Advent Calendar
by Jeff
Image text: I think you could get up to about 11:59:57 before you'd have trouble swallowing the chocolates fast enough. At that point, you'd need some kind of a liquify-and-chug apparatus to get up over the 11:59:59 barrier. Anyway, Merry Christmas!
This comic is about an Advent Calendar, which is a calendar for only the period of Advent, which is considered by the Christian calendar to be a month before Christmas. In a lot of traditions, this calendar has little doors with a piece of chocolate behind each door. Other people give very small presents on each day of Advent.
This Advent Calendar belongs to Zeno, who was a Greek philosopher who lived from ca. 490 - 430 BC. (So, obviously, he would not have had an Advent Calendar, as this tradition was not in place yet and according to the Christian calendar, Jesus was not even born yet!)
Zeno was the master of the dichotomy of motion. His dichotomy paradox indicated that something that is traveling in a direction, must arrive at the half-way point before it arrives at the destination. Since any finite destination can be divided in half (as seen in the Advent Calendar above) - the paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.
So, if you are Zeno, your Advent Calendar means that it will never be Christmas, just smaller and smaller milliseconds until Christmas. Bummer!
Don't forget, we are soliciting help from you on filling out explanations for XKCD's back catalog. Perfect for something to do while you have time off around the Holidays!
December 23rd, 2011
1st poster
Woo Hooo
I was waiting when this would be uploaded
December 23rd, 2011
Zeno was parodied in Terry Pratchett’s “Pyramids”, using the example of shooting arrows at tortoises moving away from the shooter – logically, the arrow could never hit the tortoise, it would just get closer and closer. The end result was a selection of tortoise kebabs
December 23rd, 2011
That is a reference to Archimedes and the Tortoise, not quite the same problem but one that illustrates the same dilemma.
Either way this gets fixed by Planck Time.
December 23rd, 2011
*Achilles and the tortoise
December 23rd, 2011
indeed…
or I am screwed…
December 23rd, 2011
In principle, one could invoke Planck Time (or, more precisely, any notion of a smallest, indivisible time-unit) to address either paradox in an engineering or “real world” sense. But the simpler, more direct solution involves infinite series.
That is, the basic argument of the Paradox can be paraphrased as follows: Any motion can be shown to take an infinite number of intermediate steps. Therefore it it must take an infinite amount of time.
In other words, the assumption here is that the sum of an infinite number of items must, necessarily, be infinite.
However, this logic is flawed, as is known to anyone who has studied “infinite series” in an advanced high school math class (i.e. the sum of an infinite series of numbers can be finite. E.g. the sum of the infinite series 1 + 1/2 + 1/4 + 1/8 + 1/16 + [...] is equal to 2.)
So, although it is true that the motion of the arrow involves an infinite number of steps (in the sense given by Zeno), it still reaches the target.
(In short, the real problem here is that the technique of infinite series was unknown to Zeno and his contemporaries.)
December 23rd, 2011
By my back of the envelope calculations, there would be 161 doors in the last day (the last one only a single Planck time short of midnight). Also, assuming you could keep your mouth only an inch away from the door (to give it room to open), that last piece of chocolate would need to be moving about 1 600 000 000 000 000 000 000 000 000 000 000 times the speed of light.
December 23rd, 2011
The whole paradoxon is an illusion.
For the tortoises’ case:
Idealize the world as frictionless, then the horizontal speed of the arrows is constant. That means: after half its way, some time has passed. After three quarters, its 1.5x of that time.
In that view, the arrow apporaches the tortoise asymptotically, as does the total time required to take the distance.
But in reality, time goes on linearly (in Newtonian physics, at least).
December 23rd, 2011
Even as a child it seemed clear to me that this was the flaw in Xeno’s reasoning. He seems to presume that the time needed to move each smaller and smaller chunk of distance is the same (to get to target 16 feet away, travel first 8 feet in one second, then next 4 feet in one second, then next 2 feet in one second,…), that the moving object is slowing down.
December 23rd, 2011
Zeno doesn’t need a constant unit of time. Just as his unit of distance is 1/2 the previous unit of distance, so too can his unit of time half at each step. So 8 feet in 1 second, 4 feet in .5 sec, 2 feet in .25 sec… It is the fact that an infinite number of numbers can still add up to a finite sum (see the infinite sum comment above) that freaked the poor Greek philosopher out.
December 23rd, 2011
What is cool is that 1 + 1/2 + 1/4 + 1/8 + 1/16 + … (with out end) actually = 2.
December 28th, 2011
Which is akin to saying “if you can’t cover the distance in a fraction of the time you need I guess you can’t get there at all”.
Xeno/Zeno always bugged me. Big deal being made of something that has reasoning flaws.
December 23rd, 2011
Even as a child you completely misunderstood the concept. In order to do a task you first have to do half that task (say walking across a room). This becomes your new task. Unfortunately to do that task you have to first complete half of it. THAT now becomes your new task. In order to complete that task you must first complete half of it. If you follow this train of thought and us reductio ad absurdum (sort of), you now end up with an infinite number of progressively smaller tasks.
If you assume that it must take some finite amount of time to complete any of the tasks, the conclusion is that you will take an infinite amount of time to complete them all. This is patently false, however the philosophers of the day held logic in higher esteem than observation and so concluded that the observable universe was flawed.
There are several ways around it using more contemporary thought processes. One can posit that there is a smallest possible distance or smallest unit of time, both positions somewhat supported by current theories of physics (both known as planck units). You can also posit that an object in motion has no position (part of Uncertainty Principle) or you can conclude that space and time are simply dimensions of each other (pretty much every attempt at unifying physics assumes this, so superstring theory to pick one at random)
December 28th, 2011
As you said, reductio ad absurdum. Xeno argues you can’t do something (that you obviously can d)o by making an absurd argument.
December 29th, 2011
Even as a child, TKTAP, I recognized the ILlogic and The inconsistency of Xeno’s “argument”. He’s reducing to a “infinite number of progressively smaller tasks” without reducing to an infinite number of progressively smaller time increments.
December 23rd, 2011
As a Swede, we do our Christmas celebration on the 24th December. So for us, this calendar would instead mean eternal Christmas. Yey! Although, one would probable become a bit sick of it after a while…
December 28th, 2011
It is the same in the rest of the Europe. Except for Britain, of course.
January 2nd, 2012
I do not know where your Europe stops, but Orthodox Christians celebrate Christs birth at Epiphany, January 6. Orthodox churches cover the whole of Eastern Europe and a good part of South-eastern Europe.
December 23rd, 2011
Fractions of a second not milliseconds. Minor point.
The image text also references the impossibility of completing supertasks. No matter how fast you can complete the finite tasks, it is impossible to complete an infinite number of them.
Posit a lit lamp with a switch that can be flipped in an infinitesimal amount of time and then start flipping it on and off at the rate described in the comic. On December 25 at 00:00:00 will the lamp be on or off? What if you start with the lamp off?
The answer is that there is no answer, you always flip the switch one more time.
December 23rd, 2011
The Christian calendar actually considers Advent to begin four Sundays prior to Christmas, rather than ‘one month’, this year having the longest Advent as Christmas itself is on a Sunday. The ‘Advent starts on December 1st’ tradition is down to having consistent calendars each year.
For the most part however, people get it wrong – much in the same way as the majority of people consider Christmas to be just the 25th, or from the 1st to the 25th, and tear down their decorations immediately after, forgetting that the Christian holiday runs until January 6th (hence ‘the 12 days of Christmas’). At least that’s what it’s like around me.
Minor point-turned-rant over, Merry Christmas all
(Atheist, but brought up Catholic)
December 23rd, 2011
The Spanish make this a bit more correct as far as I knwo, having presents on january 6th because the 3 kings/ wise men/ magi brought presents for Jesus then. ( or so we are to believe if Jesus was born in december )
December 23rd, 2011
oops, minor point, the twelfth day of Christ Mass is actually Jan. 5. Jan 6 is Epiphany.
December 25th, 2011
That is the funny part. Advent calenders have nothing to do with advent. At least they make more sense in the nordic countries, where they are Yule calenders which covers the Yule-month (december until the 24th).
Advents calenders in the nordic countries have only 4 parts, one for each of the four advents until Christmas.
I suspect someone made a mistake when translating the nordic or german tradition to English.
December 23rd, 2011
The title-text about liquifying chocolate refers to how most advent calendars’ small gifts are usually a piece of chocolate. Since you couldn’t eat them fast enough when there gets to be less than second left, you’d have to liquify them an try to drink them really fast.
December 23rd, 2011
However, if the pieces of chocolate also get smaller by half each door, then if you started with one piece of chocolate at the “end” of the advent calendar you would only have eaten 2 pieces of chocolate. A gyp if you ask me.
December 26th, 2011
That’s okay, I consume too many sweets in December anyway.
December 23rd, 2011
By the way: To my shame I have to admit, that I needed Jeffs explanation to make sense of that comic, although we did Zenon`s stuff in at least 3 different lessons back in first or second term.
December 23rd, 2011
“and according to the Christian calendar, Jesus was not even born yet!”
Not just according to the Christian calendar…. That is, if he existed at all..
December 24th, 2011
I’m pretty sure it’s documented (in non-biblical sources) that he exsisted, it’shis magic tricks which are contested.
December 25th, 2011
Not really. I thought so as well, but a few historian friends have told me that the earliest known documentation is actually only letters about the first Christians and what they claim.
January 1st, 2012
Actually, no. Documents centuries after he was (supposedly) around, nothing contemporaneous.
December 26th, 2011
According to Wikipedia, “Most critical historians agree that Jesus was a Galilean Jewish Rabbi who was regarded as a teacher and healer in Judaea”
December 26th, 2011
Wikipedia, my most trusted source for contested historical points.
December 23rd, 2011
Assuming the Advent Calendar starts on December 1st (although Advent itself does not), there would only be 4 doors preceding the ones shown in the comic : December 1st, 12th, 18th and 21st.
December 26th, 2011
Would it land cleanly on the 23rd next as shown in the comic?
December 27th, 2011
Yes, because 23 is halfway between 21 and 25.
December 25th, 2011
correct me if im wrong- im a non-math person- but i always thought the problem w/zeno’s concept was that it didn’t take into account the physical space occupied by the approaching object. If you are walking, at some point the size of your foot and the length of your step will be larger than half of the distance left, and then larger than all of it.
December 26th, 2011
It is a problem, but Zeno works if you consider 1/10 of the distance instead of 1/2 the distance. You then approach .1111111 recuring of the distance… the math still works and the physical space problem is removed.
January 1st, 2012
Doesn’t have to. The very tip of the arrow has to cover half the distance, then half the remainder, etc. The length of the arrow behind that tip point is irrelevant.
December 26th, 2011
The way my philosophy teacher framed the problem wasn’t about approaching distance, but that in the time it takes a fast object to move, a slow object still moves a little bit. Ex: If the tortoise is five feet ahead, when the arrow/Achilles moves five feet the tortoise will have moved just a bit. If the arrow moves to THAT distance, then the tortoise would have moved further still. Repeat as needed for the asymptote. One thing about the Zeno problem that always irked me was that it assumed an object was going at a constant speed (though I can now see from the comments that there are several issues). I know that when I walk, I don’t move at a constant speed. In fact, I feel like I’m stopped in the moment before I move my next foot forward. I’m assuming the effect would be much stronger on a smaller/slower object (the tortoise)…if you’ve ever seen them move, it’s pretty much a stop/start the whole way.
December 27th, 2011
I just love it that a Greek philosopher, a fantasy writer, Christ and chocolate can all be interconnected over a medium only (relatively) recently created and discussed with exactly the same passion by people who I have never met, but admire.