# Zeno's “Stadium” with the same metaphysical assumptions as his other paradoxes

The fourth argument is that concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This, he thinks, involves the conclusion that half a given time is equal to double that time. – Physics VI, Part 9

If it is not granted that Zeno was implicitly speaking about atomistic time and space here and instead used the “default” position, i.e. time and space being infinitely divisible – which he also implicitly assumed in his other three paradoxes

• “Achilles and the Tortoise”
• “The Dichotomy”
• “The Arrow” (here maybe less obviously)

– can we interpret “The Stadium” so that it is not trivially fallacious? That it reaches a paradoxical conclusion with the same implicit metaphysical assumptions as his other paradoxes?

PS: Okay this question keeps getting misunderstood. Again, please note: The question is NOT if Zeno assumed atomistic time and space here (there is no evidence for that, anyway!). The question is, how “The Stadium” can reach a paradoxical conclusion WITHOUT assuming atomistic time and space.

• See Zeno's Paradoxes: in Achilles' one there is the assumption that the run needs the travelling of an infinite number of finite distances, and this is a metaphysical assumption about space and time. Commented Aug 30, 2017 at 11:09
• Right, if space is atomized, the common thread that permits all of them vanishes.
– user9166
Commented Aug 31, 2017 at 4:06
• @MauroALLEGRANZA in his three other paradoxes, Zeno implicitly uses the default position, that time and space are infinitely divisible. The other assumptions he uses are explicitly mentioned and it can be assumed that most people, who haven't thought about it seriously, would grant them. Maybe they aren't as innocuous as they seem, but Zeno tricks us into believing they are. That's what makes his paradoxes work. “The Stadium” OTH doesn't seem to reach its paradoxical conclusion on the basis of naive “default” assumptions about space and time. That's why it seems so weak. So can it be salvaged? Commented Aug 31, 2017 at 8:37
• So, this one may be looking at atomized space, and saying "If we atomize space and time, and neither divides infinitely, how would speeds add?" They cannot add by using intermediate positions, because they are supposed to be atomic. He wants both theories of space to lose.
– user9166
Commented Aug 31, 2017 at 16:48
• @jobermark what's the evidence that Zeno was assuming atomized space here? Commented Sep 1, 2017 at 8:04

Opinions vary, but I suspect that Zeno was arguing against an atomic theory of the continuum. This is one possibility discussed by the entry in the SEP. His argument may not be overwhelming, but he was clearly correct to say that our usual idea of the continuum is paradoxical.

The SEP entry makes the interesting point that the Stadium paradox can be resolved by assuming that moving objects are like light bulbs in a series, where each bulb lights up in turn but nothing travels from one bulb to the next. I believe this is what Zeno was trying to say, that in order to make sense of motion we must abandon our usual idea of objects and how they move. That is to say, I read his paradoxes as an argument against naive realism and materialism.

If we run his thought-experiments using Plank-lengths and imaging how they move we soon run into trouble.

• We've had a couple millenia to sort things out -- in this day and age, the assertion that our "usual idea of the continuum is paradoxical" is an incredibly bold statement, not something you can pass off as truth in an off-hand remark. Even the best reasons to question the limits of the application of the continuum to descriptions of the real world are still in the realm of speculative physics.
– user6559
Commented Aug 31, 2017 at 17:03
• this doesn't answer the question at all. Commented Sep 1, 2017 at 11:20
• @Hurkl It is not a bold statement at all. The paradoxicality of our usual idea is common knowledge. You might like to read Hermann Weyl on the topic. Our whole idea of time and space is paradoxical. Welcome to metaphysics!
– user20253
Commented Sep 1, 2017 at 13:10
• @wolf-revo-cats Yes, you're right, But it seemed worth saying, and the question seems inadequately expressed for me to answer. At any rate, I cannot understand it well enough to answer directly. The problem for me is that if time and space are not 'atomic' then they are continuous, and in this case cannot be extended. That is, paradoxes remain whether or not we assume an atomised continuum. But I don't know if the Stadium paradox remains since this seems to rely on an extended, thus not continuous, time and space. I'm aware this comment may raise some objections.
– user20253
Commented Sep 1, 2017 at 13:17
• @Hurkyl: They're speculative physics because we can't get close to energies that probe the Planck distance but it's not at all bold, there's plenty of work that theorises a different structure to the continuum, for example in LQG the area & volume operators give discrete values. Commented Sep 5, 2017 at 10:38