Where is Waldo?

A New Argument for a Finite Past from Strange Locations

Could the past stretch back forever? In this post, I’d like to share a new argument for a finite past that centers on something we all care about deeply: Waldo’s location.

This argument unfolds through three scenarios, each stranger than the last, in which Waldo attempts to perform an infinite task in finite time. If the past could be infinite, one could argue that such scenarios should in principle be possible. But as we’ll see, they lead to strange outcomes. The final outcome is so strange, indeed, that it appears to imply a flat contradiction.

Toward the end, I’ll share why this new argument from strange locations may be better than the recent grim reaper arguments for a finite past.

Here's the general structure of the Strange Locations 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.

Here I will grant (1) for sake of argument. The more fundamental focus of this post is about the prospect of an infinite series.

Let’s walk through the scenarios.

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Scenario 1: Waldo Runs to Infinity

Waldo begins a 2-minute run. In the first minute, he runs at 1 meter per second. In the next half-minute, he doubles his speed to 2 meters per second. Then in the next quarter-minute, he doubles it again to 4 meters per second. He continues this pattern indefinitely.

At the end of 2 minutes, Waldo has traversed infinitely many meters. So, where is Waldo?

He’s not any finite distance from where he started. That much is clear. But is he anywhere? Perhaps we could simply say, “He’s infinitely far away,” and leave it at that. Strange, but not clearly impossible.

Let’s press further.

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Scenario 2: Bob Tries to Find Waldo

Waldo’s friend Bob is determined to retrace every step. Like Waldo, he accelerates through each segment of the path and completes the infinite run in 2 minutes.

But when Bob finishes, Waldo is nowhere to be found. If Waldo had a location, there should be evidence—footprints, perhaps—somewhere finitely close to him. But there are none. Every previous step is infinitely far from his current position. After all, every step Waldo took was only finitely far from his starting point, while Waldo is not finitely far from his starting point. There are no steps "behind" him in space. Therefore, Waldo has no definite location at all.

Now the scenario is very strange indeed. It seems metaphysically suspect. How can a spatial object be nowhere?

It gets worse.

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Scenario 3: The Tape Measure Paradox

Waldo tries again, this time with a plan. His friend holds the base of a tape measure while Waldo holds the extending end. As Waldo runs infinitely far, the tape stretches behind him.

But here's the problem. While the tape extends without bound, each segment of the tape can only display a finite number. That’s how measurement works. Like the natural numbers, it extends infinitely without ever reaching an “infinite” number.

So when Waldo finishes his run and checks the tape, what number does he see? It must be some finite number. But if it's finite, then his location is finitely far from the start, contradicting the fact that he ran infinitely far.

You can try to flip the setup and let Waldo leave numbered markers as he runs, but the problem remains. There is no final marker near him. No last step. So how did he leap infinitely far with no final jump?

This isn’t just strange. It’s contradictory: Waldo’s tape measure can only display a finite distance, but Waldo has run an infinite distance.

(If someone has worries about the possibility of making an infinitely extending tape measure, even with infinite time, instead imagine that Waldo simply numbers his steps as he runs. Then after he has finished, there is no final, previous step with a number on it finitely close to him. How then did he leap infinitely far without any steps?)

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Objection: Couldn’t Some Infinite Histories Avoid This?

Perhaps one might try to distinguish between types of infinite causal chains. Perhaps some infinite histories entail strange locational scenarios while others don’t. Maybe the past could be infinite without allowing these sorts of completed infinite actions.

But then what kind of infinite chain avoids the Waldo problems and why exactly? It is not obvious how to draw that line. If there could be an infinite sequence of prior events, why couldn’t Waldo (someone or something) travel infinitely far?

One idea is that the problem is with completing an infinite series in finite time—that’s what leads to the contradiction. In that case, perhaps an infinite series could be completed in infinite time, just not in finite time.

On the other hand, it is not clear to me why spreading the events out in infinite time would make the difference. Note that each of the paradoxical scenarios involves infinitely many events that occur prior to a completion point. Whether those events were spread across finite time or infinite time, the same strange results occur after they are complete. So the problem seems to be in completing the infinite series, not the timescale—finite or infinite. I expand upon this idea in “From Supertasks to Infinite Time.”

Note also that there is nothing explicitly contradictory about an infinite series in finite time. The contradiction is drawn out by considering what else would be possible if an infinite series could come to a completion. While I’m still thinking about it, my present inclination is to think that the paradox draws an implicit contradiction to the surface. The contradiction lives inside the concept of completing an infinite sequence. It sprouts into view via principles of construction that seem possible if an infinite series is possible. (Compare: it's like if there were infinitely many apple tree seeds. Then the resources would be in place for there to possibly be infinitely many apple trees. If someone proved there couldn't be infinitely many apple trees, then we could infer there also couldn't be infinitely many apple tree seeds. The infinite seeds here represent an infinite sequence of past events, and the infinite trees represents the paradoxical scenarios that could then sprout.)

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Why This Argument Might Be Stronger than Grim Reaper Arguments

Some arguments for a finite past, such as Grim Reaper paradoxes, involve controversial assumptions about causal dependence, such as the premise that a certain event occurs only if a certain other event doesn’t. These arguments have generated a lot of discussion over the root of the paradox: is infinity the root of the problem, or is the root is to be found in a contradiction arising from the causal conditions in place?

But the Waldo Argument seems to turn the spotlight more directly on infinity itself. This argument does not require the “if only if not” causal dependence premise found in grim reaper arguments. Instead, the contradiction seems to emerge directly from the structure of infinite motion in finite time. So the arguments are importantly different.

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Conclusion: Looking for Waldo

This is just a sketch of a new argument that got me wondering again about the strangeness of infinity. Feel free to build upon, subtract from it, tinker with it, or attempt to destroy it. Wisdom often springs from playing with strange ideas.