A New Typology of Temporal and Atemporal Permanence

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Philosophical literature reveals little by way of a detailed typology of permanence. The traditional typology is bare in its outline; it distinguishes between temporal and atemporal permanence, the former being vaguely characterized as existing at all times or existing now and forever and the latter as existing outside of time in unchangeable self-sameness. According to this standard typology, which originated with Plato (37-38), God and abstracta exemplify atemporal permanence and the universe or elementary particles exemplify temporal permanence, Atemporal permanence is sometimes further divided into eternality (which belongs to God alone) and mere timelessness (which belongs to abstracta). For example, the timeless subsistence of universals conceived by Husserl (1970, pp. 351-53), Frege (1950, P. 15) and Russell (1959, pp. 98-100) is a different type of atemporal permanence than the eternality conceived by Plotinus (III, 7), Boethius (V, 6) and Aquinas (a: I, 66; b: 1A, 10). Temporal permanence is often divided into two sorts, depending on whether or not time is bounded. If time is bounded, then temporal permanence is represented as existence from the beginning to the end of time or from now to the end of time. Advocates of the reality of this sort of temporal permanence include Bonaventura (1.1.1.2.1-6), Heidegger (1962, pp.476-79) and Vilenkin (1982). If time is unbounded, temporal permanence is conceived as temporal existence without beginning and end or from now into the infinite future. Champions of this sort of temporal permanence include Aristotle (203b), Swinburne (1981, p. 172) and Hoyle (1963, pp. 320-21).

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I believe this standardly (but not universally[1]) assumed typology of permanence is deficient both in precision and cogency, Regarding the former deficiency, the notion of temporal permanence and its division into bounded permanence and unbounded permanence conflates a number of different concepts. For one thing, the notion of existing now and at every earlier and later time is not the same notion as existing at every time (see sections 2-3). Secondly, the notion of having an infinite past or future is not the same notion as having a beginningless or endless temporal existence (see sections 5-6). Further, the notion of existing now and at every future time is not the same notion as existing now and in a future that contains no time that is later than every time at which one exists (see sections 4 and 7). These types of temporal permanence must be distinguished from one another.

Of equal importance are the problems of cogency. The conceptions of atemporal permanence are the most unsatisfactory in this regard. I will argue that the standard conceptions of the eternity of God and of the timeless existence of abstracta are incoherent. Eternity needs to be redefined as a manner of possessing a certain property F that temporal entities also possess, albeit in a different manner. As will become apparent, this redefinition entails that eternality is exemplifiable not only by God but also by flowers and humans (see section 8). Another problem with the standard typology is that it allows that atemporal permanence is actually (or at least possibly) co-instantiated with temporality; i.e., that there are (or can be) atemporally permanent entities and temporal entities in the same world. This view, I will maintain, is false; something can be atemporally permanent only if time does not exist (see section 9).

For these reasons, a new typology of temporal and atemporal permanence is needed. The unifying element of this typology is a general definition of permanence to which each of the types of permanence conform:

(D1) x is permanent = Df. There is no time t that is both later than x’s existence and separated from x’s existence by a finite number of nonoverlapping temporal intervals of equal length.

Admittedly, it is not intuitively obvious that D1 captures the essence of permanence; support for and clarification of D1 will appear in the following sections. In particular, it needs to be explained why the three qualifications ‘by a finite number of nonoverlapping temporal intervals of equal length’ are introduced (see section 6). But I can briefly indicate here why the definition is not “there is no time t that is both later than the time of x’s existence and separated from the time of x’s existence by a finite number of nonoverlapping temporal

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intervals of equal length”. If the phrase ‘the time of’ were included in the definition, the definition would be inapplicable to atemporal entities, which do not exist at a time. D1 as it stands is applicable to atemporal entities, since the existence of such entities is not related by a relation of earlier or later to any time.

The types of permanence distinguished in the following sections are expressed by the words or phrases ‘sempiternality’ (section 2), ‘omnitemporality’ (section 3), ‘everlastingness’ (section 4), ‘having an infinite past and future’ (section 5), ‘beginninglessness and endlessness in dine’ (section 6), ‘endless recurrence’ (section 7), ‘eternalness’ (Section 8) arid ‘mere timelessness’ (section 9). Theses words or phrases are used in a technical sense and do not necessarily con form to previous philosophical usage.

Besides defining these different types of permanence, I will (in regard to some of them) indicate whether there is empirical reason to think they are instantiated.

An entity x is sempiternal iff x exists at some time t and at every time earlier and later than t. X’s existence at t and at all earlier and later times may be understood in one of two ways, depending upon whether things are conceived as wholes of temporal parts or as continuants that lack such parts. On the former conception, a thing exists at t and at every earlier and later time iff for each of these times there is a distinct temporal part of the thing that is located at that time. If the thing is conceived as a continuant, it is sempiternal iff it was located at the present time, was located at each past time and will be located at each future time.[2]

A sempiternal entity begins and ceases to exist iff time simultaneously begins and ceases to exist. This can be understood more exactly if three notions of beginning and ceasing to exist are distinguished:

(a) X begins to exist if there is a time at which x does not exist and a later time at which x does exist, and x ceases to exist if there is a time at which it exists and a later time at which it does not exist. Neither time nor any sempiternal entity begins and ceases to exist in this first sense.

(b) Y begins to exist iff it is located at the earliest time and ceases to exist iff it is located at the latest time. If any sempiternal entity begins and ceases to exist, it does so in this second sense. But time does not begin and end in this sense, at least not if we wish to reserve the expression “located at a time” for items in time rather than time itself.

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(c) Z begins to exist iff (i) one of its parts is an interval I₁ such that every other interval of the same length is later than I₁ and (ii) prior to any one of the intervals that comprise its parts there are at most a finite number of nonoverlapping intervals of the same length. Z ceases to exist iff (i) one of its parts is an interval I₂ such that every other interval of the same length is earlier that I₂ and (ii) subsequent to any one of the intervals that comprise its parts there are at most a finite number of nonoverlapping intervals of the same length. If time begins and ceases to exist, it does so in this third sense.[3]

A sempiternal entity may begin and cease to exist in sense (b), but it need not; it may have an infinite past or future and may be beginningless or endless in the sense demarcated in section 6; but it will have such extensions 1ff time has the same extensions.

An entity x is omnitemporal iff x exists at every time. It might be thought that omnitemporality (as I here define it) is the same thing as sempiternality (as defined in the last section). It is not, for

Suppose that time branches and that time t is located on one of the branches; in that case, x will exist at t and at every time that is earlier or later than t but x will not exist at every time since it will not exist at any of the times on the other branches.[4] Analogous considerations hold for a converging time (in which at least. two separate time-series converge to become one time-series), a branching-reconverging time (in which time branches and the branches subsequently converge), and parallel time-series (in which no time in any of the time-series is earlier or later than or simultaneous with any time in any of the other time-series).

Is it possible for anything to be omnitemporal if time branches, converges or consists of several parallel series? I think it is plausible to hold (on one theory of properties) that some properties will be omnitemporal in such a case, but that no particulars will be. If size (or having a size) is a universal property that is spatiotemporally located wherever and whenever its instances (particulars that have a size) are located, and there exists at least one particular with a size at each time, then this property will exist at every time. For example,

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if time branches from the series S into the two series S₁ and S₂, then the property size will branch into S and S as well. But no particular will branch into both series or exist at all the times in S, S₁ and S₁.

This difference between particulars and universals has impor tant theological implications, at least if the relevant assumptions are granted. It may be argued

Suppose (3)-(6) are true and that in the late 19th century time branched into the two series S₁ and S₁, such that the branch containing ourselves—humans—is S₁ and God branches into S₂. Then Nietzsche’s statement that ‘God is dead’ would be literally and not merely sociologically true for us: God used to exist simultaneously with humans but no longer does. (This fanciful example is not merely a philosophical amusement; it arguably obtains in some possible world.)

But reflections upon branching time need not be confined to philosophical speculation on mere logical possibilities. It is not recognized or at least not widely recognized—not even by philosophers of science—that there currently exists physical evidence for branching time or at least for multiple time-series. This evidence is found in the vacuum fluctuation cosmogonies developed in the 1970s and 1980s. The cosmogonies of Tryon (1973), Brout, Englert and Gunzig (1978), Grishchak and Zeldovich (1982), Atkatz and Pagels (1982), Gott (1982) and Vilenkin (1982) postulate that the temporal history of our universe is only one of many—perhaps infinitely many—temporal histories that exist. Most of these theories postulate a background space (the vacuum) from which there emerges (via spontaneous ‘fluctuations’ of the background space) different and spatially separate universes, each with its own temporal history. The time of the background space is the stem of the tree of time and the temporal history of each of the emerging universes is one of its branches. Gott’s theory allows that some of these branches converge to form a single time-series—thus constituting (in respect to these branches) a branching-reconverging time. Vilenkin’s theory is unique in that it does not represent time as branching but as consisting of several (perhaps infinitely many) parallel time-series, each being the temporal history of a universe that erupts into being without cause “from literally nothing” (1982, p. 26) and subsequently returns to nothing.

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These vacuum cosmogonies are not idle constructions; we have good reason to believe that some type of vacuum cosmogony is sound, or at least so I have argued elsewhere (Smith 1986b). Given that this is the case, this creates new mysteries for theism. If the above- stated argument (3)-(6) is sound, and God exists, then the problem arises about which time-series God exists in and why he exists in that one rather than in some other one. And supposing that it could be shown that God does not exist in our time-series (e.g. by an ‘argument from the existence of evil’ in our time-series), but does exist in some other time-series, should we then live our lives at atheists or theists?

Something is everlasting if it exists at some time t and at every time later than t. Something can be everlasting without being scempiternal or omnitemporal, for if an everlasting entity exists at t it exists at every time later than t but need not exist at every time earlier than t. To say of an (alleged) indestructibly fashioned artifact or an (alleged) divinely fashioned immortal soul that ‘it is everlasting’ commits one to the thesis that there is no time later than every time at which the artifact or soul exists but not to the thesis that there is no time earlier than every time at which the artifact or soul exists.

There are two types of everlastingness. An entity x instantiates the first type iff it exists at some time t and at every later time and there is no end to x’s existence (i.e. future time is endless). This is unbounded everlastingness. An entity y has bounded everlastingness iff it exists at some time t and at every later time and there is an end to y’s existence (i.e. time comes to an end).

There is no type of permanence corresponding to everlastingness that is instantiated by entities that exist at t and at every time earlier than t but are succeeded by a time that is finitely later than the last time at which the entities exist. (The reason I say ‘finitely later’ is explained in section 6.) If something exists at each past time but will end in a finite time from now such that there is time after its end, then this entity is not permanent in any sense. This is entailed by the definition of permanence given in section 1, which implies that x is permanent only if there is no time fmitely later than x’s existence, This also conforms to our intuitions. Suppose that some particles have always existed but will come to an end in the uniquely high temperatures of a future phase of our universe, such that this phase is not the last one but is succeeded by others. It would then be counterintuitive to say of these particles that they exist permanently.

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There are some entities that actually instantiate the category of everlastingness but which are neither sempiternal nor omnitemporal, viz., photons. Photons did not exist during a very early phase of our universe, the so-called ‘Planck era’, which occupies the earliest interval of 10^-43 second that belongs to the temporal history of our universe. But there is good reason to think that they exist at every time later than the Planck era, even if the future of our universe is infinite (see Barrow and Tipler 1978).

Of course everlastingness does not exclude sempiternality and omnitemporality. Every entity that is omnitemporal is also sempiternal and everlasting. The results of the last section suggest that the property of size is an example of such an entity.

Something has an infinite past and future iff some or all the times at which it exists can be mapped onto a set with the order type w+ w*. The set of negative and positive integers in their natural

order has this order type and is mapped onto the entity’s temporal history so that the number 0 corresponds to the present interval, as follows:

If the temporal history of an entity maps onto this set, then each hour (or day, year, century, etc.) of the entity’s existence is earlier than some immediately adjacent hour and later than some other immediately adjacent hour, and before and after each hour of its existence there are an infinite (aleph-zero) number of hours at which it also exists. The mapping of the entity’s temporal history onto this set obeys the rule that when the present hour h₀ that corresponds to 0 becomes past each hour earlier than h₀ is reassigned to the next largest negative number, such that the hour h_-1 corresponding to -1 is reassigned to -2, the hour h_-2 corresponding to -2 is reassigned to -3, and so on. This opens up a position corresponding to -1 to which the newly past hour h₀ can be assigned. All the hours later than h are likewise assigned to the next smallest positive number, such that h is reassigned to 0, h to 1, and so on. It has been objected by some philosophers, most articulately by Craig (1979, pp. 65-110), that this mapping procedure is in coherent or at least conceptually problematic, but for reasons given in (Smith 1987b) I believe their objections can be met.

Is there reason to think any entity has an infinite past and future? There are no compelling reasons, but if there is reason to think

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that our universe fluctuated from a background space that extends infinitely into the past, and if there is reason to think that our universe will expand forever (there is evidence tentatively suggesting this; see (Schramm and Steigman 1981) and (Davis, Jonry, Huchra and Latham 1980)), then we may allow (on the view of properties discussed in section 3) that there is reason to think that the property of size has an infinite past and future. I think it is much more doubtful, however, that there is any particle that has this type of permanence, for cosmogonic theory disallows particles from continuing from the time of the background space into the time of our universe. The particles in our universe come into existence only after the beginning of the fluctuation. What possesses the property of size at times when there are no particles are regions of space.

An interesting claim about the property of having an infinite past and future is that this property does not entail the property of being sempiternal or the property of being everlasting, at least if these two properties are defined in the way I defined them in the previous sections. Something can exist at the present time t and at an infmite number of past times and at an infinite number of future times without existing at every time earlier than t and without existing at every time later than t. This claim is supported in the next section.

It is usually assumed that a temporal entity is beginningless and endless if it has an infinite past and future and hence that the fifth type of permanence (beginninglessness and endlessness in time) collapses into the fourth type (having an infinite past and future). I believe this is mistaken and that something can have an infinite past and future and nonetheless begin or cease to exist. It is logically possible for time to have the order type w*+ w*+ w + w, that is, the order type of the set S₀:

Suppose that some entity x exists at all and only the times that map onto the subsequence S₁ of S₀:

such that all the times that correspond to the members of S₁ are later than all the times that correspond to the members of the subsequence

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It will then be true of x, which presently exists at the time corresponding to 0, that it has an infinite past and future and yet both begins and ceases to exist. This does not mean that x begins and ends in the sense D2:

(D2) x begins and ends = Df. (a) There is some interval T of a certain length at which x exists such that T is the first interval of that length at which x exists; (b) there is some interval U at which x exists such that U is the last interval of that length at which x exists.

X does not begin or end in this sense since for each interval of a certain length at which it exists there is both an earlier interval and a later interval of that length at which x also exists. The intervals at which x exists map onto S₁ and there is no first or last member of S₁. However, x does begin and end in a second sense:

(D3) x begins and ends Df. (a) There is some interval V at which x does not exist and which is earlier than every interval at which x exists; (b) there is some interval W at which x does not exist and which is later than every interval at which x exists.

X exists at all the intervals corresponding to S₁ but at none of the earlier intervals corresponding to S₂ and at none of the later intervals corresponding to S₃.

The category of beginninglessness and endlessness in time is thus a different category of permanence than the category of having an infinite past and future. Something is beginningless and endless in time iff it exists in time and is such that neither D2 nor D3 apply to it. This is the proper or absolute sense of beginninglessness and endlessness in time, since, absolutely speaking, something is not beginningless and endless if D3 applies to even though D2 does not. One may say, relative to some temporal subsequence with the order type w*+ w in which some entity x is located, that x is beginningless and endless if it exists at every time in this subsequence (even though it does not exist at infinitely earlier or later times), but this is merely a beginninglessness and. endlessness in that subsequence, not in the entire temporal sequence itself.[5]

It is because of this distinction between having an infinite past and future and beginninglessness and endlessness that the definition of permanence in Section 1 included the qualification that some entity is permanent only if its existence is not earlier than a finitely distant time t. The definition D1, we recall, is

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(D1) x is permanent = Df. There is no time t that is both later than x’s existence and separate from x’s existence by a finite number of nonoverlapping temporal intervals of equal length.

If an entity has an infinite past and future but will end, then there is a time t later than its existence but t is separated from its existence by an infinite number of nonoverlapping intervals of equal length. Manifestly, there is an aleph-zero number of nonoverlapping equal intervals separating each interval corresponding to a member of S₁ from each interval corresponding to a member of S₃. This qualifica tion conforms, I believe, to our intutitions about permanence. We would ascribe permanence to any entity that exists for an infinite number of nonoverlapping equal intervals subsequent to the pre sent interval, regardless of whether or not there were later times not including this entity that were separated from each interval at which it existed by an aleph-zero number of intervening nonoverlap ping and equal intervals.

The introduction of ‘finite’ into D1 requires that the qualifications ‘nonoverlapping’ and ‘equal length’ also be introduced, for other ways of measuring intervals allow for the unwelcome inference that lightning flashes and mosquitoes are permanent. If time is dense or continuous, it is true of a lightning flash “that there is no time t that is both later than its existence and separated from its existence by a finite number of temporal intervals” if intervals are measured by certain standards of unequality or overlappingness. For example, the time t₁ that is 10 seconds later than the lightning flash is separated from the flash by an infinite number of nonoverlapping intervals each of which is ½ the length of its precedessor. Subsequent to the flash there is a 5 second interval, subsequent to this a 2 ½ second interval, subsequent to this a 1 ¼ second interval, and subsequent to this an infinite number of proportionately decreasing intervals before t₁. Further, there is also an infinite number of intervals separating t₁ from the flash on the following way of measuring over lapping but equal intervals. Consider a 1 second interval A that begins 2 seconds after the flash. Each instant belonging to A but for the one constituting the end of A is the endpoint of a distinct 1 second interval that partly overlaps A. Since there is a denumerably (if time is dense) or nondenumerably (if time is continuous) infinite number of instants in A, there is an infinite number of partly overlap ping 1 second intervals that separate t₁ from the flash. (D1) excludes these unwelcome sorts of ‘infinite temporal distances’ by stipulating that the intervals be equal and nonoverlapping.

A few words may now be said in defence of the coherency and empirical applicability of this fifth category of permanence, beginning lessness and endlessness in time. The main difference between a

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time-series with the order type w*+ w*+ w + w and a time-series with the order type w*+ w is that, although both series include the same number of nonoverlapping intervals of equal length, aleph zero, the former series alone includes intervals that are separated from other intervals by an aleph-zero number of nonoverlapping intervals of equal length. For example, there are an infinite number of nonoverlapping equal intervals separating the interval correspon ding to -6 in S₀ from the interval corresponding to -5 in S₀ but between any two intervals in S₁ there are only a finite number of nonoverlapping equal intervals. With this admission, I am opening myself to an argument frequently heard in discussions of infinity and time, that it is impossible for there to be an infinite number of nonoverlapping equal intervals separating the present interval from any past interval, since the present interval could never have been reached from an infinitely distant past interval. I believe this argument, which has never been analysed or defended but instead taken to be ‘self-evident’ by its exponents, such as Craig (1981), Kant (1929, p. 396) and Bonaventura (1.1.1.2.1-6), is a petitio principli since it assumes a sense of ‘never’ that presupposes the very thesis it is endeavoring to establish. It is true that an endlessly existing being that is present when the time corresponding to -6 in S₀ is present would ‘wait forever and never see the infinitely later time corresponding to -5 in S₀ become present if ‘never’ means not after an arbitrarily large finite time. But this being would not ‘wait forever and never see the time corresponding to -5 becomes present’ if ‘never’ means not after an infinite amount of time, for it an infinite amount of time elapses he would be in a position to see the time corresponding to -5 becomes present, for this time would then be finitely distant from him. If the argument of the above- mentioned authors is to get off the ground, some reason must be given why ‘never’ cannot be given the latter interpretation, and to my knowledge no reason has been given.

Actually, this last remark is not entirely fair, since the most careful exponent of this argument, Craig, has attempted, after a fashion, to support the assumption that ‘never’ cannot mean ‘not after an infinite amount of time’. As Craig sees it, the reason is not essentially temporal. “The impossibility of such a traversal [of an infinite series] has nothing at all to do with the amount of time available: it is of the essence of the infmite that it cannot be completed by successive addition” (Craig 1979, p. 104). “The reason is that. for every element one adds, one can always add one more. Therefore, one can never arrive at infinity” (Ibid., my italics). This argument is valid only if the phrase “every element” in the clause “for every element one adds, one can always add one more” refers to the ele-

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ments in any arbitrarily large finite subset of the infinite set of elements. Clearly, for every element in such a subset that is added, there always remains one more in the infinite set that can be added. But if “every element” has unrestricted scope, so that it refers to every element in the infinite set, the argument is invalid. For if one adds every element in the infinite set, then by definition there are no more elements to be added and the infinite set has been completed by successive addition. Thus this temporally neutral rephrasal of the standard argument is at bottom no less question- begging than the familiar statement of it in terms of time.

I pass now to some empirical considerations relevant to the category of beginninglessness and endlessness in time. The most pertinent considerations concern the theory of the infinite future in current cosmology. Contrary to the ‘received opinion’ (of philosophers of science), we cannot assume as a matter of course that the infinite future time postulated in cosmological theories of the unceasing expansion of our universe has the order type w, In deed, the most recent and detailed accounts of this expansion imply that the order type of the time-series that stems from the beginning of our universe into the infinite future has the order type w + w (0 1 3 5 . . . 2 4 6 . . .). Current predictions envisage an asymptotic approach to a state of maximum entropy, the ‘heat death’ of the universe, such that this state is reached only after an infinite amount of time has elapsed. In the words of Barrow and Tipler, “the asymptotic state is reached only after infinite proper time in an ever- expanding universe’ (1978, p. 458; my italics). As this final state is approached, fewer and fewer physical changes take place, until the final state is reached and change ceases. The final state is reached at the earliest time in the second temporal subsequence namely, at the time corresponding to 2 in the temporal subsequence that corresponds to 2 4 6 . . . Physical changes occupy all and only the times in the first temporal subsequence, the subsequence corresponding to 0 1 3 5 . . . Barrow and Tipler indicate that time continues to elapse after the last physical change takes place, such that the state of changelessness is reached and not merely approached. This entails that the time of our universe has at least the order type & + 1 (0 1 3 5 . . . 2). But Barrow and Tipler implicitly suggest, and there is every reason to assume, that time continues to elapse for the rest of the subsequence corresponding to 2 4 6 . . . For if there is a time corresponding to 2, at which there is no change, then there is time without change and time elapses independently of change. The cessation of change does not bring an end to time, which flows on of its own accord. This notion of a change independent time is compatible with theories of time recently argued

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to be empirically possible or probable by Newton-Smith (1980), Swinburne (1981), Shoemaker (1969) and Smith (1986a, Chapters IV and VI), theories that imply time is composed of moments that elapse regardless of whether or not they are occupied by physical or mental changes.

The scenario envisaged by Barrow and Tipler provides us with an example of something that has an infinite future but nevertheless comes to an end. It is true at the beginning of our universe, at the beginning of the temporal subsequence corresponding to 0 1 3 5 . . ., that the property of physically changing will be possessed for an infinite amount of time in the future but that it will not be possessed endlessly. For there are times later than every time at which this property is possessed, viz., the times corresponding to

If our universe fluctuated from a background space that extends infmitely into the past, then we can say that the time-series that includes the history of our universe has at least the order type w* + w + w (… -3 -2 -2 0 1 3 5 … 2 4 6 ), where 0 marks the birth of our universe from the background space. There is no reason at present to assume the temporal history of the background space has a higher order type, say w* + w But it is logically possible that the order type of the total temporal sequence in which the time of our known universe is embedded has a higher order type, indeed an infinitely higher type, such that the total temporal sequence has the order type. . . w* + w* + w* + w* + w + w + w + w.(fro m nd to infinity).

In section 1 it was indicated that the concept of x’s existing now and at every future time is a different concept than x’s existing now and in a future that contains no time later than every time at which x exists. This amounts to the difference between everlastingness and endless recurrence. Something is everlasting iff it exists at some time t and at every later time; but something recurs endlessly iff for each time to at which it exists there is a later time t at which it does not exist, and for each time t₁ at which it does not exist there is a later time t₂ at which it does exist. Manifestly, it is true for an everlasting entity but not .for an endlessly recurring entity that it exists now and at every future time.

Endless recurrence can be divided into two subtypes. Something weakly endlessly recurs iff it recurs endlessly but not in the same state and something strongly endlessly recurs iff it recurs endlessly in the same state.

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cally disintegrates and then returns to life after a lapse of time. We may think here of a person who is instantly disintegrated or decomposed such that at a later time all and only the elementary particles that composed her body at the last instant of her life are reassembled (perhaps by a technological wizard) in exactly the same order they possessed at the last instant of her previous life. At this later time, her old body is reconstituted and she has all the memories, beliefs, habits, personality traits, etc., she possessed at the last time of her previous life. However, the new life of this person would not consist of all and only the same states that comprise her previous life, since her reanimated body and mind would evolve (after the first instant of her new life) into novel states. For example, after the first instant she would remember the last time at which she was alive in her previous life and this rememberance would not have been any part of her previous life.

The most controversial assumption in the explication of the category of weak endless recurrence is that objects can retain their identity across temporal gaps. Some philosophers might think that the “reanimated person” is qualitatively similar to (is an exact replica of) the previous person but is not numerically identical with the previous person. Numerical identity, they might say, entails continuity in time. If these philosophers are correct, then there is no type of permanence consisting of endless recurrence. However, I believe (but will not attempt to support this belief here[6]) that under certain conditions objects can retain their identity across temporal gaps and will assume this view here.

An object is strongly endlessly recurrent only if in each recurrence its temporal history consists of all and only the same states that comprised its previous temporal history. If a state of an object is defined as the object’s possession of a nontemporal n-adic property, this entails that in each recurrence the object possesses all and only the nontemporal n-adic properties it possessed in its previous existence. Naturally, the object cannot possess the same temporal proper ties, since in each recurrence it exists at a later time than it existed in its prior occurence. Strong endless recurrence requires a perfectly cyclical universe in linear time, for nothing in any universe can recur endlessly with all the same nontemporal properties unless every thing in that universe endlessly recurs with all the same nontemporal properties. This can be illustrated in terms of the standard Big Bang model of an oscillating universe. Suppose there is an infinite number of expansion-contraction cycles of our universe, such that in each cycle the Earth recurs in the strong sense but the Earth’s moon recurs in the weak sense. In the present cycle the moon has n number of craters but in the previous cycle it had n + 1 craters. This entails

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that in the present cycle the Earth has the relational property of being distant from a moon with n craters but that in the last cycle it had the relational property of being distant from a moon with n +1 craters—and thus that the Earth itself is only weakly endlessly recurrent.[7]

Is the notion of strong endless recurrence coherent? I believe it is coherent iff there is a means of distinguishing the times in one cycle from the times in other cycles. There is such a means iff at least one of three conditions is met: (a) times are not sets of simultaneous states[8] but are occupant-independent items (“moments of time”), or (b) time consists of A-properties (futurity, presentness and pastness) as well as B-relations (simultaneity, earlier and later), or (c) the principle of the identity of indiscernables is not necessarily true.

(a) Suppose that times are not sets of simultaneous states but are occupant-independent moments. If the universe is perfectly cyclical, then each set of simultaneous states S in each cycle C is identical (by the axiom of extensionality: two sets are identical iff they have all and only the same members) with the corresponding set S’ in each of cycle C’. A set S’ corresponds to S iff S’ occupies the same temporal position in C’ that S occupies in C. For example, the set of states that occupies the first instant in C corresponds to the set of states that occupies the first instant in C’. Now if times are occupant-independent moments, that provides a means for distinguishing the temporal position of a set of states from the temporal positions of its corresponding sets. S’ is later (earlier) than S iff S’ occupies a different moment M’ than S occupies, such that M’ is later (earlier) than the moment M occupied by S. If, on the other hand, times are set of simultaneous states, then we lose the ground for differentiating between the time of S and the later (earlier) time of S’ and must appeal to condition (b) or (c) if we are to maintain that S’ exists at a different time than S.

(b) If times are sets of simultaneous states but nevertheless possess A-properties of futurity, presentness or pastness, it is still possible to distinguish the temporal position of S from that of S’. S’ is later (earlier) than S iff S’ possesses presentness after (before) S possesses presentness. The possession of presentness by S may be called the temporal occurence T and the possession of presentness by S’ the temporal occurence T’. The occurence T’ is both different from and later (earlier) than the occurence T and there by enables one to distinguish the time at which S’ exists from the time at which S exists.

(c) If times are set of states and do not exemplify A-properties, and the principle of the identity of indiscernables is not a necessary

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truth, then there is a logically possible world in which S’ is numerically distinct from S despite the fact that it is qualitatively identical with S. This numerical difference provides grounds for saying that S’ is a term of a later relation to S.

But if neither (a) nor (b) nor (c) holds, then there is no grounds for distinguishing the time at which S exists from the time at which S’ exists, and consequently the two times must be identical. Thus if strong endless recurrence is a logically possible type of permanence, either (a) or (b) or (c) must hold. I and others have argued in other writings[9] that (a) holds and (b) holds, and on the basis of these arguments I believe that strong endless recurrence is logically possible. However, scientific evidence at present shows that it is highly dubious that we occupy an endlessly oscillating universe, let alone a universe that contains weakly or strongly endlessly recurrent entities (see (Smith 1985c, 1986b and 1988)).

There are two concepts of eternity, the traditional theological concept and what I shall call the ‘atheological concept.’. The traditional concept has been explicated by Plotinus (III, 7), Boethius (V, 6) and Aquinas (a, I, 66; b, 1A, 10) and has recently been given a detailed and penetrating analysis by Stump and Kretzmann (1981). Not withstanding this, I shall argue that the traditional concept is in coherent and that the eternal type of permanence must be understood atheologically.

The theological concept of eternity is summarily stated by Boethius to be the “complete possession all at once of illimitable life” (V, 6). Stump and Kretzmann disengage four parts of this concept, the concepts of life, illimitability, duration and atemporality. This complex concept is theological in that it is alleged to be applicable to God and only God. This claim may be questioned, but instead of taking this route I shall argue more fundamentally that this concept is not applicable to anything since its parts—the four concepts indicated above—are mutually incompatible, I particularly have in mind the two concepts of atemporality and duration. Boethius (V, 6) speaks of the atemporal entity as “remaining” and “enduring” and Aquinas (a, I, 66) describes the atemporal entity as lacking the “duration of succession” and as instead enduring as an “ever-abiding simultaneous whole”. Stump and Kretzmann elaborate more fully on these notions and characterize God’s atemporal duration as “an infinitely extended, pastless arid futureless duration”, as a “duration without succession”, such that “there is no past or future, no earlier or later, within its life” (1981, pp. 434-35). By emphasizing God’s durational character Stump and Kretzmann mean to deny that God’s

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eternity is to be understood as a pointlike, unextended instant. But this notion of atemporal duration strikes me as self-contradictory. A duration by definition is an extension and an extension by definition has parts. If this be denied, then one is using ‘duration’ to mean its opposite, an unextended and simple instant. Now the parts of a duration, by definition, are sequentially ordered as earlier or later. If this be denied, and it is asserted instead that its parts are simultaneous, then one is again using ‘duration’ to mean its opposite, an unextended instant, for it is true instead of an instant that it has no parts p₁ and p₂, such that p₂ is later than p₁. Thus to affirm unblushingly of the divine being that it not only has an infinitely extended duration but also is such that there is no earlier or later within its life is to embrace a straightforward contradiction.[10]

But this is not to say that there is no coherent concept of an eternal type of permanence. The key to this coherent concept is to take the phrases ‘eternal presentness’ and ‘eternal simultaneity’ as denoting conceptual abstractions from the notions of temporal presentness and temporal simultaneity. This abstractive conception of eternity is developed in two ways, in reference to A-properties (futurity, presentness and pastness) and in reference to B-relations (earlier, later and simultaneity).

The A-conception of eternity is based on the idea that the property of presentness that temporal beings possess is numerically one and the same property that an eternal being possesses, with the difference being that temporal beings possess it in a temporal manner and eternal beings in an eternal manner. Presentness is possessed in a temporal manner by some entity x iff presentness either (.a) is possessed by x and either was not or will not be possessed by x, or (b) is possessed by x and has been or will be possessed by x. The ‘or’ in (a) and (b) is an inclusive disjunction, such that (a) applies to instantaneous entities among others and (b) applies to sempiternal entities among others.

Presentness is possessed in an eternal manner by some entity y if presentness is possessed by y but not in a temporal manner. More loquaciously, presentness is eternally possessed by y iff it is possessed by y and it is not the case that (c) it either was not or will not be possessed by y, and it is not the cases that (d) it has been or will be possessed by y. In a word, y is eternally present iff it is present but neither transiently nor sempiternally. If one takes the present tensed ‘is’ to refer to temporal presentness, then one will ascribe eternal presentness tenselessly, ‘y (is) eternally present’, where the parentheses indicate tenselessness.

The B-conception of eternity is based on the idea that the simultaneity relation possessed by temporal entities is the same rela-

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tion possessed by eternal beings, with the difference being in their manner of possessing it. Simultaneity is possessed in a temporal manner by x and y if x and y are simultaneous with each other and earlier or later than something else. Simultaneity is possessed in an eternal manner by x and y if x and y are simultaneous and are not earlier or later than anything else. (I allow that x and y may be identical, in which case the entity is simultaneous with itself.)

We are now in a position to see why this conception of eternity is atheological. Any person or physical thing that is capable of exist ing instantaneously exists externally in some merely possible world. This is shown as follows, Take the present instantaneous slice S of some body-and-mind-including segment of the universe, such that S includes whatever is present in this segment at this instant. Abstract from S the other segments of the universe and whatever instantaneous slices are earlier or later than S, such that every contingent existent but S is conceived as nonexistent. We are left with a conception of S as present in such a manner that it never was future and never will be past and never was or will be present and is not earlier or later than anything. Since there is some possible world in which nothing but the parts of S contingently exist, there is some possible world in which every part of S exists eternally.

If we wanted to we could stipulate that ‘eternity’ is to express a concept that applies only to God; we could say that ‘y is eternal’ means ‘y is present in a nontemporal manner and is omniscient, omnipotent, omnibenevolent, etc.’ But this gives us a spurious theological eternity for it means no more than ‘y is eternal in the atheological sense and is divine’.

Mere timelessness is traditionally distinguished from eternality by saying that only God is eternal and that necessary truths or essences or numbers and the like have a merely timeless existence. Traditionally, the timeless existence of these abstracta has been defined negatively, such that ‘x is merely timeless’ means ‘x exists but not in time and not eternally’. The word ‘subsistence’ is often applied to such timeless abstracta. But since this definition uses the traditional theological concept of eternity we must reject it. But we can construct a formally analogous definition if ‘eternally’ is instead taken to mean atheological eternity. X is merely timeless if x exists and is neither in time nor nontemporally present nor nontemporally simultaneous with anything. It might be said that the difference in kind between merely timeless entities and eternal ones is that

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the former are abstracta and the latter concreta. Only concrete entities—God, embodied minds, physical objects—can be bearers of a simultaneity relation or a property of presentness, whereas abstract entities like numbers and necessary truths lack B-relations and A-properties altogether. It might be said, for example, that God exists in nontemporal simultaneity with his angelic cohorts, but that it is false or even senseless to say that the number three is simultaneous with the number two.

I find this difference between atheological eternity and mere timelessness the least plausible of the eight differentiations among the types of permanence made in this paper. The metaphysical out look of some philosophers permit them to make this differentiation, but if one adopts the position that ‘to exist’ means ‘to be present’ (a position I argued for in The Felt Meanings of the World[11]), then one will hold that a merely timeless existence is a contradiction in terms. I admit this identification of existence with presentness is not widely shared, but I mention it to indicate a theory of existence that must be rejected if one is to allow for this eighth type of permanence.

There is one final point I wish to make in this paper and it concerns atemporal existence, whether it is conceived as eternalness or mere timelessness. If anything exists in time, then nothing exists eternal l, or timelessly. There is no possible world in which there are both temporal and atemporal existents. By this claim I mean to deny both the theological tradition that it is at least possibly the case both that God exists atemporally and that His creatures exist temporally and the Platonic tradition that it is at least possibly the case both that abstracta exist atemporally and that concerta exist temporally. Exponents of the theological tradition were discussed in the last section and so I shall confine myself here to presenting a sample quotation from an exponent of the Platonic tradition. Russell writes (1959, p. 98) that

the relation ‘north of’ does not seem to exist in the same sense in which Edinburgh and London exist. If we ask ‘Where and when does this relation exist?’ the answer must be ‘Nowhere and nowhen’, There is no place or time where we can find the relation ‘north of’ (. . .) . It is neither in space nor in time, neither material nor mental; yet it is something.

The following argument applies to any entity that is alleged to actually exist atemporally:

(1) Whatever possesses an n-adic property at one timeand not at another time exists in time.

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(2) The semantic relation of reference is a dyadic relation whose two terms are a token of a word, phrase or sentence (what refers) and some other item (the referent).

(3) For any existent x, it is true to say ‘x exists’, such that the rele vant utterance U of this sentence refers to x.

(4) The utterance U of this sentence exists in time and refers to x at the time U exists.

(5) If U occurs at time to, U possesses at to the relational property of referring to x.

(6) Therefore, the referent of U, x, possesses at to the relational property of being referred to by U.

(8) Therefore, at ti the entity x does not possess the property of being referred to by U.

(9) Therefore, x possesses a property at one time and not at another time and is in time.

It might be objected that if x exists atemporally it does not possess at to the property of being referred to by U and lack this property at t₁, but instead atemporally and unchangeably possess the two proper ties of being referred to by U at t₀ and not being referred to by U at t₁. But this objection will not work, for if x possesses the latter two properties that entails it possesses at to but not at t₁ the property of being referred to by U. It is self-contradictory to hold both that

Being referred to by U at t₀ entails that x’s possession of the property of being referred to by U is itself at t₀, i.e. that this property is possessed simultaneously with the events at to, earlier than the events at t₁, and later than the events at t_-1. Now whatever possesses a property at one time and before another time and after a third time exists in time.

The argument (1 )-(9) applies mutatis mutandis to time-containing worlds in which no language-users exist. In these worlds, the temporal preconditions for the use of language to refer to any existent are present and that is enough to preclude anything from existing atemporally.

The argument (1)-(9) entails that all abstracta actually exist in time. This further impugns the viability of the eighth category of permanence, mere timelessness, for this category is largely based on the assumption that abstracta cannot exemplify any A-property or B-relation.

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In conclusion, I believe we can dismiss the seventh and eighth categories of permanence as inapplicable to anything that actually exists. If anything is actually permanent, it instantiates one or more of the categories of temporal permanence, either sempiternality, or omnitemporality or everlastingness or having an infinite past and future or beginninglessness and endlessness in time (but probably not endless recurrence), as has been indicated in the relevant sections of this paper.[12]

1982 “Origin of the Universe as a Quantum Tunelling Event,” Physical Review D 25: 2065-2073.

1978 “The Creation of the Universe as a Quantum Phenomenon”, Annals of Physics 115: 78-106.

1980 “On the Virgo Supercluster and the Mean Mass Density of the Universe”, Astrophysical Journal 238, L113: 64.

1985 “Stump and Kretzxnann on Time and Eternity”, Journal of Philosophy 82: 260-269.

1982 “Creation of Open Universes from de Sitter Space”, Nature 295: 304-307.

1982 “Complete Cosmological Theories”, in Quantum Structure of Space and Time, ed, M.J. Duff and C.J. Isharn. Cambridge: Cambridge University Press.

1986 “Persons, Morality, and Tenselessness’, Philosophy and Phenomenological Research 47: 305-309.

1981 “Relic Neurinos and the Density of the Universe,” Astrophysical Journal 243, 1: 1-A4.

1985a “The Mind-Independence of Temporal Becoming”, Philosophical Studies 47: 109-119.

1985b “Kant and the Beginning of the World”, The New Scholasticism 59: 339-346.

1985c “The Anthropic Principle and Many-Worlds Cosmologies”, The Australasian Journal of Philosophy 63: 336-348.

1986b “World Ensemble Explanations”, Pacific Philosophical Quarterly 67: 73-86.

1986c “The Infinite Regress of Temporal Attributions”, The Southern Journal of Philosophy 24: 383-396.

1987c “Problems with the New Tenseless Theory of Time”, Philosophical Studies 52: 77-98.

1987 “Atemporal Duration: A Reply to Fitzgerald, Journal of Philosophy 84: 214-219.

1982 “Creation of ‘Universes from Nothing”, Physical Letters 1 17B: 25-28.

1975 “God Everlasting” in God and the Good, ed. C. Orlebeke and 1. Smedes. Grand Rapids, Mich.: Eerdrnans, pp. 181-203.

1979 “Can Ontology Do Without Events?” in Essays on the Philosophy of Roderiric M. Chisholm., ed, E. Sosa, Amsterdam: Editions Rodopi, pp. 177-204.

[1] In philosophy, if any view is ‘standardly’ held, there will be those who do not hold it. Thus Nicholas Wolterstorff, for example, believes that God and abstracta are temporally permanent. (See Wolterstorff 1975 and 1979.)

[2] Although the theory of things as wholes of temporal parts is often held by those who also subscribe to the tenseless theory of time, and the theory of things as continuants is frequently endorsed by proponents of the tensed theory of time, there is no logical connection among these theories. Ronald Hoy (1978) and Delmas Lewis (1986) have recently asserted there is a logical connection; specifically, they hold that the theory of persons as continuants entails the tensed theory of time, since this theory of persons cannot be stated in a tenseless idiom. But this seems false, since the tensed statement of personal continuity entails a tenseless statement of it. For example,

(1) x is now existing and time t₂ is present

and

(2) x was also existing when time t₁ was present

entail

(3) x exists (tenseless) at t₁ which is present, and

x exists (tenseless) at t₂ which was present

which in turn entails

(4) x exists (tenseless) at t₂ and x exists (tenseless) at the earlier time t₁.

Thus if it is true in tensed terms that a continuant x successively occupies different times, this is also true in tenseless terms.

[3] Swinburne, Quinton, Moore, Alexander, Kant and others have argued that it is logically impossible for time to begin or end. For references and criticisms of their arguments, see Smith 1985d. For a critical discussion of Prior’s definition of beginning and ending time in terms of propositions, and for a discussion of the relevance of Mime’s nonstandard tem oral metric, see Smith 1985e.

[4] Newton-Smith (1980, pp. 87-95) has argued with some degree of plausibility that particulars can exist in two different time-series if they ‘go back and forth’ between the two series and do not exist at every time in each of the series.

[5] The senses (a), (b) and (c) of beginning and ending distinguished in section 2 relate to D2 and D3 as follows. Sense (a) parallels D3. Sense (b) does not parallel either D2 or D3 but if it holds of any entity E then D3 does not hold of E and D2 does hold. Sense (c) applies to time rather than entities in time so strictly speaking it has a different field of application than D2 and D3. But we can say this: if time begins and ends in sense (c) it cannot be mapped Onto a set with a transfinite order type. This distinction is important since there is a sense in which time begins and ends and yet is mappable onto a Set with a transfinite order type. In this latter sense, time begins and ends if the condition (i) of (c) holds (i.e. time contains an earliest and latest interval) and condition (ii) of (c) does not hold (i.e. if it is not the case that before or after any one of its intervals there is at most a finite number of nonoverlapping intervals of the same length. Suppose that time maps onto the following subsequence of the sequence S₀:

S₄: -4 -2… -5 -3 -1 0 1 3 5… 2 4

In this case time has an earliest interval (which corresponds to -4) and a latest interval (which corresponds to 4), but there are an infinite number of nonoverlapping and equal intervals after the one corresponding to -4 and before the one corresponding to 4.

[6] Some arguments for the thesis that there can be numerical identity across temporal gaps can be found in Aune 1985, pp. 77-104 and in Nozick 1981, pp. 27-114.

[7] If one wishes, one may demarcate a third subtype of endless recurrence that is ‘in between’ the weak and strong subtypes, viz., that something possesses all and only the same monadic nontemporal properties in each recurrence.

[8] Philosophers usually say that times are set of simultaneous events, but I believe the word ‘event’ should instead be reserved for a change of state, i.e. for a thing’s acquisition or loss of a property. For a formalization of the definition of instants (unextended temporal points) in terms of states of things, see Aune 1985, pp. 111-114.

[9] Arguments on behalf of (a) can be found in (Smith 1986a, Chapters IV and VI); (Swinburne 1981); (Newton-Smith 1980); and (Shoemaker 1969). Arguments on behalf of (b) can be found in (Smith l987c, 1987a, 1986c, 1985a); (Schlesinger 1982); and (Tichy 1980).

[10] Paul Fitzgerald makes an analogous point in reference to the alleged infinite character of God’s duration: “one cannot have literal infinite duration without ordered, finitely ex tended subphases” (Fitzgerald, 1985, p. 264). Stump and Kretzmann have recently responded to this criticism, but their response is not as illuminating as one might wish. The nearest they come to defending themselves against the charge that the thesis that eternity is a duration without succession is self-contradictory is to claim eternity can be coherently conceived as indivisible but limitless. The eternal “present is indivisible, like the temporal present, but it is atemporal in virtue of being limitless rather than instantaneous, and it is in that way infinitely enduring” (Stump and Kretzmann 1987, p. 218). But this reply reintroduces the original problem. If eternity 15 durational and extended through being limitless rather than instantaneous, that implies it is limitlessly extended, Now the only sense I can attach to this latter phrase is that eternity is composed of parts but no first or last part. But this is of course denied by Stump and Kretzmann, since eternity is also indivisible.

[11] See Smith 1986a, Chapter V. In this book I identified existence with temporal present ness. This restriction to time was made on the basis of my construal of eternity as unchangeable sempiternality (pp. 159-160). But given the sense of eternity elucidated in section 8 of this paper, existence must be identified instead with presentness, whether it be possessed in a temporal or eternal manner.

[12] I am grateful to Hector-Neri Castañeda and Susan Ament Smith for helpful comments upon an earlier version of this paper.