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denoting times that (allegedly) share temporal relations such as, temporal distance, non- simultaneity, not identical to, and so on. (The temporal relations between non-identical times, are, along with the times, typically considered to be constituents of time.) If relations between times do not exist, then times are not before or after one another, and if that is the case all times would coincide, which would indicate that there is only one time, and thus presentism would be the correct theory.

Throughout this paper I will refer to divisible (non-basic) temporal extensions as 'durations.' Non-basic durations are typically considered to be either infinitely divisible, or to be composed of basic building blocks. (Some may deem that any duration is divisible, and so I only need to use the expression 'duration,' rather than the expressions 'divisible duration' or 'non-basic duration.' But I will distinguish between divisible and indivisible durations, since many physicists, especially some quantum gravity theorists, hold that a Planck time is a basic building block of time that has a temporal size (a duration).) Regardless of which is the correct position — regardless of whether or not durations are infinitely divisible (i.e., durations are not composed of time points, Planck times, or any sort of time atoms), or involve basic building blocks of time — both positions involve relations between non-identical times. 

In this paper, I will refer to both basic times and durations (divisible or indivisible durations) as 'times.' For example, one year is a time, one hour is a time, one nanosecond is a time, and one basic building block of time is a time. I will call the relata (non-identical times) that are connected by temporal relations t₁, t₂, and t₃. In the examples I give in this paper, I will often refer only to t₁ and t₂, and only occasionally refer to three times, t₁, t₂, and t₃. The examples of t₁ and t₂ I will use in this paper are two basic times that are 10 seconds apart, or a duration that might be a part of another duration (t₁=one minute, t₂=one hour).

In sections 2 and 3, I will argue that there is a specific problem to do with any variety of a temporal relation between or among any non-

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relatedness. In this paper, I will mainly discuss relations, since monadic relatedness has been discussed far less in the literature since Russell’s Principles of Mathematics, where relations were argued to be irreducible. (One philosopher who does discuss monadic relatedness at length is Keith Campbell.) I will however refer to both relations and monadic relatedness at various places in the paper, but I will mainly be concerned with relations hereafter, only infrequently mentioning monadic relatedness.

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identical times (between or among t₁ and t₂): temporal relations between t₁ and t₂ apparently cannot be located in time, T (I will call temporal relations that are located in time non-platonistic temporal relations), and they also cannot be timeless, ~T (I will call temporal relations that are not located in time platonistic temporal relations). If temporal relations between or among t₁ and t₂ are neither non-platonistic (T) nor platonistic (~T), they apparently involve contradiction, since they would be describable as ~(T v ~T), which translates to ~T ^ T. In section 2 I discuss hitherto unnoticed problems to do with non-platonistic temporal relations, T. If my reasoning is correct, only platonistic temporal relations,[2] ~T, could be considered to exist among t₁ and t₂. In section 3 I consider platonistic temporal relations among t₁ and t₂, where I also come to serious problems when considering them.

2. Non-platonistic relations between non-identical times

In this section I discuss apparent problems to do with non-platonistic temporal relations between t₁ and t₂. In subsections 2.1 and 2.2, I discuss problems to do with noncomplex non-platonistic temporal relations between t₁ and t₂.[3] In subsection 2.3 I discuss problems to do with specific sorts of complex non-platonistic temporal relations that are not affected by the reasoning against non-platonistic noncomplex temporal relations between or among t₁ and t₂ given in subsections 2.1 and 2.2. In subsection 2.4 I discuss a problem to do with non-platonistic temporal monadic relatedness.

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[2] It is standard to consider platonistic relations as those which are not in the spatiotemporal world, whereas non-platonistic relations are not outside of the spatiotemporal world, as Loux discusses:

 

What are the issues separating the Aristotelian realists from Platonists? … Aristotelians typically tell us that to endorse Platonic realism is to deny that properties, kinds, and relations, need to be anchored in the spatiotemporal world. As they see it, the Platonist’s universals are ontological “free floaters” with existence conditions that are independent of the concrete world of space and time. But to adopt this conception of universals, Aristotelians insist, is to embrace a two-worlds” ontology… On this view, we have a radical bifurcation of reality, with universals and concrete particulars occupying separate and unrelated realms… [T]here [is a] connection between spatiotemporal objects and beings completely outside of space and time. (Loux 1998, 46) 

[3] Noncomplex relations are simple (partless), and complex relations are not simple.

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2.1 Noncomplex temporal relations of non-zero temporal size

It appears that there are two ways to conceptualize a non-platonistic temporal relation, if the relation (allegedly) connects t₁ and t₂.

 

1.      A non-platonistic relation is temporally extended between t₁ and t₂. I will discuss varieties of this sort of non-platonistic temporal relation primarily in this subsection, but also in parts of other subsections of this section.[4]

2.      The second way to conceptualize non-platonistic temporal relations between t₁ and t₂ is by considering them as not temporally extended between t₁ and t₂, but only temporally located where t₁ and t₂ are. Such temporal relations are in time, but are temporally unextended (durationless) entities.[5] I discuss this position primarily in subsection 2.2, but also in parts of other subsections.

 

In this subsection, I discuss temporally extended non-platonistic temporal relations between t₁ and t₂. In other words, I am only considering temporal relations of non-zero temporal size that connect at least two non-identical temporal locations: relations of non-zero temporal size that connect t₁ and t₂, where t₁ ≠ t₂.

It is not uncommon for philosophers to hold that non-platonistic temporal relations, in addition to the times that make up time, are not occupants of time, but rather contribute to the makeup of time, without being occupants of time. In this subsection I will argue that non-platonistic temporal relations that are constituents of time, if they are any variety of non-platonistic temporal relation (temporally extended, temporally unextended, etc.), can only be temporally located: they only can be occupants of time. I do this next in 2.1.1. In 2.1.2 I give an argument that leads to the conclusion that temporally extended non-platonistic temporal relations between non-identical times are contradictory. In 2.1.3 I will consider an objection to the argument given in 2.1.2.

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[4] It is this 'betweenness,' where relations are not merely at the locations of their relata, that monadists often reject about relations.

[5] When I refer to properties as 'entities,' I use the word “entity” in the broadest possible sense, and in the way that many other metaphysicians refer to n-adic properties as 'entities.' (For example, Esfeld (2003, 10), Lowe (2002, 16), Moreland (2001, 13), and many others. Also, a passage from Reinhardt Grossmann at the very start of section 3 below involves Grossmann referring to 'abstract qualities' as 'entities.' (Grossmann, 1990, 7))

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2.1.1 Non-platonistic temporal relations can only be occupants of time

I will next argue that non-platonistic temporal relations between t₁ and t₂ can only be considered as occupants of time. This is relevant to my reasoning in 2.1.2 where I argue that non-platonistic temporally extended temporal relations between t₁ and t₂ are contradictory.

If there are non-platonistic temporal relations that contribute to the makeup of time, since they interrelate non-identical times, the non-platonistic relations must coincide with those times (t₁ and t₂) that they interrelate. Further, the temporal relation must coincide with the entirety of the time it coincides with, regardless of whether or not the interrelated times are basic building blocks of time or durations (divisible or indivisible durations). If the temporal relation only coincided with a part of one of the times it relates, then statements such as 't₁ is related to t₂' would be false, since only parts of t₁ or t₂ would take part in the co-exemplification of the non-platonistic relation (and instead, statements such as, for example, 't₁ is related to part of t₂' would be true). For example, if one hour (t₁) is related to one minute (t₂) by the temporal relation parthood, it can only be the case that the entire hour coincides with the temporal relation in order for the hour in question to be a relatum of the temporal relation, parthood. If only part of the hour coincided with the temporal relation, then the statement “the minute is related to the hour” would be false, and the statement 'the minute is related to the forty-five minute duration' would be true, if, for example, the relation only coincided with three quarters of the hour. Similar reasoning holds for Planck basic building blocks of time. For example, it cannot be the case that, with respect to a Planck time, the relation just contacts the surface of, or a left side of, a single Planck unit of time. (Also, it is unclear that what has just been written about a Planck time is coherent, given that it is unclear if a 'side' or 'surface' of a Planck time can even be discussed at all, since “side” and “surface” may be references to parts of the Planck time, or aspects of the Planck time not identical to the entirety of the single Planck time, rather than to the entirely of the Planck unit of time, and this is not possible since there are no parts or aspects of a Planck time that are not identical to the entirety of the Planck time.) Of course, if a relation did not attach or link to its relata (where 'attach' and “link” denote the special exemplification tie that holds relations to their relata[6]), then there would be a

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[6] See Loux 1998, 38-41.

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discontinuity of some sort between the temporal relation and its relata (times), which is absurd, since the relations then would not attach or link to their relata, and thus they would be relations that do not interrelate their relata.

For reasons just given, non-platonistic temporal relations that are constituents in the makeup of time must coincide with the entirety of the times that they interrelate. Times are temporal locations and thus are not located in time. I will next discuss that this means that non-platonistic temporal relations cannot also be temporal locations, even though the temporal relations are constituents of time. If the temporal relations were also temporal locations, then times and the temporal relations that connect the times to one another would coincide (overlap), where these coinciding entities would each be temporal locations. This has obvious problems, however, since two temporal locations that temporally overlap or coincide are not at a distance from one another, and cannot each be temporal locations, unless they are identical. But this cannot be the case since a temporal relation must be distinct from its relata. This implies that if there are non-platonistic temporal relations between non-identical times, since the non-platonistic temporal relations are in time but are not themselves temporal locations, then they could only be located at places in time, in order to avoid the problems just discussed. But if that is the case, then non-platonistic temporal relations that are constituents of time would be temporally located relations that occupy time (they are located in time). Hereafter, for reasons just given, I will only discuss non-platonistic temporal relations of any sort (complex, noncomplex, etc.) as being occupants of time, regardless of the fact that they are (allegedly) constituents of time.

2.1.2 The impossibility of non-platonistic noncomplex temporally extended temporal relations between t₁ and t₂

I next give an argument against non-platonistic, temporally extended, noncomplex relations between non-identical times, t₁ and t₂. If temporally extended, noncomplex, non-platonistic relations between non-identical times occupy at least two non-identical temporal locations, then they apparently involve contradiction, for the following reasons.

If a temporally extended temporal relation is partless (noncomplex), it is a single entity. If a temporally extended, noncomplex temporal relation is describable by a statement, then the entire temporal relation is describable by the statement. For example, the entire

 

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relation would be describable by the statements, 'located at temporal location t₁,' and, 'located at temporal location t₂.' If the relation is located at t₂, and if t₁≠t₂, then by being at t₂, the noncomplex non-platonistic temporal relation is describable by the statement, 'not located at t₁.' This could be said of any non-t₁ location that the non-platonistic noncomplex temporal relation occupies. If the temporal relation occupies more than two times, and for that reason is located at three temporal locations, t₁, t₂, and t₃, at locations t₂ and t₃ the temporal relation would be describable by the statement, 'not located at t₁.' These are, however, statements that reveal the nonexistence of the temporal relation: since the relation is one, partless entity, if it is 'located at t₁,' and 'not located at t₁,' each of these statements must describe the entire noncomplex non-platonistic temporal relation, and that implies the entire relation would be describable by self-contradictory conjunction of the above statements: 'located at t₁ and not located at t₁.'  

2.1.3 Temporally extended noncomplex temporal relations only located at entire temporal locations

In this subsection I discuss an objection to the reasoning given in 2.1.2 where non-platonistic noncomplex (simple) temporally extended temporal relations between t₁ and t₂ were found to involve contradiction if they occupy two or more temporal locations.

Philosophers who hold that temporal relations are temporally extended may assert that if a relation is located at a certain time t₂, this does not imply that it therefore does not also have the property of being located at some other time, t₁. Such philosophers may assert that non-platonistic temporal relations can be wholly located at two different times.[7]  In order for a philosopher to hold this position, she would merely need to avoid my reasoning above where I held that there are statements such as 'not at t₁' that describe the temporal relation; she must hold that such statements do not describe noncomplex non-platonistic temporal relations between t₁ and t₂. This might be done by holding that the temporally extended temporal relation can only be considered at the entire time it is located is at. To hold this objection is to hold that in the previous subsection temporal relations

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[7] I am grateful to a referee at Disputatio for making helpful comments that led me to discuss this objection. Around the same time, John Dilworth also expressed an objection to this that is very similar, and thus I am grateful to him for that.

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have been inaccurately described, since it may be the case that a temporally extended noncomplex non-platonistic relation might only be accurately described as being at its entire temporal location (call it t₁t₂t₃), not at a part (sub-location) of its temporal location, such as the basic times, t₁, t₂, or t₃. According to this objection, the temporally extended non-platonistic temporal relation that connects t₁ and t₃, where t₂ is between t₁ and t₃, is not located at the basic times, t₁, t₂, and t₃, of the temporal locations, t₁t₂t₃. Rather, only the entirety of t₁t₂t₃ an be called the noncomplex, non-platonistic, temporally extended temporal relation’s location. On this scenario, the statement,

 

'The noncomplex non-platonistic temporal relation between t₁ and t₃ is located at temporal location t₁t₂t₃,'

 

is true, and statements about the temporal relation being at any non-basic sub-location of t₁t₂t₃ (i.e., sub-location t₁t₂ or sub-location t₂t₃), or at the individual basic sub-locations, of t₁t₂t₃, are all false, such as the statements,

 

'The noncomplex non-platonistic temporal relation between t₁ and t₃ is located at t₁,'

 

'The noncomplex non-platonistic temporal relation between t₁ and t₃ is located at t₂,' or

 

'The noncomplex non-platonistic temporal relation between t₁ and t₃ is located at t₃,'

 

In this subsection, I will argue that this objection fails. According to this objection, the temporally extended, noncomplex, non-platonistic temporal relation is at temporal location t₁t₂t₃, but aspects of the relation at t₁, t₂, or t₃ cannot be discussed, since there are no such aspects of the temporal relation that are not identical to the whole relation. Nevertheless, since the relation extends temporally between t₁ and t₃, it is important to note that all of the individual basic times, t₁, t₂, or t₃, can only be occupied by something to do with the temporal relation. By this I merely mean that when we consider the sub-locations of t₁t₂t₃ and when we ask the question of whether or not the sub-locations (t₁, t₂, t₃) of the entire temporal location (t₁t₂t₃) are occupied, we apparently can only conclude that they are not unoccupied with respect to the temporal relation. The reason that t₁, t₂, or t₃ must

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be occupied by something to do with the temporal relation is because the entire temporal location, t₁t₂t₃, that the non-platonistic temporal relation is at, is a time that is made up of more fundamental temporal locations, and if the temporal relation is at a non-basic temporal location (such as t₁t₂t₃) and accordingly occupies the entire temporal location, it must also be the case that the temporal relation occupying t₁t₂t₃ leads to each of the temporal locations that make up t₁t₂t₃ also being occupied.

A temporal location would not be occupied at all if none of its sub-locations that compose it were occupied. Put in slightly different words, if a temporal relation occupying a temporal location (t₁t₂t₃) does not occupy the more fundamental temporal locations (t₁t₂, t₂t₃), or any of the basic times (t₁, t₂, t₃), of the temporal region t₁t₂t₃, then the temporal relation does not occupy the entire temporal location. For these reasons, the temporal relation’s being at t₁t₂t₃ must also lead to all of the sub-locations of t₁t₂t₃ being occupied. But this poses a serious problem for the noncomplex, temporally extended, non-platonistic temporal relation at temporal location t₁t₂t₃: if the relation can be described as occupying sub-locations of t₁t₂t₃, the problems of the previous subsection ensue.

The reasoning about temporal locations just given, where non-basic temporal locations were discussed as being composed of sub-locations, and of basic sub-locations (if time is not infinitely divisible), is the case for any non-basic temporal location, since any non-basic temporal location is made up of more fundamental temporal locations. If it were the case that a non-basic temporal location, such as t₁t₂t₃, were not made up of more fundamental, or basic, temporal locations, then an extended and non-basic temporal location would not be made up of anything, and it would not be a temporal location at all. For these reasons, a non-basic temporal location is composed of more fundamental temporal locations, or basic temporal locations, and a temporal relation’s occupying a non-basic temporal location must accordingly result in the more fundamental temporal locations, or basic temporal locations, also being occupied. The noncomplex, temporally extended, non-platonistic relation, for these reasons, cannot, be located at t₁t₂t₃, since the relation cannot be located at any of the temporal sub-locations make up t₁t₂t₃. This sets up a fatal problem for the coherence of the temporal relation: no sub-locations of the temporal relation’s entire temporal location (t₁t₂t₃) can have anything to do with the relation, and for that reason, the non-

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platonistic temporal relation, which is not outside of time, cannot be a temporally located entity at all, which is a contradiction.

(It appears that the argument given in 2.1.1 - 2.1.3 apply not only to temporally extended temporal relations, but also to the temporally unextended non-platonistic temporal relations that I will discuss next in 2.2. This is because the arguments just given deal with nothing more detailed than noncomplex connections between non-identical times, which applies to any sort of noncomplex non-platonistic temporal relation, whether temporally extended or unextended.)

2.2. Temporally located, temporally unextended, noncomplex temporal relations

I will next discuss the position that (somehow) a non-complex non-platonistic interrelation of t₁ and t₂ does not involve a connection across time, extending between t₁ and t₂. Rather, the interrelation of t₁ and t₂ exists only at t₁ and t₂, and not in-between t₁ and t₂. On this scenario, an interrelation of t₁ and t₂ is in time, where t₁ and t₂ are, but the noncomplex, non-platonistic relation is temporally unextended, since on this account, the temporal relation is located where and only where t₁ and t₂ are.

One thing to note before I move into my arguments is that if t₁ and t₂ are each durations (extended) (such as in the case where t₁ and t₂ are minutes that are 10 seconds apart), but the non-platonistic relation between them is temporally unextended (temporally point-sized), it is unclear how the temporally unextended non-platonistic temporal relation can relate them, since the relation would only be able to attach to one point of each duration t₁ and t₂. The non-platonistic temporally unextended temporal relation has no extension with which it can coincide with all of t₁, or all of t₂, in its relating of t₁ and t₂. Perhaps if À₁-many unextended temporal relations were involved connecting every point of t₁ to every point of t₂ (if both t₁ and t₂ have À₁-many points), this issue is solved. But philosophers typically discuss relations as if one relation relates all of duration t₁ to all of duration t₂. I see this as a serious problem for temporally unextended non-platonistic (and platonistic) relations between t₁ and t₂ if t₁ and t₂ are durations. But I however will not discuss this issue further since it is irrelevant to my arguments.

I will next move to my arguments against temporally unextended relations between t₁ and t₂. In arguing that non-platonistic temporally

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unextended temporal relations between t₁ and t₂ do not exist, (where t₁ and t₂ are any non-identical times), I will merely consider the scenario where the (alleged) temporal relation, parthood, among t₁ and t₂, where t₁=minute (part), and t₂=hour (whole), is a temporally unextended, noncomplex, non-platonistic temporal relation. On this account, the connection among t₁ and t₂ is a connection among non-identical temporal locations (times that are interrelated across a temporal distance), since pieces of t₂ do not temporally overlap with t₁: t₁ (part) is located within t₂’s (whole’s) locations, but t₁ is not identical to many of the locations that make up t₂, such as the minute before t₁, and the minute after t₁ (if t₁ is not the first or last minute of the hour). For these reasons, the relation, parthood, between t₂ (whole) and t₁ (part), connects non-identical times, which is the very sort of temporal relation I am concerned with in this paper.

If a time t₁, for example, participates in the co-exemplification of polyadic properties (such as, the temporal relation Parthood), in such a case that instantiation of the relation in question at t₁ is only at t₁. If one of the temporally located temporally unextended relation’s relata are not identical to time t₁, then t₁ is not a relatum of the relation. Similarly, if time t₂ is a temporal location, then in such a case, that instantiation of the relation in question at t₂ is only at t₂.

These restrictions imply that any non-identical times, t₁ and t₂, could not be related by a noncomplex, temporally unextended, non-platonistic temporal relation, for the following reasons. Since t₁ ≠ t₂, and since on this account the non-platonistic interrelation of t₁ and t₂ is not being considered as temporally extended between t₁ and t₂, but only at the temporal locations t₁ and t₂, then t₁ and t₂ apparently cannot have any sort of dealings with one another (such as being interrelated by the temporal relation, parthood). It appears that in order for t₁, for example, to co-exemplify a temporally unextended relation of the sort I am discussing here, which is a non-platonistic, noncomplex, non-platonistic temporal relation shared with t₂, t₁ must also be identical to t₂, and thus must apparently take on characteristics that are self-contradictory: t₁ is identical to itself and is not identical to itself). Similarly, in order for t₂ to share a temporally unextended, noncomplex, non-platonistic temporal relation with t₁, t₂ must also be identical to t₁, and thus must apparently take on characteristics that involve contradiction.

If my reasoning in this sub-section is correct, it is apparently the case that noncomplex, temporally unextended, non-platonistic tempo-

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ral relation relations cannot account for any connection or relatedness among t₁ and t₂.

2.3 A complex temporal relation as an extended continuum of non-complex temporal relations

Since noncomplex temporal relations make up complex temporal relations, it may appear that non-platonistic complex relations between or among t₁ and t₂ are also impossible. But there may be varieties of temporally located complex temporal relations between t₁ and t₂ not susceptible to the problems discussed up to this point in the paper. In subsections 2.1 and 2.2 I discussed apparent serious problems with noncomplex non-platonistic temporal relations between or among t₁ and t₂, where those non-platonistic noncomplex temporal relations were considered as either temporally extended or temporally unextended. In the case of temporally extended noncomplex non-platonistic temporal relations, the apparent problems I discussed drew from the combination of the partlessness and temporal extendedness (extended larger than one basic building block of time) of non-complex temporally located temporal relations. In the case of temporally unextended noncomplex, non-platonistic relations, the apparent problems I discussed drew from noncomplex temporal relations not being able to connect t₁ and t₂ if non-platonistic, temporal noncomplex temporal relations are not in any way temporally extended between relata. Perhaps a complex non-platonistic temporal relation of a very specific sort can avoid these problems.

The following two sorts of temporally located, temporally extended, complex temporal relations between or among t₁ and t₂ may avoid the problems of noncomplex non-platonistic temporal relations I discussed in subsections 2.1 and 2.2.

 

1.      A non-platonistic relation composed of an extended continuum of durationless (point-sized), non-complex, non-platonistic temporal sub-relations between t₁ and t₂. (Any one of these non-platonistic durationless sub-relations are temporal since they are in time (they are non-platonistic), but they are durationless in that the location in time that any one of them occupies is temporally unextended.)

2.      A non-platonistic relation composed of discrete temporal sub-relations in tandem between t₁ and t₂, where the sub-relations have a basic (irreducible) duration (a basic temporal size, such as the size of a Planck time).

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Points 1 and 2 describe a temporal relation between t₁ and t₂ that is a succession, or chain, of temporal sub-relations in tandem, linked one after the other, by analogy as chain links are linked to give rise to a chain. (Interestingly, Loux uses “link” to denote the tying of relations to other relations in one particularly interesting passage. (Loux 1998, 38-41)) This is not the sort of relation that I have seen discussed often in the literature, other than for a few specific cases.[8] In this subsection, I will consider continuous complex temporal relations (point 1 above) (I do this in 2.3.1 and 2.3.2), and I will consider a complex temporal relation as being composed of discrete noncomplex Planck-scale-sized temporal sub-relations (point 2 above) (I do this in 2.3.3). If some of the current leading theories of quantum gravity are correct (such as some of the string theories, which might be described by noncommutative geometries), there are no point-sized entities involved in the makeup of space or time, since at the Planck scale, the smallest entity is a Planck length (1.6 x 10^-35 m) or Planck time (10^-43 s).[9] I will only consider the noncomplex sub-relations to be Planck size or smaller, since if the noncomplex sub-relations were larger than that, they would occupy more than one location of time, and the problems of subsections 2.1 and 2.2 would ensue. Physicists and philosophers take each position seriously: the position that (1) time can involve durationless (temporally point-sized) items, such as time points, or perhaps durationless temporal sub-relations; and the position that (2) time only involves discrete items, and the basic building blocks of time are discrete times, and for that reason sub-relations must be discrete sub-relations of an irreducible non-zero duration (10^-43 s). Since both position are taken seriously, I will consider each scenario: the position that the noncomplex sub-relations that compose the complex relation between t₁ and t₂ are durationless (point-sized), and the position that there are noncomplex sub-relations that must be the size of a Planck time. I will find that in either case, such continuous or discrete non-platonistic noncomplex sub-relations cannot compose a complex non-platonistic relation between t₁ and t₂.

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[8] Some accounts of causation are described as this sort of a relation.

[9] To my knowledge, even though quantum gravity theories are not verified by experimental data, many physicists are very confident that there is a Planck level. There are, however, conceptual problems with it, as Zeno showed, in the paradox of the stadium.

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2.3.1 A complex temporal relation as a continuum of durationless noncomplex sub-relations, part 1

I will next discuss reasons why a non-platonistic complex temporal relation (allegedly) connecting t₁ and t₂ that is composed of À₁-many durationless noncomplex sub-relations apparently cannot constitute a temporal relation between t₁ and t₂.

It might seem that À₁-many noncomplex sub-relations constituting a temporally located complex relation between t₁ and t₂ would be a complex relation that consists of durationless sub-relations that directly link to one another, in order to give rise to a temporally extended relation between t₁ and t₂. But if that were the case, the temporally located complex relation would be denoted by a statement that describes an infinite regress of durationless sub-relations: 't₁ is related to a sub-relation that is related to another sub-relation that is related to another sub-relation…' This may, however, imply that t₁ and t₂ are not related, since there is no last step in this regress of durationless sub-relations between t₁ and t₂, and thus t₁ and t₂ would be unrelated. This infinite regress attempts to complete a task by an infinite sequence of steps, where the 'completion' 'at infinity', some might claim, in fact never occurs, since an infinite set of items has no last item. Chisholm considers this sort of regress vicious; Moreland has lucidly written about Chisholm’s position: 

 

There are at least three forms of infinite regress arguments… [One form] involves claiming that a thesis generates a “vicious” infinite regress. How should “vicious” be characterized here?... Roderick Chisholm says that “One is confronted with a vicious infinite regress when one attempts a task of the following sort: Every step needed to begin the task requires a preliminary step”. [Chisholm, 1996, p. 53.] For example, if the only way to tie together any two things whatever is to connect them with a rope, then one would have to use two ropes to tie the two the two things to the initial connecting ropes, and use additional ropes to tie them to these subsequent ropes, and so on. According to Chisholm, this is a vicious infinite regress because the task cannot be accomplished. (Moreland 2001, 24.) (Emphasis added.)

  

2.3.2 A complex temporal relation as a continuum of durationless noncomplex sub-relations, part 2

Some philosophers consider infinities to involve paradoxes, and for that reason, they make a point to avoid infinities when describing

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collections. But others may object to such a position and to the reasoning given in 2.3.1, and may hold that infinite collections can exist in nature. Examples of such collections might be, for example, the collection of spatial locations, the collection of time-instants before this present moment,[10] or, perhaps, the collection of noncomplex durationless sub-relations constituting a temporally extended complex temporal relation between or among t₁ and t₂.

An extended continuum of durationless temporal sub-relations resembles an extended continuum of topological spatial points. Such a complex temporal relation consists of À₁-many temporally unextended, temporally located, temporally non-collocated sub-relations, that give rise to an extended continuum (the complex relation between t₁ and t₂). For these reasons, hereafter I will consider a complex relation that is composed of À₁-many durationless temporally non-collocated sub-relations to be a complex temporal relation that is a continuum of durationless sub-relations. Points in a continuum do not directly contact one another, since any point in a continuum is not immediately next to any other points. This reasoning would apply to an extended continuum of temporally located durationless temporally non-collocated sub-relations extending between t₁ and t₂: none of the À₁-many durationless temporally non-collocated sub-relations are immediately next to one another. For this reason, a complex relation composed only of durationless temporally non-collocated sub-relations cannot give rise to a complex connection between t₁ or t₂.

Continuums of points are, however, typically considered to be composed of interrelated points.[11] Perhaps, as with the point-set topological account of space, the complex relation between t₁ and t₂ could consist of À₁-many interrelated temporally non-collocated point-sized sub-relations. If so, perhaps the reasoning of the previous paragraph, where À₁-many temporally non-collocated sub-relations were considered to be the only constituents of a continuum is misguided.[12] Instead of discussing the durationless temporally non-collocated sub-

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[10] This is a position discussed extensively by Quentin Smith (1995, 1993).

[11] Grünbaum (1952, 2001a, 2002b) is one of the philosophers who has argued for this commonly held position.

[12] This is typically held to be the error that Zeno made in his Measure Paradox (unextended points somehow compose an extended line, plane, or volume). See Pyle (1995, 1-7).

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relations as directly attached to one another (which is impossible), the durationless temporally non-collocated sub-relations instead should be considered as interconnected by a relation, topological connectedness, which is perhaps analogous to point-set topological accounts of connectedness of spatial points in the spatial manifold.

If a continuum is extended and interconnected, since the durationless temporally non-collocated sub-relations of the continuum cannot account for the interconnectivity of the continuum, there are two constituents of the complex temporal relation between t₁ and t₂: (1) the À₁-many durationless temporally non-collocated sub-relations, and (2) the topological relation, interconnectedness, between or among the À₁-many durationless temporally non-collocated sub-relations. I will next argue that a non-platonistic interconnectedness relation between or among the durationless temporally non-collocated sub-relations that compose the non-platonistic complex temporal relation between t₁ and t₂ cannot connect the À₁-many durationless temporally non-collocated sub-relations.

Since none of the non-platonistic durationless temporally non-collocated sub-relations are immediately next to one another, the interconnectedness relation between or among the durationless temporally non-collocated sub-relations is a relation between or among non-identical sub-relations (the sub-relations are at a temporal distance from one another). If connectedness is a relation between or among the temporally non-collocated sub-relations, and if the connectedness relation is not also a complex non-platonistic temporally extended relation composed of a À₁-many durationless sub-relations, in order to interconnect the durationless sub-relations, the connectedness relation would be a non-platonistic noncomplex relation between non-collocated sub-relations, which is for that reason located at more than one temporal location. But this is exactly the sort of relation found to apparently involve contradiction in subsections 2.1 and 2.2.

For these reasons, the relation, connectedness, connecting the À₁-many durationless temporally non-collocated sub-relations must also be a complex relation consisting of À₁-many durationless temporally non-collocated sub-relations that are not directly linked to one another. If the connectedness between or among the durationless temporally non-collocated sub-relations was also composed of durationless sub-relations, the relation, connectedness, would itself provide no continuous connection between the non-collocated durationless temporally non-collocated sub-relations that compose the complex

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relation between or among t₁ and t₂. Only if the durationless temporally non-collocated sub-relations that compose connectedness were also interconnected by a complex relation, connectedness₂ (where connectedness₂ is also composed of continuum-many durationless temporally non-collocated sub-relations), would connectedness provide a continuous connection of the durationless sub-relations between or among the complex temporal relation connecting t₁ and t₂. Connectedness₂ would need connectedness₃, and an infinite regress would ensue, where each connectedness relation would require another instantiation of connectedness. At any stage of the regress, each instantiation of the connectedness relation is composed of À₁-many durationless temporally non-collocated sub-relations that do not directly link to one another, which require another instantiation of the connectedness relation. The problem, however, is that any stage of the regress only consists of unconnected À₁-many durationless temporally non-collocated sub-relations, none of which are in contact. At any stage, the unconnected sub-relations require another distinct relation at the next stage of the regress to hold it together, but where the relation at the next stage is also composed of À₁-many unconnected durationless sub-relations that are not in contact. Every stage of the regress is only composed of unconnected À₁-many durationless (point-sized) elements (sub-relations), and for that reason, nowhere in the regress is there any contact or connection between any sub-relations, and there is no interrelating at all between t₁ and t₂. In other words, since we never arrive at a stage in the regress where there are anything but À₁-many durationless sub-relations that are not linked to one another, the temporal connectedness among the À