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BY 4.0 license Open Access Published online by De Gruyter August 12, 2022

Time Does Not Pass if Time Began from an Infinite Past

Kunihisa Morita

Abstract

Philosophers have long discussed whether time really passes. Simultaneously, they have also discussed whether time could have begun from an infinite past. This paper clarifies the relationship between the reality of time’s passage and an infinite past. I assert that time cannot have an infinite past if time really passes. This argument is based on a proposition that an infinite series of events cannot be completed if time really passes. A seemingly strong objection to this proposition is that no movement is possible if an infinite series cannot be completed. However, movement is clearly possible; thus, an infinite series can be completed. I argue that although it is sure that an infinite series can be completed in the static view, an infinite series cannot be completed (and thus, time could not have begun from an infinite past) in the dynamic view.

1 Introduction

This paper focuses on the relationship between the reality of time’s passage and an infinite past. I argue that time does not pass if time began from an infinite past. To the best of my knowledge, no research has focused on this relationship, although many arguments have been proposed about whether time could have begun from an infinite past (e.g., Craig 1994, 2000; Craig and Sinclair 2009; Puryear 2014; Sinnott-Armstrong 2004; Whitrow 1980). I maintain that it is important to consider whether time passes when discussing the existence of an infinite past.

Let us consider what the assertion ‘time passes’ means. The philosophers of time often refer to the view that time passes as ‘A-theory’ or the ‘dynamic view of time’. For example, Deasy states, according to A-theory, ‘there is an absolute, objective present moment’ (Deasy 2017, p. 378).[1] ‘Time passes’ means that this absolute present (called ‘NOW’ in what follows) changes (Leininger 2015; Price 2011; Tallant 2018).

According to the static view of time or B-theory, there is no NOW. Therefore, time does not pass. The moment we feel ‘now’ is not absolute. According to the static view (together with the four-dimensionalism), I am a four-dimensional entity extending to the time dimension as well as three spatial dimensions; each of my temporal parts feels ‘this moment is now’. In other words, such a present moment is relative to the speaker; it is not absolute but indexical, like ‘here’ in space.[2]

How, then, does the NOW change? Here, three major ontological models of the dynamic view of time have been proposed: presentism, the growing block universe theory (GBU), and the moving spotlight theory (MST).[3] Presentism insists that only present things exist.[4] Presentists typically consider that the passage of time is identified by changes to physical properties (Prior 1962). This means that which physical properties are real depends on when is NOW. GBU maintains that a four-dimensional block universe, including both the present and the past, grows as time passes (NOW is the edge of the block universe). Broad (1923) and Tooley (1997) are representative proponents of this model. The block universe changes depending on when it is NOW. MST argues that the future and the past exist, as well as the present, and that the NOW moves along the time dimension of the block universe (the NOW literally moves on the real time axis).[5] In sum, ‘time really passes’ means that ‘physical properties change’ in presentism, the ‘block universe grows’ in GBU, and the ‘NOW moves’ in MST.

This paper aims to show that if time really passes, time could not have begun from an infinite past. My argument is based on the proposition that an infinite series of events cannot be completed in the dynamic view. It could be objected that if an infinite series of events cannot be completed, movement is impossible. Yet, there evidently exists movement; therefore, an infinite series of events could be completed. I call this objection ‘no movement objection’ (NMO). I argue that the NMO does not successfully show that an infinite series of events can be completed in the dynamic view.

2 The Argument for the Impossibility of Infinite Past

The typical argument to show that time could not have begun from an infinite past is a variation of the one formulated by Kant (2007, A426 = B454). Kant tried to prove that the world had a beginning in time by showing that the assumption that the world had no beginning is irrational because an infinite temporal series of events cannot be complete.[6] Given that Kant did not consider the possibility that time is cyclic, the assumption that the world had no beginning involves the concept that the world could not have begun from an infinite past. In addition, Kant considered that the beginning of the world and that of time are different because there could be a temporal vacuum in which there is no physical change. ‘The world’ indicates all things and physical events at all times but does not include time itself. There is a possibility that time passes even when there are no things and no events—in other words, there can be a temporal vacuum (at least, we cannot rule out this possibility).[7] For the sake of the discussion, I assume that there is no temporal vacuum and, thus, that the beginning of the world and that of time are equal (however, below, I discuss a case in which a temporal vacuum exists).

Puryear (2014) formulates the type of proof suggested by Kant as follows. Like Kant, he considered that if the universe did not have a beginning, it had an infinite past (Puryear did not consider cyclic time). In addition, the word ‘universe’ is vague. In contemporary physics, the universe includes space and time. However, in Puryear’s argument, ‘universe’ seems to be used like ‘world’ in Kant’s argument; indeed, Puryear used the phrase ‘temporal series of past events’ (Puryear 2014). If there was a temporal vacuum before the beginning of the world (in Kant’s sense), there was a period in which no event occurred. However, I presume that there was no temporal vacuum; thus, the beginning of the universe and that of time are equal. Hence, I reformulate Puryear’s argument as the ‘argument for the impossibility of infinite past’ (AIIP).

The argument for the impossibility of infinite past

  1. If time began from an infinite past, then the past would comprise an infinite temporal series of events.

  2. An infinite temporal series of past events would be actually and not merely potentially infinite.

  3. It is impossible for a series formed by successive additions to be actually infinite.

  4. The temporal series of past events was formed by successive additions.

  5. Therefore, time could not have an infinite beginning.

3 The No Movement Objection

According to Puryear, premise (3) is controversial. For example, Sinnott-Armstrong (2004) points out that accepting the possibility of completing an infinite series of events seems to lead to the conclusion that no movement is possible. This is because the acceptance of actual infinity entails that space and time can be divided into an actually infinite number of parts. Given this, these actually infinite parts must exist between two given time points. Therefore, if completing the actually infinite series of events is impossible, no movement is possible. However, as movement exists, the actually infinite series of events can be completed. I call this type of objection the ‘NMO’.

The no movement objection

  1. It is impossible to complete an actually infinite series of events.

  2. Space and time constitute a continuum.

  3. Space and time comprise actually infinite subsets because of (7).

  4. One cannot move from a given point A to another given point B because of (6) and (8).

  5. Movement actually exists.

  6. Therefore, premise (6) is false.

One way to rebut the NMO is to deny premise (7) by arguing that neither space nor time is a continuum (Craig 1994; Whitrow 1980). Ardourel (2015) also insists that the discreteness of space and time can be embedded in a formulation of classical mechanics and cannot be falsified experimentally. However, for example, Salmon (1998) points out that Zeno’s paradox of ‘Stadium’ shows that space and time cannot be discrete, and Loke (2016) states that there is little ground that space and time are really discrete. Therefore, whether premise (7) is false is still controversial.

Another possibility is to deny that (8) can be concluded from (7) (Craig and Sinclair 2009). If space and time are a continuum, they could be divided into an infinite number of subsets. However, this division exists only in the human mind. In other words, space and time are not accumulations of their parts; they exist as a whole. Therefore, one can keep dividing space and time infinitely, but this division cannot be completed.

Puryear (2014) addresses this latter rebuttal arguing that space and time exist as a whole prior to their parts. In other words, time is not an accumulation of its parts. Therefore, this point entails both that premise (1) of the AIIP must be denied and, by extension, that the AIIP is invalid. Further, according to Puryear, (2) and (4) of the AIIP are also denied. Therefore, the proponents of the AIIP also have to accept the NMO.

Loke (2016) and Dumsday (2016) criticise Puryear’s argument, and Puryear (2016) replies to their criticism. Although this controversy is relevant to the philosophy of time, it is beyond the scope of this paper. Rather, this paper argues that the lesson from the NMO is that it is sure that an actually infinite series of events can be completed in the static view (thus, premise (6) is false, and conclusion (11) is true); thus, the AIIP is invalid. However, in the dynamic view, premise (6) is true, but the premise (10) is doubtful (or, more precisely, the dynamic view itself is doubtful); thus, the AIIP is valid.

As mentioned in the introduction, the static view of time does not accept the existence of the NOW. This means that space-time exists as a whole. Therefore, the meanings of ‘completing a series of events’ in dynamic and static views are different. For example, consider that Achilles runs from a spatial point A to another spatial point B (suppose that Achilles starts from A at time t 1 and arrives at B at t 2). Consider also that space and time are a continuum and that they comprise actually infinite points. To arrive at B, Achilles has to complete actually infinite series of events. In the static view (combined with four-dimensionalism), Achilles’ four-dimensional trajectory from A to B exists as a whole. In other words, Achilles standing at A at t 1 and Achilles standing at B at t 1 are different temporal parts of Achilles as a four-dimensional entity. Achilles’ four-dimensional trajectory in space-time is like a bar in three-dimensional space. Even if this bar comprises actually infinite parts, it can exist. Likewise, even if Achilles’ four-dimensional trajectory comprises actually infinite temporal parts (this means that Achilles’ running from A to B is an actually infinite series of events), the trajectory can exist. Subsequently, when such a four-dimensional trajectory of something comprises actually infinite parts, it can be stated that an actually infinite series of events is completed in the static view. Therefore, although Craig and Sinclair (2009) argue that time cannot be divided into actual infinite parts since time exists as a whole, I insist that time can consist of actual infinite parts, and it does not pose a problem for the existence of movement since time exists as a whole (which means things statically exist as four-dimensional entities).

However, ‘time exists as a whole (time is not an accumulation of its parts)’ does not mean ‘things exist as four-dimensional entities’. What I mean by ‘time exists as a whole’ is that time is static, and things exist as four-dimensional entities. Some might insist that in MST, together with four-dimensionalism, time can exist as a whole because the future and the past exist as well as the present. However, the point is that the NOW has not achieved the future; in other words, the future has not arrived yet, even if the future exists. Therefore, time does not exist as a whole in MST as discussed below.

In sum, according to the static view, ‘movement’ means that a spatial location of a temporal part of a certain four-dimensional entity, E, is different from a spatial location of another temporal part of E. Thus, the conclusion of NMO is surely true. In addition, as premise (4) is false, AIIP is not valid (thus, time could have begun from an infinite past). I assume that there can be an actual infinity in the physical world; space and time are a continuum in this argument. However, as discussed below, even if these premises are false, my argument that time cannot have an infinite past if time really passes is not undermined.

Some might state that if the movement in the static view of time is what is discussed above, it follows that movement is not real in the static view. This is because, as some might insist, real movement is not such a phenomenon. However, accepting this statement does not undermine the above discussion. This means that premise (10) is false. This is just a problem about words. Even if premise (10) is false, it is the fact that we can observe movement-like phenomena (a spatial location of a temporal part of a certain four-dimensional entity, E, is different from that of another temporal part of E) in the static view.

4 Dynamic View and Infinite Series of Events

Let us examine the dynamic view of time. As mentioned above, the meaning of ‘completing a series of events’ in the dynamic view is different from that in the static view because time does not exist as a whole; time is an accumulation of its parts (its parts exist prior to the whole). First, for simplicity, consider the dynamic view combined with three-dimensionalism and suppose that space and time are a continuum. In this view, Achilles standing at both A and B is the same three-dimensional individual. Therefore, to run from A to B (to complete a series of events), Achilles must pass on all spatial points existing between A and B. Thus, if there is an actually infinite series of events (space and time are a continuum) between A and B, Achilles cannot move in the dynamic view.

In the static view, that Achilles completes an actually infinite series of events while running from A to B during finite time is like that there is actually an infinite number of dots within a finite line. If one accepts the existence of actual infinity in the physical world, an actually infinite series of events can be completed in the static view. However, in the dynamic view, the existence of actual infinity in the physical world and the possibility of completing an infinite series of events are different issues. Even if the former is true, it is not guaranteed that the latter would also be true. Even if there is an actually infinite number of dots within a finite line, it is impossible that a finite set of dots changes (in the dynamic sense) into an infinite set of dots (it is impossible that Achilles dynamically traverses all dots in the line).

Nevertheless, some might insist that the above discussion is valid in three-dimensionalism. The dynamic view of time can combine with four-dimensionalism. However, I argue that even within four-dimensionalism, movement is impossible in the dynamic view if time is a continuum (and there is actual infinity in the physical world) because time does not exist as a whole in the dynamic view. More precisely, time cannot flow. In the dynamic view, the NOW must pass an actually infinite number of time points within finite time; nonetheless, this is impossible.

According to MST, the past and the future also exist, and the NOW literally moves on the real time axis. The period from a finite past to the present comprises the accumulation of infinite time points that were NOW; there is an infinite number of time points that have existed until the time we consider as present became NOW. Therefore, the NOW cannot arrive at the present because an infinite series of events cannot be completed. Thus, time cannot flow in MST. According to the GBU theory, the proposition that time passes indicates that the four-dimensional block universe grows. In the GBU, the future parts are successively added to an existing block universe. This means that if time is a continuum, the infinite series of events have to be completed for finite time to pass. Finally, in presentism, there is only NOW. In presentism ‘time passes’ means physical properties change as discussed in Section 1. Therefore, if time is a continuum, an infinite number of infinitesimal change is needed for finite change. Accordingly, time cannot pass even during a finite period.

In the above discussions, I assume that time is a continuum, and there is actual infinity in the physical world. However, these assumptions can be false, as mentioned. Nevertheless, as the aim of this paper is to show that time cannot have an infinite past in the dynamic view of time, whether these assumptions are true is irrelevant. First, if there is no actual infinity in the physical world, time cannot have an infinite past in the dynamic and static views (I focus only on that time cannot have an infinite past in the dynamic view; whether time can have an infinite past in the static view is beyond my scope). Second, even if time is discrete, there is an actually infinite series of events if time began from an infinite past (if actual infinity exists).[8] Therefore, for my purpose, whether these assumptions are true is irrelevant (even if both are false, time cannot have an infinite past in the dynamic view).

In the static view, if there is actual infinity in the physical world, time could have an infinite past. However, even if there is actual infinity and time is discrete, time could not have an infinite past in the dynamic view (although finite time can flow). Some might state that an infinite series of events can be completed after infinite time passes, even in the dynamics view. Nevertheless, in this paper, I insist that it is unintelligible to state that time began from an infinite past in the dynamic view. Consider there is no end of time (time has an infinite future). If infinite time passes in the future, can time end? I am afraid that this statement is meaningless; how can time end in a case where time has no end? Likewise, if time began from an infinite past, then the NOW cannot arrive at, for example, 1st November 2021. However, again, taking the static view and four-dimensionalism, time can have an infinite past and future; this is like space can be infinitely large (at least, this is intelligible).

In the dynamic world, potential infinity can only exist with temporal events, while actual infinity can exist spatially since, in the dynamic view, time itself is potentially infinite. According to Linnebo and Shapiro (2019), Cantor’s view that ‘potentially infinite’ is dubious is now dominant in mathematical domains. I consider this to be so because mathematics deals with the static world.[9]

I have assumed thus far that there is no temporal vacuum. However, does my argument remain valid even if there was a temporal vacuum before the beginning of the world (in Kant’s sense)? The point of the argument lies in the fact that time consists in the accumulation of its parts in the dynamic view and the argument that an actually infinite series of events cannot be completed. Accordingly, if some process proceeds even in the temporal vacuum, in which there are no physical changes, the same line of the argument can be applied.

Actually, both GBU and MST still allow for change even if there is no physical change. Even during the period in which there is no physical change, the block universe can grow; thus, according to GBU, the temporal location of the NOW can change. Likewise, according to MST, the NOW can move even during the period in which there is no physical change. Therefore, time cannot have an infinite past in GBU and MST, even if there had been a temporal vacuum before the world began. Nevertheless, as presentism identifies the passage of time with physical change, the existence of a temporal vacuum falsifies presentism (Tallant 2010). However, some presentists, such as the leading presentist Markosian, do not identify the passage of time with physical change; thus, they might not consider that presentism is false even if a temporal vacuum really exists. Markosian introduces the concept of the ‘pure passage of time’, which can exist even in a temporal vacuum (Markosian 1993). For example, we measure the passage of time by cyclical physical changes, such as clocks and the apparent revolution of the sun. According to Markosian, these periodic physical changes are proxies of the pure passage of time. Nonetheless, the precise meaning of ‘the pure passage of time’ is unclear. One suggestion is that a non-physical and unobservable change exists, which could be referred to as a ‘metaphysical change’, identical to the pure passage of time. If this interpretation is correct, there are some kinds of changes even in a temporal vacuum; thus, time could not begin from an infinite past.

5 Conclusions

This paper argued that time could not have begun from an infinite past if it really passes. If time passes and begins from an infinite past, an infinite series of events must be completed, which is impossible if time really passes. Some authors insist that the impossibility of completing an infinite series of events leads to the impossibility of movement. This argument is surely correct (if a series of events is formed by successive additions and space and time are a continuum). However, I argue that an infinite series of events can be completed in the static view of time, although it is impossible (thus, movement is impossible) in the dynamic view.

In the static view combined with four-dimensionalism, things also extend along the time dimension (the fourth dimension), and there is no absolute present (NOW). Therefore, if there is actual infinity in the physical world, time could have an infinite past; this is similar to the idea that space can be infinitely large if there is actual infinity in the physical world. Nevertheless, even if there is actual infinity in the physical world, time cannot have an infinite past as an infinite series of events cannot be completed in the dynamic view. Even if there can be an actual number of dots within a finite line, it is impossible to dynamically change a finite set of dots into an infinite set of dots in the dynamic view. Some might insist that time can arrive at the present (for example, 1st November 2021) after infinite time has passed, even if it began from an infinite past. However, consider that time has no end. It is also possible that time ends after an infinite time passes if the above line of criticism is correct. Nevertheless, how can time end in a case where time has no end? In the first place, an actually infinite time cannot pass in the dynamic view (so that time can be endless); time is only potentially infinite. Therefore, if time has no beginning (time began from an infinite past), the NOW cannot arrive at the present.

Finally, there is another case in which time has no beginning: when time is cyclic. I consider cyclic time to be also impossible in the dynamic view of time. However, this issue was beyond the scope of this paper, and it will be addressed in my future work.


Corresponding author: Kunihisa Morita, Graduate School of Human Sciences, Osaka University, 1-2 Yamadaoka, Suita, Osaka, 565-0871, Japan, E-mail:

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: JP19H01187

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Published Online: 2022-08-12

© 2022 the author(s), published by De Gruyter, Berlin/Boston

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