Zeno of Elea was a Greek philosopher who lived almost 2400 years ago. He is famous for the creation of paradoxes at the juncture of mathematics and the real world.

Today Ken and I want to talk about a type of Zeno paradox.

The paradoxes are claimed to be due to Zeno, although given the time gap, there is some discussion whether they are due to him or others. The central paradox has been claimed to be resolved by modern mathematics by some, and claimed to be still an important paradoxical notion by others.

I assume that you probably know this paradox well: Here is a version from our friends at Wikipedia.

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise’s starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.

I think this is answered simply in math by “so what?—the real line has lots of convergent subsequences, but they don’t prevent you from graphing objects in motion as far as you like.” Ken, however, agrees with those who say a satisfying resolution requires quantum mechanics. In a discrete time instant, Achilles either moves some multiple of a discrete unit of space, or he doesn’t. With overwhelming probability it’s the former, and so in some instant he lurches into a sudden lead and never looks back. There has been argument about further paradoxes that emerge with this view, but Ken believes they are rectifiable in several possible ways, not just in string theory’s marriage of relativity and quantum, but also in the relational model of spacetime advanced by Lee Smolin.

I knew the above paradox well, but was not aware of another variant due to James Thomson, a British philosopher, who created it in 1954. A bit after Zeno. Of course Zeno could not have imagined the following, since he did not have lamps and switches. Again from Wikipedia here is the paradox:

Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of this infinite series of time intervals is exactly two minutes.

So far no issue. The key question is:

Is the lamp switch on or off after exactly two minutes?

Good luck.

Zeno’s Arrow

Perhaps you have seen the effect of a rapidly flashing strobe light making dancers appear to stay momentarily still. If the flash itself is just an instant, nothing in what you see will suggest motion. So how can the dancer be moving? Always one to go to extremes, Zeno let both the duration of a flash and the interval between flashes approach zero.

Because he didn’t have strobe lights either, Zeno imagined a flying arrow rather than a dancer. His logic was: the arrow isn’t moving to where it is not, because it doesn’t have any time in which to do so, and it’s not moving to here because it’s here already. So it can’t be moving. It sounds like something Yogi Berra would say, and maybe being able to picture a baseball as standing still is why Yogi was such a good hitter.

My humble observation is about trying to prove hard theorems. The Zeno proof effect is based on my current difficulty. I am trying to solve an open problem—not that one. But it’s still a hard open problem, as most open problems are. Else why would they be open?

I have a collection of ideas that all seem to shed light onto the problem at hand. Yet the set of ideas has yet not yielded a proof. But each time I am about to give up and work on something else, I seem to always get another small insight not the problem. The insight seems to take me a bit closer to my goal—but not enough to actually get me there.

Thus, I feel a bit like Achilles trying to reach the Tortoise. Each insight say halves my distance to the goal, the Tortoise. But I am only closer: the goal seems nearer, but it is still not quite here. If I get another insight then I am again a bit closer. Alas, did I say that?, alas I keep getting insights, keeping getting closer. Yet I never have been able to catch the Tortoise, never yet been able to finish the proof.

The worst part is that I have the illusion that I am making progress. Should I quit the “race” to find the proof? Or should I keep trying for more and more insights? Will the insights go on for ever? Or will they eventually yield a proof? Very frustrating, indeed.

This reminds me of a quotation from Stanislaw Ulam’s book “Adventures of a Mathematician”:

Quantum Zeno Effect

As usual strange things are standard in quantum mechanics, so what is a paradox becomes an effect. The quantum Zeno paradox, now an effect, is analogous to Zeno’s arrow. I (Ken writing this section) think the name is actually unfortunate, because the point is that the effect is felt already when you’re close to 0 but not necessarily approaching 0.

Indeed it has been seen in experiments. It involves the quadratic difference between amplitude and probability, between ${\epsilon}$ and ${\epsilon^2}$. The experiments were first performed in the 1970’s, but the idea was remarked on by Alan Turing in 1954, and is implicit in the mathematical formulation of quantum mechanics laid out in a text by John von Neumann in 1932.

Here’s how it works in a standard situation. Consider a population of unstable particles that follow an exponential decay rule: after ${t}$ time units one expects only an ${e^{-t}}$ portion to be left in the original state. Near ${t = 0}$, both ${e^{-t}}$ and its derivative are close to 1 (or -1). Hence the expected number of decays over a short timespan ${t = \epsilon}$ according to the exponential decay rule is well approximated by a linear function of ${\epsilon}$. At least this is what the rule says to expect.

However, the transformation taking a particle from the unstable state ${\phi}$ to a stable state ${\psi}$ is also linear and evolves over time. For small time intervals ${\epsilon}$ again we may approximate the exponential involved in unitary evolution ${e^{-iHt}}$ by a simple linear function, so as to describe the system as being in the state

$\displaystyle (1-\epsilon)\phi + \epsilon\psi \qquad \text{normalized by}\qquad \frac{1}{\sqrt{(1 - \epsilon)^2 + \epsilon^2}}.$

Hence after a measurement, it is found to be in state ${\psi}$ with probability only

$\displaystyle \frac{\epsilon^2}{(1 - \epsilon)^2 + \epsilon^2} \simeq \epsilon^2 \ll \epsilon.$

(See this for a more proper derivation via unitary evolution.) Thus repeated measurements over tiny time intervals tend to preserve the unstable state ${\phi}$ with probability much closer to 1 than the model provides, even though the model is correct for larger times.

This is like saying if our strobe light performs such a measurement and we flash it fast at our decaying particles—regarding them as like a hail of flying arrows—a lot of them will appear to stay still. The ratio of those that do, instead of scaling as ${1 - \epsilon}$ as ${\epsilon \longrightarrow 0}$, approaches 1 much faster. The argument over this seems to go into murky questions over: what exactly is a quantum measurement?

Open Problems

When you are stuck on a proof, is it more like:

1. you are moving but never getting there like Achilles with the tortoise,

2. you think you have it but you don’t, now you have it again…oops…, like Thomson’s lamp, or

3. you are standing still and never get anywhere at all—each time you try to measure your progress you “collapse” back to square 1.

—?

April 6, 2013 4:07 am

I believe that Thomson’s paradox is not actually a paradox due to physical laws
(ie., due to the bounded speed of electrons, etc).

• April 11, 2013 1:58 pm

This sounds like the correct answer to me. Unlike the Zeno Paradoxes, there is no phenomenon to be explained here. We have simply been taken in by a formal description with no objective referent.

2. April 6, 2013 8:32 am

Meno Meno Tekel Upharsin

April 6, 2013 2:06 pm

I have a resolution of Zeno’s Paradox of Achilles and the Tortoise which I haven’t seen put forward anywhere. I’d like to know what people think.

We can consider the Tortoise to be at rest, at the origin of a reference frame and Achilles to be approaching the Tortoise at some fixed speed, v. Let’s assume that the initial distance between Achilles and the Tortoise is 100 units. Let’s also assume that Achilles is running at a rate of 1 unit per second. Then, we would expect Achilles to reach the Tortoise in 100 seconds.

The Paradox is normally set up like this: Achilles must first travel half the distance (50 units), then half of the remaining distance (25 units), ad infinitum, and we wonder how Achilles can ever reach the Tortoise.

What is not usually mentioned is that time is implicit in our construction of the paradox (Achilles must be running at some speed). In order for Achilles to cover a distance of 50 units, 50 seconds must elapse; then 25 seconds, then 12.5, ad infinitum. Thus, in this construction, 100 seconds is not allowed to ever elapse. What Zeno’s Problem of Achilles and the Tortoise is really saying is that Achilles cannot cover those 100 units, given that 100 seconds cannot elapse, which is true and thus not a paradox. Conversely, 100 seconds cannot elapse, given that Achilles, at speed v, never covers 100 units. Again, not a paradox.

Thoughts?

April 6, 2013 3:23 pm

For zeno’s paradox, if one accepts that there are two relative boundries, one for Achilles and another for the tortoise, that both have the property that there is a minimal time length such that any smaller length means that forward momentum has reached zero, then it simply becomes a race condition between the two competitors with respect to smaller time lengths to see who gets there first. Being slower, the tortoise’s momentum arrives at zero earlier than Achilles, so Achilles goes shooting right past.

To me what this shows is that our intuitive perpective of the world being continous is incorrect. It’s actually a discrete system, but just down to a depth that is beyond our perception. It’s always made me wonder if things in our universe can actually be continuous, or is that concept just an artifact of our thinking process. A perhaps similar example comes from simulating real numbers on a computer with discrete floating point data types. When we are far above the levels of precision, things work as expected. But if we hit the interior limits, it gets strange pretty quickly.

Paul.

April 6, 2013 3:29 pm

Three different situations…
1. Trying to prove something false
2. Trying to prove something undecidable
3. Trying to prove a true statement using wrong methods
Is the feeling exactly the same? Or is it possible to make a distinction with the sole help of intuition?

April 6, 2013 7:02 pm

I remember a conversation between two mathematicians (one quite distinguished, the other not so lucky…) at IHES, outside Paris, in 1977. The less fancy guy exclaimed, “What is the point? I work so hard to reach the mountaintop [prove the theorem of interest or find a counterexample to an interesting hypothesis] and then arrive at a complete impasse. I will never get there–as each time I have to start from scratch again.” The more successful man replied, “All is not lost because you’ve learned that THAT route is not the way to the top.” I was simply thinking, “The view, while climbing, is probably quite beautiful AT ALMOST ANY LEVEL OF THE MOUNTAIN–ESPECIALLY IF YOU ARE LUCKY ENOUGH TO BE PAID TO THINK!”

This example, of course, does not apply to the above Zeno-like paradoxes, but it points to another kind of sociological/psychological paradox–at least in my view. Virtually anyone can generate a new idea and so the basic situation is often extremely “ripe” for finding a way to JUMP OVER an infinite number of unsuccessful paths (and, therefore, possibly arrive at a vast number of extremely ambitious destinations).

April 6, 2013 7:19 pm

When I’m stuck on a proof, it’s either case 1 or 3. The latter is frequent. It ends up with the conclusion that x = x. Well, at least the long winding argument was entirely consistent.

8. April 7, 2013 12:28 am

Regarding my QZE shortcut, the normalizing term makes little difference for the ${\epsilon^2}$ side of the argument, but does apply to the other side. Without it one might think the probability of staying in the original state is approximated by ${(1 - \epsilon)^2}$, which is of course less than ${1 - \epsilon}$. But with it, the probability is just a tad under ${1 - \epsilon^2}$, as required.

I’d be curious to know how “kosher” my shortcut is, as it interpolates between the longer derivations in main sources and offhand descriptions I read in other sources. Amid a busy week with travel and tiring events, one doesn’t expect to write original formulas when it first looked like just filling in one section of a post mostly by Dick.

9. April 7, 2013 2:33 am

I don’t know about equations, but Quantum Zeno Effect is measurable even in appropriate 3D theater. If you place 2 linear polarizers orthogonal to each other, then no light is passed through the system (as in LED monitors, take 2 3D glasses and rotate one with respect to the other until minimal transparency [is obtained] ), but if you introduce more and more slightly rotated polarizers the system become more and more transparent, and if make adjacent angles equal then it is quadratic with the number of polarizers. Of cause, you need many hands in 3D theater to see the effect.

Zeno proof paradox – if stalled way is left with you only this is waste of time. If with just few friends, then with small probability it will propagate, if it dropped to the internet, it is again small probability, if it is systematized as a muscles around the skeleton of the math, with appropriate connections, than it can possibly move something. But this is different culture of publishing, and completely different publishing infrastructure, not yet (either discovered or deployed), and different granting concepts. So we are waiting for quantum publishing paradigm, so you difficulties can be entangled with others, and end up somewhere, after error correcting protocol, in the proof.

10. April 7, 2013 5:33 pm

Partly motivated by this charming post, and partly motivated by an thought-provoking weblog comment of Scott Aaronson’s regarding paradoxical aspects of P versus NP — a comment that (regrettably to my mind) now appears to have been  edited-out  — I have posted to TCS StackExchange the following paradoxical yet (hopefully) well-posed question:

Aaronson’s extended question: do natural generalizations of P versus NP exist?   Are complexity-class inclusions of the general form A ⊂ B and NA ⊇ B provable — in ZFC or any finite extension of ZFC — for any “natural” complexity classes whatsoever? (if “yes” construct examples; if “no” prove the obstruction).

The Leprechaun King has conceived an amusing quasi-historical narrative in regard to this question … but he wants to see some thoughtful TCS StackExchange replies before finalizing the narrative arc! So if you’ve got em, post em 🙂

• April 7, 2013 9:17 pm

Although the TCS StackExchange discussion is still evolving, apparently no strict inclusion A ⊂ B ⊂ NA is known, and even the simpler inclusion A ⊂ NA is not known for any language accepted by TMs whose runtime exponent is greater than unity. Tracing back, certain TCS StackExchange comments refer (admiringly!) to the GLL column We Believe A Lot, But Can Prove Little.” 🙂

11. April 8, 2013 10:19 am

In regard to the Quantum Zero Paradox, here is a Quantum Riddle:

Riddle What “obvious” physical principle exposes the flaw in the following four assertions, and the amazing conclusion that follows from them:

A1 Radioactive nuclei exist in the interior of the sun.

A2 The interior of the sun is a hot thermal environment.

A3 Hot thermal environments are Lindblad-equivalent to continuous weak measurement processes.

A4 Continuous weak measurements Zeno-suppress decay-rates.

Conclusion Radioactive decay-rates in the sun are Zeno-suppressed.

Solar neutrino observations establish that the above quantum reasoning is flawed … but wherein is the flaw?

April 9, 2013 9:46 am

Hmmm … the wrong assumption in the above riddle is this one:

A4 Continuous weak measurements Zeno-suppress decay-rates.

The A4 assumption is valid for finite-dimensional Hilbert spaces, but it is invalid for the infinite-dimensional state-spaces of field theory. In particular, the processes by which nuclei/atoms emit photons/beta particles/neutrinos into a vacuum ground-state cannot be suppressed by weak measurements.

The invalidity of A4 is good news … because otherwise quantum measurement effects would extinguish the chain of nucleosynthesis by which our own sun burns, that creates all of the elements in the universe that are heavier than helium … the elements of which our own bodies-and-brains are made!

Conclusion: The dynamical trajectories of relativistic gauge field theories (on an infinite-dimensional state-space) differ generically, physically, and drastically from the dynamical trajectories of quantum information theories (on finite-dimensional Hilbert spaces).

Fortunately! 🙂

• April 9, 2013 12:27 pm

And maybe I should mention that the mathematics underlying the failure of A4 can be simply understand in physical terms. The same weak measurement processes (Lindblad unravellings) that are responsible for the quantum Zeno effect act concomitantly to broaden transition linewidths, such that if we double the strength of the Zeno observation, we double the linewidth-broadening too.

However, whenever state-transitions couple to a field-theoretic vaccua, doubling the line-width also doubles the decay rate. These two effects cancel exactly, such that (for example) the QIT effects associated to hot stellar interiors do not act to Zeno-suppress the chain of nuclear state-transitions that is essential to carbon synthesis.

This is why (cosmologically speaking) it’s a very good thing that Gil Kalai’s point-of-view has been observationally verified: Nature’s dynamical state-manifold is observed to be not (even effectively) a finite-dimensional Hilbert state-space!

April 8, 2013 2:01 pm

This never occurred to me before, but if you have Achilles run with the intention
“not to pass the tortoise”, you cannot be surprised that he doesn’t pass it. This is true even if they moved in discrete steps, or when the tortoise doesn’t move at all in some intervals.

If you say that Achilles has no such intention and he is actually trying to run at 2m/s, but the “universe” is hell bent on computing/updating his progress by means of waypoints in continuous space, well, you’ve just declared the universe as a non-terminating algorithm that also slows down time to a halt as Achilles approaches the tortoise. This clearly is not the universe that we know, and it is quite an unusual view of the universe: most people would imagine that, if the universe is a form of computation, then it’s over waypoints in time as opposed to waypoints in space.

13. April 10, 2013 2:24 am

One thing to like about philosophical paradoxes is that they sometimes give a view well beyond the horizon. The ancient “Liar paradox” can be seen as a very early manifestation of the phenomena behind, Cantor’s theorem, Russel paradox, and Godel’s theorem. Zeno’s paradoxes are of similar nature. I had a post on “Google Buzz” regarding it where I wrote

“I talked in my “Beauty of Mathematics” class about Zeno’s paradox of Achilles and the Tortoise. I used this paradox an an introduction for infinite series, the first item about “infinity”. It is quite remarkable how relevant 17th century calculus is to Zeno’s three paradoxes. In fact, it looks that in a different universe these paradoxes could have started calculus.

I always regarded the geometric series explanation as a resolution of the Achilles and the Tortoise paradox. I was surprised and happy to learn an entirely opposite perspective by a philosopher friend of mine [Avishai Margalit]: Namely that the geometric series just put the paradox on a formal ground, but cannot be regarded as providing a solution to the paradox.

There were several interesting follow-up comments. Terry Tao wrote: “Zeno’s arrow paradox can be reinterpreted in the light of the theory of differential equations that the equations of motion must be second-order in time rather than first-order, since one has to specify initial velocity in addition to initial position in order to have a well-posed system. So the arrow paradox may well be the earliest precursor of Newton’s famous equation F=ma.”

Avishai Margalit to whom I referred won the Isreal prize in philosophy a few days after my class, and when I congratulated him and told him about the class he wrote me “I started my way in philosophy with Zeno’s paradox and I can assure you that I will catch no turtle in my life.”

• April 13, 2013 12:44 pm

I can see how a philosopher would say that.

If I remember my long ago readings well enough, Jimmy the Ancient Greek could lay odds as well as any modern bookmaker on the outcomes of Olympic contests, but that was not really the point of Zeno’s humble homilies. Read in philosophical context, it really had to do with a contention between two schools of thought about the relation of eternal being to secular becoming. Followers of Parmenides like Zeno would say that whatever it is that really is, is one, eternal, and unchanging. They would not be impressed that it took us a couple of millennia to “save the appearances” of change, since the appearances are only illusions anyway.

14. April 10, 2013 3:18 pm

We could hardly complete this course without mentioning the logical version of Zeno’s Paradox given by Lewis Carroll —

15. April 10, 2013 9:30 pm

April 11, 2013 2:54 am

Zeno’s paradox is just a non-sequitur. It starts by claiming that the two runners are moving at constant speed. But thereafter there is nothing at all that enforces that condition. Achilles is faster than the tortoise, but there’s nothing in the sampling that confirms that he’s moving at constant speed. He may or he may not. And therefore we can’t conclude that he will reach the tortoise. It’s as if Zeno’s lying to us with the empty words “constant speed”.

There’s a way to defeat Zeno. Simply put a third actor called Zorba initially at the same starting point as the tortoise and running at the same speed as Achilles. It’s clear that Zorba can outrun the tortoise infinitely. And Achilles will remain at a constant distance from Zorba. Therefore, so can he outrun the tortoise infinitely.

• April 11, 2013 6:28 am

Maybe. I took it that Pip was trying to tweak the classical paradox in a way more fast and loose, so I was trying to play along with that.

The classical paradoxes of change and motion really have to do with a disconnect between two realms —

On the one hand we have the phenomenology. There is no problem there since we obviously observe all sorts of Achilles passing all sorts of tortoises, the handicaps of their heels and hulls notwithstanding.

On the other hand we have the logical theories and mathematical models that we bring to bear on the phenomena by way of trying to describe and explain them.

There’s the rub. Get a model or theory that “saves the appearances” (solves the phenomena) and the para of the dox disappears.

Transpose the phenomena from a classical mode to a key more quantum mechanical, information theoretic, or just plain logical, and the note that resolves the chord is a trifle harder to find.

April 12, 2013 8:31 am

As stated in this article Zeno’s paradox is really lacking the rigor I pointed out. That is to say that there’s no confirmation at all that Achilles is truly running at a constant speed (and could therefore be approaching the tortoise forever asymptotically). However I just read an account by W. W. Rouse Ball that is more rigorous. In it, it is clear that Achilles is running at constant speed.

It still remains a non_sequitur. Zeno does provide an infinite and consistent sampling. However, he cooks this up by selecting spacetime events that are increasingly closer in the time direction (as others here, and Rouse Ball have pointed out). Zeno would apparently countered with the arguement that space is indivisible. But my argument that adds the third actor by the name of Zorba totally smashes Zeno in my opinion. If anyone still thinks that Zeno has a point, I would be curious to hear how you’re dealing with the Zorba + Achilles team.

April 11, 2013 4:30 pm

The whole premise of the “paradox” rests upon the condition that the tortoise is ahead of Achilles. The conditioinal statements, it seems, are only valid within the time period closed on the start and open on the time where Achilles pulls up even with the tortoise. Each iteration of Achilles closing in on the tortoise, say halving it, requires a smaller unit of time, until you’re not looking beyond the point at which the race changes leaders.

17. April 16, 2013 8:15 am

Re; Open Problems

Other archetypes that come to mind in this connection go by the following names:

1. Hansel & Gretel
3. Meno