Travels in Four Dimensions: The Enigmas of Space and Time
LePoidevin (Robin)
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  1. Space and time are the most fundamental features of our experience of the world, and yet they are also the most perplexing.
    → Does time really flow, or is that simply an illusion?
    → Did time have a beginning?
    → Are there parallel worlds?
    → What does it mean to say that time has a direction?
    → Does space have boundaries, or is it infinite?
    → Is there space beyond the universe?
    → Might time travel2 be possible?
    → Could time go backwards?
    → Is change really possible?
    → Could space and time exist in the absence of any objects or events?
    → Would time go on if everything else stopped?
    → What, in the end, are space and time?
    → Do space and time really exist, or are they simply the constructions of our minds?
  2. Robin Le Poidevin provides a clear, witty, and stimulating introduction to these deep questions and many other mind-boggling puzzles and paradoxes. He gives a vivid sense of the difficulties raised by our ordinary ideas about space and time, but he also gives us the basis to think about these problems independently, avoiding large amounts of jargon and technicality. His book is an invitation to think philosophically rather than a sustained argument for particular conclusions, but Le Poidevin does advance and defend a number of controversial views. He argues, for example, that time does not actually flow, that it is possible for space and time to be both finite and yet be without boundaries, and that causation3 is the key to an understanding of one of the deepest mysteries of time: its direction.
  3. Drawing on a variety of vivid examples from science, history, and literature, Travels in Four Dimensions brings to life some of the most profound questions imaginable.
  4. No prior knowledge of science or philosophy is required to enjoy this book. The universe might seem very different after reading it.

In-Page Footnotes ("LePoidevin (Robin) - Travels in Four Dimensions: The Enigmas of Space and Time")

Footnote 1:

OUP, 2003

"Bricker (Phillip) - Review of LePoidevin's 'Travels in Four Dimensions: The Enigmas of Space and Time'"

Source: Website

Full Text
  • The author writes in the preface: “This book arose out of a course of lectures I have given for many years … My purpose in writing it was primarily to introduce the reader to the classic paradoxes and problems of space and time.” (x-xi) On the whole, the book succeeds admirably in this goal, and should appeal to a wide audience. Three audiences, in fact.
    1. First and foremost, the book can serve as an introductory text for an undergraduate course in the philosophy of space and time. Discussion questions are included after each chapter, and further problems are provided in a separate section at the end of the book. Key arguments are usefully laid out in premise-conclusion format. Because the problems discussed all involve classical, rather than relativistic1, concepts of space and time, very little physics is called upon throughout the book. That could be a plus or a minus. Instructors who prefer to include some space-time physics and its philosophical ramifications will want to supplement the book, or turn elsewhere.
    2. A second prime audience is the educated layperson with a penchant for puzzles and paradoxes. Of these there should be many. Books on space and time, often by celebrity physicists, fly off bookstore shelves (only, one suspects, to collect dust in home libraries). Most of these readers would do better to start with this book. I can’t but agree with the author that “conceptual analysis of the classic paradoxes and problems [is] an important preliminary to thinking about space-time physics.” (xii)
    3. Finally, the book should appeal to professional philosophers and graduate students of philosophy who seek a lightweight but informative overview of issues in the philosophy of space and time. For, as the author rightly notes: “A great many philosophical problems are affected by views on space and time … these two lie at the heart of metaphysical inquiry.” (x)
  • All three audiences will appreciate the author’s clarity of expression and engaging style of writing. The frequent use of literary or historical anecdotes to introduce a topic, which might be out of place in a more scholarly philosophical treatise, contributes here to making the book a pleasure to read.
  • The book covers a lot of ground. Here is a list, chapter by chapter, of just some of the topics discussed.
    1. Conventionalism vs. objectivism concerning the temporal metric.
    2. The relation between time and change.
    3. Absolutism vs. relationism about space.
    4. Non-Euclidean and four-dimensional space2.
    5. The topology of time (could time have a beginning? could time be circular?).
    6. The edge of space (could space have a boundary?).
    7. Zeno’s paradoxes of plurality and motion.
    8. Time’s passage (McTaggart’s argument and the presentist and the B-theorist responses).
    9. Zeno’s Arrow (static vs. dynamic accounts of motion).
    10. Backwards causation3 and time travel4.
    11. Other times and spaces (the fine tuning argument for multiple universes; the two slit argument for branching space5).
    12. The direction of time.
      With so much ground to cover, the author aims only to provide the opening moves to a given debate; theories are sketched, but never developed in detail. Through it all, the author serves as an informed guide, pointing out avenues of research that are promising, occasionally opinionated, but never doctrinaire.
  • There are some sections of the book, however, that I do not think are adequate to the task at hand. Most of these, I suspect, result from the author’s wish to avoid too much mathematical precision or overly complex argumentation. And no doubt there have to be trade-offs in a book of this sort. But, at least in the four cases mentioned below, I think the author could have been more careful or more thorough.
  • Consider, for example, the author’s response to Zeno’s paradox of plurality (called Parts and Wholes by the author). The paradox arises on the supposition that a finite-length rod is composed of infinitely many (ultimate) parts. “If we say that each part has a definite, non-zero size, then, since the rod consists of an infinite number of parts of that size, the rod itself must be infinitely long. … if each part has no size, then the rod itself can only be of zero length, since even an infinite number of parts of zero size cannot add up to something of non-zero size.” (p. 103) The standard mathematical solution, ensconced in modern measure theory, is to reject additivity for non-denumerably many parts: a rod composed of non-denumerably many points, each of size zero, can have any finite length, or be infinite in length. But the author responds, instead, as follows: “if we say there are an infinite number of parts [in a rod of unit length], what length has each part? That is, what, when multiplied by ∞ (infinity) results in 1? The answer is 1/∞. Each part, in other words, is infinitely small, and the sum of an infinite number of infinitely small magnitudes is a non-zero but finite magnitude.” (p. 104). This reasoning is spurious. If we had started instead with a rod of length two units, parallel reasoning would have us conclude that its infinitesimal parts have length 2/∞. But since each half of that rod is a rod of length one, the very same parts must also have length 1/∞. Is, then, 1/∞ = 2/∞? If multiplying and dividing by ∞ is permissible, why can’t we conclude that 1 = 2? This is not to deny that there are non-standard measure theories that resolve the paradox and allow for infinitesimal lengths. But the author seems to be invoking, instead, what might be called the “naïve” theory of infinitesimals6, a theory that is demonstrably inconsistent. (The author also invokes infinitesimals in presenting his solution to Zeno’s paradoxes of motion, such as the Achilles and Dichotomy; but infinitesimals play no role in these paradoxes, since all lengths and durations considered are finite.)
  • Later in the same chapter, the author considers a puzzle about transition. A train is waiting at the station, and then begins to move. If time is dense, so that between any two moments of time there is a third, then there cannot be both a last moment of rest and a first moment of motion. How, then, shall we classify the moment of transition? The author argues that the moment of transition is a last moment of rest. He reaches this conclusion by assuming that it must be either a moment of rest or a moment of motion, and by arguing quite generally that there can never be a first moment of motion. For, he claims, there is no motion until there is a displacement from a prior position; and, if time is dense, there cannot be a first moment of displacement. This conclusion has some untoward consequences. For one thing, it would follow that if time has a first moment, then everything is at rest at that moment. Worse, the author’s view would seem to be incompatible with Galilean relativity. For consider a train that is in motion and then comes to a stop. Presumably, by parallel reasoning, the author will say that the moment of transition is a first moment of rest. But an object in motion coming to a stop, as viewed from one reference frame, may be an object at rest beginning to move, as viewed from another reference frame. And, presumably, the moment of transition cannot share its (specific) state of motion with preceding moments according to one reference frame, and with succeeding moments according to another.
  • If, instead, one invokes the standard mathematical response to questions of (instantaneous) rest and motion, the transition puzzle is easily resolved; whether or not the author’s account can stand, the reader should be introduced to the standard resolution. On that account, the state of motion of the train at a given moment is fully determined by the (first and higher-order) time derivatives at that moment. How our ordinary concepts of rest and motion relate to the resulting mathematical classifications may be indeterminate. But, plausibly, to be at rest at a moment is just to have the first derivative defined and zero at that moment; to be in motion is just to have it defined and non-zero. (That allows a projectile at the top of its trajectory to be at rest, which seems right.) Whether the transition moment is a moment of rest depends on how the transition takes place: if the transition is abrupt, the first derivative is undefined, and the transition point is neither a moment of rest or of motion. There is nothing arbitrary, or unduly revisionary, in explicating our ordinary concepts of rest and motion in this way.
  • Another place where oversimplification leads to an unsatisfactory result is the author’s presentation of the fine-tuning argument for multiple universes. As the author understands the argument, it is based on a general principle of probabilistic confirmation theory, roughly: “you would be wise to prefer a hypothesis that makes the observed result very likely to one that makes that result very unlikely.” (p. 186) That our universe contains life (the “observed result”) is very unlikely on the hypothesis that there is only one universe (and no creator), but, according to the author, is made very likely on the hypothesis that our universe is but one of billions and billions of universes, all with different physical constitutions – the multiverse hypothesis.
  • To motivate the argument, the author considers the following supposedly analogous case. Suppose there is a computer designed to print out pages of random numbers; and suppose, on the one page you see, the first thousand digits of the decimal expansion of pi are printed out. The author writes:
      “you do not know whether this is the one and only page that the computer has produced, or whether it is one of millions of pages, the computer having been producing its numbers non-stop for years, and this page has been deliberately selected by someone for your attention. … Given the general principle appealed to a moment ago, that we should choose the hypothesis that makes our observation more, rather than less, likely, we have reason to suppose, just on the basis of what we have before us, that this page is not unique – that it is one of many such pages.” (pp. 186-7).
    Now – although this is suppressed at the end – it is important to note that the general principle does not support the many-page hypothesis by itself; rather, it supports the conjunction of the many-page hypothesis and the hypothesis that only7 a page with the printout of the expansion of pi would be selected for our attention. For the many-page hypothesis by itself does not make it more likely that we are looking at a printout of the expansion of pi8 if the page we are looking at was randomly selected from all the pages that have been printed, no matter how many pages have been printed.
  • When the argument for the multiverse hypothesis is presented, however, the “selection effect” has totally dropped out:
      Postulating a multiverse is like postulating that the page of random numbers that just happens to match the expansion of pi is just one of many such pages, produced by many machines, running over many years. As long as our universe is unique, the fact that it contains life is (the hypothesis of a creator aside) remarkable. But once we see it as one of billions of universes, each with a different physical makeup, the fact becomes less remarkable. Indeed, we may even be tempted to say that, given enough universes, it was inevitable that one should contain the conditions necessary for life. (p. 188-9)
    But the multiverse hypothesis does not make the fact that our universe contains life less remarkable if we could equally well have found ourselves in any9 of them. As the fine-tuning argument is usually presented, the observational selection effect – that we could only find ourselves in a universe that contains life – plays an essential role. But it is not at all obvious how or whether the general principle of confirmation theory is supposed to apply10 in this case. The “selection effects” in the multiverse case and in the computer case do not seem to be analogous11.
  • Perhaps the author thinks that the argument for the multiverse hypothesis does not rest upon there being an observational selection effect. In the passage quoted above, the author shifts between applying the general principle to the claim that our universe contains life and the weaker claim that some universe contains life. Indeed, the multiverse hypothesis does make it more probable that some universe contains life. But the general principle is fallacious when applied to an existential proposition entailed by our observation rather than to the observation itself. Suppose12 the “observed result” is that I am hungry. Then I also know by observation that someone is hungry. Consider the hypothesis that you have not eaten all day. That hypothesis makes it very likely that someone is hungry. But it would not be rational for me to believe that you have not eaten all day on the basis of my observation that I am hungry.
  • One final example. At the end of an otherwise nice discussion of time travel13, the author leaves the reader with a puzzle, and a misleading suggestion as to its import. The case considered is this:
      Peter and Jane, both 20 years old, are out for a walk one day in 1999 when suddenly a time machine14 appears in front of them. Out steps a strangely familiar character who tells Jane that he has an important mission for her. She must step into the machine and travel forward to the year 2019, taking with her a diary that the stranger hands to her. In that diary she must make a record of her trip. Obligingly, she does as she is asked and, on arrival, meets Peter, now aged 40. She tells Peter to travel back to 1999, taking with him the diary she now hands him, and recording his trip in it. On arrival in 1999, he meets two 20-year olds called Peter and Jane, out for a walk, and he tells Jane that he has an important mission for her. (pp. 180-1)
    The puzzle comes from asking: how many entries are there in the diary when Peter hands it to Jane? According to the author, “there does not appear to be a consistent answer.” Whatever number of entries one says the diary contains – say, n – it seems there must be two more than that; for Jane records an entry when she travels forward in time, making n + 1, and then Peter records an entry when he travels back in time, making n + 2, and then hands the very same diary to Jane that we supposed has only n entries.
  • Now, the puzzle certainly raises some interesting questions. (Where did the diary come from? Does the diary age like ordinary objects? From a four-dimensional perspective, the diary is what might be called a “roundworm.”) But it is misleading to suggest that there is any problem of (logical) consistency. It is easy enough to fill out the story in ways that, though odd, preserve consistency. Perhaps the entries are cleanly erased prior to the hand-off to Jane, so that the diary is blank when Jane gets it. Perhaps every entry is perfectly written over a previous entry, leaving no added trace, so that the diary has two entries when Jane gets it. The story could also have been filled out in a way that ruled out these and all other fixes. Then the story would indeed become an inconsistent time travel15 story. But, of course, there are inconsistent stories that that have nothing to do with time travel16, for example, inconsistent space travel stories. In either case, it is not the time travel17 or the space travel that is to blame for the inconsistency, but the incompetent storyteller18.


In-Page Footnotes ("Bricker (Phillip) - Review of LePoidevin's 'Travels in Four Dimensions: The Enigmas of Space and Time'")

Footnote 1:
  • This is a great deficit, and it’ll be interesting to see how Le Poidevin justifies this omission.
Footnote 2:
  • Space-time, surely!
Footnote 5:
  • What can this be? And how could it work? The two-split experiment is an interference pattern – how would the interference occur if the universe had split at the quantum event?
Footnote 6:
  • A research student was pursuing a thesis in this area (supervised by Ian Rumfitt) when I was last at Birkbeck – it struck me as utter tosh, and not something that a non-mathematician should be allowed to pursue.
Footnote 7:
  • I don’t know what the reviewer is on about here. Le Poidevin’s analogy seems fair enough – though I’m not sure it adds much as it’s a rather odd situation. The numbers are supposed to be randomly-generated, so we’re trying to explain how a sequence of digits that happened to be the first 1,000 pi-digits came to be there. Pure chance as a one-off is impossible. So, we’re supposed to think it had been rambling on for years, and eventually an interesting page popped out which was handed to us.
Footnote 8:
  • But that’s not what Le Poidevin is claiming – it’s not a printout of the expansion of pi – it’s a string of random digits that just happens to coincide with the first 1,000 digits of the expansion. Mind you, I expect you’d need to wait a long time to randomly produce the first 1,000 digits in sequence (1094 seconds or so – say 1087 years. And selecting an interesting sequence would take even longer. Probably not a helpful example!
Footnote 9:
  • The whole point is that we can only find ourselves in a life-supporting universe. The “computer analogy” is supposed to be analogous – an “interesting” page has been selected for us to look at.
Footnote 10:
  • I don’t understand the objection here.
Footnote 11:
  • Well, it seems analogous to me, but not very practical. It’d still be a miracle if such a page could be selected should the computers rattle on until the heat-death of the universe.
Footnote 12:
  • This analogy of the reviewer is even more obscure than that of Le Poidevin. The “selection effect” has nothing to do with a particular individual, or a particular form of life. It’s just that life can only be observed in a location suitable for life – whatever that life is.
Footnote 18:
  • I don’t think our reviewer is taking this seriously. The issue is presumably that a closed “time loop” is infinite. So, at the one and only time (in the sense of “date”) the diary is presented, how many entries has it got in it? But the reviewer is right that TEs need to be written up explicitly. It might be the case that as you travel back in time events are “unwound”, so you get younger and eventually coalesce with your younger self, so that no-one notices anything at the time (though they do 20 years later, when you decide to travel back in time … ). But that’s a “factual” issue of how backwards time-travel is supposed to work. That’s not how the scenario is presented – and presumably that’s just to show that “naïve” backward time travel is incoherent.

"LePoidevin (Robin) - Travels in Four Dimensions: Preface"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Preface

  1. In the house where I grew up was a copy belonging to my mother of the seven-volume Newne's Pictorial Knowledge, an encyclopaedia for children published in the early 1930s. As a child I would spend hours with these books. There was an item in the back of each volume that held a particular fascination for me, a series of leaves one could lift up thus revealing the internal structure of an oyster, a frog, a dog rose, a bee (especially frightening), and various other things. In one of the volumes, containing an account of famous scientists and inventors, was a rather improbable story of the friar Roger Bacon. Bacon, the story went, had after many years' labour constructed a marvellous human head in brass, which, he said, would presently speak of wonderful things. Tired of watching over the head and waiting for it to speak, he set another friar to guard it, who was instructed to fetch Bacon the moment the head spoke. After some considerable time, the lips of the brass head began to move, and spoke the words ‘Time is'. Not thinking this sufficiently significant to fetch Bacon, the friar waited to see what else the head would say. After half an hour, the head spoke once more: ‘Time was'. Again, the friar sat still. After another half-hour, the head spoke for a third and final time: ‘Time is past'. It then dashed itself to pieces on the floor. The friar went at once to tell Bacon of the calamity. Bacon, dismayed to find that the head had spoken in his absence and was now destroyed, made many other heads in brass, but none of them spoke.
  2. The contributor of this part of the encyclopaedia wisely warned its readers that the story was a legend, but even if not true, showed what esteem Bacon was held in by his contemporaries and succeeding generations. I happily ignored this warning, captivated as I was by a story I felt quite sure was true. It convinced me that there were mysteries of time that held the key to life, but also that knowledge of these mysteries was possibly dangerous, even forbidden, for human minds. So began my fascination with time, although I had little idea where to turn for enlightenment. My interest was rekindled a few years later when my father happened to mention to me, out of the blue, J. W. Dunne's Experiment with Time, a book that had enjoyed enormous popularity on its first publication in 1927, and which had influenced the time plays of J. B. Priestley. It was, my father explained, something to do with a dream about the face of a clock and appeared to demonstrate that one could see into the future, but had since been disproved. (Dunne's account of this dream appears at the end of this book.) I am not quite sure what disproof my father had in mind, but when, sometime afterwards, I found a copy of the book, I was both excited by the accounts of dreams, but also disappointed by my inability to understand Dunne's theory to explain them—a theory that still strikes me as decidedly peculiar.
  3. It was not until I became a research student that I began thinking about the philosophy of time, and I recall the extraordinary impact that McTaggart's proof of the unreality of time had on me when I first encountered it. It convinced me, first, that there was in reality no absolute distinction between past, present, and future, and secondly, that in consequence our view of ourselves as observers moving through time was radically mistaken. The intimate connection between time and the self is, surely, one of the sources of the fascination the philosophical paradoxes discussed in this book have for us. A great many other philosophical problems are affected by views on space and time, and I believe it is no exaggeration to say that these two lie at the heart of metaphysical inquiry.
  4. This book arose out of a course of lectures I have given for many years at Leeds, entitled Space, Time and Infinity. My purpose in writing it was primarily to introduce the reader to the classic paradoxes and problems of space and time, where our philosophical thinking about these two elusive ideas begins. Introducing theories was very much a secondary aim. Although I offer in the pages that follow some theoretical apparatus, which I believe is helpful in structuring our first thoughts about the problems, I have tried to keep this fairly light. It is the problems themselves that stimulate independent thought, and my aim will be fulfilled if the reader is as excited about the problems as I have been and feels just as keenly the need to seek solutions to them. I have sketched some possible solutions, but I am no evangelist, and would encourage anyone to treat the results of my attempts with deep suspicion. To further stimulate independent thought, I have added some questions for the reader at the end of each chapter, and a set of problems at the end of the book. Most chapters are more discursive and open-ended than would be tolerable in a journal article, but where I think a certain line of thought is mistaken, I say so. And where I have a particular angle on a debate, I pursue it. Since the book begins with Kant's question ‘What are space and time?', the reader naturally expects some kind of answer. The Concluding Thoughts section, however, should be taken as a summary of some of the key ideas expressed in the book, rather than a definite conclusion, which would not be appropriate in so introductory a volume. Those looking for a more thorough grounding in theory, less elementary introductions to the subject, or less compromising defences of a particular position, will find some suggestions in the section on Further Reading at the end of the book.
  5. This is, let me emphasize, a philosophical introduction to space and time, one that is concerned throughout with the conceptual questions and difficulties that our ordinary views of space and time throw up. I have had to introduce a very modest amount of physics, as it is difficult to get far in discussing these problems without some reference to physics, but this is emphatically not a popular science book, nor is it an introduction to the philosophy of space-time physics. I do not, for example, discuss the Special or General Theories of Relativity. I regard conceptual analysis of the classic paradoxes and problems as an important preliminary to thinking about space-time physics. Again, anyone looking for books on the philosophy of space-time physics will find suggestions in Further Reading1, but I should particularly like to mention in this context Barry Dainton's excellent Time and Space, which appeared just as this book was being completed. Dainton's book also takes some of the issues discussed here several stages further2.
  6. Those familiar with the literature will be only too aware of the influences on my approach. I gratefully acknowledge those whose writings have provided particularly important influences and sources of inspiration: Bas van Fraassen, Graham Nerlich, Bill Newton-Smith, Hugh Mellor, Huw Price, and Richard Sorabji.
  7. Conversations with many friends and colleagues have affected my thinking about space, time, and related issues, and I would like to thank James Bradley (from whom I first learned about the Greenwich incident), Jeremy Butterfield, Peter Clark, Heather Dyke, Steven French, Jonathan Lowe, Hugh Mellor (to whom my greatest debt is owed), Mark Nelson, Sharon Ney, Nathan Oaklander, Peter Simons, Quentin Smith, and the late Murray MacBeath. [ … snip … ]

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

In-Page Footnotes ("LePoidevin (Robin) - Travels in Four Dimensions: Preface")

Footnote 1: This is useful – there are two sections:- Footnote 2: May be worth following up – but I’ve other books to read first – in particular:

"LePoidevin (Robin) - The Measure of All Things"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 1

    1.1 Incident at Greenwich
    1.2 Metric, Convention and Fact
    1.3 Time and the Laws of Nature

"LePoidevin (Robin) - Change"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 2

    2.1 Time as Change
    2.2 Time without Change?
    2.3 Everything has a Reason

"LePoidevin (Robin) - A Box with No Sides?"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 3

    3.1 Where Two Worlds Meet
    3.2 Aristotle against the Void
    3.3 Jars, Pumps, and Barometers
    3.4 Lessons of the Vacuum
    3.5 The Redundancy of Space
    3.6 The Search for Absolute Motion

"LePoidevin (Robin) - Curves and Dimensions"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 4

    4.1 Euclid Displaced
    4.2 Space Makes Its Presence Felt
    4.3 The Lone Hand
    4.4 More than Three Dimensions?

"LePoidevin (Robin) - The Beginning and End of Time"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 5

    5.1 Echoes of Creation, Portents of Armageddon
    5.2 The Limits of Reason
    5.3 Can the Past be infinite?
    5.4 The Great Circle

"LePoidevin (Robin) - The Edge of Space"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 6

    6.1 Archytas at the Edge
    6.2 Is There Space beyond the Universe?
    6.3 The Illusion of Infinity

"LePoidevin (Robin) - Infinity and Paradox"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 7

    7.1 Zeno: How the Tortoise Beat Achilles
    7.2 Two Responses to Zeno: Infinitesimals and Finitism
    7.3 Thomson’s Lamp
    7.4 A Puzzle about Transition
    7.5 Democritus’ Cone
    7.6 Atoms of Space and Time

"LePoidevin (Robin) - Does Time Pass?"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 8

    8.1 The Mystery of Passage
    8.2 McTaggart’s Proof of the Unreality of Time
    8.3 First Response: Presentism
    8.4 Second Response: the B-Theory
    8.5 Why Is There Only One Present?

"LePoidevin (Robin) - The Cinematic Universe"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 9

    9.1 Muybridge’s Horse and Zeno’s Arrow
    9.2 No Motion at an Instant?
    9.3 No Motion in the Present?
    9.4 Zeno and the Presentist

"LePoidevin (Robin) - Interfering with History"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 10

    10.1 The Lost Days
    10.2 The Alterability of the Past
    10.3 Dilemmas of the Time-Traveler
    10.4 Causation1 in Reverse

"LePoidevin (Robin) - Other Times and Spaces"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 11

    11.1 Probability and the Multiverse
    11.2 Branching Space
    11.3 Objections and Consequences

"LePoidevin (Robin) - The Arrows of Time"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Chapter 12

    12.1 The Hidden Signpost
    12.2 Three Arrows, and Why Things Fall Apart
    12.3 The Mind’s Past
    12.4 The Seeds of Time
    12.5 Parallel Causes
    12.6 Is Time Order Merely Local?
    12.7 Are Causes Simultaneous with Their Effects?
    12.8 A Sense of Direction in a Directionless World

"LePoidevin (Robin) - Travels in Four Dimensions: Concluding Thoughts"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Conclusion

Full Text (cut as indicated) [… snip …].
  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:
    1. 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.
    2. 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?)
    3. 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.
    4. 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 …].

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

In-Page Footnotes ("LePoidevin (Robin) - Travels in Four Dimensions: Concluding Thoughts")

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: 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?

"LePoidevin (Robin) - Mr Dunne's Dream and Other Problems"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003, Appendix

    A.1 Mr. Dunne’s Dream
    A.2 Measuring Time
    A.3 A Directionless World?
    A.4 Alien Numbers
    A.5 Representing the Fourth Dimension
    A.6 Archytas at Time’s Beginning
    A.7 Democritus’ Trick
    A.8 The Plattner Case
    A.9 The Elusive Present
    A.10 Suspicious Success
    A.11 An Infinite Regress
    A.12 Does Time Fly in Two Dimensions?

COMMENT: A set of puzzles for the reader

"LePoidevin (Robin) - Travels in Four Dimensions: Further Reading"

Source: LePoidevin - Travels in Four Dimensions: The Enigmas of Space and Time, 2003

Further Reading1 (Full Text)
  1. The Measure of All Things
    A gripping account of the pursuit of accuracy in timekeeping, centred on John Harrison's long wrestle, between 1727 and 1773, with the problem of devising a timepiece that would remain perfectly accurate at sea (thus allowing sailors to calculate their longitude) is Sobel (1996). O. K. Bouwsma's story is reprinted in Westphal and Levenson (1993). A classic defence of conventionalism about metric is Reichenbach (1958). A very helpful introduction to the debate between conventionalism and objectivism, which dearly sets out the different responses to the threat of inaccessible facts and proposes a novel solution is Newton-Smith (1980, Ch. VII). The connection between the debate over metric and the laws of motion is discussed in Van Fraassen (1980, Ch. III.2).
  2. Change
    Aristotle's treatment of the relationship between time and change, and his definition of time, can be found in his Physics, Bk. IV: Hussey (1983). For a helpful discussion of Aristotle's account, see Lear (1982). For Leibniz's views on time, see his correspondence with Samuel Clarke: Alexander (1956), and his New Essays on Human Understanding: Remnant and Bennett (19812). A well-known and fascinating thought-experiment3 designed to show the limitations of verificationist attacks on the notion of temporal vacua is Shoemaker (19684). For a discussion and extension of this thought-experiment5, see Newton-Smith (1980, Ch. II). Another argument for the possibility of time without change is discussed in Lucas (1973). A detailed discussion of relationism about time is Hooker (19716). Butterfield (19847) is salutary and required reading for anyone who is tempted to analyse times in terms of possible events.
  3. A Box with No Sides?
    Descartes's letters to Princess Elizabeth and Henry More can be found in Cottingham, Stoothoff, Murdoch, and Kenny (19918). An excellent historical survey of theories of space is Jammer (1969); Ch. 2 explores the theological dimensions of the debate over absolute space. For Aristotle's arguments concerning the void see the Physics, Bk. III (Hussey 19839). For discussion of Aristotle's and other ancient views on the void, see Sorabji (1988). A collection of classic readings on space is Huggett (1999); see especially Ch. 4 for extracts from Aristotle and commentary, and Ch. 7 for extracts from Newton's Principia (including the famous bucket experiment) and commentary. For Leibniz's views on the relationship between space and the objects within it, see again his correspondence with Samuel Clark: Alexander (1956); extracts and commentary in Huggett (1999, Ch. 8). The relationist — absolutist debate, and its relationship to the debate over absolute motion, are discussed in van Fraassen (1980, Ch. IV.I), and in greater detail by Nerlich (1994a) and Dainton (2001). See also Hooker (1971). A very detailed, but difficult, examination of the debate is Earman (1989).
  4. Curves and Dimensions
    For an accessible introduction to non-Euclidean geometry, see Sawyer (1955, Ch. 6). For the history of geometry see Boyer (1968), especially Chs. VII and XXIV. See also van Fraassen (1985, Ch. IV.a). The significance of non-Euclidean geometry for the debate between the absolutists and relationists is very clearly articulated in Nerlich (1991) and Dainton (2001). Kant's argument concerning incongruent counterparts is presented in Kant (1768); an extract is reprinted in Huggett (1999, Ch. ii), which also provides commentary. See also Earman (1989, Ch. 7) and Walker (1978, Ch. IV), which provide useful exegeses of Kant's text, and Nerlich (1994a, Ch. 2), a particularly important discussion that reconstructs the argument in terms of chirality (he uses the term enantiomorphism), and on which my account is based. A very entertaining discussion of chirality and its philosophical and scientific interest is Gardner (1982).
  5. The Beginning and End of Time
    For a brief introduction to Big Bang cosmology, see Hawking (198610). Ancient arguments concerning creation and the beginning of time are discussed in Sorabji (1983, pt. III). Newton-Smith (1980) exploits number analogies in defining the beginning and end of time, discusses a number of arguments against a beginning, including ones put forward by Aristotle and Kant, and also considers the causal anomalies involved in a beginning and end of the universe. Kant's argument concerning infinity is explored in Moore (1990). Craig and Smith (1993) is an extended, and highly dialectical, exploration of the interaction between Big Bang cosmology, philosophy, and theology, including an assessment of Kant's arguments. For some background to Eliot's Four Quartets, see Tamplin (1988). Closed, or cyclic time is discussed at some length in Newton-Smith (1980, Ch. III). For historical sources for cyclic time and history, see Sorabji (1983, Ch. 12).
  6. The Edge of Space
    Archytas' argument, and related arguments, against the edge of space are presented and discussed in Sorabji (1988i, Ch. 8). He also discusses arguments concerning extracosmic space. The difficulties the notion of an edge pose for our understanding of motion are pointed out in Nerlich (1994b). For discussion of Kant's First Antimony, see Broad (1978, Ch. 5) and Bennett (1974, Ch. 7). For readings on Aristotle's distinction between the actual/potential infinite distinction, see references under Chapter 7 below. An attempt to make sense of Kant's obscure argument from incongruent counterparts to the ideality of space is made in Broad (1978, Ch. 2) and van Cleve (1999). Poincare's thought experiment11 is presented in his (195212); extracts and discussion in Huggett (1999, Ch. 13). For a discussion of closed space see Sorabji (1988, Ch. 10).
  7. Infinity and Paradox
    The sources of Zeno's arguments can be found in Kirk, Raven, and Schofield (198313), Barnes (197814) and Huggett (1999). For discussion, see Huggett (1999), Owen (1957-815), Salmon (1970), Sorabji (1983), and Sainsbury (198816). For Aristotle's actual — potential infinite distinction, and its application to Zeno's paradoxes, see Lear (198817). The source of Thomson's Lamp is Thomson (195418). For discussion, see Sainsbury (1988). See also Clark and Read (198419), which argues against the possibility of completing uncountably many tasks in a finite time. Moore (1990) is a particularly helpful and wide-ranging discussion, covering all the above. It also presents an historical overview on theories of infinity and defends a version of finitism. The transition paradox is discussed at length in Sorabji (1983). Part V of Sorabji also discusses Ancient atomism, on which see also Barnes (1978) and Kirk, Raven, and Schofield (1983), for sources and commentary. Lloyd (1982, Ch. 4) shows how ancient atomism developed as a response to the problem of change (the problem being that change appears to involve something coming into being from nothing).
  8. Does Time Pass?
    Broad's views on time underwent an interesting development. Compare his (1923) discussion, from which the passage quoted is taken, with his encyclopaedia article (1921) and his exhaustive discussion (193820) of McTaggart's proof. McTaggart's21 proof originally appeared in his (190822), but see also the revised presentation in his (192723). For discussion and partial defence of the proof see Dummett (196024). A more critical discussion, which focuses on the legitimacy, or otherwise, of expressions such as ‘is present in the past', is Lowe (198725). A reconstruction in terms of token-reflexives is presented in Mellor26 (198127) and (199828). An unusual solution to the paradox, in terms of possible worlds, is presented in Bigelow (199129), and criticized in Oaklander (199430). Early statements of presentism are Lukasiewicz (1967) and Prior (197031). Versions of presentism are defended in Bigelow (199632) and Zimmerman (199833). The doctrine, in its various forms, is examined in Dainton (2001, Ch. 6). The B-theory is defended in Smart (1980), Mellor (1981, 1998), and Oaklander (1984). The most extensive critiques of the B-theory to date are Smith (1993) and Craig (2000). See also Teichmann (1995). For an important collection of papers on McTaggart's argument, the B-theory, and related issues, see Oaklander and Smith (1994). One issue not discussed here is the extent to which the A-theory, and the associated view of the future is unreal, is compatible with the Special Theory of Relativity. For arguments in favour of the view that they are incompatible, see Putnam (196734), Mellor (197435), and Nerlich (1998). For attempts at reconciliation, see Smith (1993, 199836), Mauro Dorato (1995), and Craig (2001). See also Dainton (2001, Chs. 16 and 17) for an introduction to Special Relativity and its philosophical consequences.
  9. Change again: Zeno's Arrow
    For sources and commentary on the Arrow, see Lee (1936), Barnes (1982), Kirk, Raven, and Schofield (1983), and Huggett (1999). For discussion, see Ross (1936), Owen (1957-8), Vlastos (196637), Grunbaum (1967), Salmon (1970), and Sorabji (1983). The suggestion that the Arrow is best understood in terms of motion in the present was made by Jonathan Lear, and his discussion is particularly helpful: see Lear (198138, 1988). The second reconstruction presented here is based (with a few details altered) on his account. For discussions of presentism, see the references under Chapter 8 above.
  10. Interfering with History
    A very readable history of the calendar, with an account of the shift from the Julian to the Gregorian calendar, the problems that it was intended to solve, and the consequences, is Duncan (1998). For an introduction to the issues involving the reality/unreality of the future, and its bearing on our status as free agents, see Smith and Oaklander (1995). Discussion of these issues as they arose in ancient times is provided by Hintikka (1973), Sorabji (1980), and Lucas (198939). The last of these defends a version of the ‘open future' account. On the reality of the past, see Dummett (196940), who defends an 'anti-realist' account. For a discussion of the philosophical dimensions of the passage from Orwell's Nineteen Eighty-Four quoted here, see Wright (1986). On time travel41, see Harrison (197142), Lewis (197843) (introduces the external — personal time distinction, and argues that the time-traveller is a free agent in the sense that anyone is a free agent), MacBeath (198244) (explores the causal anomalies of time travel)45, and Ray (1991, Ch. 8) (puts time travel46 in the context of spacetime physics). For a defence of the coherence of backwards causation47, see Dummett (196448), and arguments for its incoherence, see Mellor (1981, 1998). Mellor's argument in his (1981) is criticized in Riggs (199149) and Weir (198850) (who puts the issue in the context of closed, or cyclic, time).
  11. Other Times and Spaces
    An excellent and very readable discussion of the multiverse hypothesis, its theistic rival, and the probabilistic reasoning used to motivate them, is Leslie (198951). Leslie also briefly discusses the two-slit experiment. The branching spaces interpretation of the experiment is closely related to what is often called the ‘many worlds' interpretation of quantum mechanics52 (although there is considerable dispute over the correct interpretation of the interpretation). For a philosophical introduction to quantum mechanics53, see Lockwood (1989). The many worlds, or ‘many minds', interpretation was the subject of a symposium, the (highly technical) papers being published in the June 1996 issue54 of the British Journal for the Philosophy of Science. Disunified space is the subject of a classic paper by Quinton (196255). In that paper Quinton puts forward a fantasy designed to show that, under certain conditions, we might regard our experience as providing evidence for disunified space. He rejects the temporal parallel, however, as does Swinburne (1981, Ch. 10), which nevertheless considers it in sympathetic detail (see also Ch. 2 on disunified space). The temporal parallel is given support in Newton-Smith's (1980) discussion of Quinton's thought experiment56. See also Hollis (196757).
  12. The Arrows of Time
    An important recent discussion of the arrow(s) of time, which clearly presents the view that the classic way of articulating the problem of the direction of time is ill-posed, and which defends a novel treatment of causation58, is Price (1996). The issue of what kind of reduction is appropriate in reductionist theories of direction is taken up by Sklar (198159), who argues in favour of a theoretical reduction rather than one in terms of meaning. For a discussion of the psychological arrow, see Newton-Smith (1980). The thermodynamic and causal arrows are discussed in Dainton (2001), which also tackles the tricky issue of how the asymmetry of causation60 is to be explained. The causal analysis of time is subject to lengthy and detailed criticism in Sklar (197461). Tooley (1997) is an intriguing attempt to explain temporal and causal asymmetry in terms of the unreality of the future, but appealing only to B-series facts. The importance of causation62 in explaining our experience of time, and in particular its direction, is well articulated in Mellor (1981, 1998). On the issue of dimensionality: a two-dimensional model for time's passage is presented in Schlesinger (198263) and criticized by MacBeath (1986). For an ingenious discussion of what might count as evidence for two-dimensional time, see MacBeath (199364).
  13. Concluding Thoughts
    The human significance of time, as revealed by the history of Western and Eastern thought about time, is discussed in Fraser (1968, pt. I). On the significance of the A-theory / B-theory debate for our views on death, see Le Poidevin (199665). The reconciliation of human freedom with the B-theory is the subject of Oaklander (1998). The ethical dimension of the debate is explored in Cockburn (1997, 1998). The issue of whether the B-theory requires a revision in our ordinary views of persistence through time has been the subject of much debate. For arguments that it does, see Lowe (1998a, 1998b); that it does not, Mellor (1981, 1998).

In-Page Footnotes ("LePoidevin (Robin) - Travels in Four Dimensions: Further Reading")

Footnote 1:
  • This requires reference to the Le Poidevin’s Bibliography.
  • Where I have the book / paper, I’ve provided links to my database (on the first occasion).
  • Where I don’t, and don’t care, you’ll have to refer to the book to find the full reference.
Footnote 2: See "Leibniz (Gottfried), Remnant (Peter), Bennett (Jonathan) - New Essays on Human Understanding".

Footnote 4: See "Shoemaker (Sydney) - Time Without Change".

Footnote 6: See "Hooker (Clifford A.) - The Relational Doctrines of Space and Time".

Footnote 7: See "Butterfield (Jeremy) - Relationism and Possible Worlds".

Footnote 8: See "Descartes (Rene), Cottingham (John), Stoothoff (Robert), Murdoch (Dugald), Kenny (Anthony) - The Philosophical Writings of Descartes Vol III - The Correspondence".

Footnote 9: See "Aristotle, Waterfield (Robin) & Bostock (David) - Physics", as an alternative.

Footnote 10: See "Hawking (Stephen) - A Brief History of Time - From the Big Bang to Black Holes".

Footnote 12: See "Poincare (Henri) - Science and Hypothesis".

Footnote 13: See "Kirk (G.S.), Raven (J.E.) & Schofield (M.) - The Presocratic Philosophers".

Footnote 14: See "Barnes (Jonathan) - The Presocratic Philosophers".

Footnote 15: See "Owen (G.E.L.) - Zeno and the Mathematicians".

Footnote 16: See "Sainsbury (Mark) - Paradoxes".

Footnote 17: See "Lear (Jonathan) - Aristotle - The Desire to Understand".

Footnote 18: See "Thomson (James F.) - Tasks and Super-Tasks".

Footnote 19: See "Clark (Peter) & Read (Stephen) - Hypertasks".

Footnote 20: I only seem to have "Broad (C.D.) - McTaggart's Arguments Against the Reality of Time" by Broad on this topic. This is an extract from An Examination of McTaggart's Philosophy.

Footnote 21: Some unmentioned items on McTaggart are:- Footnote 22: See "McTaggart (J. McT. E.) - The Unreality of Time".

Footnote 23: Footnote 24: See "Dummett (Michael) - A Defense of McTaggart's Proof of the Unreality of Time".

Footnote 25: See "Lowe (E.J.) - The Indexical Fallacy in McTaggart's Proof of the Unreality of Time".

Footnote 26: Why not "Mellor (D.H.) - McTaggart's Proof".

Footnote 27: See "Mellor (D.H.) - Real Time".

Footnote 28: See "Mellor (D.H.) - Real Time II".

Footnote 29: See "Bigelow (John) - Worlds Enough For Time".

Footnote 30: See "Oaklander (L. Nathan) - Bigelow, Possible Worlds and the Passage of Time".

Footnote 31: See "Prior (Arthur N.) - The Notion of the Present".

Footnote 32: See "Bigelow (John) - Presentism and Properties".

Footnote 33: See "Zimmerman (Dean) - Temporary Intrinsics and Presentism".

Footnote 34: See "Putnam (Hilary) - Time and Physical Geometry".

Footnote 35: See "Mellor (D.H.) - Special Relativity and Present Truth".

Footnote 36: See "Smith (Quentin) - Absolute Simultaneity and the Infinity of Time".

Footnote 37: See "Vlastos (Gregory) - A Note on Zeno's Arrow".

Footnote 38: See "Lear (Jonathan) - A Note on Zeno's Arrow".

Footnote 39: See "Lucas (J.R.) - The Future - An Essay on God, Temporality and Truth".

Footnote 40: See "Dummett (Michael) - The Reality of the Past".

Footnote 42: See "Harrison (Jonathan) - Dr. Who and the Philosophers or Time-Travel For Beginners".

Footnote 43: See "Lewis (David) - The Paradoxes of Time Travel".

Footnote 44: See "MacBeath (Murray) - Who Was Dr Who's Father?".

Footnote 48: See "Dummett (Michael) - Bringing About the Past".

Footnote 49: See "Riggs (Peter J.) - A Critique of Mellor's Argument against 'Backwards' Causation".

Footnote 50: See "Weir (Susan) - Closed Time and Causal Loops: A Defence against Mellor".

Footnote 51: See "Leslie (John) - Universes".

Footnote 54: Footnote 55: See "Quinton (Anthony) - Spaces and Times".

Footnote 57: See "Hollis (Martin) - Times and Spaces".

Footnote 59: See "Sklar (Lawrence) - Up and Down, Left and Right, Past and Future".

Footnote 61: See "Sklar (Lawrence) - Space, Time and Spacetime".

Footnote 63: See "Schlesinger (George N.) - How Time Flies".

Footnote 64: See "MacBeath (Murray) - Time's Square".

Footnote 65: See "LePoidevin (Robin) - Arguing for Atheism: An Introduction to the Philosophy of Religion".

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