A Commentary on Stephen Puryear's response to Andrew Loke and Travis Dumsday

Puryear's response essay is called Finitism, Divisibility, and the Beginning of the Universe: Replies to Loke and Dumsday. 

In this commentary, we'll see if Puryear's response to Loke and Dumsday holds up, but I also want to see if what I said in the previous blog is still defensible. 

Puryear says:

My argument has elicited replies from Andrew Ter Ern Loke [2016] and Travis Dumsday [2016]. Here I address the three basic objections to emerge from those replies.

The first two, due to Dumsday, concern the distinction between infinite magnitudes and infinite multitudes, and the distinction between extensively and intensively infinite progressions. 

The third objection, which both Loke and Dumsday urge in one form or another, concerns the possibility that time might be continuous yet naturally divide into smallest parts of finite duration.

That third objection I'm especially interested in. But I am very interested in the first two objections as well. Let's dive in. 




2. Infinite Magnitudes vs. Infinite Multitudes (Puryear's section title). 

Dumsday brings up something that I had actually been wondering about: that traversing infinite magnitudes are just as (or at least as) "problematic" as traversing infinite multitudes. Dumsday says: 

[I]t is counterintuitive to suggest that those worried about the possibility of traversing an infinite multitude would have no plausible corresponding concerns regarding the possibility of traversing an infinite magnitude…. [I]f one thinks it impossible to cross an infinitely large lake (for instance), it is not immediately clear why that intuition would be overthrown by being told that the lake has no real proper parts, being instead a single extended simple. For if the lake is a single infinite extended simple, it still seems prima facie impossible to swim across it.

We'll get to Puryear's response in a second. Dumsday's point seems to be that an infinitely large lake (an infinite spatial extent) would remain impossible to traverse even if the arbitrary, but equal, non-zero, finite intervals haven't yet been conceptually applied to "the lake" such that there'd be an actually infinite number of subdivisions. While I appreciate Dumsday's point here, I think it might veer away from Craig's initial project of demonstrating that traversing an infinite multitude of intervals isn't possible. But the parity of this multitude with magnitude might open up other avenues through which the impossibility of traversal might be motivated. Puryear seems to agree with what I'm saying regarding Craig's initial project: 

The main problem with this objection is that misgivings about infinite traversals—particularly those with an end but no beginning—have consistently been rooted in arguments concerning infinite multitudes, namely, of rooms, books, orbits, days, numbers, and so forth. This is no accident. 

The arguments all proceed by generating some alleged absurdity from the idea that one infinite multitude can be put in one-one correspondence with another. Craig [Craig and Smith 1993: 33–34; Craig and Sinclair 2009: 120–22], for instance, appeals to examples in which infinitely many past days stand in one-one correspondence with the negative integers, and again with the days recorded in an inverse Tristram Shandy diary. 

The principle that such multitudes are equal in size plays an essential role in the arguments (cf. Craig and Smith [1993: 23]). Yet for that very reason these arguments lose their teeth when we shift from infinite multitudes to simple infinite magnitudes, for which there is no analogous principle of correspondence. 

If beginningless traversals of such magnitudes are impossible, then the finitist owes us more of an argument for that conclusion than we have been given thus far.

I do think Dumsday has such an argument lurking in there, but let's first look at what Puryear has to say about the "lake example". 

As for Dumsday’s lake example, if the body of water is supposed to be infinitely large in the sense that it extends without end in both directions, then I grant that one cannot possibly swim across it, regardless of whether it has an actual infinity of parts or not. 

But suppose instead that the person has been swimming forever and merely reaches a certain point in the lake. (Alternatively, suppose the lake is infinite only in the direction opposite the direction of the swimming, and that the person, having been swimming forever, reaches shore.) 

Unlike the case of swimming across a lake that is infinite in both directions, it is far from clear that this sort of beginningless progression is impossible. Indeed the only substantive arguments we have been given for the impossibility of such progressions depend essentially on the assumption of an infinite multitude of distinct steps, and thus have no obvious bearing on simple infinite magnitudes.

Agreed that there could be two different kinds of infinite magnitudes: (1) the lake extends without end in both directions, and (2) the lake extends in the direction opposite the swimming. But I'm not sure why Puryear claims that it's not clear that successfully traversing a beginningless progression is impossible. A lake, ontologically extending into the past and the future, and a beginningless lake with a shore, implies that traversing both lakes is a metaphysical impossibility because both lakes involve traversing an actually infinite multitude (for swimming at a rate, I suppose, involves successive interval traversal). It's a multitude that slices the magnitude into an ever-growing number of intervals, intervals that didn't exist prior to the application of the swimming rate (a metric) to the magnitude attempting to be traversed. Perhaps what's inevitably introducing a multitude into the magnitude is this metrical application of the swimming rate. Could we attempt to rid the magnitude of any dynamically progressing multitudes? Perhaps perdurance would be introduced. But let's avoid that, as Craig is a presentist. The best that I can get out of Dumsday's suspicions is the possible notion that magnitudes are non-metric extents and multitudes involve metric extents. Perhaps the relationship between the two could be cashed out counterfactually. That is, if an infinite multitude (dynamically traversing or not) were to be applied to infinite magnitude, it would be the case that such a magnitude couldn't be possibly traversed. 

3. Extensively vs. Intensively Infinite Progressions (Puryear's third title) - In this section, Puryear responds to another criticism from Dumsday against utilizing Zeno to demonstrate the reality that actual infinities are traversed constantly. Puryear says: 

According to Dumsday, however, it is not at all clear that these are the same kind of infinite progression. 

The Zeno case involves an extensively finite but intensively infinite progression: for instance, a finite period of time with an actual infinity of parts. 

In contrast, the argument for temporal finitism requires only the rejection of extensively infinite progressions that have an end but no beginning. 

Since progressions of the latter sort may raise an additional difficulty by dint of the lack of a starting point, Dumsday [2016: 4] concludes, ‘It is not obvious that these two cases of traversing an infinite are sufficiently close, such that it would be obviously irrational to affirm the possibility of one while denying the possibility of the other’.

The Zeno case could also apply to a finite magnitude involving an infinite multitude (where the multitude is a multitude of unequal intervals). Anyhow, this overlaps with the last blog a bit so let's consider Puryear's reply:

If beginningless (and thus extensively infinite) progressions face a special difficulty in virtue of their lack of a starting point, which does not beset the extensively finite but intensively infinite Zeno progressions, then this needs to be shown. 

As it stands, the finitist argument I have criticized does not even purport to do this. It simply denies the possibility of forming an actual infinite by successive addition, without regard to whether the infinite thus formed is extensively or intensively infinite. 

First, I haven't read Dumsday's essay but I think I've 'shown' that the lack of a starting point for an extensively infinite multitude is an issue that doesn't plague intensively infinite Zeno progressions (not the least of which is that the intensive infinities seem to me to be potentially infinite and involve intervals that aren't equal). Second, Puryear is just mistaken when he claims that those who promote the arguments against traversing an actual infinite (or forming it via successive addition) don't take into account the distinction an extensive and an intensive infinite. Here's my contention. Extensive infinite multitudes must supervene on extensive infinite magnitudes. When such extensive infinite multitudes supervene on extensive infinite magnitudes, the metrical intervals that constitute such multitudes are arbitrary, but equal, non-zero, finite intervals. In fact, I think this sub-point is important enough to be abbreviated since I bring it up constantly (hereafter, let's abbreviate it as AENF). The infinite Zeno progressions don't involve AENF. Thus, intensive infinities are not only finite (because the point about continuousness implies that the subdivision of ~AENF intervals is potentially infinite), but only supervene on finite magnitudes, which doesn't seem to be the kind of conceptual magnitude involved with bracketing the history of the universe. The consequent conceptual subdivisions of such an infinite magnitude in terms of AENF-type-multitude of intervals is conceptually possible (because mathematically legitimate) but metaphysically impossible due to the paradoxes. 

Second, it seems obvious (to me) that the type of infinity being considered when talking about the kind formed by successive addition is an extensive infinite due to the AENF-type-intervals progressively denominated as a conceptual, metrical index for how temporal events have progressed. The argument is that if this conceptual index is posited as metaphysically possible, paradoxes result. 

4. A Tertium Quid?

In this section, Puryear considers another view of continuous time and its divisibility offered by Loke and Dumsday. 

On this view, time is continuous and hence infinitely divisible, but is not actually divided to infinity. 

Rather, time naturally divides into smallest actual parts of finite duration, with any further divisions being only potential or conceptual. 

On such a view, events of finite duration never resolve into more than a finite number of actual temporal parts, thus never involve the sequential occurrence of an actual infinity of subevents. 

But reaching the present in a universe without beginning would involve such an occurrence. So the Zeno objection falls away, without imperilling the finitist argument or abandoning the continuity of time.

At this point, I'll just say that Craig would seem to deny this proposal because it seems to revert back to the chronon-view, which we saw Craig has problems with. Puryear has issues with the coherency of this proposal, as Craig might. I agree with Puryear that this view does seem committed to the idea that time is discrete since the chronons (what I'm calling 'the smallest actual parts of finite duration') are 'there' independent of conceptual subdivisions. Loke disagrees, arguing that if the chronons are "joined together", the continuousness of time is preserved. This seems implausible but Loke illustrates the idea in this way: 

To illustrate one possible way of joining temporal parts, consider Aristotle’s view that things neither move nor rest at a point, but instead move or rest only during an interval (Physics 232a32–4). The extensionless ‘point of transition’, which we conceptualise between ‘moving interval’ and ‘resting interval’, can be understood as the coincidence of the boundaries of the two intervals at which they join together; and, as Aristotle explained, there is neither motion nor rest at this (or any) point.

At this point, it'll be helpful to consider what Craig said in the Reasonable Faith podcast we looked at in this previous blog. Craig says: 

Now, suppose instead, then, we take the view that the “now” does have a certain non-zero finite extent. In that case time would be composed of little time atoms and these would be incapable of further division. 

Well, the difficulty with this, although it's possible, is that it would mean that temporal becoming precedes by fits and starts. 

Reality is sort of jumpy, it jumps ahead, rather like the way a movie does. In a film you have these separate frames, and the separate frames involve jumps in the action from one to the next. But when the film is shown they go by so quickly that you can't discern the discontinuities. 

Craig's 'separate frames' might be akin to Loke's (or Aristotle's) 'intervals'. Loke's 'extensionless point of transition' might be the 'discontinuities' Craig says we can't 'discern'. Though we can't discern them (in the sense of perceiving the transition), Loke argues that perhaps we can conceive the transition (from moving to resting interval) in terms of boundary-coincidence. Motion/rest don't occur at the point of transition, or any point. Motion/rest only apply to intervals, not points. The northern border of South Dakota is coincident with the southern border of North Dakota. The ending of SD is identical with the beginning of ND. Puryear unpacks Loke's thought thus: 

The thought here appears to be that the two intervals join together in virtue of the coincidence of their boundaries at an instant. 

In other words, the moving interval ends at the same instant the resting interval begins, as on option (i). 

However, Loke also describes this ‘point of transition’ as something we conceptualise, which seems to imply that, apart from our conception, there is no point of intersection, no coinciding boundaries, and thus no division. 

The implication appears to be that the moving and resting intervals are in reality just one interval that we divide in thought, whereas they were supposed to be really distinct. 

Indeed, Aristotle himself seems to hold that any such transition point could be only a potential division in time, since on his view actual divisions introduce discontinuities [Physics, VIII.8, 263a23–30] and time is continuous [Physics, V.2, 232b22–25]. 

Whether time could be both continuous and really divided into parts therefore remains to be seen.

I think I agree with Puryear here. He seems to side with Craig that actual divisions are actual, non-conceptual discontinuities, which doesn't seem compatible with time's actual continuousness. Loke's illustration imports conceptual metrics, such as points and intervals and boundaries onto actual, continuous time, while still trying to preserve its continuousness, which seems like importing links in a chain onto a continuous wire. It just doesn't seem to be a continuous wire anymore; it would then be a discrete chain. 

Puryear offers another issue with the proposal:

Assume for the sake of argument that these doubts about coherence can be answered. If the goal were merely to show that time could be continuous and have smallest parts of some finite duration, or in other words, to show that the argument for temporal finitism is compatible with the continuity of time, then perhaps Loke and Dumsday would have succeeded. But this still does not fully vindicate the finitist argument. 

For as long as the conceptualist (or conventionalist) approach suggested by Craig’s remarks also remains a viable option, then the finitist has not yet established the crucial premise that a beginningless past would consist in an infinite sequence of events (rather than one simple event that we divide in thought). 

In other words, in order for the finitist argument to go through without falling prey to the Zeno objection, the finitist needs time to have smallest natural parts. But if all that has been shown is that time could have smallest natural parts, not that it does have such parts, then temporal finitism has not been established. 

In order to fully vindicate the finitist argument, then, Loke and Dumsday not only need to say more in support of the coherence (and indeed the plausibility) of their alternative conceptions of time; they also need to show that the conceptualist alternative is not plausible, or at least that it is comparatively implausible. Until that has been done, the case for the finitist argument remains at best incomplete.

As my previous blog makes clear (I hope), I don't take the tact of Loke and Dumsday. I take the tact that Craig takes, but which isn't confuted by any of Puryear's issues. This dichotomy between an actually infinite number of events and one simple event misses the point. This one simple event is (if beginningless) capable of an actually infinite number of conceptual sub-divisions. The very presence of this capability is what renders this one simple event absurd, if beginningless. Thus, Craig can approach this as a reductio. Assume the universe is beginningless (at this point the history of the universe is an infinite magnitude, but not an infinite multitude if one conceptualizes such a history in terms of being one simple event). Next, make the argument that infinite magnitudes are capable of the sort of conceptual sub-divisions involves with an actual infinite number of AENF-type-intervals, and so a specific sort of infinite multitude is capable of supervening on any infinite magnitude. Once such multitudes are conceptually postulated, we can see that the traversal of such a multitude involves absurdities. Such absurdities ensure that the infinite magnitudes on which they supervened are also thereby absurd. Hence, whether the history of the universe involves the conceptualization of one simple event or an actually infinite number of conceptualized events, the absurdity still seems to linger.