Travels in Four Dimensions: Concluding Thoughts
LePoidevin (Robin)
Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Conclusion
Paper - Abstract

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  1. What are space and time? Are they real, or do they exist only in the mind? And if they exist without the mind, are they objects in their own right? Or are they collections of relations between things and events? What are their features, and what explains why they have these features? Could they have had different ones? Are they, for example, infinite, or only finite? If finite, do they have boundaries? Are they infinitely divisible, or are they composed of ‘atoms'? How does time differ from space? Does it really pass? Is the future real? And what accounts for time's direction?
  2. These are some of the questions we have tried to address in the preceding pages, and we have done so, for the most part, through a study of the difficulties and paradoxes that our ordinary views of time and space throw up. This has really just been the start of an investigation, rather than an exhaustive enquiry, and I do not propose to offer a set of definite answers to the questions just posed. Instead, this last section of the book summarizes1 the preceding discussion, attempts to draw some of the threads together, and raises some further questions, questions to do with the human significance of our philosophical views.
  3. Space and time have an unrivalled capacity to generate paradox. It is hardly surprising, in the light of this, that many of the philosophers who have written about them have concluded that they are unreal. Parmenides, Zeno (arguably), Kant, and F. H. Bradley denied the reality both of space and of time. Even Aristotle, who does not dispute their existence, acknowledges that they raise so many difficulties that it is quite reasonable to suppose that they do not exist. St Augustine concludes his long and searching exploration of time with the judgement that time is in the mind. McTaggart argued that there was a contradiction in the very notion of time. The consequences of denying the existence of space and time, at least outside the mind, are, however, significant. So much of our conception of the world is bound up with its apparent spatial and temporal features that to deny the reality of those features would imply that in investigating the nature of the world we are simply investigating the contents of our own minds. Kant embraces this consequence, arguing that only this explains how we come to have knowledge that is both a priori (i.e. such that recognition of its truth does not depend on particular experiences) and synthetic (i.e. not just a matter of definition).
  4. One of the lessons of our inquiry is that the reality of space and time need not be an all-or-nothing matter. There is a considerable variety of features we ordinarily ascribe to space and time, and it is always possible to hold that one feature is unreal but another not (provided the features are not logically related). For example, we may2 believe in the objectivity of spatial and temporal relations without also believing in the objectivity of their metric. It may be true, independently of any mind, that event C occurred after event B, which occurred after event A, without there being any fact of the matter as to whether or not the interval between A and B was equal to that between B and C. According to conventionalism over metric, it is only according to a particular system of measurement, and not absolutely, that one interval is as long as3 another. Whether this system is the correct one is, for the conventionalist, an improper question (although one could legitimately compare the usefulness of different systems of measurement). But, as we saw in Chapter 1, conventionalism over metric has serious consequences for our view of physical law. How, for example, can we regard the laws of motion as being objectively true if the facts about metric implied in the concepts of velocity and acceleration are merely conventional?
  5. Assuming that space and time are not merely our projections onto the world, what are they? The first place to look will be among the objects of our direct experience, for a failure to perceive space and time contributes to the sense of their unreality. Indisputably, we perceive change. Equally indisputably, we perceive objects as being certain distances from other objects. So perhaps the best strategy for keeping them out there in the world, rather than locked up in the mind, is to identify time with change, and space with the collection of all spatial relations between objects. These are (versions of) relationism about time, and relationism about space, respectively. We seem, however, to be able to conceive4 of the idea of time continuing in the absence of change, which obviously would imply that time and change are two different things. What we need is some method of judging the legitimacy of this conception. We may think of a period of time without at the same time thinking of the changes that take place in it, but it does not follow from this that we can think of a period of time that contains no changes, or that such a thought is coherent. In Chapter 2, we looked at three influential arguments against the intelligibility, or possibility, of ‘empty' time. The strongest of these appealed to the idea that empty time would be causally inert, implying that we would never have any reason5 to posit its existence.
  6. The existence of spatial vacua is less contentious6, but then it does not immediately defeat the view of space as a collection of spatial relations between objects, for these relations are not incompatible with regions of empty space. The difficulty for the relationist, however, is to explain apparent reference to unoccupied spatial points in apparently true statements, for if we can refer to unoccupied parts of space, it seems that these places are objects in their own right, not obviously reducible to facts about things in space. The relationist may be able to dodge this objection, at least temporarily, by distinguishing between truths about the physical world, and abstract geometrical truths. Apparent reference to facts involving unoccupied spatial points could be construed in terms of the latter. These issues were the topic of Chapter 3, in which we also looked at arguments in favour of absolute7 motion, an idea that implies the existence of space as existing independently of objects.
  7. Part of what makes attempts to reduce space and time to features of objects and events tempting is the conviction that, considered as the absolutist considers them, namely as objects in their own right, they would be entirely featureless. In a completely empty and so changeless universe, nothing would distinguish one place or one moment from any other. And such a featureless medium would really explain8 nothing of what we observe, but would be merely a theoretical abstraction. But would an independently existing space and time really be featureless? In Chapter 4 we looked at the implications the discovery of consistent non-Euclidean geometries had for our understanding of space. First, it undermined9 the distinction between physical truths and geometrical truths, and so strengthened the argument for the existence of unoccupied spatial points. Secondly, the fact that space has a certain shape (curvature, dimensionality, presence or absence of boundaries) can make a real difference to how things can move in space, and so suggests that space can be a cause, and not just an impotent medium. The existence of space may be explanatory in another way, too. The spatial properties of certain asymmetric objects such as hands appear to depend on some global property10 of space itself.
  8. We are pulled in two directions. On the one hand, we feel uncomfortable with the idea of space and time existing in the absence of any concrete objects or events — a discomfort that is particularly strong when we try to imagine time going on in a completely empty universe. On the other hand, treating them just as abstract ways of talking of objects does not do justice to everything we want to say about space and time. Is there any way of resolving this tension? One compromise we canvassed was to treat space as nothing other than the fields of force11 around and between objects. These would not exist in the absence of any objects, but on the other hand they are something other than those objects, and can exhibit a certain shape that explains the behaviour of objects moving about in apparently empty space. Similarly, we do not have to treat time as wholly reducible to changes. Collections of states of affairs, some of them perhaps unchanging states of affairs, would provide alternative building-blocks for time. Combining these two, we have a picture of space and time as an ordered series of states of affairs concerning the properties of and relations between, concrete objects and their fields of force.
  9. The extent to which we think of space and time as independent of their contents will affect our view of their boundedness (or unboundedness). Did time have a beginning? Will it have an end? Is there an edge to space? Evidence for a ‘Big Bang', in which our universe had its beginning, is, we suggested in Chapter 5, at best very equivocal evidence for a beginning to time. First, the hypothesis of the Big Bang does not necessarily rule out a preceding12 ‘Big Crunch', in which a previously existing universe collapsed. Secondly, to identify the beginning of the universe with the beginning of time is tacitly to make some contentious conceptual presuppositions. This is not to say that those presuppositions are unwarranted, merely that they need to be made explicit and to be justified. What if we do not identify the beginning of the universe with the beginning of time? Then we invite the picture of aeons of empty time preceding the Big Bang, leaving it inexplicable why the Big Bang occurred just when it did, and not earlier or later. Indeed, the Big Bang itself would appear to be an uncaused event. But causal anomalies are also implied in other accounts of time and the universe: if the universe had no beginning in time, but extends infinitely into the past, what then explains its existence? And what if time is cyclic, and so has neither a beginning nor an end? Does it not then follow that every event, ultimately, causes itself13?
  10. The idea of an edge of space is as difficult to conceive as a beginning of time, but perhaps particularly difficult for the absolutist, who cannot explain the idea in terms of the finitude of the physical universe. As an ancient paradox nicely brings home to us, we find it very hard to say just how things will behave at the edge of absolute space. But, equally, the idea of space going on indefinitely also causes intellectual discomfort. Discussion of these problems in Chapter 6 ended with the suggestion that space may be both finite and unbounded, a view that may be less problematic14 than its temporal counterpart.
  11. Chapters 5 and 6 were concerned with the infinite extent of time and space. In Chapter 7, we turned to their infinite divisibility. Intuitively, we think that there is no limit to the extent that an interval of time or region of space can be divided. This seemingly innocuous idea led to a plethora of paradoxes, including two of Zeno's famous paradoxes of motion, the Achilles and the Dichotomy. The essential idea on which these two are based is that, if space and time are infinitely divisible, then any moving object will have to achieve an infinite number of things in a finite time: i.e. pass through an infinite number of sub-distances. Treating these problems simply as mathematical conundrums, requiring for their solution only the technical notion of the infinitesimal15, does not do justice to their philosophical interest and importance. Two important philosophical solutions to the paradoxes present themselves: finitism and atomism. The finitist asserts that there is no set of actually existing concrete objects that is infinite. An interval of time or region of space does not therefore actually contain an infinite number of points. There may nevertheless be no natural limit to the process of dividing an interval or region, and it is this that justifies us in talking of space and time as infinitely divisible. The infinite exemplified by space and time is therefore, in Aristotle's terms, only a potential infinite. The problem with the positive part of this proposal is that it leaves it mysterious what grounds the fact that the process of dividing has no natural limit. It is not enough to say that there is nothing that prevents us dividing further: we naturally want to know what it is that enables us to go on dividing, and this, surely, is something to do with the structure of space and time. The doctrine of the potential infinite seems an exhortation just to be silent on this structure. Atomism (which is compatible with the negative part of finitism) is not so silent: it asserts that there are spatial and temporal minima, of non-zero magnitude, which represent the limit of any division. This theory has the merit of solving a range of paradoxes: Zeno's Achilles, Dichotomy, and Parts and Wholes paradoxes, Aristotle's conundrum concerning the first and last moment of motion, and Democritus' paradox of the cone16. Admittedly, it involves a revision in our ordinary conception of change, and requires us to adopt a non-Euclidean geometry, but we were unable to detect any contradiction in the idea.
  12. So far in our investigation, we had been concerned with problems common to both space and time. But from Chapter 8 onwards, we turned to features that arguably distinguish time from space: the passage, and direction, of time. Although both these features (and perhaps they are not distinct) are deeply familiar ones, pervading our experience as they do, they are not particularly easy to define. The passage of time is often represented in metaphorical terms, typically in terms of a river. The problem with these metaphors is that they typically have time built into them, and so already presuppose a grasp of what the passage of time amounts to. Two ideas are particularly important in articulating the notion: the first is of the changing pastness, presentness, and futurity of events; the second is of events coming into existence and so adding to the total stock of reality. Much of our thought about the passage of time has been dominated by McTaggart's important distinction between two ways of ordering events in time: as an A-series — which orders them according to whether they are past, present, or future — and as an B-series — which orders them in terms of earlier and later. The key question here is this: given that the facts underlying the two orderings cannot be different, which determines which? Is it the A-series positions that determine the B-series positions, or the other way around? The natural answer is that it is the A-series positions that determine the B-series positions, but this leads straight to McTaggart's famous paradox, which attempts to show that the notion of a real A-series is self-contradictory. This obliged McTaggart himself to deny the reality of time.
  13. Two strategies for coping with the paradox were outlined in Chapter 8: one was presentism, the view that only what is present is real, the other was the B-theory of time, which regards the B-series as more fundamental than the A-series. Presentism may perhaps articulate our intuitive conception of time, as we naturally regard the past as no longer real and the future as not yet real. It nevertheless faces some formidable difficulties. First, it is by no means clear that it can explain how statements about the past can be true or false. One mechanism presentists might appeal to concerns the causal traces the past leaves on the present: it is these present causal traces, they could argue, that make statements about the past true. But what, in presentist terms, does it mean for there to be causal relations between past and present times? Secondly, presentism creates difficulties for our understanding of motion. In one reconstruction of Zeno's Arrow paradox, the topic of Chapter 9, the following problem arose: the presentist is committed to the idea that a moving object must be conceived as moving in the present, but is unable to reconcile this with the fact that motion essentially involves facts about an object's position at times other than the present.
  14. In so far as presentists hold that past truths are determined by present fact, it might seem to follow from their position that the past is alterable, in a manner reminiscent of Orwell's dystopian vision in Nineteen Eighty-Four. However, one conclusion from our discussion in Chapter 10 was that the presentist is not committed to this dubious position. Indeed, the very notion of the alterability of the past seems to lead inexorably to contradiction. However, there is an important distinction to be made between altering facts and affecting them. This allows us, both to avoid the fatalist conclusion that, since we cannot change the future, we cannot affect it, and also make sense of time travel17, in which present decisions have a causal influence on past events. However, whether time travel18 is really a coherent notion depends on our understanding of the direction of time
    (on which more below).
  15. Presentism is one version of the A-theory of time, which holds that B-series facts are determined by more fundamental A-series facts. Not all A-theorists are presentists (although those that are not, we suggested above, will have difficulties in escaping from McTaggart's paradox). But whether or not it is combined with presentism, the A-theory faces another problem. If space and time are just the products of our minds, as Kant thought, there are good grounds for thinking them both unified: that is, there is just one time and one space, which, if time and space are in the mind, has to be interpreted as meaning that every object of experience is presented as spatially and temporally related to every other. But what if space and time exist independently of our minds? Is there any reason then to think of them as being essentially unified? Here, being unified means that every object and event is really spatially and temporally related to every other. In some contexts, we suggested in Chapter ii, the idea of multiple spaces and time-series (something like the idea of ‘parallel universes' in fiction) may have a useful application. Two such contexts are the multiverse hypothesis, entertained by some cosmologists, and the two-slit experiment with light. Now, the idea of multiple spaces, although perhaps surprising, does not, arguably, raise any serious conceptual difficulties. But the idea of multiple time-series is not readily reconcilable with the A-theorist's assertion that B-series facts are determined by A-series facts.
    The second strategy we canvassed for dealing with McTaggart's proof of the unreality of time was the B-theory. According to this theory, there is no A-series in reality, only a B-series. As a consequence, since the B-series positions of events do not change, there is no passage of time, at least in the way we ordinarily conceive of it. This raises many questions: if there is no A-series, does this entail that statements such as ‘The post has just arrived' are false? If there is no passage of time, what becomes of our intuitive belief that the future is unreal? And what accounts for the obvious fact that things change, for does, e.g. a cup of tea changing from hot to cold, not require the tea's being hot to have receded into the past? Finally, how can the B-theorist explain the direction of time, for surely direction and passage are inextricably linked? These questions raise profound issues, whose surface we have no more than grazed in this discussion, but here is a summary of provisional answers from the B-theorist's perspective:
    • (i) A-series truths. Despite the absence of an A-series in reality, we indisputably have A-series beliefs and give voice to them ('The train has just left', ‘The War ended years ago', Aunt Jane will be arriving tomorrow'). What makes these beliefs true (or false) are B-series facts. Thus, if the train leaves just prior to 7 a.m., and at 7 a.m. I have the thought that it has just left, then my belief is true. We do not need to appeal to the pastness of the train's departure.
    • (ii) The reality of the future. To describe what is past or present on the one hand as real and what is future on the other as unreal, seems to require a real distinction (and not merely a distinction in thought) between past, present, and future. Since the B-theory denies that there is such a distinction in reality, it follows that, on the B-theory, all times are equally real. (Some B-theorists, it should be pointed out, have attempted to retain something of our ordinary belief in the unreality of the future by relativizing what is unreal to B-series times. So, at any given time, later events are unreal. Is this coherent?)
    • (iii) Change. Change, in B-series terms, is just an object's possessing one property at one time, and an incompatible property at a later time. However, for this to be a completely convincing answer, we need to be able to explain how it is that one and the same object can exist first at one time, exhibiting one property, and then at another time, exhibiting a different property. If time did indeed pass, as the A-theorist holds, then objects could, by simply staying in the present, move from one B-series moment to another. But there is no room for such movement in the B-universe.
    • (iv) Direction. In B-series terms, the fact that time has a direction is neither more nor less than the fact that events form a B-series: that is, that they are ordered by the asymmetric earlier than relation.
  16. This last answer requires some expansion. Time, experience tells us, has an intrinsic direction, space not. But what does this actually amount to? Can it really be no more than the asymmetry of the earlier than relation? There are asymmetric spatial relations, too, so we need to say a little more than this to explain the difference between time and space. In particular, we need to be able to answer the following questions: why do we experience time as having a direction, from earlier to later? Why does the arrow of time point in the same direction as the causal arrow (from causes to effects), the psychological arrow (from experiences to memories), and the thermodynamic arrow (from order to disorder)? We tackled these questions in Chapter 12, and much of our discussion was taken up with the causal analysis of time order. If this analysis is successful, then we have the prospect of being able to solve a number of conundrums concerning the direction of time, as follows:
    • Q: Why do causes occur before their effects?
      A: Because ‘earlier than' is defined in causal terms.
    • Q: Why is the ‘earlier than' relation asymmetric?
      A: Because the causal relation is asymmetric.
    • Q: Why do memories never precede the experiences of which they are memories?
      A: Because the experiences are the causes of those memories.
    • Q: Why do we have a sense of the direction of time?
      A: Because we remember the past, perceive only the present (strictly, the very recent past), but never remember or perceive the future.
    • Q: And what explains these facts?
      A: Perception and memory are causal processes. Perceiving or remembering the future would entail backwards causation19.
  17. However, unless all events are causally related, the causal analysis appears to imply the possibility that time may go in different directions in different parts of the universe.
  18. So, finally, what of the human significance of these issues? Our view of ourselves is intimately bound up with space, time, and causality20: we take up space and move about in it, we are affected by change and are the instigators of change, we persist through time. We think of ourselves, in short, as spatial and temporal agents. What, then, if on investigating the matter, we found that space and time could not, on pain of insoluble paradox, be thought real features of the world? This would have a revolutionary effect on what we think ourselves to be. In particular, we would have to reassess the idea of ourselves as physical beings if to be physical is to occupy space. We would have to take seriously the idea that we were unembodied spirits. Or what if instead we concluded only that a particular feature of space and time was unreal, a result of our projecting a feature of our experience onto the world? For example, suppose we decided that the balance of argument was against our common belief that time passes. How would this affect our view of death? For do we not ordinarily see life as an inexorable movement towards extinction (in this world, at least)? If there is no such movement, what does death amount to? What becomes of our belief that our present existence is somehow more real and significant than our past or future existence? Can we continue to see ourselves as free agents if denying the passage of time implies that what we call the future is as fixed as what we call the past? And what becomes of our conception of ourselves as persisting through time being the same person from moment to moment, irrespective of any change we suffer?
  19. Let me dwell a little longer on this last point. A problem for the B-theory, we noted, was to explain how it can be that one and the same object can exist at one time and exist at another time, without it being the case that the object somehow moves from one time to another, implying that time itself passes. Addressing this issue head on, let us now, for the first time, introduce the following radical thought: maybe one and the same object cannot be at different times. What we ordinarily think of as the same object, persisting through time, is in fact a succession of different (though very similar) objects, each unchangingly locked into their own time. Change is then the having of different and incompatible properties by different (but suitably related) objects. Perhaps the best way to imagine this is to think of time as another dimension of space, and treat objects' apparent persistence through time as just extension in this fourth dimension. We imagine, then, a four-dimensional object, which has different parts at different places in these four dimensions. Although we can see the different parts in three of the dimensions as different, we experience different parts in the fourth dimension as if they were one and the same thing, moving through time. There is much that is misleading in this description. Time is not just a fourth dimension of space, and the B-theorist does not have to say that it is. But this picture nevertheless gives us an intuitive sense of how much we may need to revise our ordinary conception of the persistence of objects through time if the B-theory is correct. (Again, some B-theorists do not accept that we have to give up the idea that one and the same object persists through time. But what alternative account can they offer?) And if we do revise our conception of how ordinary objects persist through time, then we must also revise our conception of our own persistence through time. This may mean giving up the idea that we are the same person from time to time. As Wells's Time Traveller21 puts it:
      For instance, here is a portrait of a man at eight years old, another at fifteen, another at seventeen, another at twenty-three, and so on. All these are evidently sections, as it were, Three-Dimensional representations of his Four-Dimensional being, which is a fixed and unalterable thing.
  20. Apart from these particular ways in which the philosophy of time and space impinge on our self-conceptions, simply contemplating these difficult and abstract issues widens our view of the world. Recalling Roger Bacon's legendary head of brass, we may find the mysteries of space and time unsettling, but there is reason to hope that part of their solution, at least, lies within the compass of human understanding. [… snip …].


Annotated printout filed in "Various - Papers on Identity Boxes: Vol 09 (L)".

In-Page Footnotes

Footnote 1: This is useful – I’ll add some comments on the summary, but need to look at the full text in due course.

Footnote 2: As this is a philosophy book, it is unconstrained by scientific theory. But (in my view) this may make some of the discussions idle. It implies that we can discuss these things from our armchairs without observation. Now, while all Einstein needed was pencil and paper, he was responding to the observations of others (eg. the Michelson-Morley experiment). Le Poidevin just seems interested in internal consistency, when “natural philosophy” needs to be consistent with what’s known about the physical world.

Footnote 3: Surely this is tosh? “as long as” is a relational matter, and nothing to do with the any arbitrary metric (eg. set of units?). But it’s true in SR that lengths (irrespective of units) are relative to the observer’s motion – but then so are matters of simultaneity. All this shows how constrained philosophy of space and time (and of any other real-world matter) ought to be by the facts.

Footnote 4: Conceivability arguments have a bad history in philosophy; as Le Poidevin goes on to note, how can we be sure we can conceive of any particular state of affairs – especially if we describe it vaguely).

Footnote 5: This sounds to me to be somewhat in the spirit of verificationism.

Footnote 6: But, again, what about the impact of physics? What about the “quantum vacuum”, and Heisenberg’s uncertainty principles? If we’re doing philosophy of the physical world we need to have cognizance of physics. If we’re talking about any possible world, that’s another matter, but we need to watch out lest we slide between the possible and the actual.

Footnote 7: Again, this is Newtonian mechanics, which appears to have been shown to be inadequate. Some day I may look at the “philosophical” books I’ve got (supposedly) confuting Einstein. Eg.
"Aspden (Harold) - Modern Aether Science",
"Nordenson (Harald) - Relativity, Time and Reality",
(and another with a more abusive title that I seem to have lost).

Footnote 8: What’s wrong with space-time being the “stage” on which the play of matter is enacted – that was the historical supposition.

Footnote 9: Not at all – it was presupposed that space was Euclidean because this was the only geometry envisaged – but when it was appreciated that there could be other geometries, the distinction between geometry and the world it models was appreciated. This is the opposite of what Le Poidevin says.

Footnote 10: This seems to be a muddle. In GR, space-time and matter are interlinked. Matter warps space-time to create the geodesics in which matter moves by free-fall.

Footnote 11: This doesn’t sound right – it seems too “localist”, and is not “just” this anyway. Again, we need to take GR (Big-Bang cosmology) and QM into account.

Footnote 12: But, time “re-starts” at a ‘Big Crunch', so the “earlier” phase isn’t “earlier” in the same timeframe.

Footnote 13: This seems to be a bit of a throw-away line. Just how would this “ultimate causation” work?

Footnote 14: Why should it be more difficult to envisage finite but unbounded time than finite but unbounded space? Lack of imagination?

Footnote 15: Bah! Use of infinitesimals ignores all the nineteenth-century mathematical advances in the theory of limits that did away with infinitesimals altogether.

Footnote 16: What is this paradox? Whatever, it strikes me that mathematical “paradoxes” should be solved using mathematics, and not by muddying the water with unquantified philosophy. What does "Sainsbury (Mark) - Paradoxes" have to say on the matter?

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