Zeno’s famous Paradoxes involve the impossibility of arriving somewhere as well as the impossibility of even starting to go somewhere. And that flying arrows have to be an illusion. [Time flies like an arrow, but fruit flies like a banana.]
If Zeno were alive today, he’d be over 2500 years old and would have seen his paradoxes explained in a variety of ways by a lot of very smart people. Yet at heart they still have some metaphysical oomph. And the thing is, at least in some contexts, Zeno was (sort of) right. There is something of a paradox here involving space and time.
Or at least something interesting to think about.
Zeno of Elea apparently created many paradoxes (paradoxii? paradoxen?), but only nine survive, and some of them are different views of essentially the same thing. Which is that infinity is funny. Strange things happen when infinity joins the game (you get magical hotels).
Part of the paradox comes from the notable absence of infinity in the physical world. Nothing we know of is infinite. Some think the universe is infinite, but that seems unlikely as it implies infinite energy and matter (not to mention space).
It also goes the other way, to the infinitely small. We end up with two big questions about space: Firstly, what happens if we go in a straight line for a long time — does the universe just keep going or does it wrap around somehow? Secondly (the issue here), what happens if we divide space into smaller and smaller bits — can we do that indefinitely or is there an end to how small reality is?
Simply put, what are the small and large size limits of the universe?
Zeno’s paradoxes assume reality is infinitely divisible. They depend on it!
Three of the nine paradoxes are well-known and much discussed:
The first says that you can’t start going someplace because to do that you need to first get halfway. To get halfway, you need to get halfway to halfway. And to get there, you need to get halfway to halfway to halfway. And so on, infinitely. So, motion is impossible — an illusion.
The second says that you can’t arrive someplace because at some point you’re only halfway there. At some point afterwards, you’re half that remaining distance away. Later, you’re half that distance. And again, so on infinitely. You never arrive because you’re forever cutting the remaining distance in half. Again, motion is impossible and must be an illusion.
The third says that the motion of a flying arrow is an illusion because, at any given instant in time, it’s frozen in space at that moment — not moving. Imagine a snapshot of a car on the street. There’s no way to tell if the car is moving or just sitting there. And since nothing is moving in the instant, nothing is moving, period.
On some level, these do make sense. The logic isn’t trivially refuted. They’re paradoxes because, duh, motion is real.
Over the twenty-five centuries since Zeno devised these, they’ve been well explored and explained. Many believe calculus fully accounts for them because:
Or in the language of calculus:
Which says that, as i approaches infinity, the sum of the fractions approaches two (and effectively it is two).
This leads to the one about the infinite number of mathematicians who walk into a bar. The first one says, “I’d like a beer!” The second one says, “I’d like half a beer!” The third one says, “I’d like one-quarter of a beer!” And so on, infinitely, each halving the amount. After an infinite time, the bar tender says, “You guys are just plain crazy!” and serves them two beers.
True story! Happened to an infinite number of guys I know.
Yet Zeno really does apply to the real numbers because every real number is an island. Movement from one real number to the “next” real number isn’t possible because there is no notion — given any real number — of the “next” real number.
What’s the next real number after zero, or 2.5, or pi? There is no way to answer. There are an infinite number of other numbers between the given number (the island) and any possible “closest” number we could name. This is exactly Zeno’s first paradox.
That any two numbers we pick, no matter how close together they seem, have an infinite number of other numbers between them embodies Zeno’s second paradox. No matter how close two numbers are, there’s always another number halfway between them.
This property of being an island also applies to the rational numbers — it is not due to the continuum but to how, for any two given real or rational numbers, there is always another number halfway between them. The island property is due to plain old countable infinity, not the uncountable infinity of the continuum (see Cantor’s Diagonal for details).
Zeno’s arrow paradox is a little different. At least since Newton, if not earlier, we’ve had a picture of the momentum of an object. It’s an important and fundamental property of everything (it’s one of the conserved properties along with energy and mass).
So, even considered in an instant where no motion takes place, the arrow has momentum — the product of its mass and velocity — that embodies its motion.
There is also a question of whether a “frozen instant of time” is a meaningful concept. We have no way to measure an “instant” of time. We measure its duration or interval. Cameras, which appear to capture instants, have shutter speeds — for instance 1/500 second — reflecting the duration of time the shutter is open.
The only way to find out if something happened at precisely 10:00:00.0 is to start checking just before and stop checking just after. Even if it’s one-trillionth of a second before and after, it’s still two-trillionths second in duration.
Bottom line, Zeno’s paradoxen have long been resolven in terms of the real world, but Zeno had it right in terms of the real (and rational) numbers.
Which isn’t exactly a revelation. Zeno’s paradoxii come from a classical view of a reality described by the real numbers. And we’re still not sure if reality is infinitely divisible. We do know we don’t know what happens below the Planck Length (of 1.616×10⁻³⁵ meters).
It all raises some interesting questions about the math that describes our reality!
Stay real (or at least rational), my friends! Go forth and spread beauty and light.
January 26th, 2023 at 8:32 am
A different take on Zeno, but this is one of my favorite articles ever, which your post reminded me of😀
January 26th, 2023 at 8:52 am
I had never heard of the analogy of the closed fist versus the open hand! So, I went poking around. I can only access the preview page of the article you linked to, but looking at Wikipedia I think we’re dealing with two different guys named Zeno!
The guy with the math paradoxes is Zeno of Elea. His Wiki page makes no mention of the closed fist analogy.
There is also a Zeno of Citium. Who seems linked with the idea of stoicism, and under Logic, Wiki has this:
Zeno said that there were four stages in the process leading to true knowledge, which he illustrated with the example of the flat, extended hand, and the gradual closing of the fist:
So, I’m thinking this is the Zeno from the article you linked?
I had no idea there were two of them, so I learned something this morning, thanks!
January 27th, 2023 at 5:59 am
I had no idea there were two Zenos either! Yes the Zeno you found is the one I was thinking of.
January 27th, 2023 at 8:20 am
There’s a “mistaken identity” comedy there waiting to be written, I’m sure! 😁
January 26th, 2023 at 8:59 am
I didn’t explain in the post, but for those without the math background, the summation in the post expands like this:
At least in theory, but we can never sum an infinite series. The thing some might question is the first terms in that expansion, but remember anything to the power of zero is just one (and anything to the power of one is just that thing), so it evaluates to:
For as large an N as you care to calculate for.
January 26th, 2023 at 10:57 am
Ahhh, paradoxes!They make you intellectually humble. I especially like the one in set theory by Bertrand Russell: ‘The barber of Seville shaves all those, and those only, who do not shave themselves.’ So, Bertie asks, who shaves the Barber of Seville? Does he shave himself? Nope, that would be shaving himself and he only shaves those who do not shave themselves. Does his wife shave him? Nope, the Barber of Seville shaves all those who do not shave themselves. Mmm?
January 26th, 2023 at 12:38 pm
Absolutely! And, unlike some paradoxes (such as the Twins paradox in Special Relativity), these are true contradictions that cannot be resolved. They’re like the Liar Paradox that way (“Everything I say is a lie.”).
As you no doubt know, the formal version of Russell’s Barber paradox blew up the mathematics program of the time. They were thinking set theory fully explained arithmetic, and it was all wrapped up in a neat package with red bow. And then that troublemaker Russell asked, “Does the set that contains all sets that do not contain themselves contain itself?”
If it doesn’t, it should, but if it does, then it is no longer the set of sets that do not contain themselves. So, it can’t, but then it has to… 🤯
Not only humbling, but I rather like that reality is complicated enough to not be easy to figure out, and that there are some things that remain, in principle, forever beyond our grasp. Russell’s troublemaking lead to Gödel’s Incompleteness theorems and Turing’s Halting theorem, both of which are statements about what’s beyond our grasp.
So, yeah, great paradox!