Is it correct to formulate Zeno's arrow paradox as follows?

1) If the arrow is still, it is not moving. 2) The flight of an arrow can be broken into instances, in all of which the arrow is still. 3) Hence, putting all of these instances of time together, the flying arrows is motionless.

Is this a paradox because the conclusion, 3), is false, since nothing can be in motion and at rest at the same time? Also, does this necessarily imply that one or both of the premises must be false?

Finally, was Zeno forming these paradoxes using premises assumed to be true by everyday people? With the goal of showing that "normal" reality is actually wackier than the reality posited by Parmenides?

  • See Zeno's Paradoxes : The Arrow : "This argument against motion explicitly turns on a particular kind of assumption of plurality: that time is composed of moments (or ‘nows’) and nothing else. Consider an arrow, apparently in motion, at any instant. First, Zeno assumes that it travels no distance during that moment—‘it occupies an equal space’ for the whole instant. But the entire period of its motion contains only instants, all of which contain an arrow at rest, and so, Zeno concludes, the arrow cannot be moving." – Mauro ALLEGRANZA Jul 10 '17 at 6:36
  • Correct: it is called a "paradox" because the conclusion is contradictorty: the arrow is at the same time in motion and still. – Mauro ALLEGRANZA Jul 10 '17 at 6:38
  • The fact that the conclusion is false implies: either that the inference is "logically flawed" or that one of the premises is false. Presumably, the second one: in every "instant" the arrow is motionless. – Mauro ALLEGRANZA Jul 10 '17 at 6:40
  • The Arrow is one of Zeno's Paradoxes regarding Motion aimed at defending Parmenides' views by attacking his critics. "Parmenides rejected pluralism and the reality of any kind of change: for him all was one indivisible, unchanging reality, and any appearances to the contrary were illusions," Thus, also motion (a form of change) must be illusion. – Mauro ALLEGRANZA Jul 10 '17 at 7:19
  • The infinite divisibility of space and time and the nature of division have been left out of your analysis. I don't think Zeno's understanding can be conveyed without them. So I would say that your formulation is not correct. From the POV of Newtonian physics, the entire problem really is about our notion of division when the denominator approaches zero. – jobermark Jul 10 '17 at 15:57

Two helpful texts:

Zeno's paradoxes. Salmon, Wesley C., compiler. Indianapolis, Bobbs-Merrill. 1970.

The motion paradox : the 2,500-year-old puzzle behind all the mysteries of time and space. Mazur, Joseph. New York : Dutton, c2007.


With the benefit of millenia of hindsight, I believe the point of the arrow paradox can be summed up, for example, as the observation that if

  • you consider the discrete topology on the real numbers
  • you only work with the values of a function at points

then you can't do any nontrivial calculus; in particular, there is no reasonable notion of derivative, and no fundamental theorem of calculus to let you compute how much a function changes value over an interval by integrating its derivative.

It's only a paradox if:

  • you look only at point sets and underestimate the importance of some notion of topology to glue all of the points together (e.g. the ordering and distance function between real numbers)
  • you don't consider retaining more information about a function at a point than just its value

Regarding the second point, you can do quite a bit of calculus completely local to a single point if you remember not just the value of a function at a point, but the value of its derivative (or all of its derivatives!).

Or, from a more physical point of view, the fundamental variables of motion should include both position and momentum at all times, rather than position alone!


The summary of the paradox in the question seems correct to me as far as it goes. Zeno's proofs were presumably intended to support his master Parmenides' view that relative motion implies a background or source that is absolute stillness and changelessness.

The same argument is just as valid today, when formulated carefully, and it supports the view of time and change proposed by the perennial philosophy. If you read Nicolas de Cusa you'll see that he does indeed propose that things move and move-not at the same time.

I don't think Zeno knew this other philosophy well or he would have made more developed arguments. The idea is not that motion is impossible but that it is mind that moves, not intrinsically-existing objects (there would be no such thing).

Time and motion would be paradoxical only when we reify objects and space-time. If we follow Kant and de-refify them then the paradoxes evaporate. But we then have to say the things both move and move-not. They move to the extent they exist (as appearances) but for an ultimate view motion and change would be impossible, just as logical analysis proves.

A common Buddhist argument runs - change cannot happen in the past or the future and there is not enough time for it to happen in the present. Ergo it does not happen. For the experience or perception of change we have to combine past, present and future, and this can only be done by a mind.

You ask whether we are supposed to see his paradoxes as 'true'. It's probably better to say that we are supposed to see them as 'holding', thus proving that our usual idea of motion is metaphysically absurd.

The reason the paradoxes of motion (Zeno's or any others) are usually ignored seems to be that if they 'hold' then they render materialism absurd and lend credibility to the perennial philosophy. It is an odd thing that the philosophy of the mystics pays greater respect to logic than what Russell calls 'Rational' philosophy but it's always been this way.

This so-called 'Rational' philosophy cannot make sense of metaphysics because it ignores these paradoxes on the grounds that naive-realism (the wysiwyg universe) is true even it it is logically absurd. Zeno was trying to poke holes in naive-realism and so is not a popular philosopher among scholastics.

His goal was not "showing that "normal" reality is actually wackier than the reality posited by Parmenides." There is only one reality. His goal seems to have been to show that Parmenides was right, reality is a lot whackier than naive-realism would have us believe, and lot less paradoxical.

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