Episode 265 - The Deep-Set Boundary Stone - Epicurus and The Perils of Applying the Principles of Geometry to Ethical Philosophy
Date: 01/26/25
Link: https://www.epicureanfriends.com/thread/4268-episode-265-the-deep-set-boundary-stone-epicurus-and-the-perils-of-applying-the/
Summary
Section titled “Summary”Another special episode in which co-host Joshua presents a full prepared talk titled The Deep-Set Boundary Stone: Epicurus and the Perils of Applying the Principles of Geometry to Ethical Philosophy. Cassius opens with an extended introduction drawing on Carl Sagan’s Cosmos episode “Backbone of the Night,” which traces the Pythagorean and Platonic elevation of mathematical abstraction over sense-based inquiry — framing the Epicurean position as a principled rejection of rationalism divorced from nature.
Joshua, drawing on his background as a land surveyor in Florida, examines three scholarly approaches to the Epicurean critique of geometry: (1) Epicurus rejected geometry’s axiomatic foundations, particularly infinite divisibility, which contradicts the indivisible atom; (2) geometry contributes nothing to the art of living; and (3) Epicurus was reacting to contemporaries who misused geometry to make unfounded ethical claims. Key sources include Proclus’s Commentary on the First Book of Euclid (translated by Glenn R. Morrow), Cicero’s On Ends (Torquatus defending Epicurus), Principal Doctrines 11 and 12, the Letter to Herodotus, Seneca’s Letters to Lucilius, Norman DeWitt’s Epicurus and His Philosophy, Plato’s cave allegory from the Republic, a Britannica article on Pythagorean numerology, John Maynard Keynes’s essay Newton, the Man, and scenes from the 2012 film Lincoln that illustrate the seductive but fallacious use of geometric analogy in ethics.
Transcript
Section titled “Transcript”Cassius:
Welcome to episode 265 of Lucretius Today. 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 text, and we 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 discuss this and all of our podcast episodes.
This week we have another special episode in which our co-host Joshua will give a talk entitled The Deep-Set Boundary Stone: Epicurus and the Perils of Applying the Principles of Geometry to Ethical Philosophy. This episode is going to provide a general introduction to the relationship between Epicurean philosophy and science. Ironically enough, there has been a criticism of Epicurean philosophy over the years that the most famous exponent of Atomism — the most famous exponent of an evidence-based approach to life and ethics — is somehow anti-science.
Nothing could be further from the truth, so that is an important fiction to be dispelled, because it is through the study of nature — which is what true science is all about — that Epicurus reached all of his conclusions about the nature of the universe and how to live. This is a very contentious topic, however, and one reason is that people get tripped up over what actually is science. There are many ways to approach that question, but fundamentally the approach that Epicurus took is that your senses are the root of all of the information you can get from the outside world. The senses are your contact with reality, and the information that senses provide is the basis which your mind will then take and develop and assess theories — but always coming back to the evidence of the senses as the test by which we determine whether a theory is accurate to nature or not.
And so the senses have to be taken into account, and your theories about the way the world works have to be brought into conformity with the senses, not the other way around. We do not make our senses conform to our theories. We recognize that theories derive from the reality we are able to gather from the senses.
In today’s episode, Joshua will be able to give only a brief presentation of the topic, but one additional source I would recommend for anyone interested in this topic would be to locate on the internet Carl Sagan’s Cosmos series and look for episode seven, which is entitled Backbone of the Night. In that episode, Carl Sagan — one of the most famous proponents of scientific inquiry in the last hundred years — went back into the history of Greek philosophy, visited the island of Samos where Epicurus was from, and produced a very helpful summary of where this theory of a geometrical or mathematical approach to reality came from in the first place.
The episode does not mention Epicurus, but does mention Democritus, and sets out how a series of philosophers — pioneered by Pythagoras but expanded and implemented further in Platonism, and implicit as well in Stoicism — in all of these rationalistic-based philosophies which attempt to argue that through the mind alone, separated and freed from the body, there is somehow a higher truth than what Epicurus was pointing to in looking to nature and the senses as the ultimate guide for how we should live. If you look around, you can find episode seven on the internet, and while I recommend the entire episode, if you want to fast-forward past the introduction, you can start around the twenty-minute mark when Sagan goes to Greece and begins discussing the history of the Greek schools. Then when you get to around the thirty-nine-minute mark, you will find that Sagan takes Plato apart using an approach and a perspective very similar to that advocated by Epicurus.
Here is a section worth quoting in full, starting with the Pythagoreans where this problem developed and then continuing into Plato. Sagan says this — and it is very interesting to hear Sagan say it in his own voice:
The Pythagoreans had discovered in the mathematical underpinnings of nature one of the two most powerful scientific tools. The other is, of course, experiment. But instead of using their insight to advance the collective voyage of human discovery, they made of it little more than the hocus-pocus of a mystery cult. Science and mathematics were to be removed from the hands of the merchants and the artisans. This tendency found its most effective advocate in a follower of Pythagoras named Plato. Plato preferred the perfection of these mathematical abstractions to the imperfections of everyday life. He believed that ideas were far more real than the natural world. He advised the astronomers not to waste their time observing the stars and the planets. It was better, he believed, just to think about them. Plato expressed hostility to observation and experiment. He taught contempt for the real world and disdain for the practical application of scientific knowledge. Plato’s followers succeeded in extinguishing the light of science and experiment that had been kindled by Democritus and the other Ionians. Plato’s unease with the world as revealed by our senses was to dominate and stifle Western philosophy.
Now, that is what I would submit to you is essentially the attitude that Epicurus takes toward geometry and mathematics as ethical tools. To the extent that those tools are useful in producing a better and happier life, Epicurus is going to endorse them — just as he would endorse any tool that leads to pleasure and to the avoidance of pain. But that is not what Pythagoras and Plato, and through them later versions such as the Stoics, are doing. They are looking at a disembodied science separated from the senses as if it is an end in itself — as if it is more real than nature, according to Carl Sagan.
Now, this indictment of Plato can seem very harsh to those who have been taught all their lives that Plato, Socrates, and even Aristotle are the highlights of Western civilization. But Epicurus saw that there were profound and damaging consequences to their rejection of the senses and their elevation of rationalism — as if the mind alone, separated from the body, was the source of truth. And Epicurus, being the practical philosopher that he was, is always going to focus on the real-world consequences. He is not going to let his mind go off into flights of fancy that are disconnected from the senses and which lead to extremely harmful and damaging conclusions when one thinks that somehow mind and body can be separated.
So today Joshua will bring an introduction to this topic using his own personal experience, including his background as a surveyor in which he had to work with — as part of his job — the relationship between mathematics and the real world. That will introduce you to the topic, after which you can go to the Carl Sagan episode of Cosmos if you have time. But whether you have time for Carl Sagan or not, Joshua’s talk will give you a very good introduction to this topic. So without further ado, here is Joshua on The Deep-Set Boundary Stone.
Joshua:
The Deep-Set Boundary Stone: Epicurus and the Perils of Applying the Principles of Geometry to Ethical Philosophy.
Two or three years ago, I was employed as a crew chief in a land surveying and civil engineering firm in the state of Florida. My day was spent in the field, and while the work we did was diverse — woodland, swampland, and coastal topography, boundary surveys, section breaks, and so on — nearly all of it was reducible to geometry. Using the traditional method, land surveying starts by occupying a given point, taking a backsight to a second point to establish a baseline, and then pulling angles and distances off that line to measure the land, the improvements made to it such as buildings and fences, and — most importantly — the boundary lines that circumscribe it. Even though the field computers do most of the math these days and GPS-based systems have introduced an added layer of complexity, land surveying is rooted in a branch of mathematics using axioms largely unchanged since the days of Euclid in the fourth century BC.
But I was also spending much of my free time in the study of the ancient Epicureans, and there came a moment when I began to realize that there might be a fundamental conflict between my passion and my work. While Lucretius was happy to evoke the land surveyor’s language of the “deep-set boundary stone” no fewer than four times in his poem — as a metaphor for the limits of nature, limits to what can and cannot be — Epicurus, it seemed, had little patience for the discipline of geometry.
In the fifth century, a Neo-Platonist scholar by the name of Proclus makes the point explicit:
Now that we have summed up these matters, it remains for us to examine the propositions that come after the principles. Up to this point we have been dealing with the principles, and it is against them that most critics of geometry have raised objections, endeavoring to show that these parts are not firmly established. Of those in this group whose arguments have become notorious, some — such as the Skeptics — would do away with all knowledge, like enemy troops destroying the crops of a foreign country, in this case a country that has produced philosophy; whereas others, like the Epicureans, propose only to discredit the principles of geometry. Another group of critics, however, admit the principles but deny that the propositions coming after the principles can be demonstrated unless they grant something that is not contained in the principles. This method of controversy was followed by Zeno of Sidon, who belonged to the School of Epicurus, and against whom Posidonius has written a whole book and shown that his views are thoroughly unsound.
— Proclus, Commentary on the First Book of Euclid, translated by Glenn R. Morrow
Cicero — and this is typical of him — is even more scathing: “Nor is it proper,” he wrote, “in a natural philosopher to believe in a least possible body — a hypothesis Epicurus certainly never would have formed if he had chosen to learn mathematics from his friend Polyaenus rather than to make him actually unlearn what he knew himself.” — Cicero, On the Ends of Good and Evil, translated by James Reed.
It is difficult to know what Epicurus really thought about geometry, because it was a very long time ago and the texts that he and his friends wrote on the subject are sunk without trace. Diogenes Laertius in Book 10 of his Lives and Opinions of Eminent Philosophers records that Epicurus wrote a work on the angle of the atom, and that Hermarchus wrote an essay on mathematics.
There is no royal road to geometry, as Euclid was said to have quipped in response to Ptolemy the First when the latter complained that it was too difficult to learn. Fair enough, Euclid — but when it comes to ascertaining what Epicurus thought about it, there does not appear to be any road at all.
In attempting to explain the Epicurean response to geometry, scholars studying the question have basically come up with three different approaches.
Approach number one: as proposed a moment ago by Proclus, it was suggested that Epicurus rejected the axiomatic foundations or first principles of geometry. This disagreement partially turned on the question of infinite divisibility and its opposing claim, the least possible body — that is, the atom, as mentioned by Cicero.
The second approach suggested was the problem that geometry contributed nothing to the art of living, or what Professor Norman DeWitt calls the conduct of life. If the purpose of philosophy was to teach us how to live, geometry could add nothing to that. Understanding how to solve for the hypotenuse of a right triangle is very useful, but its use is practical and mathematical, not philosophical.
The third approach: on the other hand, some of Epicurus’s contemporaries really did seem to think that geometry could inform ethics. Perhaps it was the misuse of geometry — a practical science for portioning land and building roads and aqueducts and digging tunnels and so forth — now being corrupted to cover for unfounded ethical claims. I suspect that would really rankle. I do not have a great deal of evidence from the texts because again, most of them are lost, but I really think that Epicurus would have disapproved.
Let us take a closer look at each of these proposals. I quoted earlier from Cicero’s On Ends, and in the first book of that dialogue, Cicero has his Epicurean interlocutor Lucius Manlius Torquatus touch on the question of geometry in his final rousing paragraph:
And though you think Epicurus ill-educated, the reason is that he held no education of any worth but such as promoted the ordered life of happiness. Was he the man to spend his time in conning poets — as you and your school do on your advice — when they afford no substantial benefit and all the enjoyment they give is childish in kind? Or was he the man to waste himself like Plato upon music, geometry, mathematics, and astronomy — which not only start from false assumptions and so cannot be true, but if they were true would not aid us one whit towards living a more agreeable, that is, a better life? Was he, I ask, the man to pursue those arts and thrust behind him the art of living — an art of such moment, so laborious too, and correspondingly rich in fruit? Epicurus then is not uneducated; but those persons are uneducated who think that subjects which it is disgraceful for a boy not to have learned must be learned through life into old age.
— Cicero, On the Ends of Good and Evil
Now, as you may have noticed, Torquatus actually addresses two out of the three points under consideration. Geometry, he says, starts from false assumptions and so cannot be true — but even if it were true, it would not lead to a more agreeable life.
If you think he is being harsh on geometry here, I suggest taking a look at Principal Doctrines 11 and 12. Principal Doctrine 11 reads: “If we had never been troubled by celestial and atmospheric phenomena, nor by fears about death, nor by our ignorance of the limits of pains and desires, we should have no need of natural science.” And Principal Doctrine 12: “It is impossible for someone to dispel his fears about the most important matters if he does not know the nature of the universe but still gives some credence to myths. So without the study of nature, there is no enjoyment of pure pleasure.”
Epicurus’s magnum opus was a work called On Nature, written in 37 books. I regret to say that this text survives only in fragments, but in his Letter to Herodotus he says this: “Wherefore, since the method I have described is valuable to all those who are accustomed to the investigation of nature, I — who urge upon others the constant occupation in the investigation of nature, and find my own peace chiefly in the life so occupied — have composed for you another epitome on these lines, summing up the first principles of the whole doctrine.”
Epicurus was a devoted student of nature, but as he says right there in Principal Doctrine 11, there would be no need to study nature at all if doing so failed to relieve our anxiety about death and the gods. It would not be bad to study nature necessarily — it would just be fruitless. It would furnish nothing for the art of living. This is his baby, the study of nature — but if it were not real, he would have no problem dumping the metaphorical bathwater.
Somewhat ironically, neither would the Stoic Seneca. There is a surprising level of agreement on this point: geometry is useful and all, but let us not lose our heads. The following is a passage from Seneca’s Letters to Lucilius, translated by Margaret Graver and A.A. Long:
The geometer teaches me to figure the size of a plantation, but he does not teach me how to figure what quantity of wealth is sufficient for a human being. He teaches me to do computations, adapting my fingers to the purposes of greed, but he does not teach me that such computations are beside the point — that it does not make one any happier to have accountants wearing themselves out over one’s income. Indeed, that a man who would find it a misfortune to have to compute his own net worth possesses nothing but superfluities. How does it help me to know how to divide up a field if I do not know how to divide it with my brother? How does it help me to figure out the precise size of a garden plot down to the tenth part of a foot if it upsets me that my unruly neighbor is shaving a little off my land? He teaches me how to keep from losing that strip near my property line — but what I want to know is how to lose all my property and yet remain cheerful.
Now, while Epicurus would probably disagree with some of the direction that Seneca is taking, it does help to address the second point under consideration. But back to the first point: why did Epicurus think geometry was based on false assumptions in the first place?
Well, as I said before, this primarily turns on the question of infinite divisibility. In the Epicurean cosmos, there is — according to Lucretius — a deep-set boundary stone, a least possible size, a minimum. And that minimum is the atom. The idea of infinite divisibility is directly antithetical to atomism, partly because the word atom is Greek for “uncut.” Under no circumstance can the Epicurean atom be divided. This aspect of geometry — where a line segment can be subdivided continually without end — meant that geometry was incompatible with nature. And if geometry’s most basic rules do not work in nature, what possible recourse could we have to her higher-level conclusions?
The third point under consideration: Epicurus may have been annoyed that his fellow philosophers were making frivolous ethical claims based on their study of geometry. The following excerpt from an Encyclopædia Britannica article gives some idea of where things were heading:
The Pythagoreans invested specific numbers with mystical properties. The number one symbolized unity and the origin of all things, since all other numbers can be created from one by adding enough copies of it. The number two was symbolic of the female principle, three of the male — they come together in two plus three equals five, as marriage. All even numbers were female, all odd numbers male. The number four represented justice. The most perfect number was ten, because ten equals one plus two plus three plus four. This number symbolized unity arising from multiplicity. Moreover, it was related to space: a single point corresponds to one, a line to two, a triangle to three, and space or volume to four. Thus ten also symbolized all possible spaces. The Pythagoreans recognized the existence of nine heavenly bodies — sun, moon, Mercury, Venus, Earth, Mars, Jupiter, Saturn, and the so-called central fire. So important was the number ten in their view of cosmology that they believed there was a tenth body, a counter-earth, perpetually hidden from us by the sun.
Now I am familiar with the story about John Maynard Keynes purchasing a trunk of papers belonging to Isaac Newton at a Sotheby’s auction, as well as his later poetic description of what those papers implied:
Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than ten thousand years ago. Isaac Newton, a posthumous child born with no father on Christmas Day 1642, was the last wonder child to whom the magi could do sincere and appropriate homage.
— John Maynard Keynes, Newton, the Man
Look — it is all fine. But if Cicero and others are going to mock Epicurus for his alleged ignorance of mathematics, I think we should allow ourselves a moment to step back and examine the disparity between two widely divergent claims. On the one hand, you have Euclid making proposals in an abstract mental space about how many lines can be drawn through a point parallel to a given line. On the other hand — and yes, I think it matters — you have a plausible-seeming philosopher in Pythagoras using numerology in what basically sounds like a religion.
In his book Epicurus and His Philosophy, Norman DeWitt furnishes another example of the crossover between geometry and ethics:
Geometry in particular, though itself a positivistic study, inspired in the minds of men a new movement that was genuinely romantic. It was the romantic aspect of the new knowledge that captivated Plato. He was no more than up to date as a mathematician himself. In geometry he seemed to see absolute reason contemplating absolute truth — perfect precision of concept joined with finality of demonstration. He began to transfer the precise concepts of geometry to ethics and politics, just as modern thinkers transferred the concepts of biological evolution to history and sociology. Especially enticing was the concept which we now know as definition. This was a creation of the geometers. They created it by defining straight lines, equilateral triangles, and other regular figures. If these can be defined, Plato tacitly reasoned, why not also justice, piety, temperance, and other virtues? This is reasoning by analogy — one of the trickiest of all logical procedures. It holds good only between sets of true similars. Virtues and triangles are not true similars. It does not follow, therefore, because equilateral triangles can be precisely defined, that justice can be defined in the same way. Modern jurists warn against defining justice: it is what the court says it is, from time to time.
— Norman DeWitt, Epicurus and His Philosophy
But what does Plato himself say about these matters? Well, in Book Seven of his Republic, Plato has Socrates continue the dialogue by presenting the well-known allegory of the cave. In this allegory, we are to imagine a group of people chained to the wall of a cave. Out of sight of these people — perhaps on a ledge high up in the cave wall behind them — are two things: first, a fire far back in the recess of the ledge; and in front of the fire, on the edge of the ledge, there are figurines and other objects. The light of the fire casts the shadows of these objects onto the cave wall in front of the people on the floor. This little world of flickering light and shadow is all they know.
Now, we are to suppose that one of these people manages to escape the microcosm of the cave and ascend up into the world of sunlight and fresh air. For Socrates, this person represents the philosopher — one who has seen through and escaped the illusory world of sense perception, with its constant change and flux, like so many flickering shadows on the wall. He has now left the little world of endless becoming and risen up into the pure world of perfect being.
Socrates suggests that if this man — enlightened as he has become by his experience of truth — were to descend back down into the cave, he would never be able to express what he has seen. Indeed, his eyes, which were dazzled by the sun above, would take some time to adjust to the darkness of the cave, and those down below would think him blind in comparison to themselves.
It is an engaging little story. In the world of the allegory, the senses are not merely unreliable — they are presenting an illusion, indeed an outright lie. Things really start to get weird when we see that only those who can escape the cave and ascend to the world of pure being — that is, the world of the Good — should be permitted to rule in this ideal state. But here is the rub: those who are permitted to rule, having ascended to the Good, must now make a descent back into the world of the cave:
Wherefore, each of you in his turn must go down to the general underground abode and get the habit of seeing in the dark. When you have acquired the habit, you will see ten thousand times better than the inhabitants of the den, and you will know what the several images on the wall are and what they represent, because you have seen the beautiful and just and good in their truth.
For Socrates, descending into the darkness signifies a willing sacrifice on the part of the ruler — like a man with sight going among the blind in order to help them. The ruler reluctantly sets down his own desire for the upper world in order to serve the needs of those in the lower.
Socrates then moves on to the all-important question: how does one learn the way out of the cave? With his interlocutor Glaucon, he discusses several branches of study among the Greeks and judges whether each one will help the philosopher to make the ascent. The arts of war, gymnastics, and music are each discussed in turn, but all of them are found wanting on the most essential question — none of these arts may lead the philosopher from the flickering cave of becoming into the world of pure being.
It is Socrates himself who supplies the answer. Arithmetic and geometry are the arts which lead onto truth. Dispensing almost entirely with the practical aspects of these arts, Socrates sums up the whole discussion in a well-known quotation: “The knowledge at which geometry aims is knowledge of the eternal, where the senses lie. Geometry leads onto truth, where the pleasures of the body are like lead and weights dragging us into darkness.” Socrates says that geometry is a ladder to the pure realm of perfect being.
In the 2012 film Lincoln, directed by Steven Spielberg and starring Daniel Day-Lewis in the title role, there is a scene in which the president is mulling over an important decision in the telegraph room. While he is thinking, he falls into conversation with the telegraph operators about Euclid and the ancient axioms of geometry. The script was written by Tony Kushner, and in this scene Lincoln says:
Euclid’s first common notion is this: things which are equal to the same thing are equal to each other. That is a rule of mathematical reasoning. It is true because it works — has done, and always will do. In his book, Euclid says this is self-evident. You see, it is even in that two-thousand-year-old book of mechanical law: it is a self-evident truth that things which are equal to the same thing are equal to each other. We begin with equality. That is the origin, isn’t it? That balance — that is fairness. That is justice.
Now, remember what I quoted above from Norman DeWitt: “This is reasoning by analogy — one of the trickiest of logical procedures. It holds good only between sets of true similars. Virtues and triangles are not true similars. It does not follow, therefore, because equilateral triangles can be precisely defined, that justice can be defined in the same way.”
I will admit that the analogy Lincoln uses in that scene makes for really engaging cinema — but using geometry in service of ethical claims like human equality puts him on shaky ground. Justice cannot be defined in the same way as the transitive property in mathematics can be defined, and pretending that these are similar when they are not just muddies the waters around important ethical questions.
In a separate scene from that film, the president seems to agree with that view when his interlocutor in a negotiation makes an analogy by reference to a moral compass. Lincoln explains the flaws in this analogy from his own experience:
“A compass — I learned when I was surveying — it will point you true north from where you are standing, but it has no advice about the swamps and deserts and chasms that you will encounter along the way. If in pursuit of your destination you plunge ahead heedless of obstacles and achieve nothing more than to sink in a swamp — what is the use of knowing true north?”
Now, if you spend any time in the land surveying profession, you are likely to hear a canned joke about the four faces carved into the rock at Mount Rushmore. Those faces, the joke goes, represent three surveyors and one other guy. Like most jokes that circulate professionally, it is not particularly funny — it is not supposed to be. It circulates because it is a point of pride. The map is not the territory, and no one knows that better than the people who map the territory. It is one thing the Lincoln film got right. It is something we should all keep in mind.
Thank you for watching. I hope you have enjoyed this video as much as I have enjoyed making it. I have barely scratched the surface on this topic, and we will be discussing this and much more at EpicureanFriends.com.