Episode 244: Zeno's Paradoxes - Profundity Or Gaslighting?

Transcript of Episode 244. Note: The following transcription was prepared with speech to text software, and has been only lightly edited for accuracy. This file has been prepared primarily as an aid in searching for material in the audio episode. This transcript likely contains errors, and should not be relied on for anything other than searching for discussions relevant to particular topics. Please consult the original audio episode for accurate information. If you come across egregious errors in the transcript below, please let us know in the Epicureanfriends.com forum!

This is a podcast dedicated to the poet Lucretius, who wrote On the Nature of Things, the most complete presentation of Epicurean philosophy left to us from the ancient world. Each week, we walk you through the Epicurean texts and discuss how Epicurean philosophy can apply to you today. If you find the Epicurean worldview attractive, we invite you to join us in the study of Epicurus at EpicureanFriends.com, where we have a discussion thread for this and each of our podcast episodes.

We're continuing in Cicero's On the Nature of the Gods. We're now in approximately Sections 26 and 27 of Book 1, and we're going to continue this week by following up on a few aspects of last week's episode. In particular, we're going to keep in mind how Cotta, who is speaking against the Epicurean position here, is an Academic skeptic.

Last week, one of the issues we discussed at length was an article by Dr. David Sedley entitled "Epicurean Theories of Knowledge." Dr. Sedley raised in that article Philodemus's On Methods of Inference (or On Signs, depending on how you translate the title of that work). The issue being discussed was starting with the Epicurean reliance on the senses always being true in the sense of being truly reported and moving forward to a form of logical reasoning which emphasizes similarity and analogy. Those things that you're not able to observe directly, you infer how they really are based on things that you do have experience with—either by similarity (in the sense of: all men you have come into contact with in the past are mortal, and so, therefore, it's safe to presume that all men that you will come into contact with will also be mortal) and also by analogy (the example there being that in our level of perception, we see that empty space is necessary for movement to occur, and therefore we infer that at the atomic level there must also be a void if the atoms are going to be able to move themselves). That's not something that we've ever seen any examples of because we can't see atoms, but by analogy, we can reasonably infer that that's the way this operates. That form of reasoning gives us a rational explanation for how things can occur the way they do, which does not require us to suspend judgment or come up with an imaginary god that controls everything or that violates our reliance on the senses as the primary means of knowledge.

In last week's episode, Joshua was referencing Zeno and his paradoxes, and those paradoxes deserve additional attention today because they give us what might be one of the most dramatic contrasts between the perspective that Epicurus was taking and the perspective of other philosophers of his time and prior to his time. We often use the example of Plato and his cave and how Plato argued that the senses are not reliable and that human beings are basically looking at reflections on a wall, and that what we really need to be doing is learning geometry and the other logical arts that Plato suggested as the only way to truly have knowledge of the truth—the view that it's possible to have firm knowledge of the idea of a horse, but that what your eyes are telling you about any particular horse that may be directly in front of you is not to be trusted.

We're familiar with that contrast from our prior discussions, but even more dramatic are those examples suggested by Zeno in his paradoxes—the Achilles example and the different other paradoxes which boil down to this point: that even more radically than Plato, Parmenides and Zeno were willing to argue that motion itself is impossible, that change itself is impossible, and that all the motion we see in our lives, all the change that we see in our lives, is but an illusion. This strikes me as being dramatically worse than Plato's suggestion that what we are seeing are reflections on the wall, which, to some extent, you can see a linkage between that and some form of reality. But Parmenides and Zeno were going to the extent of saying that basically everything we see in front of us, to the most dramatic extreme of motion and change, are illusions. And in such a viewpoint, the senses are obviously going to have to be discarded.

When we compare what they were saying and doing with what Epicurus was suggesting, I think we can really fruitfully see the choices that people have to make when they look at ancient philosophers and decide between the alternatives being offered to them. So let's drop back and review the background and the assertions of the Eleatics.

Joshua: Right, so this conversation branched off of something we were talking about last week. You mentioned just a moment ago this article by David Sedley, and it was one that you quoted extensively in our last episode. It deals with questions related to skepticism, and we're dealing a lot with skepticism in this text. For example, last week, you quoted from Sedley talking about the problem of whether the relative is in conflict with the material. I think I used the transparency of a diamond versus the diamond itself and whether those two things are in conflict.

But really, to get to the heart of skepticism as a problem for Epicurus in the ancient world, probably the best place to start—and where I went next in that conversation last week—was Zeno of Elea, part of this Eleatic school with Parmenides and Zeno of Elea. The challenge they pose to a philosophy in the ancient world, which views sensation as fundamentally reliable, is significant. Zeno and Parmenides viewed sensation as not just unreliable but as totally and completely illusory to the extent that the sensation that objects change place in our environment—that they change place in space—that that itself is an illusion. I brought that up last week, and that has generated, I think, more conversation than the discussion of the text, which was this issue of quasi-body and quasi-blood. So we're going to spend most of today dealing with this on a level that we did not last week. Last week, I just kind of breezed through it, but today we're going to set this up as an interesting challenge to Epicureanism.

We've pulled together for the thread for this episode a number of sources that are interesting on this question, and two of the ones we're going to be dealing with extensively are actually lecture notes from two universities: the University of California, Davis, and the University of Pittsburgh.

And I'll start with the University of Pittsburgh. This text was put together by John D. Norton, and he opens the conversation for us in a way that is helpful. He says:

"Zeno was an associate and student of Parmenides and undertook the work of defending his mentor's views. Parmenides argued that things in the world are an unchanging unity without parts and that all change is impossible. This may at first seem like an abstruse position that requires great philosophical sophistication to understand. It is not. It is an extreme fantastical form of skepticism."

We've been talking quite a lot, Cassius, about skepticism as we've gone through Cicero's text, because both Cicero himself, to some extent, and Cotta, his interlocutor, speaking for him in a sense, take skepticism to an even greater extreme. Cotta will only talk in terms of probabilities—he cannot say what is or what is not; he can only say what is probable.

With Zeno, we have an approach that combines geometry and skepticism to demonstrate, as far as he's concerned, that change is impossible. As John D. Norton indicated there, this is partially because Parmenides and Zeno viewed everything that exists in nature as an unchanging unity. This philosophical view is called monism. Monism is a theory or doctrine that denies the existence of a distinction or duality, and in the ancient world, we see, going all the way back to Thales, that some of these Greek philosophers were trying to find the one single substance that underlies everything that we experience with our senses. Thales thought that he found that substance in water, but other thinkers thought that fire was the one substance of nature, and others thought that earth or air or ether was the one substance of nature.

In viewing everything that exists as a single undifferentiated whole, Parmenides and Zeno further developed this into the idea that any change within this single unified whole becomes impossible. Now, this stands in an interesting contrast, I think, to the Epicurean view, which posits that there are two things that exist in nature at the most foundational level: matter and void. You have indivisible, irreducible atoms—when you go far enough down, you get to the level of the atom, which cannot be divided, cut, or destroyed. Then you have the void, which is the idea of nothingness, but it's more than just nothing—it is space, empty space.

Several weeks ago, Cassius, we had a conversation about the properties of the void in Epicurean physics. If the atoms have three properties—weight, shape, and size—what properties does the void have? The fact that the void creates space for the atoms to move around in is central to understanding Epicurean physics, and it's central, I think, to understanding the difference between Epicureanism and its approach, even before we get into the senses and this kind of proto-empiricism among the Epicureans. This dualism stands in contrast to the monism of Zeno and Parmenides, and this is going to be at the heart of everything we're going to be talking about today.

Cassius: Let me say a couple of things before you proceed, Joshua, about monism and the issue of the void. That's one of the many clear examples of the difference in approach that Parmenides and Zeno were taking. They argued that void and empty space do not exist, and therefore everything is close-packed. When you think about the basis for their reasoning, it certainly wasn't that they observed everything to be close-packed and not moving, because they observed the opposite in our world—they observed things not being close-packed and things moving and changing all the time. But they were taking the position that "that which does not exist does not exist."

For example, one quote we have from Parmenides is:

"Whatever can be spoken of or thought of necessarily is, since it is possible for it to be. But it is not possible for nothing to be, for never shall this be proved that there are things that are not."

Through using verbal and word analysis, defining empty space as something that does not exist, they reasoned that "that which does not exist does not exist," and so therefore, there is no empty space. Now, you're probably better equipped to explain that than my feeble effort there, Joshua, but the point of it is that they were following the path of thinking to themselves that what they speak of or think of can create its own reality—that words are sufficient to conclude that empty space does not exist because "that which is not cannot be, for never shall this be proved that there be things that are not." This is totally opposite to Epicurus's approach of using the senses to observe that we see motion going on around us and inferring by analogy that at a lower level, where atoms exist, there is also space through which the atoms move.

In Epicurean philosophy, the void does have certain properties, primarily the property of allowing atoms to exist or move. Epicurus does not take the position that the void does not exist; he takes the position that the void absolutely exists, but it only has the single quality of providing space within which matter can exist or move.

The reason I wanted to jump in at this moment was that as we dive into the further details of these mathematical proofs and the responses to them, we always want to keep in mind the implications of what we're talking about. For example, in the Norton article that you've just cited, Dr. Norton starts out with, I think, an excellent characterization of the entire episode. He says:

"Parmenides argued that things in the world are an unchanging unity without parts and that all change is impossible. This may at first seem like an abstruse position that requires great philosophical sophistication to understand. It is not. As you said, Joshua, it is an extreme fantastical form of skepticism."

I would emphasize as well that Dr. Norton says:

"Parmenides held that all motion is an illusion; nothing changes. We just have the mistaken impression that it does. This view is an instance of a perennially appearing form of skepticism—that reality is other than what our senses overwhelmingly tell us."

He further states:

"The enduring and fatal difficulty of this form of skepticism is that the evidence of our senses is powerful. Perhaps, with tenacity and some clever sophistry, a skeptic can shake our confidence in the strength of that evidence. However, this word game falls far short of what skeptics need—they need to provide positive evidence for their alternative fantastical conceptions that nothing changes. And they don’t."

In my characterization added on there, Dr. Norton says:

"It is hard to believe that Parmenides and Zeno really believed that motion is impossible. The evidence of our senses is powerful, unrelenting, and, I believe, irrefutable. Someone who genuinely believes that all change is illusion would seem to be massively deluded and in the grip of a mad fantasy."

Those words are similar, in my mind, to what Epicurus uses to describe other philosophers to whom he was so adamantly opposed. I think Epicurus likely had a similar view of this argument that Parmenides and Zeno were making—it’s a massive delusion or a mad fantasy.

Now I want to go ahead and lay this out as well. Dr. Norton says:

"If we choose, we can do this: we can cast a kinder light on Parmenides and Zeno’s project if we understand them not to be challenging change, but to be challenging the account we give of change. If that was Zeno's goal, then his efforts have met with great success."

I'm going to turn it back over to you, Joshua, but I wanted to emphasize that as we're having our background thoughts while listening to this discussion, we can take a charitable position that Zeno and Parmenides were just really challenging us to be better thinkers and to sharpen our analysis and understanding of the way things really are. However, I don't know that that's really justified. The quotations we have from Aristotle discussing these issues don't give any indication that Parmenides or Zeno were just setting these things up as thought constructs to be knocked down by their students. They seem to be really suggesting that motion is impossible, that change is impossible, and that we should discard reliance on the senses. Those have lots and lots of very damaging results if you do those things.

So again, we're going to dive deeper into the details, talk about the different examples, and how to deconstruct them and then reconstruct them. But for those who don't have time to listen to the whole episode or might get interrupted, the important thing to remember about Zeno's paradoxes from an Epicurean point of view is that they are closer to the fantastical conceptions of madmen—people who are massively deluded or in the grip of a mad fantasy—than they are serious philosophical questions. They are mathematical questions, but as to whether they are seriously related to reality, for people of an Epicurean persuasion who consider the reality that we live in to be of primary importance, they're very similar to what Lucretius says about the person who is a radical skeptic: "He who says that nothing can be known essentially is saying something that is self-contradictory and needs to be just discarded." You don't even talk to such a person. The person who says that change and motion are an illusion is awfully close to a person who you just can't listen to at all, other than to be very suspicious about where he's coming from in the first place and then use your mind to deconstruct and understand where he's coming from so that you can refute him and not be confused by their argument.

Yeah, I think that's all well said, Cassius. It's important to keep in mind the broader conclusions and how we should respond to them when we get to the detailed level. It does get rather interesting. You were just quoting, Cassius, from the John D. Norton page from the University of Pittsburgh website. I'm going to go to the UC Davis website now.

The UC Davis website is interesting because it specifically pits this Eleatic approach up against Epicurus's physics. The title of these lecture notes is "The Physics of Epicurus," and the first subheading is "The Origins of Greek Physics," with the second one being "Monism."

Now, what we learn from both of these pages is that Aristotle is hugely important here as both a transmitter and a critic of what Zeno is saying. We don't have Zeno's writings; we have Aristotle describing Zeno's thought. So under the subheading of Monism on the UC Davis page, the author writes this:

"Aristotle remarked that Parmenides seems to have conceived of reality as one by definition. The idea is that if there were something other than being, it could only be what is not being, i.e., not being. But as not being is nothing, whatever might be other than being is also nothing, so there is nothing other than being."

This gets into the ontology side of this question, and we were talking before the recording about St. Anselm and the ontological argument, and some of this is related to that approach. The author of these lecture notes continues:

"Moreover, being is incapable of change. Being itself could not come to be, as that would mean that it passes to what it is not, i.e., non-being. As Parmenides is quoted as putting it: 'How could what is be something of the future? How could it come to be? For if it were coming to be, or if it were going to be in the future, in either case, there would be a time when it is not. Thus coming to be is quenched, and by similar reasoning, destruction is also unthinkable.'"

Then the lecture notes continue:

"By the same reasoning, no change is possible within being. The final alternative, that being changes its place, is also ruled out."

This is where we get the idea that motion is impossible, and Plato is the source for this quotation:

"Parmenides and his associates maintain that everything is one and stationary and entirely self-contained since there is no empty place in which to move."

The lecture notes continue:

"An empty space or void would be not-being, which therefore does not exist. If there is no void, being cannot move, nor can anything move within it."

As Aristotle noted:

"Certain earlier thinkers maintained that what is must necessarily be one and immovable. They argue that since the void does not exist, what is cannot be moved, and that there cannot be a plurality of things because there is no void to keep them apart."

Cassius: Let me jump in for one moment. I just want to emphasize, at least from my perspective, how outrageous this is. There is no citation or pointing to an observation in nature as an example of what they're talking about. It's purely a matter of putting together a string of words and defining it in a way that seems to be consistent, and then using that self-consistency to speculate about another conclusion that is internally consistent within the words but which has never been brought back to nature through observation of the senses for a foundation. You can prove anything in the world that you wish to prove if you're willing to string together words with specific definitions that are self-consistent, and if you don't find yourself bound by verifying what you're saying as connected to reality. That's what these guys are doing, and I would suspect that the irritation in my voice would have been mirrored in the ancient world by Epicureans dealing with this kind of argument because it is the absolute opposite approach to living a practical life based on the senses, the anticipations, and the feelings. Everything turns on how you're defining those terms, and none of it turns on taking it back for confirmation to the evidence of the senses. I'm sorry for the interruption—go ahead.

Joshua: No, I think that's well said, Cassius. So the argument so far is this: everything that exists exists as an undifferentiated whole. There is no empty space, void, or vacuum, and any change or motion in this undifferentiated whole is impossible.

Now, how thinkers in the ancient world grappled with this does differ based on their own position, philosophy, and biases. Aristotle does confront some of these problems directly, for example, the problem of whether there can be motion or if bodies can change place without there being a void in which they move. Aristotle's answer was this: motion can take place in a totally full space—a plenum is the word for that—insofar as bodies push what is ahead of them aside, initiating a chain reaction that results in the displacement of bodies, which reaches to the rear of the moving body.

The image I always think of for this is a fish nosing its way through the water. The water is tightly packed around the fish, or so it appears to be to our senses, and yet the fish is still able to move. Aristotle thinks the answer must be that when the fish pushes its way forward, it's displacing the water to its side, and that in turn displaces the water that was to its side to behind the fish. Aristotle's view is that this works with everything—that everything in nature works this way. When you walk forward, when you move your hand, you are displacing an entirely filled-up space of matter with your hand, and it just moves to allow your hand to move through it. The air that we breathe and that we inhabit is, in the same way as the water, totally filled up.

Many people will have heard this famous view from Aristotle that "nature abhors a vacuum"—there is no empty space. This is his approach, which, of course, is rejected by Lucretius and the Epicureans, specifically in the opening books of Lucretius's poem. The foundation of their argument against it is that this compression and expansion, which seems to allow this kind of movement you're talking about, implies that whatever is being compressed and expanded has void within it. Because if it were, in fact, solid, it could not be compressed. So there's a much longer attack on that theory, and we'll cite it in the show notes for today.

The next place we go in these lecture notes is to the subheading of Atomism, and then under Atomism, you have Democritus and then Epicurus. That's where we start to really get into the Epicurean response to this problem. Again, I suggest people go look at these sources because I find this to be very interesting.

Before we get anywhere near that, though, I wanted to quote from this letter from Thomas Jefferson. I'm sure everyone's heard me quote it before—dated the 15th of August, 1820, to John Adams. This is what Thomas Jefferson has to say, and this, to me, is the crux of the problem here because what Jefferson is about to do is to build up an understanding of nature using sensation, while what the Eleatics are trying to do is invalidate sensation and leave us muddled in our understanding of nature. It’s part and parcel of an approach that thinks that geometry is a better way to understand ourselves and the world we live in than sensation, and that, to me, is the great problem with everything we’re talking about here. So, Thomas Jefferson starts this way:

"Let me turn to your puzzling letter of May 12th on matter, spirit, motion, etc. Its crowd of skepticism kept me from sleep. I read it and laid it down, read it and laid it down again and again, and to give rest to my mind I was obliged to recur ultimately to my habitual anodynes: I feel, therefore I exist. I feel bodies which are not myself. There are other existences then. I call them matter. I feel them changing place; this gives me motion. Where there is an absence of matter, I call it void, or nothing, or immaterial space. On the basis of sensation of matter and motion, we may erect the fabric of all the certainties we can have or need."

And then at the end of the letter, he goes on:

"Rejecting all organs of information, therefore, but my senses, I rid myself of the phantasm with which an indulgence in speculations hyperphysical and anti-physical so uselessly occupy and disquiet the mind."

Another quote along the same line of thinking comes from Joseph Conrad in the author’s note to The Shadow Line. He says this:

"All my moral and intellectual being is penetrated by an invincible conviction that whatever falls under the dominion of our senses must be in nature and, however exceptional, cannot differ in its essence from all the other effects of the visible and tangible world of which we are a self-conscious part. The world of the living contains enough marvels and mysteries as it is, marvels and mysteries acting upon our emotions and intelligence in ways so inexplicable that it would almost justify the conception of life as an enchanted state."

Okay, and then back into the lecture notes from UC Davis, we have a description of Epicurus's physics in which the author starts this way:

"Let us now turn to the atomism of Epicurus. His system is laid out extensively in his Letter to Herodotus. The letter begins with a distinction between what is evident to the senses and what is not evident to them. The senses observe objects which have various properties, such as size, shape, color, and motion. What is not evident to the senses is the unity underlying these objects."

And several paragraphs down, the author continues:

"Having spoken of the totality—this Greek phrase to pan, the all, the whole, ta holon, the whole cosmos, the whole universe—having spoken of the totality as such, Epicurus breaks decisively from Parmenides and monism by dividing it into bodies and void. He appeals to sense perception as proving the existence of bodies, and he repeats the arguments of Lucretius that sense perception also reveals that bodies move."

That’s that reference to the letter of Thomas Jefferson that I think is useful—motion is possible only in a void; a void exists is the conclusion.

Cassius: Yes, Joshua, and that section you're reading ended with:

"Unlike Lucretius, Epicurus recognized that to say that [the void] exists requires that he deny that a void is non-being, and indeed he equates void with space and intangible nature. In this way, Epicurus avoided making the apparently contradictory claim that non-being exists."

So, that’s what we were referring to earlier—how Parmenides and Zeno were taking the word-based position that empty space does not exist because "that which is not, is not." They were using a definition approach—a word logic approach—to reach a physics conclusion, whereas Epicurus is basing his conclusions on observation through the senses. His observation through the senses tells him that there is a space through which things move. I don’t think you could come up with a much more dramatic way of stating the difference in reasoning. Epicurus is going to come down on the side of observation and perception, even if his ability to explain the details in words is not possible to him. Even if he has to talk about things like quasi-blood and quasi-bodies, he’s still going to come down on "I observe things." He’s not going to use his words—his definitions—as his ultimate explanation of the way he thinks things work, like these guys who are willing to take their words and conclude that all motion and all change are impossible. That’s the radical difference between these approaches.

Now, let’s go into some of the specific examples of these paradoxes that Zeno was suggesting, and I think as we look at the specifics, we’ll get an even clearer understanding of the difference in approach.

Joshua: Yes, exactly, and if we go back to the University of Pittsburgh website, John D. Norton’s website, this, I think, is one of the better resources for a deeper understanding of what Zeno and Parmenides are actually saying. He says:

"We do not have Zeno’s wording for his paradoxes. Rather, we are in the unfortunate position that our best account of the paradoxes comes from a critic of them—Aristotle. Nonetheless, that is what we have, and that is what we must work with."

Then he goes on to give the reference in Aristotle that contains all of the essential citations for what Zeno thought. Now, the paradox that I mentioned last week was the so-called "Achilles." This is the second of Zeno's paradoxes, and Aristotle summarizes it like this:

"It amounts to this: that in a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point where the pursued started, so that the slower must always hold a lead."

This argument is the same in principle as that which depends on bisection, though it differs in that the spaces with which we have successively to deal are not divided into halves. The result of the argument is that the slower is not overtaken, but it proceeds along the same lines as the bisection argument. For in both, the division of the space in a certain way leads to the result that the goal is not reached, though the Achilles goes further in that it affirms that even the runner most famed for his speed must fail in his pursuit of the slowest. So the solution, too, must be the same, and the claim that "that which holds a lead is never overtaken" is false.

This is Aristotle’s response to the paradox—that "that which holds the lead is never overtaken" is false. It is not overtaken while it holds the lead, but it is overtaken nevertheless if it is granted that it traverses the finite distance. It will be helpful for anyone listening to actually go to this website because there is heavy reliance on graphs and images to explain what is being discussed, and I find that very helpful. But the problem with all of this centers around the idea of infinities—infinities in space and infinities in time. With Achilles and the tortoise, as I mentioned last week, the argument is this: Achilles will never overtake the tortoise because, before Achilles gets to the tortoise, he has to complete an infinite number of actions. In order to do that, he has to get first halfway there, and then three-quarters of the way there, and dividing it on and on and down the line. And before he’s even done the first action in any of that, the tortoise, meanwhile, will have moved some distance beyond its original point.

And so the conclusion here is actually twofold. One, Achilles will never overtake the tortoise, and he’ll never finish the race. But even more importantly than that—and even more central to Zeno’s argument—is that Achilles will never even start moving. This is where this becomes a problem for the idea of change and motion. It’s not just that he won’t overtake the tortoise; it’s that he can’t even move at all. Because before you can move halfway to the tortoise, you have to move halfway to halfway to the tortoise. And before you can move halfway to halfway to the tortoise, you have to move halfway to that point. You can infinitely subdivide this distance, and there will always be more halfway points that Achilles needs to go before he can make that first step. This is the real problem of the argument.

I didn’t really start to get this until I found this website that explains it very well—probably better than I’m explaining it now. But do you see the point there, Cassius? For all of these paradoxes, the central problem is that you have to perform an infinite number of actions, and between each one of those infinite number of actions, there is an infinite number of other actions that have to be performed. So the real problem is not only can you not finish the footrace, not only can you not beat the tortoise in the footrace, you can’t even start the footrace. And you can’t start it because there will always be another infinite number of tasks to complete before your next task, and there will be an infinite number of tasks to complete before your first task.

Cassius: Joshua, I think you’ve done a great job of expressing the issue. There’s a lot of misdirection going on in the discussion of these illustrations—talking about Achilles and getting the picture in your mind of some great Athenian warrior, talking about a tortoise and getting a picture in your mind of a turtle, and all the associations that go with Athenian warriors and tortoises, and there’s a race going on and one of them gets a little bit of a lead. All of these different implications that are not really stated in the example but that a normal person is going to start thinking about—those don’t get to the heart of the issue.

You have just stated the heart of the issue. And I don’t even like calling it a paradox. The heart of the problem is the assertion by Parmenides and Zeno, with no evidence to base it on, that matter—that the universe—is infinitely divisible. This is one of the major reasons why Epicurus insisted that atoms are not divisible, because when you create these fantasies in your mind that, "Oh, in my mind, I can divide the distance down into halves, and I can divide that half down into quarters, and I can divide that quarter down into eighths," those are mental projections that you are creating in your mind, and those mental projections do not create their own reality.

Now, we’ll talk further as we go forward about the mathematical responses that people have come up with to these suggestions by Zeno, but I’m going to suggest that it’s not even necessary to get to that mathematical issue once you realize that the planted presumption of Zeno here is that matter is infinitely divisible. You’ve talked about how important infinity is to these examples, and I think there’s another really important point to add to that. Infinite divisibility is contradicted by our senses, and so we can reject arguments like Zeno is suggesting because infinite divisibility is a mental construct as opposed to a reality.

On the other hand, infinity of space—the lack of boundlessness to the universe—is something that we do see evidence of through our senses, and therefore Epicurus embraces infinity at the universal scale. Because at the universal scale, that’s where infinity comes into play to allow us to understand how the things that we see around us can come into being without the need for supernatural gods. Anything multiplied by an infinity creates an infinity. Anything that does exist and can exist in an infinity of space and time, we can infer, will exist an infinite number of times.

But dividing things down an infinite number of times is something that our senses fight against—our senses tell us cannot be true. And this Zeno paradox illustrates for us why it cannot be true, because if you do try to divide these distances down infinitely and suggest that, "Well, you have to make each incremental step before you can even move at all," your senses prove to you the absurdity of such a result.

So, one of the things I think is a takeaway from all of this discussion for us as we attempt to understand Epicurean philosophy is that, just as firmly as we would reject infinite divisibility, we would embrace—and Epicurus would embrace—infinity at the macro scale. Because it is that infinity of void and infinity of matter in the universe that allows us to understand how, in an infinity of space and time in which nature does not create only a single thing of a kind, but in fact we see these isonomic relationships going on between these things, we can use a proper view of infinity to construct a view of the universe that makes sense, while rejecting an improper view of infinity that says that everything can be subdivided an infinite number of times.

Zeno was apparently using this to try to disconcert people and try to convince them to doubt the power of their senses. But if you maintain the force of your senses and your confidence in the senses, you can take the same type of analysis here and use it to illustrate the proper versus improper uses of infinity. But to turn it back over to you, Joshua—this issue of subdividing things indefinitely is a mental construct. It leads to the absurd results that Zeno is suggesting about the absence of motion and the absence of change, and that’s why it is so important to grasp this aspect of Epicurean philosophy. This is, again, what Epicurus tells us in the Letter to Pythocles:

"Most of all, give yourself up to the study of the beginnings and of infinity and of things akin to them."

And that’s the way you begin to understand how the universe operates—by separating out absurdities versus things for which there are, in fact, evidences available to us through the senses.

We have many things linked in the thread for this week's episode, one of which is an example from a movie with Meg Ryan and, I think, Tim Robbins called I.Q., where Meg Ryan gives an example illustrating the contention of Zeno that it's impossible to walk across the room, as she, in fact, walks across the room. It’s the reality of what we observe that we go with, and not the logical absurdity that comes from definitions that do not connect with reality.

We're already pretty close to our normal point for an episode, and we haven't even scratched the surface here, Cassius. I'm going to suggest that people go to this University of Pittsburgh website maintained by John D. Norton because he goes into a great deal of detail. But really, one of the interesting aspects of this is the way that Aristotle solves the problem. John D. Norton says:

"There is something discomforting about the infinity involved here and that is supposed to precipitate a paradox. Is it not the essence of an infinite task that it cannot be completed?"

Then he says:

"Aristotle presented several responses. The most influential response depended on Aristotle's distinction of two types of infinities: the potential and the actual."

He goes on to say that Aristotle's view was that we can employ potential infinities only and not actual infinities, and that Zeno's arguments fail because they depend on a disallowed actual infinity. Aristotle says that this disallowed actual infinity is introduced by Zeno when Zeno represented the completion of the run as the completion of infinitely many shorter runs. According to Aristotle, we should instead conceive the runner's task to be this: the runner has to traverse a single continuous distance in a single continuous action. Such a distance can be traversed without problems.

John D. Norton does go into further detail about this as well and why that raises its own issues. Then later on, we get into the real heart of the mathematical problem, which I am not well equipped to deal with, but he discusses it algebraically and using calculus. I have read in several places that it wasn't really until the invention of calculus independently by both Newton and Leibniz that there was a real mathematical solution to the problem. But as you've been saying, Cassius, if you're Epicurus in the ancient world, waiting around until the mathematicians come up with the solution to the problem is not an adequate proposal because we have to be able to deal with these problems even if we don't have the mathematics to prove them wrong necessarily.

For Epicurus, his solution to the problem includes a number of principles, some of which derive from his physics, including the dualism of atoms and void, the void allowing the atoms to move freely through it. But most importantly, as you've been saying, it's the reliance on sensation—that walking across a room is sufficient. It may not, in the most perfect sense, be sufficient to disprove the dichotomy, but at some level, it has to be sufficient for us to get on with our lives.

Cassius: Yeah, let me jump on that for just a moment. Yesterday, in preparation for this episode, I was reviewing what is out there on YouTube, and there are many videos that talk about how this paradox was unsolvable until the 20th century and how modern mathematics has finally resolved the paradox. Well, people have been continuing to walk and move, and change has been continuing to happen all the way through the 2,000 years that Zeno's paradoxes have been out there. What has been solved, if you want to look at it that way, is the lack of mathematical formulas that would explain what Zeno is saying. If anything has been saved or resolved, it’s mathematics, and it's not the reality of the world that has always existed before and after Parmenides.

As you said, Epicurean atomism, Epicurean physics, doesn't even consider these to be a problem in the first place because not only does the void exist, as you just said a moment ago, and not only do we rely on our senses, but we infer that one of our bedrock conclusions is that atoms exist and that atoms are uncuttable. When you start with a premise of uncuttable, irreducible atoms, you're not even going to entertain a problem of infinite divisibility because that is a contradiction to what we observe as a universe based on atoms and void.

So, there are lots of really interesting videos out there, but I tell you what—these two articles that you found, Joshua, are just excellent. We could almost spend weeks reading through each one of these articles word by word, and we'd get a lot out of their discussion of Epicurean physics and its relation to this issue. The California article is extremely good in a more general way, and Dr. Norton's article has amazingly good diagrams and a really easy-to-follow explanation of a number of these problems that he portrays in visible form. I really can't recommend them highly enough or thank you enough, Joshua, for finding them because these were new to me when you posted about them.

But here's one thing that I want to emphasize. As I referenced earlier, Dr. Norton says that Zeno's paradoxes have forced us to think with great rigor about these infinities, but I didn’t, I think, quote this part:

"These paradoxes have forced us to think with great rigor about these infinities, and in immunizing ourselves from his paradoxes, we have brought greater clarity to these notions."

That's what we need to be doing—I hope we are doing—in this episode today. We are building an analysis which immunizes ourselves from being concerned about these apparent problems with the senses and with these apparent suggestions that reality is totally different than what we observe it to be. This issue of immunizing ourselves against errors from other schools is, I think, one of the really key threads of Epicurean philosophy.

When you read through the Principal Doctrines and start thinking about how Epicurus is immunizing against fear of the gods, immunizing against fear of death by showing you the reasons for his conclusions, immunizing against allegations that being good or virtuous is the goal of life by pointing out the true relationship of pleasure and pain to choice and avoidance—all of these things that Epicurus is doing are immunizing us against these errors that allow people to manipulate us into thoughts and conclusions, into deferring to them, to let them tell you what to do, rather than following your own happiness as the goal of life that nature has set for you.

That’s why these things are important. It’s not the mathematical consistency or the ability of anybody to be a mathematician that is really important to your happiness. But for those of you—and many of us do—who go to college, we get confronted with Zeno's paradoxes. These are the things that discourage people from thinking that philosophy is of practical use. I won’t go into quoting the whole thing today, but there’s an excellent section from Seneca about the importance of using philosophy correctly—not getting caught up in word games about whether mice eat syllables or cheese or whatever. Philosophy is critical to living a happy life, and it’s critical that everyone come to a basic understanding of a philosophy that does allow them to live a happy life.

People who get fascinated and mesmerized by the intricacies of logical problems such as Zeno's paradox can do a very harmful thing if they turn people off to the necessity of having a philosophy of their own and understanding the basics of the way they should live. Hopefully, Dr. Norton is correct, and we can be charitable and think that Parmenides and Zeno were really just trying to help us out by showing us the absurdity of logical conclusions that are not based on reality. But unfortunately, I think one of the things we do observe is that there are many people, many philosophers, many movements, and many religions that will assert things without evidence, and they do not have our best interest at heart.

That’s why I think there’s so much emotion and sincerity in Lucretius, in Diogenes of Oenoanda, and in the ancient Epicureans who were describing Epicurus as basically a savior to free us from these oppressors—basically, not the false gods who never really existed, but the priests and the people who peddle these notions, such as Zeno and Parmenides are doing here, that use these methods to confuse people and cause all sorts of harm by doing so.

Joshua, I know we’ve gone on long today, but any closing thoughts?

Joshua: I just want to end with a quote from Edward Abbey. Edward Abbey was a park ranger, wilderness advocate—I think he had a degree in philosophy from the University of New Mexico. Anyway, he says this:

"As for the solitary confinement of the mind, my theory is that solipsism, like other absurdities of the professional philosopher, is a product of too much time wasted in library stacks, between the covers of a book, in smoke-filled coffee houses—bad for brains—and conversation-clogged seminars. To refute the solipsist or the metaphysical idealist, all that you have to do is take him outside and throw a rock at his head. If he ducks, he's a liar. His logic may be airtight, but his argument, far from revealing the delusions of living experience, only exposes the limitations of logic."

This, to me, is one approach, and maybe the only approach we need when we're dealing with people like Zeno of Elea who say that motion is impossible: throw a rock at his head, and if he ducks, he's a liar.

Cassius: You're exactly right. This has been a great discussion of Zeno of Elea and Parmenides today. I wish we had a lot more time, and maybe we will spend more time in the future on these issues because these really help us to explain things with greater clarity. Let's go ahead and close for today. As always, we thank you for being with us. Please drop by the forum and let us know if you have any questions or comments about anything we've discussed in this episode or anything else you'd like to discuss about Epicurus. We'll be back next week. We'll see you then. Bye.