Supertasks, or: The Lamp That Won't Turn Off
You have a lamp. At time zero, it's off. One minute in, you flip it on. Half a minute later, you flip it off. A quarter minute later, on. An eighth of a minute later, off. And so on.
If we line up all the flips, they pack tighter and tighter as we approach the two-minute mark. The intervals get arbitrarily small, but they sum to a finite time: 1 + 1/2 + 1/4 + … = 2. So in exactly two minutes, you've flipped the lamp infinitely many times.
Here is the question: at the two-minute mark, is the lamp on or off?
The honest answer is that we don't know — not because we lack information, but because there doesn't seem to be a principled answer. Every "on" flip is followed by an "off" flip, and every "off" is followed by an "on." Whatever state you'd like to assign the lamp at the end, there is no last flip that put it there.
This is a supertask: an infinite series of actions completed in finite time. And it's where the previous puzzles in this series come home to roost.
How We Got Here
In my earlier post on Zeno's Achilles paradox I set up the puzzle of motion: how can a runner catch a slower one when there are always more sub-distances left to cover? In the follow-up on infinity by division I tried to show that the puzzle isn't really about racing — it's about any continuous magnitude. A finite line seems to contain infinitely many parts, and no candidate for what those parts could be holds up.
This third video makes the move that turns the puzzle from a curiosity into a crisis: time is a magnitude too. If a line is infinitely divisible, so is an interval of time. And if time is infinitely divisible, then between any two moments there are infinitely many moments. So how do we ever get from one moment to the next?
Augustine's Question
About 800 years after Zeno, Augustine asks the same question in Book XI of the Confessions, and famously notes that he can't answer it: "What then is time? If no one asks me, I know; if I wish to explain it to him that asks, I know not."
Augustine's puzzle goes like this. The present, if it has any duration at all, has a past part and a future part inside it — so the "present" we were trying to isolate already isn't really a single present. But if the present has no duration, it's just a point, and no collection of durationless points can add up to the flowing thing we experience as time.
This is exactly the puzzle of length from the previous post, run on temporal instead of spatial magnitude. Time can't seem to be made out of lengthless moments, but it also can't seem to be made of moments with length, because each of those would have a sub-past and a sub-future and we're off to the races.
Back to the Lamp
This is why the lamp matters. The lamp is a stress test for what we say about infinite divisibility. If time really is infinitely divisible, the lamp setup is fully describable — every flip is at a precise moment, and the sequence converges to two minutes. There's nothing wrong with the setup. But the conclusion has no answer. So either:
- Time is not really infinitely divisible, and what looks like a coherent setup is actually impossible, or
- Time is infinitely divisible, but the concept of "the state of an object at the limit of an infinite sequence" is not well-defined, or
- There is some other concept buried in here that we're using badly.
Notice that this is the same pattern we kept hitting in the earlier videos. The math is fine. The intuitive concepts are fine. Combine them and something breaks.
Aristotle, Again
In the previous two posts I leaned on Aristotle's distinction between potential and actual infinity, and I'm going to lean on it once more here, because supertasks are exactly where the distinction earns its keep.
A supertask is defined by indexing one action to each of infinitely many moments. That's a description of an actually-completed infinite series — every step done, in sequence, leading to a final state. But Aristotle's whole point is that the infinite divisibility of time is potential, not actual. We can keep dividing intervals as small as we like, but it doesn't follow that we can perform an action at each of those nested intervals, as though they were already laid out in a sequence waiting to be checked off.
If Aristotle is right, the lamp paradox isn't really a discovery about lamps. It's a discovery about the limits of treating potential infinities as if they were actual ones.
And It Gets Worse
The lamp is the cleanest supertask paradox, but it isn't the worst one. The Ross-Littlewood paradox involves a jar and an unlimited supply of marbles. At each step you put ten marbles in and take one out. After infinitely many steps, how many marbles are in the jar? The "obvious" answer is infinitely many — you're netting nine per step. But depending on which marble you remove at each step, you can construct a version where the answer is zero. Same procedure, different bookkeeping, completely different "final state."
That isn't a fun riddle. It's a sign that the concept of a "completed infinite process" doesn't have the kind of structure we keep treating it as having.
Watch the Full Video
In the video I walk through Augustine's puzzle, set up the lamp carefully, and look at why both "on" and "off" have arguments behind them but neither survives scrutiny — and what that should make us suspect about the assumption that started us down this road.