# 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. – Michael May 14 '18 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. – Not_Here May 14 '18 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 ... – Bram28 May 14 '18 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. – Conifold May 14 '18 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. – Cort Ammon May 14 '18 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. – Conifold May 15 '18 at 4:26
• I accept that Zeno was working at a low base of understanding, based on the philosophy of the pre-Socratic Greeks. (The same excuse does not apply to people living now who still have trouble with Zeno's "paradoxes".) Nevertheless, there is a difference between quantitatively formalising an idea, and inventing it. Calculus has formally mathematized the meaning of change over time, but I disagree that pre-calculus understanding could not conceive of the idea of motion as "change over time" (not just at a point in time). Even under Heraclitus idea of flux (pre-Zeno) this was understood. – Ben May 15 '18 at 5:07
• Zeno's understanding was no more "low" than the current one is "high", history is not a climb to a summit. You remarks about the history of formalization and calculus also show lack of historical awareness, what one can conceive today could happen is not what actually happened, and what actually happened modern folk often fail to conceive. But in the case of motion and calculus at least it is well covered in the historical literature. – Conifold May 15 '18 at 5:16
• 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 '18 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 '18 at 5:34

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.