[Xmca-l] Re: Verismo and the Gothic

[David H Kirshner]

Henry and Annalisa, sorry for not providing more of an explanation. 

By zigs and zags, I was referring to the changes in direction of the fly as it travels in a straight line first from the tortoise to Achilles, then back to the tortoise, then to Achilles, etc. What we know about the fly's flight is that it lasted one hour, and at 10 mph amounted to exactly 10 miles. 

What's more obscure is that the beginning of the flight involves an infinite number of these zigs and zags. Partly, the difficulty of understanding comes from our physical intuitions which conflict with the mathematical idealization the problem needs. For instance, because flies have physical dimension, eventually, the fly would be touching both Achilles and the tortoise at the same time. So we have to idealize the fly as a point which has location, but no dimension. 

With this in mind, if we run the simulation backward over the course of that hour, then the dynamics of the infinite legs of the journey can come more into focus. At first, Achilles and tortoise are 9 miles apart (recall, the tortoise travelled 1 mile, and Achilles travelled 10 miles). So the fly has a long trip to make on the first leg of its journey. However, as the backward simulation continues, the distance between the characters shrinks, so each trip from one character to the next is shorter than previous one. 

To understand how this could result in an infinite number of zig zag legs of the journey, we need to come to terms with a somewhat paradoxical calculation--how could an infinite number of legs of the journey not require an eternity to accomplish? This is the very calculation that stymied the Greek philosopher Xeno, and inspired Plato's belief that the changing world we experience around us, is just an illusory image of the true, unchanging world of ideal forms. 

To simplify the numbers, let's assume each leg of the journey is 50% of the preceding leg in both time and distance, and that the first leg in the backward simulation takes 1/2 of an hour. Then the second leg would take 1/4 of an hour, then 1/8 of an hour, etc. So the total time T in hours would be given by:

T = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... and onward without end.

Interestingly, the sum of all of these times onward to infinity is 1 hour. To see this, think of the unit as being a cake, instead of a unit of time. First eat half the cake; that leaves the other half on the cake tray. Then cut that half into two halves (each 1/4 of the original cake) and eat one of them. So you've eaten 1/2 + 1/4 of the cake and still have 1/4 cake left. Next cut that piece into two equal parts and eat one of them. You will now have eaten 1/2 + 1/4  1/8 of the cake and still have 1/8 cake left. As you can see at the infinite end of this process you will eaten exactly the whole cake. In the same way, traversing the infinite number of legs of the flies journey takes 1/2 of an hour plus 1/4 of an hour plus 1/8 of an hour, and onward through an infinite number of legs for a total journey of one hour. 

Bringing this back to the discussion of what is a moment, we can see the problematic nature of trying to make sense of a point in time in relation to a continuum of time. The infinite oscillation of the fly's path means the ending position of the fly after one hour is indeterminate. From a mathematical point of view, there's simply too much that goes on the level of the infinitesimal point to be able to relate it to our more familiar and comfortable continuous intervals. 

I hope this helps, a bit.

David


-----Original Message-----
From: xmca-l-bounces+dkirsh=lsu.edu@mailman.ucsd.edu [mailto:xmca-l-bounces+dkirsh=lsu.edu@mailman.ucsd.edu] On Behalf Of HENRY SHONERD
Sent: Wednesday, February 18, 2015 1:53 PM
To: eXtended Mind, Culture, Activity
Subject: [Xmca-l] Re: Verismo and the Gothic

David,
I was going to hide my ignorance, but Annalisa has provided cover: I am assuming that the zigs and zags in the word problem indicate that the fly need NOT fly in a straight line between tortoise and hare. Is that what makes for an infinite number of solutions?
Henry

> On Feb 18, 2015, at 12:22 PM, Annalisa Aguilar <annalisa@unm.edu> wrote:
> 
> Hi David,
> 
> I am terrible at math word problems! As a result I got dizzy thinking about infinite trips by the Trojan Fly.
> 
> Is the answer that the fly is facing up?
> 
> In one hour the tortoise and Achilles are 4 miles apart. But I can't determine where the fly would be because it is flying back and forth at 10 mph. Knowing the life of flies, wouldn't it be likely to be dead from flying that fast for that long?
> 
> Kind regards,
> 
> Annalisa
> 
> 
> On Wednesday, February 18, 2015 12:05 AM David H Kirshner <dkirsh@lsu.edu> wrote:
> 
>> From a mathematical point of view, there are some interesting paradoxes associated with the notion of a moment in time. Here's a particularly nice one building on the race between Achilles and the tortoise, the latter having been given a head start.
> David
> 
> The Trojan Fly:
> 
> Achilles overtakes the tortoise and runs on into the sunset, exulting. As he does so, a fly leaves the tortoise's back, flies to Achilles, then returns to the tortoise, and continues to oscillate between the two as the distance between them grows, changing direction instantaneously each time. Suppose the tortoise travels at 1 mph, Achilles at 5 mph, and the fly at 10 mph. An hour later, where is the fly, and which way is it facing?
> 
> We can find the answer by running the scenario backward, letting the three participants reverse their motions until all three are again abreast. The right answer is the one that returns the fly to the tortoise's back just as Achilles passes it. The paradox is that all solutions do this: Place the fly anywhere between Achilles and the tortoise, run the race backward, and the fly will arrive satisfactorily on the tortoise's back at just the right moment. [The reason this is so is because there are an infinite numbers of zigs and zags in the fly's path--you can get a sense of the infinite oscillation as you move the actors backward in your mind ever closer to the point at which the Achilles and the tortoise are abreast of one another.]
> 
> This is puzzling. The conditions of the problem allow us to predict exactly where Achilles and the tortoise will be after an hour's running. But the fly's position admits of an infinite number of solutions.
> (From University of Arizona philosopher Wesley Salmon's Space, Time, and Motion, after an idea by A.K. Austin.)
> 
>