# Does the uncertainty principle resolve Zeno’s arrow paradox?

Zeno’s arrow paradox says that motion is impossible. Does quantum mechanics say that the underlying assumption is wrong?

Assumption: in any given moment, an arrow in flight is motionless. Then it remains stationary at every moment. Thus the arrow never moves. Mazur, Joseph; The motion paradox (New York: Dutton), p. 4-5.

Here is quantum mechanics:

One striking aspect of the difference between classical and quantum physics is that whereas classical mechanics presupposes that exact simultaneous values can be assigned to all physical quantities, quantum mechanics denies this possibility, the prime example being the position and momentum of a particle. According to quantum mechanics, the more precisely the position (momentum) of a particle is given, the less precisely can one say what its momentum (position) is. This is (a simplistic and preliminary formulation of) the quantum mechanical uncertainty principle for position and momentum. “The Uncertainty Principle”, SEP. https://plato.stanford.edu/entries/qt-uncertainty/

It appears that quantum mechanics says that the initial assumption is wrong. The arrow paradox assumes certainty of both position (stationary) and momentum (none). That premise allows the distances over a range of moments to add up to zero. But quantum mechanics says that part of this assumption can never be known.

(1) If the position of the arrow is known to a certainty, then its momentum is unknown. The arrow might be moving at that moment. The possibility of movement resolves the paradox by allowing for momentum at any given instant.

(2) If the momentum (zero) is known to a certainty, then its position is unknown. The arrow might be in any of a range of places. If the arrow might be anywhere over a range of places, then it must be moving.

I am neither a physicist nor a mathematician. But I have questions.

• As far as I understand it, the point behind the thought experiment of Zeno's Paradox was that our underlying assumptions about physical mechanics are wrong. So I don't think that your example would really resolve it so much as confirm it. May 14, 2018 at 18:57
• Zeno's paradox talks about a classical object, objects behave classically at large (i.e. pretty much larger than molecular) scales, so an arrow would behave classically when measured in the way the thought experiment is positing. I think the more interesting take on Zeno's paradoxes in light of QM is that QM says spacetime should be discrete and not continuously valued, so the paradox of walking halfway before halfway before halfway, and so on, reaches a bottom at the Planck scale. May 14, 2018 at 19:00
• "If the arrow might be anywhere over a range of places, then it must be moving." ... I am not sure how that follows ... May 14, 2018 at 19:40
• No, it does not, and Zeno lived long before classical mechanics, so its differences with the quantum one did not matter to him. The paradoxes are about conceptualizing change under Parmenidian law of identity, they can not be resolved by mathematical or mechanical means, see Papa-Grimaldi, Why Mathematical Solutions of Zeno's Paradoxes Miss the Point. May 14, 2018 at 22:46
• Also worth noting is that QM is based on calculus, which was basically invented to resolve issues like Zeno's paradox. So using QM to refute Zeno's paradox is begging the question. May 14, 2018 at 23:33

Zeno's arrow paradox is a redefinition of "motion": Quantum physics is not required to deal with Zeno's arrow paradox. The statement of the "paradox" works by invoking the idea of "motion" while only ever considering instants of time, and thus not considering motion as a concept that applies with respect to change over time. All that happens in Zeno's statement of the "paradox", and similar restatements by other authors, is that an assertion about "motion" is made on the basis of the position of a thing at a single instant in time; because a thing occupies a single space at an instant in time, it is "motionless", and since this applies at all instants in time, it is "motionless" at all instants in time --i.e., it is always motionless, and motion is impossible.

Zeno's argument rests on a persuasive definition of "motion" which is different to its real meaning. In Zeno's argument, the concept of "motion" is a property of an object at a single instant in time; it bears no relationship to actual motion, as the concept is used by anyone. Striped of its persuasive definition of "motion", all the argument says is: at any instant in time, everything occupying space is in the same space it is in, and not some other space. (If there is any branch of physics that disputes Zeno's argument, it is not quantum physics, but regular classical mechanics, which quantitatively defines the concept of motion. Simple use of classic physics equations show the ridiculousness of trying to measure motion by position data at a single instant in time. However, even a pre-physics understanding of "motion" is sufficient to refute the argument, so long as you recognise that motion is conceptually describing change in location over time.)

Zeno's argument is a classic case of a philosophical argument that tries to bamboozle people by simply redefining a concept to have a completely different meaning. Since the argument invokes the idea of "motion" but does not ever consider changes of position with respect to time, it is similar to (but not exactly the same as) the stolen concept fallacy. Once "motion" is correctly defined as change in position with over time, it is not correct to say that (at any given instant) an arrow is "motionless" merely because it occupies one space at that point in time. (Whether it is motionless or in motion cannot be determined by its position at a single point in time, but by the rate of change of position with respect to time, taken relative to some other existent used as a reference point.)

A little rant about quantum physics and philosophy: This little rant is not a negative comment on the OP, or his question, but just something that needs to be said in the context of this question. People seem to have this ridiculous fetish for quantum physics, where they act like it solves all the philosophical problems of the world. (And no-one seems to have such a fetish for this as non-physicists.) Theory of mind? Quantum physics will solve it! Zeno's paradox? Quantum physics! Moral laws concerning lifeboat situations? Hell, let's try to apply quantum physics!

This is a dead end --- quantum physics solves exactly zero philosophical problems. It is philosophy that is required to help interpret the data from experiments in quantum physics, to avoid making stupid mystical conclusions from this data. ("Oooh, the cat is both dead and alive - I have transcended the law of non-contradiction!") The vast majority of what is written about quantum physics and philosophy is mystical horse-shit, dressed in fancy pseudo-mathematical verbiage.

With regard to the "uncertainty principle", it is a principle of epistemology, not metaphysics, and it merely circumscribes limits of our ability to measure things. Not only does it have no application to the existence or non-existence of motion, but it is a principle that makes reference to motion, and therefore pre-supposes that motion is a thing.

• Zeno did not redefine the concept as understood by ancient Greeks, he and Parmenides rather showed that the intuitive notion behind it was incoherent. The idea of "motion" as change of position in time is a calculus based idea that only emerged in 17th century, and even that not so much redefined a folk notion as extracted a coherent fragment out of it. It is charming that this fragment has now been culturally ingrained enough to acquire the status of "real meaning", but, as Zeno paradoxes (along with detractors of "mechanical time") still show, it fails to capture the intuitive notion too. May 15, 2018 at 4:26
• None of what you say raises an argument against the position in my answer - it is just ad hominem remarks about my presumed lack of understanding of history. You have no idea of my "background", so smarmy remarks like that add nothing to the discussion other than a pathetic "argument-from-intimidation".
– Ben
May 15, 2018 at 5:32
• To be clear, my answer is assessing Zeno's argument on the basis of what we now know about the concept of motion. I am pointing out to the OP that he needn't appeal to quantum physics to deal with this argument. Discussion of the history of the pre-Socratic Greeks is tangential to this at best.
– Ben
May 15, 2018 at 5:34
• @Ben - It seems to me your idea does not address the problem. If it does then you should write a paper and settle this matter once and for all. I prefer the ideas of one Australian physicist who uses Zeno to show that the idea of an 'instant' is incoherent.
– user20253
May 15, 2018 at 9:42
• That zero over zero is anything you like remains a problem in math. We would like to think of the continuum as both made of points and infinitely divisible, and we can't. If the continuum is just points, the 'point' zero should be usable like any other number, but it isn't. The fact it is motion, and that one of the two things being divided is time is from that point of view completely irrelevant. So Zeno is not 'redefining' anything. The paradox is implicit in our model of anything continuous whatsoever, and you are mischaracterizing the problem
– user9166
Oct 23, 2019 at 21:17

Zeno's paradox was formulated after Parmenides theoriesed about the ontological status of being:

One path only is left for us to speak of, namely, that It is. In it are very many tokens that what is, is uncreated and indestructible, alone, complete, immovable and without end. Nor was it ever, nor will it be; for now it is, all at once, a continuous one.

The emphasis here is on what it means to be 'is'. Whereas relativity questioned the relations between space, time and motion; quantum mechanics questioned the ontological status of what it means to be. Heisenberg noted this, and theorised that quantum mechanics, was in its essence, had revived the old aristotelian categories of potentiality and actuality (these were in fact how Aristotle phrased his answer to Zenos paradox), and this was further refined by Popper in his theory of propensity.

Zeno's paradox exists because of an incorrect statement. Imagine the arrow as it passes from a to b. Imagine you could take a picture of the arrow at time c, an instant in time which has no length. The arrow would appear stationary, occupying a space. But for the phrase it is at rest to be true, one would have to take another measurement at another instant, some time from the first instant. Only then could you say the arrow was at rest or not.

Looking at the arrow at any isolated instant just tells you where it is at that instant, and everything is somewhere at any instant. And any object could be at rest or moving at that instant, but any movement would be impossible to know because the instant has no length. So you can say all objects could be at rest or moving at that instant, we do not know.

And yes this does seem to fit very well with quantum physics. This maybe why at the very small on the border of instants, the indetermined state of things actually exists.

Whilst some said it is resolved like Russell by defining clearly the concept of motion as over time, it really does not. The problem is that even if you accept motion over time, the Achilles vs tortoise still there. Yes but a and t move over time. But by taking snapshot infinity small in time a can never take t. Even worst qm has a quantum Zeno effect about this frequent observation make the quantum state “motionless”.

The classical world is really a problem. Unless you adopt Aristotle quantum like idea that time cannot be continuously divided may be you have a bit of chance.

In quantum world if it is a two particles a (photon)and t (electron which has mass and hence slower), may be you can resolve by assuming an ever moving particle of probability over the field. As there is no strict a is exactly on t position in last moment, hence even if you measure very quickly you always has a chance to observe a is ahead of t. The probability will increase over time.