## Paradoxes That Made Us Laugh Out Loud

Cidu Bill on Dec 28th 2012

- Morris Keesan: hovertext: The prosecution calls Gottfried Leibniz.
(In case this is a CYDU: Zeno’s paradox says that you can never get from point A to point B, because first you have to go halfway, and then you have to go half of the remaining distance, etc. Leibniz was one of the inventors of calculus, which effectively answers and explains away Zeno’s paradox.)

Filed in Bill Bickel, Comics That Made Us Laugh Out Loud, comic strips, comics, humor, lol, xkcd | 19 responses so far

CaroZ Dec 28th 2012 at 09:08 pm

1Indeed… but personally I think the comic would have been funnier without the “but never reach it” line. If you don’t know Zeno’s paradox, you won’t get it even with the extra text. And if you do know it, I think you’ll chuckle more when you have to think about it for just that extra second.

(Wow — I’ve just said an xkcd cartoon explains too much!?)

James Pollock Dec 28th 2012 at 11:12 pm

2I disagree. Zeno isn’t exactly a common name, but I wouldn’t have jumped directly to Zeno’s paradox just from a reference to “Zeno” (especially since I’m more familiar with the “Achilles and the Tortoise” formulation). Approaching (but not reaching) the bench made me go back, and then I noticed “Zeno” in the text above it, and THEN I got it.

So put me down for the “xkcd got it just right this time” camp.

Kilby Dec 29th 2012 at 02:06 am

3I agree with CaroZ (@1): “never reach it” is superfluous. The name Zeno and the word “approach” are more than sufficient to carry the joke, and the Leibniz reference nails it down. All of this presupposes that the reader can identify Zeno and Leibniz, of course, but if you don’t have that knowledge, then this cartoon will always be a CIDU.

Keera Dec 29th 2012 at 05:18 am

4I’m with James Pollock on needing “never reach it”. That was the line that got me thinking.

Ooten Aboot Dec 29th 2012 at 07:00 am

5Two yeas, two nays. Is this a fiscal cliff parable?

Dave in Asheville Dec 29th 2012 at 08:25 am

6People keep explaining it, and I think I get closer to understanding it, but I just can’t quite get there…

Ian Osmond Dec 29th 2012 at 09:50 am

7James Pollock: technically, “Achilles and the Tortoise” is a slightly different, though very closely related, paradox than the “arrow” paradox, although I believe they’re both by Zeno. The arrow paradox is that an arrow goes half the distance, then half again, then half again, forever, so can’t reach its target. The Achilles and the Tortoise paradox is that the sprinter Achilles, when racing the tortoise, gives it a bit of a head-start, and when he reaches where the tortoise was, the tortoise has moved just a little bit, and when he reaches where the tortoise was again, the tortoise has moved just a little bit, and so forth, so he can never pass the tortoise.

Very closely related, but slightly different.

I felt the “but never reach it” line was unnecessary, but I still liked it, because “approach but never reach” is part of the formulation of the paradox. So I didn’t see it as an explanation so much as a reference to how the thing is phrased.

heather Dec 29th 2012 at 09:56 am

8I’m very familiar with the paradox, but not with the name of the guy who made it up (Zeno, apparently). And since “approach the bench” is just standard courtroom lingo, that would NOT have been enough for me to figure out the joke - it would not have been obvious at all that “approach the bench” was part of the joke.

So I fall on the side of “but never reach it!” is totally necessary. And not just for the sake of those of us who don’t know Zeno. It also shows the silliness of the defense’s position. The judge is not saying ‘approach the bench’ because it’s Zeno, he’s saying ‘approach the bench’ because he wants the guy to come see him. Then Zeno, who is obsessed with his idea, blurts out ‘but never reach it!’. His defense is based on the idea that he’ll never reach ANYTHING, including the bench. Without that phrase, the “approach the bench” line would be just left hanging. “But never reach it” is indeed the punchline.

fj Dec 29th 2012 at 11:55 am

9The prosecution though about calling Isaac Newton, but figured that no one would understand his notation. ;^)

The arrow paradox is not the 1/2 + 1/4 + 1/8 + …. problem. That’s Zeno’s dichotomy paradox (which is the basis for the “approach the bench/never get there” part of the joke). The dichotomy paradox says if something is moving, it must reach its halfway point before it can reach its goal. However, before it can make the rest of its journey, it must travel half the remaining distance. Therefore there are an infinite number of these progressively smaller journeys to make. Since there are an infinite number of journeys, it must be impossible to reach a destination.

I’m pretty sure you can disprove the dichotomy paradox and provide a satisfactory explanation without calculus using the ancient Greek method of exhaustion technique as employed by Eudoxus and Archimedes. Or at least you could come close… ;^)

Or you could just approach the bench and get there…

The arrow paradox basically says that the arrow can never move: motion is impossible. In any given durationless instant of time, the arrow must be moving either to where it is, or to some place it currently is not. However, it cannot move to somewhere it is not, because it has no time to move. If the arrow is not moving to where it is not, it can only be moving to where it already is. However, if it is only moving to where it already is, it is, by definition, motionless.

By the way, the Achilles and the tortoise is a third Zeno paradox that says that the speedier Achilles ought never be able to catch the slow tortoise, because before he can catch the tortoise, he must first reach the point to where the tortoise was when he started his pursuit. Once he reaches that point, he must then reach the tortoises new location, and so on. It’s similar to the dichotomy problem, but differs in that the destination keeps changing, and reasoning behind why it should be impossible for the goal to be achieved.

Anyway, for me, the “But not reach it” adds to the joke, as it adds a second absurd so-called paradox, and even if Zeno never reaches the bench, he destroys his arrow “proof” if he moves at all!

Mark in Boston Dec 29th 2012 at 02:23 pm

10Lewis Carroll wrote a story in which Achilles actually does catch up to the Tortoise.

Then the two of them get into a long discussion of the rules of logic. Suppose you have a rule of inference: “If A is a B, and all B are C, then A is C”. If Achilles is a man, and all men are mortal, then Achilles is mortal. So have we proven that Achilles is mortal? Not quite, says the Tortoise. We are assuming that there is a rule that tells us how to apply the rule. We need to make that rule explicit: “If something is known to be A, and there is a rule ‘if A is a B, and all B are C, then A is C’, then we can apply the rule and prove that the something is C.” But how do we know we can apply THAT rule? We need another rule, “If something is known to be A, and there is a rule ‘if something is known to be A, and there is a rule “if A is a B, and all B are C, then A is C”, then we can apply the rule and prove that the something is C’, then we can apply the rule …” etc. etc. etc.

Mark in Boston Dec 29th 2012 at 02:25 pm

11Zeno had his own opposite, Heraclitus. To Zeno, everything was always standing still. To Heraclitus, everything was flowing. You can’t step into the same river twice. The second time, it’s a different river because all the old water has moved on.

James Pollock Dec 29th 2012 at 02:39 pm

12All of Zeno’s paradoxes flow from the same mistaken notion, that the sum of an infinite series must be infinite. The Greeks knew better; Pi is the sum of an infinite series. The notion that an infinite series must be infinite is easily defeated, with the advantage of Arabic numerals. Consider the sum of the infinite series 0/1+0/10+0/100+0/1000+0/10000… the sum of all these fractions is not infinite; it converges on 0.

Arthur Dec 29th 2012 at 03:20 pm

13Not really. Half of his paradoxes attempt to show that motion can’t be continuous. The other half

try to show that motion can’t be discrete.

The version of the arrow paradox I know is a bit different. Think of a moment when an arrow is in

flight. It is a moment, therefore the arrow is not moving. How could you examine that moment in

time and deduce that the arrow is moving forward rather than dropping? What impels the arrow to

move forward to the next moment?

Less Reality, More Fantasy, No Baby Bkues Dec 30th 2012 at 12:01 am

14I heard an interesting paradox recently: Two spaceships are flying at the same speed, one below the other. One fires a light at a receiver on the other ship, and from the perspective of a guy on one of the ships, the light is traveling straight. Someone in a space station to the side of the ships sees this and perceives the light as traveling at an angle. Is the light going straight, or at an angle?

Ian Osmond Dec 30th 2012 at 07:25 pm

15Less Reality: um, neither? That seems to misunderstand the very nature of how light propogates. . .

Mark in Boston Dec 30th 2012 at 08:06 pm

16“Is the light going straight, or at an angle?”

Um, both? A straight line that’s at an angle to the perpendicular?

If the ships are not going at relativistic speeds (a significant fraction of the speed of light), then it’s a matter of visual perspective. The ships can be in a position such that angles look different, like two parallel railroad tracks appear to meet at an angle at the vanishing point.

If the ships are going at, say, one-half the speed of light and are one light-second apart, each ship will see the other as being thousands of miles behind it. Assuming the engineer has enough information to calculate that the two are “really” side-by-side, he can calculate how far ahead to angle the beam so that it eventually reaches the point where the second ship will actually be. Then he can look back and see the flash on the side of the ship that appears to be way behind him.

Mark in Boston Dec 30th 2012 at 08:35 pm

17Assume both ships blasted off at the same moment, one from the north pole and one from the south pole of their home planet, which is big enough that the poles are one light-second apart (186,000 miles, bigger than Jupiter). The ships are identical, with the same engines and fuel, and they quickly accelerate to one-half light speed and then maintain that as a constant speed, which is how the space station operator sees them.

Are the two ships in the same frame of reference? Have they been in the same frame of reference all along? i.e. if their clocks were synchronized, are they still synchronized?

And if they are in the same frame of reference, then shouldn’t each appear to be alongside the other, as if they are both at rest? So the engineer would just aim directly at the other?

Mark in Boston Dec 30th 2012 at 09:02 pm

18The more I think about it, the less sense it makes. I never studied the math required for this. I give up.

James Pollock Dec 31st 2012 at 11:46 pm

19“Is the light going straight, or at an angle?”

Not “or”. “and”. Then, yes.