Posted on
16 January 2011

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

Themes:

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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.

## Comments

## An interesting take on zeno's paradox

http://drawingboardcomic.com/index.php?comic=45

## 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."

## Infinities

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?

## re:Infinity

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.

## 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...

Peter

## 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.

## 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.

## 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

notanticipating 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.## Paradoxes

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

The link goes to the page about Zeno:

http://ocw.nd.edu/philosophy/paradoxes/eduCommons/philosophy/paradoxes/…

## 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!

## 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!

Peter

## Way of knowledge

Great work Peter! I am listening eagerly, with no background in philosophy, (and english is a second language). I experience the same pull as the person writing the previous post, which is to listen to these thinkers whith as much empathy as I can. The presocratics' schools, make me wonder if their search for an explanation of reality, was supposed to lead them to a fundamental experience. You say at one point, (paraphrasing), that "...their approach is a conceptual analysis rather than a search for empirical evidence". But is it possible that they were looking for an empirical "universal experience of beingness" through their reflexions? I studied the Bön/buddhist tradition, and a lot of what they are doing, ressemble what Bön practionners call "exhausion of the mind" exercises. The Koan version in Zen has that same function I believe. It is meant to bring you to the edge of conceptual analysis, (similar to Zeno paradoxes), to liberate you into pure beingness. This state comes from relasing the tension after creating an extreme investigation of the paradoxical nature of life, to get to its outer limits, to the "Unbounded Wholeness" that Bön mystics talk about (very similar to Anaximander). Is it possible that the writings of these philosophers had an adjoined "meditative" praxis in reality, that were maybe the real secret of the schools, and were transmitted directly form teacher to disciple? It might be that limiting the analysis and critic of these thinkers to the logical aspect of their arguments, makes us "literalists/reductionists", and that we are missing the real function of their work, which would be to teach us to get to an actual experience, (although it does not contradict more thought processes if required to reach the state). In the same line of reasoning (which is far fetched and probably difficult to prove), the real function of their thought processes, might have been to do away with them all together... I know it is probably a very biased and historically unsubstantiated approach, but there was so much (and still is), of that clear delineation between theory and practice in ancient cultures, and the importance of direct, but very often secret transmission of knowledge. I am tempted to contend, that through time and within the constraint of our era, we are all searching for a way to end the divisive nature of our mind, to reach an experience of lasting peace, and what best way to achieve this than do it than through the mental temporary breakdown we experience when confronted with

unsolvable paradoxes? Best to you and thank you for your great work.

## Meditative praxis

Thanks for your comment! That's an interesting idea and I would go with it to a certain extent: especially the Eleatics (Parmenides, Zeno, and Melissus) were in some sense trying to show the boundaries of what can be thought, or at least the falsehood of thinking as we normally do it. Of course we can't know what went on in terms of practices or teacher-student training in this tradition, as we have only the texts, or rather not even those but only fragments and reports. However I have to say that if your reading is anywhere near the truth, then Plato and Aristotle were staggeringly far off the mark in their reports of what the Presocratics were up to, since they present the Presocratics basically as cosmologists. In that sense I think what you're suggesting is not only unsupported by the evidence we have, but contradicted by that evidence.

By the way the point about using conceptual analysis and not empirical inquiry was only supposed to be about the Eleatics, I don't think it necessarily applies to, say, Heraclitus or Anaxagoras, though it would fit for the Atomists quite well.

## 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.

## infinity

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.

## infinity

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.

## 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.

## 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.

## 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?

## Can time really pass?

Hi,

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?

## 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").

## Paradox of travelling a distance (infinite series): Zeno

If there are infinte mid distances in between a finite distance then is also infinite half-time avialable within the finite time.

Does that solve that paradox, doc ?

## Zeno

Maybe I don't follow what you're suggesting, but I think that just IS the paradox: that a finite distance contains an infinity of parts, which can't all be traversed.

## Paradox of travelling a distance (infinite series): Zeno

I got that. What I was exphasizing that just like there is an infinite series of half-distance within the given distance (that has to be traversed), there is smilarly an infinite series of "half-time" within the estimated time to cover that distance. For example if that particular distance (lets say 20 m) is covered in 2 min, then half of that is covered in 1 min, 5 m with will be covered in 30 sec, 2.5 m will be covered in 15 sec and so forth. Which means as the distance can be broken into intifinte halves similarly time can be broken into infinite halves. So infinite steps were taken in simlar infinte moments of time and those infinite moments of time made it possible to traverse those infinite series of half distances. I hope I was able to explain this. English is not my mother tongue actually.

## Half-times

Oh, I see. Your solution is in fact exactly what Aristotle thinks, and how he responds to Zeno. I probably discuss that in episode 40.

## Motion is impossible

I don't see it as rediculous for Parmenides and his followers to think that motion is impossible. They obviously are not referring to the idea of motion that modern people would think based on Newton's Laws and modern physics. I can understand how the Eleatics would see me walking down the street for example as not moving. Obviously, I am moving, but that is only with respect to some other observer: the street, the sun, someone watching. If two people are moving at the same speed relative to each other, then they seem to not be moving at all to each other. I think the Eleatic notion that moving/motion is impossible is given credence due to this notion of relativity.

Though Melissus wouldn't like it, if we were to think of the sphere of reality as a clear, solid, glass ball, it would look to us that not only is the ball not moving, but nothing inside of it is moving either. However, that is not the case. From the perspective of an atom inside the ball, there is a lot of motion happening, with electrons flying around and everything shaking like crazy. If we go even smaller to the perspective of a proton on an atom in the ball, its vibrating along with all the protons and neutrons around it and can see streaks of electron flying by every so often. If we tell the proton that moving is impossible, it would just laugh because obviously it is impossible (no, laughing protons are not impossible). However, looking at the ball in its totality (oneness), we see it as not moving at all. By analogy, I think this is the perspective that Parmenides and the Eleatics had of reality. It is pretty convincing, and I can see why it lasted as long as it did in the history of philosophy. I like Aristotle's view that as one thing moves, it allows another to take its place, similar to what goes on at the quantum level in the glass ball.

There is still the whole problem of the "outside" which would imply non-being, but despite that, I think it problematic to apply the modern conception of motion (which is packed with unique meaning given the last 4 hundred years of physics) with the presocratic. I would be interested to see the original work of Parmenides and other Eleadics in the Greek to see if that would shed light on what seems obviously wrong in Eleadic thought from the modern perspective.

## Relativity and Eleaticism

That's a nice thought but it doesn't, I would say, capture what the Eleatics were claiming. They are apparently saying not just that existence as a whole doesn't move, as with your glass ball example, but that it contains no internal motion either. In fact there is no multiplicity at all in their metaphysics. So we can't ascribe to them the view that, for instance, two things are unmoving relative to one another, or that two things are moving within an unmoving larger whole: rather, there are never two things at all, only one thing, namely Being. Still I like your instinct to try to find a way to make it make sense!

## If what you say is true, then

If what you say is true, then I find it very difficult to see the world from the perspective of an Eleatic given my experience of reality. If I am going to criticize their metaphysic, I want to do it on their terms. Obviously there is a multiplicity; I am me and my computer is not me. However, the idea of a computer cannot exist independently from anything else in existence. One could argue that my computer and I being different is just a useful categorical tool used by humans in order to make sense of the world. Meaning, there is no real separation/difference/multiplicity. Everything simply is. In Eleatic terms, there is only one thing, Being.

I find it hard to believe that if I were to ask an Eleatic if he and I were different or if he and I together made two people that he would say no. Sure, ultimately there is no separation and all is one. I can get behind that. It may be the case that everything simply is and that multiplicity is an illusion, but in order to live a human life, I would argue that one (including the Eleatics) must give into the illusion at least a little bit. My experience tells me that there is separation, even if it's an illusion.

If the Eleatics were seriously telling people that they weren't moving when they were clearly walking or that there was no difference or separation between themselves and a tree, it seems to me that they were simply crazy, or they were caught up in an abstract argument that seemed to defy their immediate experience, and instead of trying to figure out the flaws in the abstract argument, they just decided that all experience is an illusion.

I think what I am trying to say is that based on my understanding of the Eleatic worldview, one cannot prove that it is correct or not in the same way that I cannot prove that I actually exist or not.

It could be that I'm misunderstanding their worldview, but I feel as if understanding their worldview from their perspective can't lead to the conclusion that they were simply dumb because they thought motion was impossible when it clearly is. It had to have made sense from their perspective.

## Silly Eleatics

Well, basically what you are implying there is that there cannot have been any monists (people who deny the reality of multiplicity) in the history of philosophy, because monism is obviously false. But in fact monism appears recurrently, and apparently independently, in the history of philosophy: for instance in Advaita Vedanta, and also in a different way in Spinoza. The Eleatics do seem to have been monists, and in fact to have said exactly what you are saying is so unbelievable: remember that Parmenides also wrote a "way of opinion" in addition to the monist "way of truth", in which he basically makes the concession of giving a theory of reality that coheres with how it _seems_ to be. An interesting question is why we have the way of truth and the way of opinion; but it seems pretty clear that the way of truth is privileged and that it does involve denying multiplicity.

More generally, lots of philosophers in history have had radically revisionary theories of reality: everything is an illusion, only immaterial things exist, moral judgments do not have any basis in reality, we cannot know anything for sure, etc. etc. I think one should approach these theories "charitably" in the sense of trying to understand what reasoning drove them to such radical conclusions but not in the sense of deciding they can't possibly be serious in putting forward their revisionary theories, because we find them so counterintuitive.

## Thanks!

Okay, I think I get it now. If I was in a room with an Eleatic, and he was explaining to me his monist theory, my critique of him would be that: the very fact that he is explaining his theory to me implies a contradiction because he would have to accept that he and I were different to even have the conversation. His responce to me would be something like, "No it is not a contradiction. Everything is simply one. These ideas of 'you,' 'I,' and 'conversation' are illusions. Therefore, there is no multiplicity, no motion, no duality." Is that more or less accurate?

Thank you, by the way, for your work on this podcast. I am really enjoying it so far. Hopefully I will be able to catch up soon, as I am only on episode 15. I'm especially excited to listen to the Islamic World talks, as I have almost no background in the History of Islamic thought. I wish you the best as you continue with this massive project.

## Talking to Eleatics

Yes, that is exactly the idea. In fact the response you suggest, that merely the possibility of putting forward the Eleatic to another person shows that it is wrong, is reminiscent of some ancient critiques of radical philosophical positions. For instance Aristotle argues that no one can assert that the law of non contradiction is false and Plato in several places points out that there are theories whose assertion is self-undermining. I guess though that, as you say, the Eleatic would happily admit that the appearance of having a conversation with him is just another illusion.

Hope you enjoy the rest of the series!

## Paradoxes

I have an issue with the contemporary view of Zeno's paradox as not giving the people of the era the credit they deserve in terms of intellectual capacity, and just is not very satisfying. However, when i read it, instead of a paradox of motion, but a paradox of discreet space, it makes more sense. I understand that discrete vs continuous was a debate at this time in mathimatics as well, which would further that viewpoint. That instead of proving motion to be impossible, it proves that motion through a set of discreet points requires an infinite number of them, so discreet motion is impossible. Thus motion must be expressed as continous. This would then draw into question, if the path from A to B is continous, not a sum of discreet points, then how can we even say the fixed points of A and B exist? This would also feed into Parmenides's theories then, because A and B would be part of a continous whole. The arrow in motion, but at a fixed point, would also then line up with this view instead of just motion. So I ask, do we know for sure that Zeno's paradox is about the impossibility of motion? Or do we just know the paradox, and infered it to be about motion?

## Is the paradox about motion?

Well, perhaps we don't know anything about it "for sure" since we rely on later testimonies but Aristotle's presentation of the Dichotomy is certainly in terms of moving across an extended space. Remember too that motion is involved in some of the other paradoxes like the Arrow and Moving Rows. But you may be right in the sense that Zeno could be using motion to critique the idea of discrete space; the ultimate target is a matter for debate. However the Eleatic background, to my mind, makes motion a more likely target than space. After all Parmenides' Being is spatial (it is a sphere), but unchanging/unmoving.

## If things are many, then things are infinite

Hi, Peter!

Many philosophers and mathematicians claim that, when Zeno says that if a distance is infinitely divisible, then it's infinitely large, his mistake lies in thinking that the sum of of infinite series is infinite, which is not the case. I think it was Vlastos who said that the only way for the sum of an infinite series to be infinite is if it has a smallest member. Why do you think he says that?

## I just found the text

I just found the text containing the Vlastos argument I was talking about: "There must have been some tacit assumption which would have made it seem obviously true that any collection of an infinite number of sizeable parts would have to be infinitely large: so very obviously. that even someone who knew all about Aristotle's theorem (as Simplicius certainly did, and some of Epicurus' associates almost as certainly) would not think of applying it to the present case, but infer forthwith inflnity of size for the container from infinity of number of the parts contained. I cannot imagine what this could be except that the collection had a smallest member. This would be quite sufficient to make the conclusion seem a matter of course: given an infinity of nonoverlapping parts the least of which has some finite magnitude, it would be obvious that the aggregate magnitude would be infinite." Try as I might, I fail to see how having a smallest member could make the sum infinite. If ½ + ¼ + 1/8 + 1/16 + 1/32 + 1/64…. (no smallest term) converges on 1, why would the sum of the members of this geometric series be infinite if, say, 1/64 were the smallest member?

## Vlastos on Zeno

Well, that probably seems obvious to you because you learned math in high school after, and not before, the invention of calculus. The notion that an infinite series could add up to a finite result is in fact staggeringly counterintutive, even though it is true: somehow, you add an infinite number of lengths of positive quantity and the result is only, say, one meter long in total. Aristotle is actually the first person to make the point, as far as I know - that this is possible because the amounts being taken diminish in size as you go. However even he doesn't think this is possible with an actual series, only that you can take arbitrarily small portions by dividing smaller and smaller, but always with a finite and not infinite number of parts. He would have agreed with Zeno and think that your "obvious truth" (namely that an infinite series of actual parts of positive size would yield a finite result) is obviously false.

One relevant point here may be that for the Greeks such quantities would not be represented with the kind of notation you used, but with the quantities as line segments, because they tended to think of numbers geometrically. So you're asking them to accept that you put an infinite number of line segments next to one another, each of which has a positive length, but the whole thing is finite. Again that may be true but it is far from intuitive.

Alternatively could it be that you and Vlastos are thinking about different paradoxes by Zeno? Maybe he is not talking about the dichotomy (travel half of the path, half of the half, etc) but another argument given by Zeno which is about adjacent bodies.

## Thank you, Peter. I now see

Thank you, Peter. I now see where I made the mistake: Vlastos is indeed talking of another paradox—the one where he says that, if things are many, then they are both infinitely small and infinitely large. I realize now that how you make the divisions matters. In the Dichotomy and the Achilles, you first divide the distance in half, then either the first or second resulting halves in half, then the quarters, ad infinitum. The result is that you never get from A to B, or that you can never leave A at all. In this paradox, the dichotomizing is done in such a way that the result is equal parts, instead of the parts represented by the notation I used (geometric series), and now I understand why Vlastos says what he says: if there is a smallest part, and it has magnitude, the sum of all members will be infinite! Thank you very, very much for helping me see this. Studying philosophy on your own is a, as they say, fraught with peril. Having the help of a scholar like you makes all the difference.

## Smallest parts

Ah, good. Glad this cleared it up!

## Praise

This site/project/enterprise is magnificent. I cannot praise it highly enough. It makes the rubbish strewn underground car park of the internet worthwhile. Please carry it on and bring it up to the present day.

## Rubbish strewn car park

Ha! Thanks, that is one of the nicer compliments I have ever gotten on the podcast. I should put that in big letters at the top of the website.

Anyway glad you're enjoying it - as for bringing it up to the present day, as I always say I have no plans to stop anytime soon, so let's see how far I can get.

## Emptiness in our modern conception of the world m

Hi Peter,

Great work of yours. I am listening to it so voraciously. Thank you for this great encyclopedia of philosophy you made.

I would just like to precise about modern concepts of vacuum. Today, we have no vacuum at all. I mean, there is energy in vacuum and a principle of uncertainty developped by Heizenberg states that particules can appear and disappear at anytime in the vacuum. Plus, the particules are not corpuscles, they are kind of waves as well and therefore they do “occupy” the full spaces of atoms and molecules. Complex to explain it, but I would just point out then that emptiness is not very acceptable in today’s physics.

## Void

Yes, I guess that must be right - my knowledge of modern physics is not what it should be! But I guess perhaps the real question, philosophically, is whether empty space is conceptually permitted in our physics or not, and I guess that in modern physics, it is?

## math and reality

I thought that was a very good point that modern math, specifically calculus, does not answer Zeno's paradox. It is a model for reality only and not necessarily reality itself. But doesn't a mathematical model for what Zeno claims is impossible at least show that it is non-contradictory? In fact, the question of the connection between math and reality is still an open question. As I understand it, general relative implies that space is quantized, that is, that there is a shortest possible distance between two points. Space is not a continuum, in other words. I am not a good physicist so I may be misunderstanding them.

## Math

Well, I think if you agree that there is an open question as to whether a mathematical model captures the physical reality, it is also an open question whether the consistency of the mathematical model shows that motion is non-contradictory (to put it another way, if the claimed inconsistency turns on our concept of physical motion or space, then math has nothing to say about it... arguably). Not sure what modern physicists would say but at least, if this line of defense for Zeno is a good one, the physicist would be the right person to show why he was wrong and not the mathematician.

## models vs reality

This seems a lot more complicated to me and not so clear. If Zeno is trying to prove that motion is inherently self-contradictory, then the existence of a mathematical model for motion refutes him. The question of the relation of a mathematical model to reality is one thing, but is it really tenable that something self-contradictory could be modeled by something logically consistent? Put another way, can self-contradiction ever be "approximately" consistent? This is really a question about the nature of logic and its relation to reality. It seems to me that if something self-contradictory can be disguised as logical consistency then we are losing our grip on the validity of logic as a whole. Or am I missing something?

## Physics vs math

I agree this is complicated and difficult. But my basic intuition here is that if we agree it is an open question whether the mathematical model represents the physical situation accurately and shows why motion is possible (if it in fact does: consider that the model involves approaching a limit getting infinitely closer as the time gets infinitely closer to the end moment of the motion... which sounds more like Zeno explaining why motion is impossible than an explanation of why it

ispossible) then the consistency of the mathematical model surely doesn't prove that a belief in motion is consistent. Zeno can say, "sure, the model is consistent, but it doesn't represent what is (supposedly) happening in the physical world: indeed it can't because motion, as I've shown, implies inconsistencies, so that alone proves that the model doesn't represent it because how can a consistent model represent an inconsistent physical scenario?" It would be worth thinking about whether this is a question begging response though.## let's turn it around

Perhaps I just need to digest this for a while to see what I am missing. But let's try one more time. Zeno attempts to prove that motion is self-contradictory by a thought experiment. Newton comes along and takes the same assumptions as Zeno and replies, "There is no inherent self-contradiction here." Newton hasn't proven that reality corresponds to the calculus model, with infinitesimals and what not, but it does seem that he has refuted Zeno's claim that motion is inconsistent. Zeno could still be correct that motion is illusory, but Zeno would be incorrect as to why it is known to be illusory, as to his proof. Zeno would need to start from different assumptions if he still wanted to make his case. I am sorry if I am getting tedious and just repeating myself. I am new to the philosophy business. I promise I will refrain from belaboring the point any further if if does strike you that I am just rehashing the same point here.

## Contradictions

I don't think that's quite right. Take any (supposed) phenomenon you like, call it P. Zeno says "I have an argument to show that P is contradictory" and Newton says, "I have a way of thinking about P that doesn't seem to involve a contradiction." Unless Newton shows exactly where Zeno's argument went wrong - that either the argument is invalid or has false premises - then he hasn't defused the argument, he's only given you a model for thinking about it that does not contradict itself.

To put it another way, positively offering a story about how something could work is not the same as refuting a negative argument that it can't work. To do that you have to address the argument directly, and the point we've been discussing is that you can't address Zeno's argument just with mathematics since his argument isn't purely mathematical.

See what I mean?

## Zeno's Paradox

Of course, both of this is assuming that Zeno whoever Zeno was trying to show was inconsistent (probably Pythagoreans) would have considered and accepted the notion that math and the real world don't correspond. Given that the models suggest a mathematical atomism (continuity is the sum of points), and that motion is a real-world phenomenon, I strongly doubt that Zeno's opponent would have granted such as suggestion. So, if the position Zeno is responding to will remain intact, it needs to respond to both.

However, that still leaves the possibility that Newton solved half the problem, though prior to calculus. However, this isn’t clear either. For one, Pythagorean atomism, which suggests that a continuum is a unity of points still has a couple of problems. After this, the doubt in mathematics wasn’t the possibility of continuum based mathematics (such as adding two line segments, or multiplying two line segments into a rectangle), but rather that a continuum was an infinite collection of points. Newton in “De Analysi Per Aequationes Numero Terminorum Infinitas” was able to show that math through infinite series of discrete numbers could be as soundly proven as algebra and geometry, but that doesn’t quite show that an infinite discrete sum is a value rather than approaches a value, prior to the operations that cancel out the infinite scale [likewise, the meaning of the sum of 1-2+4-8+16-32…..=1/3 is still debated]. We can see this debate continue in philosophy from the example given of the light switch (I don’t remember who), where we consider a light that is turned off and on each time Achilles reaches where the turtle was, will the light be on or off when Achilles finally reaches the turtle? Or will Achilles have to break the pattern of running only to where the turtle was?

Meanwhile, in the reality side, though my physics is worse, I believe at least in many circles, it is accepted that reality is discrete, not continuous. Zeno had his own response to this, which I can sum up as, let ABCD be a body that exists at point 0,1,2,3. Let EFGH be a body that exists at point 1,2,3,4. Now let both of them travel at the slowest speed possible towards one another. After one unit of time, A is now at 1, and E is now at 0. However, this means that A has travelled 2 spaces with respect to E, and essentially “jumped over it”, which implies a slower speed. This paradox, however, has been more adequately addressed. In the physical world, things can in fact move slower, because they exist in a probabilistic state. If you consider A as existing in a probabilistic cloud, it can still move slower than any theoretical slowest, by shifting its probability less than one plank length. This will result in the average expected location moving less than 1 plank length, even though it would be impossible for something to actually move less than 1 plank length. All you need mathematically to address that paradox is probability theory, and this actually addresses both reality and mathematics, rather than just solving it on the mathematical side.

I hope either of you found this helpful! (and that i hope i didn't make any mistakes!)

## Zeno's Paradox

I am not sure I do understand you, and I will clearly have to consider your response and this whole question a lot more. I really appreciate you taking time to help me out, though.

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