8 - You Can't Get There From Here: Zeno and Melissus

The paradoxes of Zeno and the arguments of Melissus develop the ideas of Parmenides and defend his Eleatic monism.

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Further Reading: 

J. Faris, The Paradoxes of Zeno (Aldershot: Ashgate, 1996).

P.S. Hasper, “Zeno Unlimited,” Oxford Studies in Ancient Philosophy 30 (2006), 49-85.

J. Lear, “A Note on Zeno’s Arrow,” in Phronesis 1981, 91-104.

R. McKirahan, “Zeno’s dichotomy in Aristotle,” in Philosophical Inquiry, 23 (2001), 1-24.

J. Palmer, “Melissus and Parmenides,” Oxford Studies in Ancient Philosophy 26 (2004), 19-54.

G.E.L. Owen, “Zeno and the mathematicians,” in Proceedings of the Aristotelian Society, 58 (1958), 199-222.

Stanford Encyclopedia: Zeno of Elea

Elephants in the forest's picture

An interesting take on zeno's paradox

Luke Cash's picture

Zeno's Dichotomy Paradox

Zeno's Dichotomy Paradox is refuted by modern day philosophy, because a distinction is now made between a potential infinity and an "actual infinity". Al-Ghazali first established this when he, amongst his criticism of Islamic philosophers who believed in a universal understanding of Platonic Forms, used similar logic to refute the idea of an actual infinity.


And I quote:


"What about set theory?

In the other discussion, it was hinted at that in modern set theory the use of actually infinite sets is commonplace. The set of the natural numbers {0,1,2,...} has an actually infinite number of members in it. The number of members in this set is not merely potentially infinite, rather the number of members is actually infinite according to set theory.

But this merely shows that if you adopt certain axioms and rules, then you can talk about actually infinite collections in a consistent way without contradicting yourself. All it does is shows how to set up a certain universe of discourse for talking consistently about actual infinities. But it does nothing to show that such mathematical entities really exist or that an actually infinite number of things can really exist.

This isn't a claim an actually infinite number of things involves a logical contradiction but that it is really impossible. For example, the claim that something came into existence from nothing isn't logically contradictory, but nonetheless it is really impossible.

The absurdities of an actual infinity

First, let's define what absurd means here:

absurd - utterly or obviously senseless, illogical, or untrue; contrary to all reason or common sense

So when we say it results in an absurdity, we don't mean to imply that it's merely "baffling", or that it is "misunderstood" or that it is contrary to our knowledge. But rather, it is because we do understand the concept of actual infinity and the implications of it existing in actuality, that such examples cannot be true (thus, absurd).

German mathematician David Hilbert used the following illustration to show why an actual infinity is impossible. It's called "Hilbert's Hotel".

Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.

Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel hasinfinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.

It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room n to room 2n, and all the odd-numbered rooms will be free for the new guests.

This of course, results in the hotel being always able to accommodate guests, even though all the rooms were full when the guests arrived. The sign outside the hotel could read: "No Vacancy (Guests Welcome)".

It gets even more absurd. What happens if some of the guests start to check out? Suppose all the guests in the odd numbered rooms check out. In this case, an infinite number of people has left...and there are just as many guests who have remained behind. And yet...there are no fewer people in the hotel! The number is just infinite. The manager decides that having a hotel 1/2 full is bad for business. This isn't a problem with an actual infinity. By moving the guests as before, only in reverse order, he converts the half-empty hotel into one that is full! Seems like a simple way to keep doing business (in this absurd reality)...but not necessarily.

What happens if guests 4, 5, 6, etc... check out? In a single moment the hotel is reduced to a mere 3 guests (1, 2 and 3). The infinite just converted to finitude. Yet, it is the case that the same # of guests checked out this time as when all the guests in the odd-numbered rooms checked out. Hilbert's Hotel is absurd. It is impossible in actuality."

Peter Adamson's picture


Without getting into Hilbert's Hotel which is above my pay grade, I just wanted to note that Aristotle is actually the originator of the actual/potential infinity distinction. He basically allows potential infinities in various contexts, but doesn't allow actual infinity in any context. (And this is indeed the core of his response to Zeno.) He's followed in that by most ancient and medieval thinkers.

Al-Ghazali was joining an ongoing debate about the eternity of the world, part of which was the question of whether a world that has already existed for an eternity would somehow involve actual infinity. For instance al-Ghazali's predecessor in the Islamic tradition, al-Kindi, argued that the world is not eternal precisely on the basis that an actually infinite number of moments would already have had to elapse. In general I'd say that the pro-eternity camp felt obligated to insist that an already eternal world would involve only potential infinity. We'll get to this in due course!

Not sure what you mean about al-Ghazali objecting to Platonic Forms, though?

Luke's picture


Right, that comment about al-Ghazali originating that idea was definitely from left field, and I didn't mean that. I was thinking of al-Ghazali in that he had some interesting things to say about the subject, and he (IMO) did the best job of defining "potential" and "actual" infinities when it comes to medieval philosophers, using those ideas to refute a static, eternal backwards and forwards, creation.

He also wrote about Greek philosophy and his issues with it, if I remember correctly. I'll try to get back to you soon with those works.

Peter Adamson's picture

Ghazali and infinity

Dear Luke,

Interesting -- well, I think Averroes would be very unhappy with your praise of Ghazali because he complains that Ghazali fails precisely to make this distinction between actual and potential infinity (which is Aristotelian). Philoponus is the most acute eternity opponent here, I think, because he argues explicitly that past infinite time would be an _actual_ infinity. Ghazali thinks that too I suppose but he is less clear, the point emerges best when he draws an analogy between infinite time and infinite spatial extent.

Needless to say this will all be covered with care in future episodes...


Luke's picture

re: Ghazali

I'll have to look into that. I'd certainly trust your opinion better than mine at this rate. Thanks for your time.

Luke Cash's picture

Also on Zeno's work

The arrow at rest actually seems reminiscient of Newtonian physics. Do you think there would be any substance to that comparison? I think it would have substance if Zeno was thinking of the arrow in the sense that it was being propelled, or a Greek might have termed it compelled, to go in the velocity it went.

Peter Adamson's picture

The arrow

This has indeed occasioned a good deal of comment about the arrow. One way of thinking about it might be that Zeno is precisely not anticipating Newton, because he is thinking of an arrow in mid-flight "at a moment" as being simply at rest, which leaves out the idea of impetus. There is an interesting anticipation of theories of momentum or impetus in the late ancient thinker John Philoponus... stay tuned for episode 93 or so.

Felix's picture


Here is an interesting resource on paradoxes from University of Notre Dame.

The link goes to the page about Zeno:


Brandon's picture

Do we truly understand the Eleatics position on the infinite?

When I imagine Zeno and Parmenidies giving discourse on "Oneness" and the paradox of movement, it strikes me that these men held onto these doctrines, and found them to hold keen to truth. It brings the thought to mind: Did the Eleatics imagine things in being were possessive of the infinite?

It seems as though they did, for by stating that "All is one", and that there is no such thing as non-being, everything that we seem to know and surrounds us must be eternal (infinite) in nature and made up of things that have no beginning or end. A constant state of being.

Upholding the paradox of movement, Zeno undoubtedly moved around each day. So what was at the core of his belief, to allow him to hold true to the paradox, though he defied it at every moment?

Enter the Eleatic perception of oneness, of infinity: if all is one, nothing is short of oneness, making us, and everything else, infinite and eternal. Thus, by being, we are eternal and infinite.

Lets go from there, and look at the dichotomy paradox. If we must be in contact with an infinite number of things on our path from the baseline to the service line on a tennis court, we'd struggle as a finite being, with not enough "Time" to reach all those points. However, does this not change once we perceive ourselves possessed of oneness, being infinite ourselves? It is but a trivial matter to reach all the points, because they are not separate from us: we are in contact and a part of everything else. We can traverse the distance because it is a part of us, a part of the oneness that pervades all things. The reduction (1/2, 1/4, 1/8, etc) always leads back to the whole, and this must have been at the core of their philosophy. I may be 1 out of 7 billion people, but we're all people, and people may be one of a certain species, etc etc until we reach the most elementary piece from which all things have their origin - the oneness that Parmenides and Zeno advocated for.

I close with a question: how close can we really get to these minds, thousands of years later, pervaded with centuries of philosophy and thought, influenced by modern science and mathematics? Can we experience as they did the intoxication of their knowledge and profound reasoning? And what steps were there beyond the writings we have? Ah, so much dashing about the shadows of history, with only morsels to sate ourselves with! But the quest through darkness, unearthing light, never loses its appeal through millennia - let us continue to add to the store of treasures to be found!

Peter Adamson's picture

The Eleatics

Hi there,

Well I certainly share your eloquently phrased worry in the last paragraph. Particularly worrying for me given that I do this for a living. But I think the goal has to be to read them as sympathetically as possible and try to come up with an interpretation that makes sense given their own philosophical concerns, insofar as we can understand them - it's crucial to figure out what sorts of pressure (philosophical or otherwise) these long-dead people were responding to with their theories.

Along these lines I would also like to agree with your point about Zeno: surely he knew he was moving around all the time? So there is a deep problem here about the Eleatics and what they would say about the deliverances of everyday experience. Are these just an illusion? Or perhaps a less-than-fully-adequate understanding of reality? Remember that Parmenides also wrote his Way of Opinion which accepts multiplicity, motion, etc. So a good answer to this question is basically a plausible interpretation of the Poem and, in particular, why it includes a Way of Opinion, not just a Way of Truth.

Thanks for listening!


Denziloe's picture

Infinity in modern mathematics

Your presentation of modern mathematics' treatment of Zeno's paradoxes, which was basically that mathematics just asserts finite answers, is really inaccurate and misleading. As far back as Ancient Greek mathematicians like Eudoxus and Archimedes, and certainly after modern developments which started in the 1800s, mathematicians have done a lot of work analysing these matters, culminating in a substantive and rigorous body of work called "Analysis". This provides real answers to questions about the nature of infinite series in mathematics and physics, definitely not just a set of assertions.

Peter Adamson's picture


I'm not sure what exact phrase you're picking up on here (this episode was a long time ago!) but I don't recall accusing mathematicians, ancient or modern, of merely asserting anything. I think I just said that with modern mathematics one can easily model infinite series that look like Zeno's paradox e.g. 1/2 +1/4 + 1/8... and there is no problem with seeing such a series as adding up to 1. Then I think I might have added that there is still a question as to whether that mathematical model in fact corresponds to what is happening in physical reality, like in space and time - and obviously that is not a question that mathematics by itself can (or needs to) answer.

Denziloe's picture


It's around 8.20 where you're talking about a mathematical approach, and you say,

"we now have no problem that 1/2 + 1/4 + ... just adds up to one; in fact, we might say that the number represented by this series just IS 1".

Well, anybody who did say this would be wrong to do so. This excerpt sounded like you were essentially presenting the aforementioned fact as an axiomatic or close-to-axiomatic fact of mathematics, which nowadays is very far from the case - it can be given a substantive deduction from some very conservative axioms of logic and sets. And I don't think the wider context provided any further clarification.

And with respects to that wider context of models, I think this becomes a kind of important point... because the fact that the deduction IS substantive means there's really no reason to think of it as a tautological question of justifying an abstract model; rather, it reduces to justifying the much more fundamental and evident axioms. Once these are granted, the resolution to Zeno's problems just becomes a matter of logical consequence.

I suppose the wider point I'm making is that I felt like that section presented the mathematical work as unequivocally independent of the arguments, when the mathematics is actually substantive and of real consequence to the philosophical and physical questions... in fact I'd go so far as to say that mathematical and philosophical arguments are essentially the same thing in this case. Zeno was, after all, in the mere act of talking about these matters, adopting a whole bunch of tacit axioms about space and time, and using standard logical inferences. All mathematics does is state what these are in a formal language... which has the advantage of throwing the wobblier aspects of his discourse into sharp relief.

But of course, you have a finite time in which to distil an infinite number of arguments and counterarguments, so I suppose truncations are, regrettably, inevitable.

Peter Adamson's picture

Math again

Right, I was definitely sketching the mathematical solution there rather than really getting into it. In fact (though again I don't really remember, since I wrote the script years ago now) I was probably trying to evade any commitment by saying "we might say that..." The point you make is an interesting one. Zeno himself was almost certainly adopting a strategy like the one you suggest here: consider various assumptions about space and time, and show that on any of the assumptions motion is impossible. Here it is important that the dichotomy is only one of an array of paradoxes, and was probably meant to be complemented by others.

Anyway I would certainly agree with your basic point, which is that modern mathematicians would trace these points about infinite series to more fundamental axioms. I do still think that there is a question about whether the mathematical modeling of a motion (however we justify the model) is going to correspond to what is really happening in physical reality. There is a basic question there about philosophy of mathematics and science, and philosophers have taken various views on it, e.g. that mathematics is only instrumentally useful for doing physics. But I wasn't, of course, trying to get very far into those issues in this podcast, just to explain that Zeno's paradox is more challenging than one might think, even in light of subsequent developments in mathematics.

Denziloe's picture

Math again

Indeed, I don't think there is any way he could avoid such a strategy. In simply using the word "space", he is communicating to us a concept with a bunch of properties; if this weren't the case and he weren't asserting any properties at all for the object at hand, then the word wouldn't refer to anything - and then we might as well conceptualise "space" as referring to giraffes - and then find the rest of the argument nonsensical. So, whilst the issue of whether various axioms behind a "mathematical" argument veritably describe motion is definitely something that needs to be considered, that's not a problem about mathematics per se; the same exact issue is the case with the tacit properties behind Zeno's argument. And so when (speaking rhetorically of course) you said, "to a mathematical resolution, Zeno would ask you why your model is correct", my first response to Zeno would be, "but you haven't even tried to specify what you take the properties of space to be - so what is YOUR model, and then why is THAT correct?".

P.S. thanks for all the food for thought Peter. It's hard to make critical objections without sounding negative; so let me say this podcast is truly fantastic. I tried a couple of other philosophy podcasts recently but they absolutely pale in comparison... in fact this applies to a bunch of books I've tried too, including Russell's famous one. Masterful from the very beginning.

Peter Adamson's picture

More Zeno

Thanks very much! Regarding Zeno, remember that he doesn't (at least in theory) even believe there is such a thing as motion or "space". Rather the dialectical situation is that he is implicitly assuming what is, for him, a false premise which is that space is extended and is infinitely divisible. If someone rejected that and said that space had other properties, e.g. is not infinitely divisible, he could and in fact did offer different paradoxes aiming at this rival assumption. In other words Zeno doesn't want to make any particular claim about the nature of motion or space, rather he wants to show that any non-Eleatic theory (any theory that makes motion possible) will have to make some such assumptions, which will lead to a contradiction/paradox. Does that make sense?

Asef's picture

Can time really pass?

I'm thinking about a paradox, which may be parallel to Dichotomy paradox.
We can divide a time period (say one hour) infinitely many times. So if one hour is going to pass, first half an hour should pass and so on.
I'm wondering why Eleatics did not conclude that time does not pass at all. Maybe because it was not among Parmenides' teachings?
Or maybe they did but I have not heard of it?

Peter Adamson's picture

Eleatics on time

There is no Zeno argument quite like the one you are describing, though the Arrow in particular looks like it is raising questions about time. However Parmenides' Poem itself does say "it was not and will not be, but is," which has often been taken to be a denial of time's applying to Being - sometimes people credit him with the notion of timeless eternity though others find that a bit much to read into the passage. It could for instance just mean that Being does not change (in other words, it is what it is now and was never different, and will never be different - this is bound up with the infamous problem about whether the verb "to be" is being used by Parmenides existentially or as implying a predicate, like "to be/exist" vs. "to be blue").