Episode 120 - Letter to Herodotus 9 - Epicurus' Rejection of Infinite Divisibility
Date: 05/06/22
Link: https://www.epicureanfriends.com/thread/2490-episode-one-hundred-twenty-letter-to-herodotus-09-epicurus-rejection-of-infinite/
Summary
Section titled “Summary”Episode 120 centers on sections 56–59 of Epicurus’ Letter to Herodotus, which address the upper and lower limits of atom size and the doctrine that atoms, though physically indivisible, can be conceptually divided in the mind. Cassius opens with a frank disclaimer: this is a subject he doesn’t understand well enough to communicate to a specialist audience, and the podcast is directed at ordinary people who need a framework for responding to arguments that would undermine their confidence in sensation. The main vehicle for that discussion is Zeno of Elea’s paradox of Achilles and the tortoise — the argument that motion is logically impossible — and the panel examines why Zeno put it forward (Aristotle called him “the inventor of the dialectic”), why Bertrand Russell took it seriously in Principia Mathematica, and how Martin explains the resolution in terms of middle school mathematics. Martin also describes how modern physics has extended Epicurean atomism from molecules through atoms to quarks — the Rutherford experiment revealed atoms to be mostly empty space — and notes that the standard model presently contains around sixteen to seventeen fundamental particles with perhaps a hundred total expected. The episode closes with Lucretius Book 4, line 500, on why it is better to be at fault in accounting for causes than to lose confidence in sensation entirely, “and to pluck up the whole foundations on which life and existence rest.”
Transcript
Section titled “Transcript”Cassius:
Welcome to episode 120 of Lucretius Today. This is the podcast dedicated to the poet Lucretius, who wrote On the Nature of Things, the only complete presentation of Epicurean philosophy left to us from the ancient world. I’m your host Cassius, and together with our panelists from the EpicureanFriends.com forum, we’ll walk you through the ancient Epicurean texts and discuss how Epicurean philosophy can apply to you today. We encourage you to study Epicurus for yourself, and we suggest the best place to start is the book Epicurus and His Philosophy by Canadian professor Norman DeWitt. If you find the Epicurean worldview attractive, we invite you to join us in the study of Epicurus at EpicureanFriends.com, where you’ll find a discussion thread for each of our podcast episodes and many other topics. Today we continue our review of Epicurus’ Letter to Herodotus and move further into fundamental physics, discussing issues related to the question of whether matter can be infinitely divided. Now let’s join Joshua reading today’s text.
Joshua:
“Moreover, we must not either suppose that every size exists among the atoms, in order that the evidence of phenomena may not contradict us, but we must suppose that there are some variations of size. For if this be the case we can give a better account of what occurs in our feelings and sensations, but the existence of atoms of every size is not required to explain the differences of qualities in things, and at the same time some atoms would be bound to come within our ken and be visible. But this is never seen to be the case, nor is it possible to imagine how an atom could become visible.
“Besides this, we must not suppose that in a limited body there can be infinite parts or parts of every degree of smallness. Therefore we must not only do away with division into smaller and smaller parts to infinity, in order that we may not make all things weak, and so in the composition of aggregate bodies be compelled to crush and squander the things that exist into the non-existent; but we must not either suppose that in limited bodies there is a possibility of continuing to infinity in passing even to smaller and smaller parts. For if once one says that there are infinite parts in a body or parts of any degree of smallness, it is not possible to conceive how this should be. And indeed, how could the body any longer be limited in size? For it is obvious that these infinite particles must be of some size or other, and however small they may be, the size of the body too would be infinite.
“And again, since the limited body has an extreme point which is distinguishable, even though not perceptible by itself, you cannot conceive that the succeeding point to it is not similar in character, or that if you go on in this way from one point to another, it should be possible for you to proceed to infinity, marking such points in your mind.
“We must notice also that the least thing in sensation is neither exactly like that which admits of progression from one point to another, nor again is it in every respect wholly unlike it, but it has a certain affinity with such bodies, yet cannot be divided into parts. But when, on the analogy of this resemblance, we think to divide off parts of it, one on the one side and another on the other, it must needs be that another point like the first meets our view, and we look at these points in succession starting from the first, not within the limits of the same point, nor in contact part with part, but yet by means of their own proper characteristics, measuring the size of bodies — more in a greater body and fewer in a smaller.
“Now we must suppose that the least part in the atom too bears the same relation to the whole as the least thing in sensation does, though in smallness it exceeds that which is seen by sensation. It has the same relations; indeed we have already declared on the ground of its relation to sensible bodies that the atom has size, only we placed it far below them in smallness. Furthermore, we must consider these least indivisible points as boundary-marks, providing in themselves as primary units the measure of size for the atoms, both for the smaller and the greater, in our contemplation of these unseen bodies by means of thought. For the affinity which the least parts of the atom have to the homogeneous parts of sensible things is sufficient to justify our conclusion to this extent, but that they should ever come together as bodies with motion is quite impossible.”
Cassius:
Joshua, thank you for reading that for us today. My introduction today is going to start off by quoting the movie line: “a man has to know his limitations.” And one of my limitations is that I do not understand infinite divisibility at a deep enough level to communicate the way I would like. This podcast is not directed towards mathematics specialists or theoretical physicists. It’s directed towards ordinary people who want to apply Epicurean philosophy to live more happily in their day-to-day lives.
And despite the pretzels you can be twisted into by certain philosophical arguments, I believe that Joshua exists, and I believe he is able to walk across a room. Even though people like Zeno or Parmenides can produce mind-bending arguments that might imply you don’t exist or can’t walk across the room, reality tells me otherwise. As Epicurus stands for the sensations as the ultimate standard of truth, my sensations tell me that Martin and Joshua exist and are capable of walking across rooms — no matter what a theoretical mathematician might want to argue.
These kinds of arguments were going on in the ancient world as well, and they’re always going to go on. People who listen to this podcast or study Epicurus need to prepare themselves that most of them are never going to be able to competently deal with some of the mathematical issues involved. But part of philosophical warfare is to be able to understand and take a position you’re confident of in response to arguments that are intending to twist you into a pretzel and cause you to doubt the reliability of your senses. So what we’re going to do today is the best we can to unwind this particular subject, which is in fact a response to those arguments. Basically, Epicurus was aware of prior arguments that motion was impossible, that mathematics can be used to point to conclusions that would leave you numb if you thought about them all the time. And so Epicurus was providing for his students, here in this basic letter to Herodotus, the outline of a response to these questions.
So let’s start at the very beginning, because we’ve been talking about atoms and now we want to talk about the size of the atoms. You’ve got two sets of questions: how big can an atom get, and how small can an atom get? The first point made in section 56 is that you don’t have to have atoms of every size in order to explain the reality of the things we see around us. Joshua, Martin — where is he going with this initial part?
Martin:
Sorry for coming a bit back to your introduction — I think we did make some progress since ancient times. No one is pushing the old paradoxes anymore because they are clearly refuted. To define Zeno’s paradox: Achilles can never catch up with the tortoise if the tortoise in a race with him has a head start, because if Achilles reaches the point where the tortoise was just now, the tortoise has again moved a bit further ahead. This is just breaking things up mathematically in a way that limits itself to the time period within which Achilles has not yet bypassed the tortoise. If you go beyond that time, then he easily runs past the tortoise.
Cassius:
Let’s explain as basically as we can what Zeno’s paradoxes are. But Josh, just to make absolutely clear — why was it important to Epicurus to say that atoms cannot be divided forever? What was Zeno arguing?
Joshua:
I have to conceptualize this within the broader context of the whole history of ancient Greek and particularly Ionian physics and the attempt of these early Greek philosophers to understand nature. So Zeno’s paradoxes are part of this problem. You’ve got these other problems, like: they’re trying to come up with what was the first substance — was it water? Is water the foundational element of all things, or is air, or fire? And you’ve got Heraclitus saying everything is in flux and therefore it’s impossible to know anything. And you’ve got Parmenides saying that not only is everything not in flux, but motion is impossible. And so we come down to this idea of Zeno’s paradoxes.
And what he’s really saying is basically what Martin just said — motion is impossible, because any forward motion will involve going merely halfway before you get the whole way. The idea is that when you go halfway to a thing — before I can go a foot, I have to go half a foot. OK, now I’m at the half-foot mark. But before I go to the foot-mark, I still have to go a quarter of a foot. And before that, an eighth of a foot, and then a sixteenth, and then a thirty-second. And so you never get anywhere. There are an infinite number of divisions that you can always make.
Cassius:
Was Zeno simply insane, or what? Let’s take him seriously for a minute. Most people realize from experience that any normal human being can outrun a tortoise. So what was Zeno doing by asserting this example?
Joshua:
He was either insane, or he was a genius — or as you suggest, maybe he’s just somebody who likes to confuse people. But Aristotle gives him a rather distinguished title: he calls him “the inventor of the dialectic.” Zeno of Elea is the one we’re talking about here — not to be confused with two other important Zenos. And so this whole Socratic method of reasoning, Aristotle says, starts with Zeno. There’s this problem because if you’re just going by sense perception, by what you can see in the observable universe, what he’s talking about is rank madness. But presumably he’s not mad. So what is he doing?
I’m going to quote Bertrand Russell here, who co-wrote a book called Principia Mathematica — he had a co-author in that. And the goal of Principia Mathematica, as I understand it, was to drill down to the most basic fundamental level in the principles of mathematics and discover the absolute foundation for believing mathematical conclusions are true in themselves — that mathematics is at least faithful to its own principles. And from that angle, Bertrand Russell viewed Zeno of Elea as, and I quote, “immeasurably subtle and profound.” So this idea that you can take mathematics seriously and explore these questions without being thought a mere fool — from that point of view, you can see there’s a kind of value in it. But I guess what Epicurus would say is that for the second or third century Greek just living their life and trying to come up with a better way to live, this kind of thing just doesn’t help.
Cassius:
I find it very difficult to get out of my mind the possibility that he’s just being a jerk and intentionally confusing people — almost in the way Socrates would tease people into questioning the things they think are true. But in this case, he’s asserting something that everyone knows in reality is not true. Martin, what do you think Zeno was after?
Martin:
The problem is I don’t know Zeno’s context, so anything I would say about what he meant would be speculation. So maybe those who know the background texts in more detail can come up with it. But I can say this: I think these Zeno paradoxes are very good for people who are learning mathematics or physics, because when you go through this and find the solution, you get a deeper understanding. You can then very easily deal with any kind of argument like this in future life. So I see at least an educational value in formulating something that looks like a paradox — because you’re no longer stunned by paradoxes once you know how to resolve them.
Cassius:
This is from a lecture by Albert Einstein called “Geometry and Experience” — I haven’t read the full lecture, but a quote I’ve always been familiar with is this: “As far as the laws of mathematics refer to reality, they are not certain. And as far as they are certain, they do not refer to reality.” Is that kind of what you’re talking about? There’s the discipline of mathematics which has its own rules, and it doesn’t always relate perfectly to what’s going on in the world?
Martin:
Yes — that may be related to this. It refers to what I said in my comment on your introduction: that fits exactly.
Cassius:
And I don’t mean in my criticism of Zeno to indicate people should just not study this. I think it’s extremely important for people to understand that there are paradoxical questions out there that they’re going to be confronted with in life, and they need to understand how to proceed when those paradoxes are pressed on them — without resorting to mystical religious angles, or without just throwing up their hands and saying we should ignore science because it’s too crazy. None of those are the right answers. Martin, what would you point to as how to unwind a paradox like Zeno’s tortoise paradox?
Martin:
You consider the speed at which the tortoise and Achilles are moving, and then you see that by dividing these steps into smaller and smaller halves, you are also dividing the time into smaller and smaller halves. That means you never reach the point at which Achilles bypasses the tortoise within the artificial limit you’ve imposed on yourself by this method of cutting — but by going beyond that artificial limit you can move beyond it.
Joshua:
Yes — and what I’d like to touch on is this: this is the legacy of natural science that Greece has left to us — stealing fire from the gods. This is what natural physics boils down to, and it starts with Thales and goes through Anaximander and Anaxagoras and Democritus and all these pre-Socratic philosophers. And while some of their conclusions might strike us as quite absurd, and most are totally contrary to Epicurean physics and also to contemporary physics, this was nevertheless the first attempt to explain nature through natural causes — not simply to refer the phenomena of nature to a divine cause, but the idea that there’s a natural explanation for things that happen and that we can discuss these things intellectually. There is a kind of value in it, even when their conclusions were wrong.
Cassius:
I’ll double down on that and say it’s extremely valuable because there are people like Zeno who, in all walks of life, are going to assert things to you that sound reasonable and who come with lots of qualifications behind them. Zeno was extremely well thought of — Bertrand Russell thinks he was extremely subtle — so you’ve got a very highly qualified philosopher presenting an argument that if you believed it, you would conclude that a tortoise cannot ever be beaten in a race by a hare. Now, if you believe that, you’ve got a problem. How do you deal with arguments that appear to be sound, coming from authoritative people who present a methodology in mathematics or some other science that you don’t understand, and tell you motion is impossible?
What I think a philosophy has to do in the end is to provide a realistic way of living in the world. And when someone comes to you with a conclusion that would lead you to adopt a position you see in your life is contrary to reality — what do you do? Do you throw up your hands and say all of philosophy is useless? Certainly not. And this is one more thing I’d like to say in defense of the Greek tradition: Zeno never claimed divine revelation for the things he was saying. He didn’t claim infallibility. So there’s always the open invitation for critique and counter-argument. And all it takes, really, is a very simple and repeatable experiment. All you need is a tortoise and a hare. You accept the authority of your senses as greater than the authority of this alleged mathematical proof, and you have your answer. If you’ve allowed this alleged mathematical proof to cause you to question the reliability of your own eyes, then you’ve got a problem. At a certain point, you just have to leave the cave, get a tortoise and get a hare, and settle the question.
There’s a reference in Lucian’s “Alexander the Oracle-Monger” where he has a statement to the effect that a person like Epicurus or Metrodorus — even though they may not understand the precise manner in which an imposition is being asserted by a magician — was confident that there was a rational explanation for what they were seeing. And I kind of think that’s where you end up with questions like this: when something has been asserted to you that just absolutely contradicts what you can observe for yourself and what other people are observing, you should have confidence that there’s a rational explanation that could eventually be shown.
Martin:
Most people are not trained in physics and mathematics. What are they supposed to do when a highly trained philosopher presents them an apparently compelling argument that motion is impossible?
Do not get confused. They can always refer to reality, but they won’t understand why what the philosopher was saying was wrong. So if you want to understand where the philosophy is at fault, you need to drill down and find it. Sit down, take your son’s or daughter’s math textbook from about the eighth year, study it, and calculate it through. This is not university mathematics. This is middle school mathematics, with which you can figure it out.
Joshua:
I think there are ways of even mathematically demonstrating that this argument is wrong, and ways of objectively doing an experiment and repeating it with different people. You could do this in six different universities and come up with the same result. And at that point, it’s not even so much an issue of trusting your senses — though certainly if you had started out by trusting your senses you would have been right. There are people today who hold views that are as patently wrong — and there will always be. I don’t think it’s always wise to chase every argument. At a certain point, you’ve got better things to do with your time. I certainly wouldn’t spend the rest of my life sitting in a cave with Zeno arguing over whether a tortoise or a hare was faster. At a certain point, you leave the cave and settle the question.
Cassius:
Let me cap that off by saying that in my experience, there are a lot of people out there in the world who have some moderate interest in philosophical arguments but don’t have a lot of confidence in their ability to deal with them. And when confronted with paradoxes or things that don’t seem to have an obvious solution, a significant number of people tend to dismiss the whole issue and say, “I’m not reading any more philosophy. I’m just going to put all of that away.” That’s, at a very basic level, one of the things I think is a mistake to do. Whether you can yourself reason through some complicated mathematics or whether you can’t, you still are faced with thousands of issues in life where you don’t have the answer and have to go out looking among competing alternatives. So the question of how to deal with things that appear to be complicated is something that has very practical application.
Let’s go make some comments on some of the more specific aspects of what Epicurus is talking about, because section 56 is also pointing to a logical argument: if you can divide things infinitely into smaller parts, but every part has some minimum size, then if you’ve got an infinite number of even extremely small particles, you’ve got a solid universe with simply no room for anything else. He seems to think that eventually you do have to come down to something that is solid and indivisible. He says: “therefore we must not only do away with division into smaller and smaller parts to infinity, in order that we may not make all things weak, and so in the composition of aggregate bodies be compelled to crush and squander the things that exist into the non-existent.” He seems to be saying that when atoms come together into aggregate bodies, if they can be cut further, the mere act of crashing into each other would cut them even further, and if that were the case you would never have aggregate bodies. So it’s a logical argument. And the upper limit on atom size is obvious — they’re not so big that we can see them.
Martin, what do you recommend for a layperson as the best way to get a handle on infinite divisibility? Can you continue cutting things in half infinitely, or is there eventually some limit at which something is uncuttable? And is that something we can actually hope to answer one day through improved microscopes, or are we talking about a logical question of words alone?
Martin:
Our science has started already in the recent few hundred years, so when we cut things down smaller and smaller, eventually we reach the level of a molecule. We take a homogeneous substance, come down to a molecule, and then if we take this molecule and break it down further, it does not behave as the substance we were analyzing. So that molecule is the smallest unit which still has approximately the properties of the bulk material. If we break up that molecule, the pieces no longer behave like what the bulk material behaves. So we’ve split it into something more elementary than this material — we may have split it into its atoms. And these individual atoms have different properties than what the molecule and the substance made of those molecules had.
Now, moving toward the issues of subatomic particles: if we take those atoms apart, you have the nucleus inside and the electrons outside. The electron is, from our current perspective, already one elementary particle — we don’t have a generally accepted model that claims the electron is further divisible. The electron counts already as an elementary particle. The nucleus is not — it consists of protons and neutrons. And then if you take it even further in experiments with a single proton or neutron, you come down to the quarks. And if you then try to take it apart — to take a neutron apart into its quarks — you will produce more particles. The quarks interact in a way that the more you try to separate them, the stronger the force becomes. Eventually you need to put up so much energy that you create new particles. So the quark which you tried to take away ends up with another quark created which attaches to it, and another quark is formed at the same time which takes the place of the one you removed. You basically couldn’t manage to cut it further individually because you come to this paradoxical thing: by trying to cut or tear this apart, you create more.
Cassius:
Is there a certain number of subatomic particles that stands out in your mind? Have they identified ten, fifty, a hundred?
Martin:
If we just go to what regular matter consists of: we have the up quark, the down quark, the electron, the electron neutrino, and standing out a bit separate from these, the photon. With these particles we can pretty much describe on a simple level everything we are dealing with. But if we want to see the bigger picture, we have two more quark families with associated other particles. So you easily get into something more than a dozen elementary particles already — and plus the interaction particles. I’m looking at a chart with sixteen or seventeen. The expectation is not that it’s infinite. I think if you go complete, you should be in the order of a hundred or something, including suggestions for further particles to explain the few things not covered by the standard model.
Cassius:
And there’s a Rutherford experiment you mentioned?
Martin:
Yes — this idea of hard bodies with a shape was experimentally verifiable. The Rutherford experiment worked by shooting particles at atoms and looking at how the projectiles were distributed coming out, and from this you could determine the size of the elementary particle and possibly even some information on the shape. That experiment was done to measure the size of the nucleus. Rutherford figured out that an atom is mostly empty — the nucleus is very small compared to the size of the atom as we see it from the electron orbit.
Cassius:
Before we brush over sections 58 and 59 — the issue of the least part of the atom bearing the same relation to the whole — what that simply means is that the atom, at least in Epicurean physics, is a solid body, perfectly dense, with no holes in it. So that will be true for any part of the atom as it is for the whole of the atom. If the atom is colorless, then any part of the atom will also be colorless. I think that’s what he’s getting at there.
And there’s a broader challenge in 58 and 59. Democritus and Leucippus, from what I’ve read, appear to have held the view that it was not possible even to conceive of something like half an atom — that not only was it physically indivisible, but you couldn’t even mentally divide it in your head. And Aristotle apparently took issue with that argument. So Epicurus is coming in after Democritus and Leucippus, and somewhat in response to Plato and Aristotle, and what he hopes is that while there is a limit to the divisibility of atoms in nature, conceptually you can divide them in your mind. It does make sense to talk about the left side of an atom or half an atom conceptually — but physically, there’s a level at which the cutting has to stop.
I think those are also extremely specialized arguments with philosophical implications, but the further we get into the specialized arguments the less applicable it’s going to be to a general audience. There will always be people who are extremely interested in talking about this, and we’ll provide additional links in the thread for this episode. We don’t want to squelch that discussion — it’s extremely helpful for those people. But just to close with the big picture: there are always going to be Zenos out there. The principle of there being never only one thing of a kind — there are Zenos living today, making arguments like this. Having a framework to respond to them is important.
Martin:
Yeah — I’m fine, I have nothing more to add.
Joshua:
If you go to the thread on the forum for this episode, I’ll post a couple of links to things that bear on what we’re talking about here — like fractals, the graphical images of mathematical sets like the Mandelbrot set, where you do get things like infinite detail in a finite space. So you can just keep going down and down and down forever mathematically. And so there’s a lot of stuff like that that you could spend a lifetime looking into. But of course, for Epicurus, we have to remember that the principal thing is he viewed the study of nature as a source of joy and happiness in his life. So while it’s very good to have an understanding of what he’s talking about in areas like infinite divisibility — and it’s worth talking about and worth thinking about — if it doesn’t give you a whole lot of pleasure, this isn’t the only thing in nature that’s worth looking at. I would say to Cassius that having a deep thorough understanding of every technical point is likewise not always the goal. It’s studying it and trying to learn those answers. And I say that only because I know there are people like me and like you in the audience who don’t know what Martin knows — and nevertheless, we can derive benefit from studying these principles.
Cassius:
That sounds good. Let me close then with this — it’s Book 4 of Lucretius, line 500: “And if reason is unable to unravel the cause why those things which close at hand are square are seen round from a distance, still it is better through lack of reasoning to be at fault in accounting for the causes of either shape rather than to let things clear-seen slip abroad from your grasp and to assail the grounds of belief and to pluck up the whole foundations on which life and existence rest. For not only would all reasoning fall away, life itself too would collapse straight away unless you choose to trust the senses and avoid headlong spots and other things of this kind which must be shunned and to make for what is opposite to these. Know then that all is but an empty store of words which has been drawn up and arrayed against the senses.”
So if you think a tortoise can outrun a rabbit, you’re in the wrong philosophy. Okay, I’ll close with that and we’ll come back next week. Thank you Joshua, thank you Martin.
Joshua:
Thanks.
Martin:
Thanks and bye.