Hey, Vsauce. Michael here. Now, one of my favorite
treats of the holiday season is Gabriel’s cake. It’s a super solid
based on Gabriel’s Horn that you can make right at home, as long as your home
is infinitely large. Okay, all right. Now, the first thing you want to do is bake a
cake. I prepared this cake earlier. It’s a real beautiful cake. It’s a little large,
but I bet you I could eat that whole thing in a day if I tried. Am I right?
Okay, now the second step is cut the cake in half. Notice that while I cut this cake in half you don’t add or create any new cake but
the surface area of the cake has increased. It used to be completely
covered and now we’ve got two regions on the inside that aren’t covered. The next
step is to cut this half, one of the halves, in half. Here we go. Again, the volume of cake on
the table is the same as it was in the beginning but the surface area is going
up. As you may be able to guess, the next step is going to be to cut one of these
quarters of the original cake in half. It’s getting pretty thin, but all you have to
do is keep this up – cutting halves in half and half and half and half forever and
once you’ve done that, well, you’re almost there. When you finish cutting, stack the
halves on top of one another in order, like this, to create a beautiful tiered
dessert. Because you have an infinite number of thinner and thinner slices
when you’re done stacking them the cake’s vertical height will be endless.
Such a cake has an interesting mathematical property. Its volume, the
amount of cake it contains, is clearly no different than the amount you started
with, but its surface area is infinite. It’s a cake you can eat but not frost.
You would need an infinite amount of frosting to cover the whole thing with a uniform
coat. An object with finite volume but infinite surface area doesn’t need to
be endlessly tall, by the way. There are bounded super solids, like this cube,
with an infinite number of smaller and smaller circular holes.
Of course, building these objects in the real world comes with some obvious difficulties.
One is the fact that the amount of steps required to complete the construction of
Gabriel’s cake or any other super solid is literally infinitely many.
And infinity isn’t a number that, you know, you eventually get to.
It means unending. There will always be a next step, another piece to slice in half again.
By definition, an infinite sequence of tasks has no last task. So you could never
finish making a super solid. Or could you? Enter the Supertask. What if instead of taking the same
amount of time to complete each step we accelerated as we worked and did each
step in half the time as the last? For example, let’s say you wanna make
Gabriel’s cake in just two minutes. That’s easy. First, cut the original
cake in half and then wait a minute before making the second cut. Wait half a minute before making the third, a quarter of a minute before the 4th and so on.
Always waiting for half of what’s left to pass before cutting again. Since you can keep
dividing in half forever, there will always be another step. Infinite actions
within a finite amount of time is a supertask. The strange thing, of course,
is that while supertasking no matter how many steps you’ve already completed
there will always be an infinite amount of steps ahead. But yet, when time’s up,
you will have finished all of them. What we’ve just described is similar to Zeno’s
famous paradox of the dichotomy in which the Greek hero Achilles runs a short race
that he can clearly finish, I mean, people finish races all the time,
but how exactly it’s finished is a mystery. Because first Achilles must
cover half of the race’s distance and then half of what’s left and half again
and again and again. Since there will always be another half way point to
reach, the number of subdivisions infinite, there is no final destination,
after which Achilles’ next stop is the finish line. But yet Achilles obviously can finish.
He somehow reaches them all despite the fact that during his journey he always
has an infinite number of steps left to reach. Think about it this way.
What if at each half way step along the way Achilles is required to hold up a flag.
At even steps a blue flag and at odd steps a red flag. Blue, red, blue, red,
blue, red, the flags will alternate faster and faster. But when Achilles finishes
the race, which flag will he be holding up? To be holding up either flag would
seem to suggest that a largest number not only exists, but is either even or
odd. Now, to this it is often suggested that the real problem here is that Achilles
isn’t actually doing infinitely many things.
He’s just taking a few strides or whatever. A true supertask requires doing
infinitely many distinct actions. Fair enough. But even if Achilles ran the
race in a staccato fashion, stopping at each half way point for half as long as he waited
to the last one, the scenario would still be sensical and logical and would
complete in a finite amount of time. It has even been shown how a staccato
runner could move to avoid discontinuous velocities and accelerations.
The point is, fine, infinite distinct actions are at least
logically possible to do in a finite amount of time, but only logically.
Only as a mathematical abstraction. Supertasks are obviously just products of our
imagination. Because in the real world there must be some smallest amount of
space and/or time that cannot be meaningfully divided in half again.
Once you’re within one of these distances of the finish line, the only possible next position you can have is the finish line. That certainly seems fair.
After all, it’s believed that in reality there is a smallest meaningful distance – the Planck length. Any interactions between or observations of particles
more accurate than this span make no sense within physics as we understand it
today. And the time it would take to travel a Planck length at the fastest
speed possible, the speed of light, is the Planck time. So these seem to be the
briefest pieces of space and time any known force in the universe could act
across. Whether that force is gravity or a force from Achilles’ leg muscles.
But does that resolve the crisis or is it just a way of avoiding the problem
altogether by saying “it doesn’t matter, because it won’t be on the test?”
As John Earman has complained, it seems to me unattractive to make the truth of
mathematical statements depend on the contingencies of space-time structure.
Maybe Zeno’s dichotomy tells us less about motion and time and space and truth
than it does about our impressive abilities to confuse ourselves.
And that’s what makes supertasks important. They’re a contact point in uneasy
handshake of sorts between the universe we live in and the brains inside us.
Let’s play around with some. They’re not all the same. Some converge, like Zeno’s dichotomy,
others converge but in ways you may not expect and still others refuse to
cooperate in any way. They diverge. A great example of a supertask whose
behavior diverges is Thomson’s lamp, a famous supertask devised by James F. Thomson. Imagine a lamp that can be turned on and
off as quickly as you desire. What would happen if you turned such a lamp on and
off Zenoianly? Well, let’s find out. Just set a timer and turn the lamp on. Then wait
one minute and turn it off. After half a minute turn it back on and then off
again after a quarter of a minute, on again after an eighth of a minute and so on. Waiting half as long to flip the switch
each time as you previously did. Now, as the number of switching grows without bound,
the total time elapsed approaches just two minutes. The lamp will be turned on
and off an infinite number of times in just two minutes.
So after two minutes, will the lamp be on or off? Well, by the definition of infinity,
there is no last step. There is never a switching not followed by another.
So the lamp can’t be on, because whenever it’s turned on, it is immediately turned off
in the next step and it can’t be off because whenever it’s turned off it’s
turned on right afterwards. What’s the answer? It’s easy to say something about
a supertask when its partial sums converge, but when they just oscillate back
and forth forever… Hmmmm… Zeno objects do this. Imagine building a meter
high cube. As with all supertasks, you construct at an accelerated pace.
First, you put down a green half metre tall slab, then a quarter metre tall slab that’s
orange, then a green eight metre slab, then an orange 16th meter slab and so on,
until there are infinitely many layers of alternating color. Now, when you look at the cube from above, what color will you see? Orange, green? Well, it can’t be orange
because every orange layer is covered by a green one. And it can’t be green
because every green layer is blocked by an orange one above it. What if we had a
machine display each digit of Pi in order at a supertask pace? After its
finite runtime, what would be on the screen?
The last digit of Pi? Well, that’s impossible, right? But how could it be anything else? A supertask allows us to exhaust an infinite sequence. Paul Benacerraf delivered what is often considered the
best response to these confusions. Is Thomson’s lamp on or off? Is the cube orange or green? The answer
is we don’t know, because these questions are incomplete. Thomson’s lamp could be
on or off or broken. The cube could appear orange or green or something else.
But the supertasks, as stated, don’t let us figure out which. I may as well ask
you if a lamp hidden in a locked room is on or off. It’s definitely one or the other,
but I haven’t given you enough information to do anything but guess. Supertasks like
these describe an endless sequence of tasks and then ask us about the end.
But we can’t determine an outcome because, although there may be an end to their
duration, there is no end, no final member of their actions.
They must be reworded or coupled with extra assumptions in order to be solved.
For example, if we assume that the switch used on Thomson’s lamp can only be all
the way on or all the way off, we can’t determine where it is after the supertask.
But if the switch is, say, a bouncing ball that completes the circuit
turning the lamp on each time it bounces on a metal plate, an outcome can be
determined. If the physics here are ideal and the ball bounces half as high and
half the time as it did on the previous balance, its sequence of
bounce heights will turn the lamp on and off an infinite number of times in a
finite amount of time. Although this bouncing ball has no penultimate state,
no second-to-last bounce, it does have an ultimate state, a final one, resting on the plate. The circuit will be
complete and the lamp will be on. You can also describe a switch on Thomson’s lamp that leaves it off. Sometimes, the next state after infinitely many isn’t
paradoxical because of lack of information, but because of a surprising,
or non-intuitive, discontinuity that occurs there. The Ross–Littlewood paradox
is one of the greatest examples. Imagine a giant urn that can hold an unlimited
number of balls. Now, imagine that you have an unlimited supply of balls, each
with a unique natural number written on it. All natural numbers, in fact, since
there’s no end to how many balls you have. Now, working at an accelerated Zenoian pace you move the balls to the urn 10 at a time, but in a weird way. At step one, you place
balls number 1 to 10 in the urn, but remove number 1. At step two, a minute
later, you place balls 11 to 20 in the urn and remove ball number 2. At step three,
you place balls 21 to 30 in the urn and remove ball 3, and so on. Upon the completion of the supertask
how many balls will be in the urn? At first the answer seems obvious. At each
step you are adding 10 balls and subtracting 1, so a net of nine balls
is added each time. 9 + 9 + 9 + 9 forever, the series grows without end. Infinite nine’s means infinite balls at
the end. But here’s the problem. At each step, the ball with that step’s number
written on it is removed. Ball 1 is removed at step one, ball 2 was removed
at step two, ball 12-googol is removed at step twelve-googol. Since there are an
endless number of steps for any ball number, there is a step number at which
it is removed. So although the urn’s ball population grows without bound during
the task, after the supertask the number drops to 0. It gets weirder. Here’s a second, seemingly identical
method. Instead of beginning with balls 1 to 10 and then removing 1, begin with
balls 1 to 9. Then write zero after the “1” on ball 1. For step two, add balls 11 to 19 and draw a zero on ball 2, making it say 20. For every finite step, both methods
results in identical earned contents, but after infinitely many steps, the first
leaves us with no balls and the second leaves us with infinitely many balls
written on which are all the natural numbers, each followed by an infinite
string of zeros. Both are discontinuous at infinity, but dang, in very different ways.
The bigger question now becomes, “so what? Who cares?” You will never have an
infinite number of balls and you will never have a large enough to urn to
hold all of them. You will never build a lamp that can turn on and off arbitrarily fast. We cannot investigate
time or space past a certain smallness, except when pretending, so what are supertasks, but recreational fictions, entertaining riddles? We can ask more
questions than we can answer, so what? Well, here’s what. Neanderthals. Neanderthals and humans, us,
Homo sapiens, lived together in Europe for at least five thousand years.
Neanderthals were strong and clever, they may have even intentionally buried their
dead, but for hundreds of thousands of years, Neanderthals barely went anywhere. They pretty much just explored
and spread until they reached water or some other obstacle and then stopped.
Homo sapiens, on the other hand, didn’t do that. They did things that make no sense
crossing terrain and water without knowing what lay ahead. Svante Pääbo has
worked on the Neanderthal genome at the Max Planck Institute for Evolutionary Anthropology
and he points out that technology alone didn’t allow humans to go to
Madagascar, to Australia. Neanderthals built boats too. Instead, he says, there’s
“some madness there. How many people must have sailed out and vanished on the
Pacific before you found Easter Island? I mean, it’s ridiculous. And why do you do
that? Is it for the glory? For immortality? For curiosity? And now we go to Mars. We
never stop.” It’s ridiculous, foolish, maybe? But it was the Neanderthals who went
extinct, not the humans. Maybe it’s only a fool who will perilously journey out to
what might not be there. And maybe it’s only a fool who will ask about supertasks,
about infinity. But if you want to solve problems, you don’t just solve the
ones that are there, you find more and make more and go after the impossible
ones; fostering a love and obsession with problems is how you solve problems. Antoine de Saint-Exupéry wasn’t a
mathematician, but his advice fits nicely here. If you want to build a
ship, don’t drum up people to collect wood and don’t assign them tasks and work,
but rather teach them to long for the endless immensity of the sea. And as always, thanks for watching. Supertasks are cool, but super gifts are
even cooler. That’s why I’m excited to announce this year’s Vsauce holiday
box. This thing comes loaded with exclusive Vsauce stuff and science gear,
plus all Vsauce proceeds go directly to Alzheimer’s research. I’m really proud of
this box. You can pick one up at geekfuel.com/Vsauce,
link down in the description. There’s a limited amount available, so
don’t hesitate. And as always, thanks for watching.