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| Alex Malpass |
Wes Morriston and Alex Malpass (MM) argue that there is no symmetry-breaker (SB) between a beginningless past and an endless future. They consider proposed SBs (potential infinity, non-actual potentiality, and non-existence) and demonstrate their inadequacy.
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| Wes Morriston |
There is no SB with the endless future being a potential infinite. If the future is endless, then the series of events that will be is an actual infinite. There is no SB with the future being non-actual because the series of potentialities that will be actual is actually infinite. There is no SB with the future being non-existent because the beginningless past is also composed of non-existent events. The endless future is composed of a series of events that will exist.
Our thesis is that MM’s case for the symmetry of the beginningless past and the endless future fails. While non-actuality and non-existence are important corollaries of the more fundamental properties of a relevant SB, we think MM’s case against the endless future being a potential infinite is misconceived. This is mainly implied by how tense differently relates to a task T1 that is happening or became completed, but never began, and another task T2 that is happening, but begins at a time, but which will never be completed. This will become evident in how we think this view answers all of the worries MM have with the proposed SBs.
Endless Future: Potential or Actual Infinite?
A potentially infinite series of events is an ever-increasing series that is always finite and increasing toward infinity as a limit. An actually infinite series of events is a complete series of definite and discrete members that is not increasing. We think the endless future is like the former and not the latter because of the nature of tense. Temporal becoming is a real and ineradicable part of reality. Things and events come into being in the present and pass out of existence as the present becomes.
MM introduce mathematical functions to illustrate the two series. A(x) is a function where the output is “everything less than or equal to the input”. For example, if x = 2, then A(2) is {0, 1, 2}. A(x) illustrates a potential infinite by constantly increasing the input without limit, thereby increasing the function’s proportional output. B(x) is a function where the output is “everything greater than the input”. For example, if x=2, then B(2) is {3, 4, 5 . . .}, where the output is an actually infinite series of numbers. How do these functions apply in the case of God’s decree that an angel sings praises once a day on into the endless future? How many praises will the angel sing? Is it true now that there is some number n of future-tense propositions specifying the number of future angelic praises? Are those propositions denumerable thereby making the praises denumerable?
MM argue against the thesis that a potentially infinite number of praises will be sung. They propose a dilemma. Either the endless future consists of an ever-growing series of events that will be (which is false) or what is ever-growing is not the future, but the past (which is irrelevant).
They illustrate the first horn through the example of presidential inaugurations.
For example, it is not as if today there will be, say, ten future presidential inauguration events, but that tomorrow there will be eleven. However many future presidents will be sworn in, that number can only go down, as each new one gets sworn in. It certainly cannot increase as time passes. (MM's essay)
Since we think this is false as well, something is going wrong here. The endless future is not an ever-growing series of events in the sense that the number of events that will be increases from the standpoint of the present. That would have the absurd results above with the inaugurations. In one reading (the divided sense), the number of events, that will be, is indefinite and indiscrete and so not denumerable. In another reading (the composite sense), the number of events that will be, is zero. In this sense, the endless future can’t be an ever-growing series of events because there are no events. In the divided sense (i.e. the number of events, that will be), the endless future serves as a definite description for an ever-growing series of events. ‘Endless series of events’ serves as a definite description for ‘the ever-growing modal shape of the past-present block’, in the sense that it will forever take on a new modal shape as new events take place (distinguishing this from the growing-block theory). This also is indefinite and indiscrete and, so, the number of shapes the block will take is not denumerable (since a potential infinite is not a number) either.
Perhaps MM will object that our block represents the future-perfect and not the simple future. In other words, perhaps the block represents all the events that will have happened. This is a good place to introduce one of our main objections to MM. MM argue that the future perfect doesn’t imply the simple future. That is, from the fact that an event will happen, we can’t infer that an event will have happened, with the important caveat that there is no time that the events stop coming to be. That is, that this is some kind of fallacious inference in the context of tensed logic. While we haven’t yet found an explicit prohibition against the inference in tensed logic, we do have the strong intuition that there is an entailment relation between the two. Unfortunately, this is just the denial of their claim. But let’s back up the denial this way.
First, there’s no reason why the future perfect can’t imply the simple future in some cases. For instance, from the fact that I will have eaten a normal loaf of bread I can infer that I will eat it. From the fact that I will have counted all my baseball cards I can infer that I will count all my baseball cards. This seems to imply that the future perfect can imply the simple future in the case of finite collections, all things being equal.
Second, why can’t this implication hold for the future perfect and the simple future regarding the endless future? We are told that the endless future needs to be cashed out in terms of an actually infinite series of events that will be. What is potentially infinite is the series of ever-growing events that will have been. We are to imagine an immortal counter who begins today and counts a natural number a day. MM suppose that the counter will count an actual infinity of natural numbers, but that the natural numbers that will have been counted is potentially infinite. But if the counter never stops counting, whence this wedge between the future perfect and the simple future? If we look grammatically at the future tense, there is a continuity between the modal auxiliaries (will) and a discontinuity between the verbal auxiliaries (be vs. have been). The modal continuity ensures that there isn’t a change of metaphysical subject. That is, both future tenses are about the future, about something that will happen. The verbal discontinuity comes in when the future perfect makes the addition that what will be will have been. But since both tenses have the same modal auxiliaries, both tenses specify an identical modality. What is it for some event to be such that it will be? The modality of it is clear enough: it is in the future. Reverting to counting again, the verb denotes that the counting-event will be such as to come into being. That is, the event will come into being. If grammatical tense supervenes on metaphysical tense, and metaphysical tense has the property of becoming, then, once that event comes into being, it will be such as to have come into being. The simple future has a necessary counterpart with the future perfect as both covary with an identical metaphysical tense, the latter of which is signified by both verbs being qualified by an identical modal auxiliary. This has the consequence that whatever will be will have been, and whatever will have been will be. In fact, it seems as if there is a biconditional relationship between the two tenses when certain conditions are in place (immortality, incessant counting, etc.).
If this is the case, then MM are subject to a dilemma. Either both tenses signify an actual infinite or they both signify a potential infinite. If the former, the MM’s thesis is trivial since it just trivially follows that if there is an actually infinite number of numbers the counter will have counted, then there is an actually infinite number of numbers the counter will count (and vice versa). If the latter, then MM are committed to the endless future being a potential infinite. The A(x) function will apply to both the future perfect and the simple future (not to mention the other two grammatical future tenses). The B(x) function will apply to neither since its output signifies a series of numbers all of which cannot be counted (it’s impossible to count to an actual infinite by successive addition).
With this in mind, we’re in a better position to understand the endless future in terms of the shape that the block both will and will have had. MM cannot confine the future shapes that the block will have had to the future perfect and disconnect such future shapes from the shapes that the block will have if the future perfect and the simple before imply each other!
This addresses MM’s second horn as well (the charge of irrelevancy), since if both tenses imply each other, then whatever is said about the future perfect implies something said about the simple future! If we specify the potential infinite for the future perfect, then we’re also specifying the potential infinite for the simple future.
Why can’t the same be said for a series that ended but never began? It is here that the proposed SB makes itself felt. Here is the dialectic. Tense is an SB because a potential infinite is an SB, and a potential infinite is an SB because tense relates to a series without a beginning differently than to a series with a beginning. If tense is real, and you begin a series of counting, it will never end. But the same can’t be said for the beginningless past, even if tense is real. Not beginning a series is a necessary condition for that temporally successive series to be actually infinite, while beginning is a necessary and sufficient condition for that series to be potentially infinite. The grammatical point that there is a symmetry between the tenses is of no avail because the direction of time is only going in one direction (from the past to the future). For MM to sustain the symmetry, they will need to have the past growing in the earlier than direction, which contradicts temporal becoming.
*Citations come from the essay Endless and Infinite, by Alex Malpass and Wes Morriston.
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