Josh Rasmussen has recently offered a “new” infinitary paradox on his blog. Josh is consistently wonderful to interact with; I deeply admire the patience and charity he extends to his interlocutors and it’s something that I work on (but rarely succeed at) emulating myself. Josh doesn’t seem to be completely convinced by his new paradox — he invites people to “attempt to destroy it” — but he does think that it is superior to the grim reaper scenarios that have recently been popular in the Kalam-related literature.

What’s the new paradox? We imagine a scenario where Waldo runs an infinite distance in 2 minutes. Josh specifies that Waldo’s speed increases so that, in the first minute, Waldo runs at 1 meter per second; in the next half-minute, Waldo runs at 2 meters per second; in the next quarter-minute, Waldo runs at 4 meters per second; and so on. We may worry that Waldo’s speed changes discontinuously and that this requires an infinite force. Not to worry: we could modify the example so that Waldo’s location is given by x = tan(t/(4pi)). In that case, for every time between t=0 and t=2 minutes, Waldo only requires finite acceleration and so finite force. Still, Waldo ends up traveling an infinite distance over those two minutes.

Now, here comes the paradox: if Waldo has traveled infinitely far, where does Waldo end up? Josh suggests that Waldo ends up infinitely far away and that, while this is strange, it’s not contradictory. In a second scenario, Josh considers Waldo’s friend, Bob, who tries to retrace Waldo’s steps to find where Waldo ends up after the two minutes. But Waldo is nowhere to be found — Waldo “has no definite location at all.” This, Josh thinks, is metaphysically suspect — how could a spatial object not have a location? But — Josh adds — things get even worse in a third scenario. Now, we imagine that Waldo extends a tape measure as he travels. Even if we imagine an endless tape measure, there is no final number for Waldo to see when he finishes his journey. Worse, it appears that Waldo has leaped infinitely far, because there is no number Waldo’s tape measure displays immediately before reaching his infinitely far away location. Josh thinks this is a contradiction: “Waldo’s tape measure can only display a finite distance, but Waldo has run an infinite distance.” Josh uses these three scenarios to defend the following argument:

1. If the past could be infinite, then an infinite series of events could occur.
2. If an infinite series of events could occur, then these “Waldo” scenarios would be possible.
3. But the Waldo scenarios are not possible.
4. Therefore, the past could not be infinite.

I’m not convinced by Josh’s argument. Importantly, I’m not convinced that there’s anything problematic about the three scenarios, though I arrive at different conclusions about them than does Josh; in turn, this means that I reject either premise 2 or 3. I reject premise 2 if the Waldo scenarios include the conclusions that Josh draws from them, while I reject premise 3 if we substitute those conclusions with the ones that I will draw. In any case, in what follows, I present two alternative ways of analyzing the Waldo scenarios.

The two alternatives depend upon how we imagine an infinitely large spacetime. On the first possibility, spacetime is infinitely large but does not include any points at infinity. On the second possibility, spacetime includes points at infinity.

Let’s unpack the first possibility — spacetime is infinitely large, but does not include points at infinity. To make our lives easier, instead of thinking about four-dimensional spacetime, let’s think about the real line. We can think about two sets of topological properties: first, the topological properties that some geometric object has intrinsically and independent of however we might (or might not) embed that object into some other object and, second, the topological properties that some geometric object has given its embedding. If we only consider the first kind of properties, then an open, finite sub-interval of the real line — like the interval (0,1) — is topologically equivalent to the entire real line. Just as there is no first or last point in the real line, so, too, there is no first or last point in (0,1). (For a proof that (0,1) and the real line are topologically equivalent, trivially modify the proof I offer in footnote 4 of this paper.)

So, in terms of its intrinsic topological properties, the real line is just an interval without boundary points. If some item ends up going through either open boundary, where does the item end up? The item ends up nowhere at all — it literally exits existence altogether.[1]

I’m not the first person to say this. Josh’s scenario is identical to one often considered in connection with so-called space invaders and well-known among philosophers of physics (Saari & Xia 1995; Malament 2008). According to the standard view, if we allow unbounded accelerations, then objects can exit spacetime and so go out of existence. In fact, we can even have a situation where this occurs with realistic forces. In Newtonian physics, a system of 5 point-like particles, interacting via the gravitational force, can fly out to infinity and exit spacetime altogether in finite time.

As far as I can tell, this reply takes away everything that seemed paradoxical about Josh’s scenario. There’s no longer a puzzle about where Waldo ends up — since Waldo ceases to exist, and non-existent items do not have locations, Waldo doesn’t have a location. There’s no longer a puzzle about why Bob cannot find Waldo — you cannot find people who have gone out of existence. And there’s no puzzle about what value the measuring tape reads when Waldo finishes — Waldo no longer exists, can no longer read the measuring tape, and the end of the measuring tape has gone out of existence along with Waldo.

Let’s unpack the second possibility, that is, we can construct an infinitely large spacetime that includes one or more points at infinity. This isn’t a possibility often considered by philosophers, but it is utterly standard among mathematicians. (There’s a similar construction in relativistic spacetimes, where one considers adding, e.g., the conformal boundary to a spacetime.) It turns out that there are multiple ways to construct spaces (or spacetimes) that include points at infinity, and how we think about Josh’s scenarios will differ depending on how we carry out the construction. However, the point is just that we can consistently construct such a thing, and that, upon doing so, we no longer reach a contradiction for the Waldo scenarios.

As a basic example, let’s consider the one-point compactification of the plane. To construct this example, consider a sphere, located above a plane, so that the plane is tangent to the South Pole. (A nice diagram is located here.) Now, consider a line from the North Pole, through the sphere, and into the plane. We identify that point of the sphere with that point of the plane — this way, every point of the sphere is mapped to a point in the plane. Where does the North Pole map to?

For the standard plane, there are no points at infinity, and so the North Pole is not mapped to any point in the plane. However, on the standard way of thinking of the plane, the plane is a set of points equipped with some additional (e.g., topological and metrical) structure; we can consider the union of the points in the plane with an additional point. This new object is called the extended plane. And then we can construct our map from the sphere to the plane so that the North Pole is mapped to the additional point. It turns out that, given a smooth function mapping the sphere to the plane, we can make all of the details work out so that the extended plane is topologically equivalent to the sphere. Go infinitely far in any direction on the extended plane and you end up at the point at infinity — which corresponds to the sphere’s North Pole. In some sense, we’ve “wrapped up” the plane into a sphere, and so the resulting object is called the one-point compactification of the plane.

What happens if we plop Waldo down on the extended plane and suppose that Waldo goes off on his accelerated journey? Well, since the one-point compactification of the plane is an utterly standard mathematical construction, we can just write out some equations and check. Since every point on the extended plane corresponds to a point on the sphere, Waldo’s trajectory on the extended plane is equivalent to some curve on the sphere. And since Waldo goes off to the point at infinity, the curve on the sphere must pass through the North Pole.

Let’s now answer the various questions Josh asks about his scenarios. At the end of Waldo’s journey, where is he? Waldo is located at the point at infinity — which corresponds to the North Pole on the sphere. Can Bob find Waldo? Sure, Waldo is at the point at infinity — hence, Josh is simply mistaken when he says that “Waldo has no definite location at all”. Where are the steps that Waldo takes immediately before he reaches the point at infinity? Well, for the curve on the sphere, all of the points the curve passes through “before” it reaches the North Pole are elsewhere on the sphere. Hence, all of the steps Waldo takes are at some finite distance from where he starts. But how could that be if Waldo ends up infinitely far from his starting place? On the sphere, we can mark out points along the curve that correspond to Waldo’s steps. These marks bunch up and become arbitrarily dense as the curve approaches the North Pole; there is no last step — so no step Waldo immediately takes before reaching the North Pole — but then that’s precisely what we should expect from an infinite process. Doesn’t this mean that Waldo’s journey somehow makes a leap out to infinity? Well, where does that “leap” happen? The curve on the sphere is continuous, has no breaks, is topologically equivalent to Waldo’s trajectory on the extended plane, and so Waldo’s trajectory on the extended plane must also be continuous.

What about the tape measure? What value does Waldo read when he gets to the point at infinity? The trouble here is that we’ve left unspecified what happens when Waldo reaches the point at infinity. We’ve only specified what happens for every point before Waldo reaches infinity. (This response parallels the standard analysis of Thomson’s lamp; see Benaceraff 1962, Grünbaum 2001, pp. 228-240, and Earman and Norton 1996.) If we want to, we certainly can add to our construction what Waldo’s tape measure reads when Waldo reaches infinity.

There’s a possibility that I’m missing a crucial piece of Josh’s argument. If I am, I’d be excited to hear what it is. But at least from where I currently am, I don’t find this new argument all that much more convincing than the previous grim reaper arguments.

After I wrote this post, I read this other excellent response to Josh’s post, which I also recommend reading.

Disclaimer: The posts on the Cosmotherium should never be taken as definitive and I am typically not completely convinced of what I post here. This is my place for working out my views without the pressure or rigor of publication.

End Notes

[1] We might think that such scenarios are impossible. At some point, Waldo’s speed exceeds the speed of light. Most philosophers think the prohibition on speeds above that of light is metaphysically contingent, but I find it somewhat attractive to think that the prohibition is metaphysically necessary. So, one might wonder if all we’ve really shown is that unbounded speeds lead to paradoxes, or at least outrageous situations, and so a new reason for thinking that, as a matter of metaphysical necessity, speeds are bounded. Unfortunately, this reply doesn’t take into account the possibility that spacetime is curved, as happens in General Relativity. In some General Relativistic spacetimes — such as de Sitter spacetime — one’s future light cone “opens up”, allowing for trajectories that, while always locally remaining below light-speed, nonetheless go out to infinity, exiting spacetime altogether, and so ceasing to exist.

References

Benaceraff, P. (1962). Tasks, Super-Tasks, and the Modern Eleatics. The Journal of Philosophy 59(24), pp. 765-784.

Earman, J. & Norton, J. (1993). Forever is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes. Philosophy of Science 60(1), pp. 22-42.

Grünbaum, A. (2001). Modern Science and Zeno’s Paradoxes of Motion. In Wesley Salmon (Ed.), Zeno’s Paradoxes. Hacket Publishing Company, pp. 200-250. Originally published in 1970.

Malament, D. 2008. ‘Norton’s Slippery Slope’. Philosophy of Science 75(5), pp. 799-816.

Saari, D. & Xia, J. 1995. ‘Off to Infinity in Finite Time’. Notices of the American Mathematical Society 42, pp. 538–546.