a l charge, on behalf of the Minkovskian proponent, that l o c a l Euclidean computability vould f a i l on the T± axiom only, vithout the Hausdorff axiom, i t can quickly be shovn that since each one-point set in X i s closed under Tj_, there i s s u f f i c i e n t l o g i c a l structure for a metric. The Hausdorff axiom has a unique l i m i t s property; on the T^ axiom, functions don't necessarily converge to a unique l i m i t . The function vhich describes the motion of the nov, i n p a r t i c u l a r , does not converge to a unique l i m i t , but instead continuously selects from a set of r e a l p o s s i b i l i a , the p h y s i c a l l y possible vorlds. Nov the cat i s f i n a l l y out of the bag. I t must be shovn that the possible vorlds d i s t i n c t in order that the possible vorlds be r e a l p o s s i b i l i a . Othervise, Quinean objections about the number of possible fat men in the doorvay vould apply and the PWM program undermined, as there vould be no determinate ontology implied by r e a l p o s s i b l i a . McCall can shov that his model i s committed to 2 C vorlds, 2 to the pover of the continuum. The radicalness of McCall's For more information, see: McCall, S. "Choice Trees." proposal i s that one must be a r e a l i s t about t r a n s f i n i t e l y many existents, rather than merely i n f i n i t e l y many on the standard non-tenser Minkowski view. But McCall's model i s not nonsensical or out of the ballpark by any means, despite the apparent size of the ontological committment. After a l l , the distinctness requirement mentioned i n the above paragraph has been met by McCall's model. McCall holds that time's arrow points the way i t does as a matter of semantic necessity, not due to empirical facts about thermodynamic processes. If the world were completely deterministic, then a state description of the world, together with the laws of nature, would e n t a i l every other state of the world at any time. If the world i s indeterministic, then the complete state description of the world i s d i f f e r e n t at d i f f e r e n t times. McCall's theory e n t a i l s that the state-description at a la t e r time i s a proper subtree of the universe at an e a r l i e r time. On McCall's model, the l o c a l Euclidean computability of the world holds. It i s a global requirement that has been weakened, while the l o c a l considerations remain the same. I submit that where a global requirement can be weakened at no cost to l o c a l considerations, then the global requirement was superfluous i n the f i r s t place. Local Euclidean computability (LEC) must be preserved; but ve need not be generous in our global a t t r i b u t i o n s . LEC holds on T± as every point set i s closed. It's just that functions behave more 'reasonably' in a Hausdorff space and hence they have been studied more (convergent functions coverge to a unique point). The t r i v i a l l e v e l of topological structure, Tn, makes a topology on X look not much d i f f e r e n t from a single point. It would be much nicer i f some set-theoretic structure of X were refl e c t e d in i t s topology. If so, then we can assimilate the t r i c k s of set theory into the topological analysis. What i s needed, apparently, i s the requirement that the topology of X contain enough open sets to di s t i n g u i s h between the points of X in some way. Increasing the levels of geometric structure from Tg to TJL and then to T 2 gives more and more richness of set-theoretic character to the topological space X. But how much set-theoretic structure on X i s enough? What do we lose in the weakening of the topology from T 2 to T ^ In a nutshell, we lose a more global r e s t r i c t i o n on the T 2 space which i s not true of a merely T^ space; the global r e s t r i c t i o n of the Hausdorff axiom. But the loss of the global r e s t r i c t i o n of H does not necessitate a lack of l o c a l l y Euclidean computability. As indicated above, the global r e s t r i c t i o n i s l o c a l l y superfluous. 3<i Hence McCall's model s a c r i f i c e s an unnecessary global r e s t r i c t i o n of the possible topology of X. That s a c r i f i c e e n t a i l s boosting the ontology of the model, as the Ideology s p e c i f i e s a greater than continuum number of p h y s i c a l l y r e a l branch nodes. So the s a c r i f i c e , such as i t i s , i s worth i t . The benefit, the unique evolution of the universe-tree in time, demarkates the nature of the privleged property of being present, or now. Therefore, the passage of time i s an i n s t r u c t i v e case showing the i n t e r - r e l a t i o n between mind and mind-independent nature, and how the model in the mind evolves to match, correspond to, or in the jargon, to be an isomorphic diffeomorphism of, the r e l a t i o n s in the objective and mind-independent world. 4.3 The soundness of McCall's model McCall has argued that his model can derive some support from the observed quantum indeterminism in our world. Certainly the idea of r e a l a l t e r n a t i v e p h y s i c a l l y possible futures derives some support from the indeterministic nature of quantum events. On t h i s point, Reichenbach argued that temporal becoming, or in my terms the a c t u a l i z a t i o n of nowness, should be understood in terms of 'becoming determined', because i t i s only now that the quantum wave function collapses and achieves a fixed, d e f i n i t e value. This causal feature defines a s t r u c t u r a l difference between the past and the future, since past events are s p e c i f i c and i d e n t i f i a b l e , whereas future events are i n d e f i n i t e . On Reichenbach's analysis, the present divides the future from the past and i t i s the moment at which p r o b a b i l i t i e s change into a c t u a l i t y . McCall's claim that his view escapes the problems of Reichenbach's view betrays a weakness of his tree model. McCall notes that Grunbaum has shown that Reichenbach confused the epistemological discovery of the actual event-properties of the once-future event out of the wider matrix of possible properties allowed by the quantum-mechanical p r o b a b i l i t i e s with the m e t a p h y s i c a l or e x i s t e n t i a l becoming of such a p r o p e r t y . The becoming d e t e r m i n e d of an e v e n t i s a l v a y s now, r e l a t i v e t o t h a t e v e n t , and hence i t appears t h a t Reichenbach has f a i l e d t o i d e n t i f y a unique p r o p e r t y t h a t i n t r i n s i c a l l y c h a r a c t e r i z e s t h e now and s i n g l e s out a unique p r e s e n t f o r each moment. Wh i l e i n d e e d M c C a l l i s c o r r e c t t h a t i t i s not a r b i t r a r y what f u t u r e p o s s i b i l i t i e s a r e open a t a t i m e , the f a c t t h a t we have no e p i s t e m o l o g i c a l a c c e s s t o t h e f u t u r e i s does not e n t a i l the m e t a p h y s i c a l c o n c l u s i o n t h a t f u t u r e i s i n d e f i n i t e . Some event or o t h e r w i l l c o n s t i t u t e the b e i n g p r e s e n t of t h e y e a r 2,000 AD, and some event or o t h e r w i l l c o n s t i t u t e t h e b e i n g p r e s e n t of the year 20,000 AD. Hence f o r a l l o b s e r v e r s t h e r e l a t i v e p a s t i s f i x e d , and s i n c e one o b s e r v e r ' s r e l a t i v e p a s t i s a n o t h e r o b s e r v e r ' s r e l a t i v e f u t u r e , t h e r e l a t i v e f u t u r e i s as f i x e d as t h e r e l a t i v e p a s t . While t h i s argument does not h o l d f o r the a b s o l u t e f u t u r e b e i n g n e c e s s a r i l y f i x e d i f t h e a b s o l u t e p a s t i s f i x e d , assuming t h a t an a b s o l u t e p a s t and a b s o l u t e f u t u r e can be d e f i n e d f o r a l l o b s e r v e r s , t h i s f a i l u r e does not d e t r a c t from the m e t a p h y s i c a l p o i n t . On M c C a l l ' s model, the f u t u r e of a i s n e c e s s a r i l y i n d e f i n i t e , and so no e p i s t e m o l o g i c a l c o n c l u s i o n s can be d e r i v e d from p r e s e n t e v i d e n c e r e g a r d i n g which f u t u r e p h y s i c a l p o s s i b i l i t y w i l l be i n s t a n t i a t e d . Even i f an event E l i e s i n the a b s o l u t e See: R e i c h e n b a c h , H. (1956) The D i r e c t i o n of Tlme r Grunbaum, A. (1974) P h i l o s o p h i c a l Problems of Space and Time, p. 324., and M c C a l l , S. (1976) " O b j e c t i v e Time Flow" P h i l o s o p h y o f S c i e n c e . future of an observer 0 at tg, in McCall's model, E i s i n d e f i n i t e u n t i l a c tualized. The fact that i t is only after E i s in the absolute past of 0 that 0 can recognize that E was i n the absolute future of 0 at tg Is a fact that can be inferred by a b s t r a c t l y reconstructing the universe-tree. Another consideration about McCall's model must be noted. McCall allows that the progressive f a l l i n g off of future branches does not take time, but 'generates' time. But 'generates' i s a metaphor that without further explanation i s incoherent. T-theor i s t s have often thought that an e s s e n t i a l part of the moving now was to see the motion, the directed nature of time flow, as a process. A process i s a continuous causal chain, with phase-changes. McCall's model represents the flowing aspect of the A-series by the evolutionary process of pruning the universe-tree. McCall states that the reason there exists a method of transforming a l a t e r universe-picture into an e a r l i e r one, within the same frame of reference, but not an e a r l i e r one into a l a t e r one, i s that e a r l i e r branches are preserved and recoverable while there i s no distinguished future branch. However, one future branch w i l l become so distinguished. Hence that set of events can be seen re t r o s p e c t i v e l y to have been the ones that were going to be distinguished. That set of events i s distinguished by having the property of being the set that would be distinguished by retrospective analysis in the future. Light Cones Tip Over in a Black Hole flat space-^Tme curved space—time figure 9 : Light Cones in a Black Hole There are s t i l l technical d e t a i l s that need to be worked out in McCall's model. For instance, in the curved space-time region of a black hole, there would be an additional complexity in the branching-structure of the dynamic model of time (see figure 9 ) . For an e n t i t y in a region of t i g h t l y curved spacetime, the motion of the now ceases because the curvature of that region e n t a i l s that no future-pointing light-cones e x i s t , hence no future-pointing branches e x i s t . After the black hole evaporates, in accordance with the Hawking-Ellis theorem, the motion of the now w i l l resume with respect to the remainer of the space-time. Considering the supermanifold containing a in which t h i s black hole e x i s t s , the motion of the now up the universe-tree w i l l have a gap or di s c o n t i n u i t y which represents the duration, r e l a t i v e to the r e s t of the space-time, t h a t the now spent i n the black h o l e . That gap w i l l appear to be i n f i n i t e , s i n c e a n a l o g o u s l y , an o b j e c t e n t e r i n g w i t h i n the S w a r t z c h i l d r a d i u s of a black hole appears to receed from view f o r e v e r . The presence of gaps complicates the p i c t u r e of the b i n a r y branching t r e e . Furthermore, i t must be noted t h a t M c C a l l ' s model must e x p l i c i t l y i n t r o d u c e the assumption t h a t there are no c l o s e d t i m e l i k e c u r v e s . Hawking has shown t h a t t h i s assumption i s e q u i v a l e n t to the assumption t h a t there i s a cosmic time 33 f u n c t i o n . I f there i s a cosmic time f u n c t i o n , then t h a t f u n c t i o n can measure the motion of the now f o r the e n t i r e u n i v e r s e . I f c l o s e d t i m e l i k e curves were i n s t a n t i a t e d i n our u n i v e r s e , then f o r an e n t i t y t r a v e l l i n g along t h a t i n e r t i a l t r a j e c t o r y , t h e r e would be continuous 'lopping o f f of the f u t u r e - d i r e c t e d branches as the now moves up the w o r l d - l i n e , yet the e n t i t y would r e t u r n to i t s own p a s t , where the branches have a l r e a d y been 'lopped o f f . Hence no such curves e x i s t on the dynamic t r e e model. See: Hawking, S. (1973) The L a r q e - S c a l e S t r u c t u r e of  Space-Time. 4.4 The conclusion of McCall's model McCall's model of a binary branching tree can specify p r e c i s e l y the number of ph y s i c a l l y possible vorlds, 2 to the pover of the continuum. Despite the fact that the number i s greater than aleph n u l l , the countable i n f i n i t y , the model i s nevertheless l o g i c a l l y v e l l - d e f i n e d , hence l o g i c a l l y possible. Furthermore, although the model requires a weakening of the global topological requirements, i t has been shovn that the global topology can be veakened and yet l o c a l Euclidean computability i s preserved. In other vords, even in the branching model, the vorld vould continue to appear to conform to a Euclidean space i n the small, v h i l e the global topology could vary from point to point ( v i t h i n some constraints). McCall's model su c c e s s f u l l y incorporates time f l o v through the fact that the now picks out a unique tree-structure. The tree evolves in time, so that the past i s the trunk, fixed and determined; and the future i s a set of branches, open and not-yet-determined. The tree-structure can di s t i n g u i s h betveen the past and the future, and hence provides a unique manner of characterizing the motion of the now. On the model, nowness i s a quadruple r e l a t i o n <P, a, R, K> vhere R i s asymmetric and vhere a, K evolve. McCall's model solves the problem of the passage of time, and avoids the i n f i n i t e regress of temporal a t t r i b u t i o n s , by positing time f l o v v i t h i n a t r a n s f i n i t e set of manifolds or vorlds. The passage of time i s explicated as a world-relative r e l a t i o n , a r e l a t i o n between a and the set of p h y s i c a l l y possible worlds at that time. The ontological committment of the model i s j u s t i f i e d as a moderate position between actualism and the complete Many Worlds Interpretation. 5.0 Conclusion The conclusion of McTaggart's paradox i s that there are no l o g i c a l l y possible models. The existence of McCall's dynamic universe-tree model contradicts McTaggart's claim, since McCall's model i s l o g i c a l l y possible. Since McTaggart's paradox i s a v a l i d argument, at least one of i t s premises must be denied. It i s clear that the premise that needs to be denied Is the premise that the A-series leads to a contradiction. McTaggart's metaphysics was fuzzy on th i s point. McCall's modal analysis of the flow of time replaces t h i s fuzziness with a d e f i n i t e function, the evolution of the universe-tree. Thus McCall's modal sol u t i o n i s a counterexample to the conclusion of McTaggart's paradox, a solut i o n which the hidden modality of the l o g i c of McTaggart's paradox permits. The existence of McCall's model e n t a i l s that the conclusion of McTaggart's paradox cannot be accepted. The passage of time i s an i n t r i n s i c part of the nature of time. The passage of time i s explained by the asymmetric evolution of the universe-tree. The passage of time and the motion of the 'now* i s thus i n t r i s i c a l l y related to something l i t e r a l l y out of th i s actual world, namely, the nonselected branches of the p h y s i c a l l y possible worlds. 6.0 References Adams, R. (1979) "Primitive Thisness and Primitive Identity" Journal of Philosophy. Vol. 76 pp. 5-26. Angel, Roger B. (1980) R e l a t i v i t y ; The Theory and i t s Philosophy. Pergamon of Canada, Ontario Canada. Bigelow, John. (1989) "A Modal Theory of Time" (forthcoming r e t i t l e d as "Worlds Enough for Time") Broad, CD. (1938) An Examination of McTaggart's Philosophy, Vol. II, Cambridge University Press, Cambridge UK. Part 1, Ch. 35. B u t t e r f i e l d , Jeremy. (1984) "Prior's Conception of Time" Aristotelean Society Proceedings, Vol. LXXXIV. pp. 193-209. de Witt, B.S. (1970) "Quantum Mechanics and Reality" Physics  Today. September, pp. 30-35. & Graham, N. (1973) The Many-Worlds Interpretation of Quantum Mechanics. Princeton University Press, Princeton USA. Earman, J. (1967) " I r r e v e r s i b i l i t y and Temporal Asymmetry" Journal of Philosophy f Vol. 64. pp. 543-549. • (1969) "The Anisotropy of Time" Australasian Journal of Philosophy, Vol. 47. pp. 273-295. (1970) "Space-Time, Or How To Solve Philosophical Problems and Dissolve Philosophical Muddles Without Really Trying" Journal of Philosophy. Vol. LXVII. No. 9. (May 7). pp. 259-277. (1974) "An Attempt to Add a L i t t l e Direction to 'The Problem of the Direction of Time.'" philosophy of Science, Vol. 41. pp. 15-47. (1989) World Enough and Space-Time. MIT Press, Mass. USA. Friedman, Michael. (1983) Foundations of Space-Time Theories. Princeton University Press, Princeton USA. 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(1976) "Objective Time Flov" Philosophy of Science, Vol. 43, pp.337-362. (1984) "A Dynamic Model of Temporal Becoming" Analysis, Vol. 44, pp. 173-176. (forthcoming) "Choice Trees" (forthcoming) "Interpreting Quantum Mechanics v i a Quantum P r o b a b i l i t i e s " McTaggart, J.M.E. (1908) "The Unreality of Time" Mind. Vol. 18. pp. 457-484. (1927) The Nature of Existence. Cambridge UK, Vol. I I . Ch. 33. (1934) Philosophical Studies, London. Ch. 5. Malament, David. (1985) "Minimal Acceleration Reguirements for Time Travel in Godel Space-time" Journal of Mathematical  Physics. Vol. 26. No. 4. A p r i l , pp. 774-777. Melberg, Henry. (1980) Time, Causality, and the Quantum Theory. D. Reidel Publishing Co. Boston USA. Vol. 2. p. 154. Mellor, D.H. (1981) Real Time, Cambridge University Press, Cambridge UK. M i l l e r , Arthur I. (1981) Albert Einstein's Special Theory of  R e l a t i v i t y , Addison-Wesley Publishing Co., Mass. USA. Oaklander, L. Nathan. (1987) "McTaggart's Paradox and the I n f i n i t e Regress of Temporal At t r i b u t i o n s : A Reply to Smith." The Southern Journal of Philosophy, Vol. XXV, No. 3 pp. 425-431. P r i o r , A.N. (1957) Time and Modality. Oxford UK. (1967) Past. Present and Future, Oxford UK. Putnam, Hi l a r y . (1967) "Time and Physical Geometry" Journal of  Philosophy. Vol. 64. p. 240-247. Reichenbach, H. (1956) The Direction of Time. Univ. of C a l i f o r n i Press, Berkeley C a l i f . USA. Salmon, W.C. (1980) Space. Time and Motion. Univ. of Minnesota Press, Minneapolis USA. % . (1984) S c i e n t i f i c Explanation and the Causal Structure of the World. Princeton University Press, Princeton USA. Schlesinger, G. (1980) Aspects of Time. Hackett Publishing Co., Indianapolis USA. S e l l a r s , W. (1962) "Time and the World Order" Minnesota Studies in Philosophy of Science. Vol. I I I . University of Minnesota Press, Minn. USA. Sklar, L. (1974) Space, Time, and Spacetime. 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