Hitting a Zeno in One - Solution to the Dichotomy Paradox
Zeno's Dichotomy paradox can be considered a more economical variant of his Achilles and the Tortoise paradox, and hence a tougher nut to crack.
This description of the paradox is from wikipedia.
Suppose Homer wants to catch a stationary bus.
Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
Because you can keep dividing the half-distances to infinity, Homer is faced with an infinite number of distances to traverse.
This is not possible according to Zeno, so he concludes that Homer is unable even to start moving.
Aristotle put it in one line:
'That which is in locomotion must arrive at the half-way stage before it arrives at the goal.' [Aristotle, Physics, v1:9 239b10]
Solution
According to Zeno and as per Aristotle's statement above, any object in motion must arrive at the half-way mark before it reaches the finish.
Since nothing can be said to 'arrive at' or 'get to' a place without having first traversed some distance to that place', to arrive at the half-way mark the object must have been able to traverse that half-distance.
(One must admit that the concept of 'arrival' necessarily implies traversal to the point of arrival, else 'motion' in general has no meaning.)
This being so, an object which is able to traverse a distance from one point to another is necessarily able to first travel half that distance.
Since the formulation of each half-distance inherently admits the ability for the object to have traversed that distance prior to considering the next half-distance, we can logically conclude that an object can indeed traverse an infinite number of smaller and smaller half-distances.
The Zeno-In-One Solution
That which is in motion must necessarily be able to arrive at the half-way stage before arriving at the goal, no matter how many successive half-stages there are.
kyy 2007-10-08 mon 0841 z+10
This description of the paradox is from wikipedia.
Suppose Homer wants to catch a stationary bus.
Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
Because you can keep dividing the half-distances to infinity, Homer is faced with an infinite number of distances to traverse.
This is not possible according to Zeno, so he concludes that Homer is unable even to start moving.
Aristotle put it in one line:
'That which is in locomotion must arrive at the half-way stage before it arrives at the goal.' [Aristotle, Physics, v1:9 239b10]
Solution
According to Zeno and as per Aristotle's statement above, any object in motion must arrive at the half-way mark before it reaches the finish.
Since nothing can be said to 'arrive at' or 'get to' a place without having first traversed some distance to that place', to arrive at the half-way mark the object must have been able to traverse that half-distance.
(One must admit that the concept of 'arrival' necessarily implies traversal to the point of arrival, else 'motion' in general has no meaning.)
This being so, an object which is able to traverse a distance from one point to another is necessarily able to first travel half that distance.
Since the formulation of each half-distance inherently admits the ability for the object to have traversed that distance prior to considering the next half-distance, we can logically conclude that an object can indeed traverse an infinite number of smaller and smaller half-distances.
The Zeno-In-One Solution
That which is in motion must necessarily be able to arrive at the half-way stage before arriving at the goal, no matter how many successive half-stages there are.
kyy 2007-10-08 mon 0841 z+10
posted by kyy at 9:06 AM
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