1 See, among many writers who make such a distinction, Grünbaum, Adolf ‘The Pseudo-Problem of Creation in Physical Cosmology,’ Philosophy of Science 56 (1989) 373-94CrossRefGoogle Scholar. He distinguishes the subclass of models with ‘a cosmic time interval that is closed at the big bang instance t = 0, and furthermore, this instant had no temporal predecessor’ (389) from the subclass of models which differ from the former ‘by excluding the mathematical singularity at t = 0 as not being an actual moment of time’ (391).

2 See the discussion in Wang, H. From Mathematics to Philosophy (New York: Humanities Press 1974), 86Google Scholar. He contrasts the ‘intuitive concept’ of a length (i.e. what we are talking about when we talk about length in normal contexts) and the ‘set-theoretical concept,’ that concept with which set theory finds it useful to operate. In terms of the intuitive concept, he writes: ‘Summing up all the points, we still do not get the line; rather the points form some kind of scaffold on the line.’ ‘When we use the set-theoretical concept and try to assign a length to any arbitrary set of points on the line, we lose touch with the intuitive concept.'

3 I shall be arguing not merely that talk about instants is reducible to talk about periods, but that talk about periods is reducible to talk about (actual or possible) events. Hence whether time is continuous, dense, discrete, or none of these will depend on the physics operative in each spatio-temporal region. It is plausible to maintain that for our region any period of time of unit length (as measured by the perfect clocks of that region) can be divided without remainder into two periods, the first of which is a real number (i.e. real fraction) of units — in as many ways as there are real fractions. It is plausible because, for any such latter period of time, there are (or, it is physically possible, can be) events which last for that period — given our normal physics. A body falling with an acceleration of 32 ft/sec² , for example, takes 2 seconds to fall 32 ft. One can certainly devise a mechanics equally compatible with observations in which all events last only rational numbers of units — Newton-Smith, W.H. (The Structure of Time [London: Routledge and Kegan Paul 1980]), 121-6Google Scholar outlined such a mechanics, which he called Notwen's mechanics, in contrast to Newton's mechanics. But Notwen's mechanics makes different claims from Newton's — it claims that there is for every finite collection of events a level of precision of measurement at which every event coincides exactly in length with some rational number of unit measurements of a perfect clock — while Newton's theory denies this. Even if, as Newton-Smith claims, it is not practically possible to make an experimental test between the two theories, they do make different coherent claims about the unobservable; and the greater simplicity of Newton's theory gives reason to suppose that Newton's theory is more probably the true one. And when we replace Newton's mechanics by post-Newtonian physics, the same consideration supports the view that time is continuous in our region.

4 For fuller description of the operation of these criteria see Swinburne, R. Space and Time, second edition (London: Macmillan 1981), 177-80Google Scholar.

5 I ignore the point made by the normal interpretation of Relativity Theory that sameness of temporal interval is relative to frame of reference, because it will not affect our topic. If the beginning of the Universe is a finite temporal interval from the present by one frame of reference it will be so by all frames moving with a finite velocity relative to the former. In any case the ‘cosmic time’ used by cosmologists limits the possible frames of reference to the galaxy-clusters which make up the Universe. For justification of their use of the cosmic time scale see ibid. 193-6.

6 See (e.g.) Maudlin, Tim ‘Buckets of Water and Waves of Space? Why Space-Time is probably a Substance,’ Philosophy of Science 60 (1993) 183–203CrossRefGoogle Scholar.

7 Philosophers of science sometimes write about ‘psychological time’ or ‘the time of our experience’ and contrast this with ‘physical time,’ which is what in effect I have been discussing. But talk about ‘psychological time’ turns out to be talk about how long experiences seem to their subjects to last, i.e. how long (by public clocks) subjects would judge their experiences to have lasted, if they had no other subsequent information — e.g. if they had not seen what the clock on the wall registered at the end of the experience. So the concept of psychological time is not a separate concept from that of ordinary (physical) time — it is simply apparent physical time.

8 ‘Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year’ (Newton, I. Principia, Scholium to Definition 8Google Scholar).

9 E.A. Milne once suggested that dynamic and electromagnetic processes were best measured by temporal scales that were related to each other asymptotically. See his Kinematic Relativity (Oxford: Clarendon 1948) passim and especially 224-5.

10 My claim that there is a true metric only where and when and to the extent to which there are perfect clocks, is open to the more radical objection that even if there are perfect clocks, it would still be a matter of ‘convention’ whether we adopt a given temporal metric. Grünbaum claims that — given the ‘intrinsic metric amorphousness’ of space and time — all (non-trivial) ascriptions of temporal equality are ‘conventional,’ so that it is as near to the truth to say that one interval is equal to another as that it is twice as large. It would follow that one would be equally at liberty to adopt a convention which had the consequence that clocks speeded up asymptotically relative to some other convention, in consequence of which by the first convention the Universe would be of finite age, whereas by the second it would be of infinite age. My answer to this radical objection is the one given in the text that our very understanding of a temporal interval is fixed by the kind of criteria which we use to determine that two intervals are equal. Grünbaum holds that the ‘intrinsic metric amorphousness’ of Space or Time is a contingent thesis, dependent on their ‘atoms’ not being ‘granular.’ I do not accept this, holding that whether or not there are ‘granular’ temporal ‘atoms’ is quite irrelevant to the issue; see Swinburne, review of Grünbaum's, ‘Geometry and Chronometry in Philosophical Perspective,’ British Journal for the Philosophy of Science 21 (1970) 308-11CrossRefGoogle Scholar; and Grünbaum's, reply in his ‘Space, Time and Falsifiability,’ Philosophy of Science 37 (1970) 469–588 (see esp. 567-76)Google Scholar.

11 ‘The beginning is an existence which is preceded by a time in which the thing is not’ (Kant, I. Critique of Pure Reason. Smith, N. Kemp trans. [London: Macmillan 1964] B 455, First antinomy, antithesis)Google Scholar.

12 Smith, Quentin makes the crucial distinction between these two ways of understanding ‘beginning’ in his’ A New Typology of Temporal and A temporal Permanence,’ Nous 23 (1989) 307-30CrossRefGoogle Scholar.

13 Since they would operate somewhat differently from normal forces, we may wish to call them ‘influences’ rather than forces. The issue whether there could really be universal ‘forces’ was the subject of some controversy in the 1960s — see R. Swinburne Space and Time ch. 4.

14 For this reason I reaffirm the soundness of a very quick argument for the conclusion that time has no beginning or end given in earlier writing of mine: ‘Either there will be swans somewhere subsequent to a period Tor there will not. In either case there must be a period subsequent to T, during which there will or will not be swans. By an analogous argument any period which has a beginning must have been preceded by another period, and hence time is necessarily unbounded,’ Swinburne, Richard Space and Time, second edition (London: Macmillan 1981) 172Google Scholar. Newton-Smith, W.H. The Structure of Time (London: Routledge and Kegan Paul 1980) 97–101Google Scholar, objected that my argument begged the question. The second disjunct of the first sentence quoted, ‘There will not be swans subsequent to T’ could, he claimed, be true either because there is a period subsequent to T in which there are no swans, or because there is no period subsequent to T. I had not, he claimed, ruled out the latter possibility. Analogously for the similar argument to time not having a beginning, Newton-Smith claims, I have not ruled out there being no period earlier than T. But my covert assumption was that there is no content to a claim that something began to exist except in terms of there being a period of time before it existed. So I reaffirm the soundness of my earlier argument, but acknowledge a shortage (in the book in which it appeared) of arguments to show its soundness which I hope that I have remedied in the present paper. An objection to my argument similar to Newton-Smith's, as well as to various a priori arguments adduced by other writers for time not having a beginning are given in Smith, Quentin ‘On the Beginning of Time,’ Nous 19 (1985) 579-84CrossRefGoogle Scholar. Quentin Smith does not mention Aristotle's discussion. Aristotle has an a priori argument for time not having a beginning, similar to mine, from the inconceivability of there being any period or instant of time not being preceded by another one; and he comments that, with one exception (Plato), ‘it seems that all thinkers agree that time did not come into existence’ — Physics 251b.

15 It is sometimes suggested that a law by itself might explain the beginning of the Universe without needing to act on any events or substances, in order to do so. The suggestion (to caricature it a bit) is that there could be a law of the form ‘nothing necessarily gives rise to something.’ In practice when a law looking something like this is seriously suggested, it turns out that ‘nothing’ never really means nothing; it is some sort of empty space in a quiescent state. If ‘nothing’ was really nothing, and the law really was ‘nothing necessarily gives rise to something,’ then since there are an infinite number of possible universes (each non-existent at some past moment of time), not spatially related to each other, then by the law they must all have come into existence. In fact an infinite number of universes must be coming into existence at each moment of time. But that is to multiply entities beyond plausibility. A plausible law to explain the beginning of the universe would at least have to have the form ‘empty space necessarily gives rise to matter-energy,’ where’ empty space’ is not just ‘nothing,’ but an identifiable particular. In an important sense there must have been something there already, if anything is to evolve from it in a law-like way.

16 That instants are the instants they are in virtue of the events which occur at them or their temporal distance from actual events is a thesis argued by Forbes, Graeme ‘Time, Events and Modality’ in Poidevin, R. Le and MacBeath, M. eds., The Philosophy of Time (Oxford: Oxford University Press 1993)Google Scholar. He advocates a theory of an instant as constituted by the events which actually occur at it, or occur in a possible world which branches from the actual world (i.e. has the same history as our world up to a certain instant) at the same temporal distance from the branching point as the actual events; and the branching point is identified by the events which actually occur at it. My account in the text allows for instants to be picked out in possible worlds which converge with as well as in ones which diverge from the actual world. But in virtue of my first verificationist thesis I must give a reductive account of what it is for an event to occur at an instant.

17 Hence I am in a position to reject Newton-Smith's ‘Leibnizian argument’ purporting to show that ‘if the Universe had a beginning, so (plausibly) did time’ (Newton-Smith, 104-6). For if there was empty time before the beginning of the Universe, there would, he claims, be no explanation for the Universe beginning at the instant it did rather than at any other instant. The antithesis of Kant's first Antinomy has an analogous argument purporting to show that, since in his view, a beginning of the Universe would involve there being empty time before the Universe, and so no explanation of why the Universe began when it did, both time and the Universe must be everlasting (op cit., 8455). I claim, contrary to both writers, that there is nothing to be explained. There is, that is, nothing to be explained with respect to why it began when it did, although there remains something to be explained about why it began at all.