--Stuart K. Hayashi
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| This is Bartolomeo Carduci's fresco of Zeno of Elea (he is the old man leading the young men). |
Chris McKenzie explains:
Zeno's Paradox is as follows. The ancient Greek philosopher Zeno stated that in order to cross a room, one must first cross half the room. In order to cross the remaining half, one must cross half the remaining distance, and so on--infinitely. Zeno concluded that one can therefore never cross the room.
Let's change Zeno's numbers but keep his intent: In order to cross a room, one must be first cross 99% of the room. In order to cross the remaining 1%, one must cross 99% of the remaining 1%, infinitely.
Zeno has a problem though, and it's one he's smuggled in. Why does Zeno think you can cross 99% of the room? After all, if you treat 99% as a whole, you must first cross 99% of the new whole. The end result is that you can't move at all because in order to move you must first move through 99% of whatever infinitesimally small space you've chosen to move through--say the first 1% of the distance.
Now to the claim that "nothing is certain" or we can only be 99% certain. It's easy to explode this claim with a single question: Are you sure? However, we can see that the 99% certainty claim suffers from the same problem as Zeno's room: In order to be 99% certain, you must first be 99% of 99% certain. Which is to say, you must be wholly certain of a certain whole (the first 99%).
Or, you could just walk across the room and proudly claim, with 100% certainty, "I did it."
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Stuart K. Hayashi adds:
As Chris said, we can turn this around on Zeno.
Every fraction is, in another context, one whole. That is, Y may be a fraction of X, but Y is still one whole of Y. If the room is 30 feet across, and I only walk "half that distance," that is 15 feet. But that "half distance" is actually "the entire distance" of another measurement: 15 feet. Therefore, every time you travel a distance, you travel the entirety of that distance. Every measurement is, in at least one context, "100 percent."
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UPDATE from the Same Day:
Robert Nasir wrote the following comments to me. I am quoting them here with his permission.
Similarly, if you can travel half the length of a given room, then you can travel what is half of twice the length of the room.
The real issue is the integration of the discrete and the continuous. Everything is discrete. Entities are discrete. The distances they travel (to the extent they're measurable) are discrete.
Mathematics treats space (and time) as continuous. That's fine, it's useful (and arguably necessary) to do so.
But to understand why apparent paradoxes arise, one must never lose sight that math is method, not metaphysics.
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UPDATE: Additional comment from Stuart on January 26, 2018:
I think I have a new way of phrasing this matter:
Zeno's Paradox is based on the false presumption that no measurement can ever be 100 percent. But the exact opposite is true: every measurement is 100 percent, at least in terms of each measurement being 100 percent of itself.
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UPDATE: Addendum from Stuart on February 9, 2020:
Only today did it occur to me that pointing out that every measurement is “100 percent of itself” is a restatement of “A is A” (here, “1 = 1”).
