Eudoxus: The hedonistic mathematician

"All beings, both rational and irrational, aim at pleasure." – Eudoxus

Eudoxus was an ancient Greek mathematician and philosopher. He was born circa 400 BCE in Cnidus, a Greek city-state on the south-western coast of today’s Turkey.

Diogenes Laertius — an important late antique biographer who came from the same area of Eudoxus — reports in Lives of Eminent Philosophers that when Eudoxus “was about twenty-three and in straitened circumstances, the fame of the Socratics attracted him, and he set sail for Athens.” Diogenes adds that Eudoxus traveled to Athens “with Theomedon the doctor, who supported him” and that “some say he was Theomedon’s beloved.” The word translated “beloved” is paidikos, so “lover-boy.” (DL 8.86)

In Athens, Eudoxus acquired the funding and recommendations needed to travel to go to Egypt for 16 months. When he returned, Diogenes tells us, he had shaved off both his hair and his eyebrows (in Ancient Egypt, it was common for both men and women to shave off all body-hair). Now he was also able to translate Egyptian hieroglyphs into Greek. After another tour, now along the coast of his native Turkey, Diogenes Laertius reports that he “then returned to Athens with a great many disciples” and adds that “some say that he did this to annoy Plato, who had originally ignored him.” (DL 8.87-8)

Although Eudoxus seems to have had a tense relationship with Plato, Aristotle apparently liked him a lot. Aristotle writes, in the Nicomachean ethics, that Eudoxus showed “excellence of character” and was considered “remarkably temperate” (EN X.2, 1172b15-16).¹

The mathematician

To the extent that Eudoxus is known today, it is mainly for his work in mathematics, and his applications of mathematics to geography and astronomy.

Eudoxus was:

• … the first to figure out that a year is 365 days + ¼ day.

• … the first to prove that the volume of a cylinder is three times that of a cone.

• … the first to show that any irrational number (e.g. the square root of two) can be approximated to any degree of accuracy by means of rational numbers only.

• … the first to develop the method of exhaustion, i.e. to approximate to any degree of accuracy the area of any shape by filling its inside up with smaller, simpler shapes.

• … the first to develop a theory of magnitudes, and to precisely define proportionality, transitivity, commensurability, and continuity.

The only thing Diogenes Laertius has to say about Eudoxus’ philosophy is that he “declared pleasure (hēdonē) to be the good (agathos).” (8.88)

All of Eudoxus’ own works have unfortunately been lost, so we have no other choice than to depend on the presentations of his ideas in the works of others. Whereas our main source for Eudoxus’ mathematical views is Euclid, our main source for his philosophical views is Aristotle.

The Aristotle and Eudoxus craters on the moon. (Wikimedia Commons)

Good vs. praiseworthy

Although Aristotle’s main discussion of Eudoxus’ ethics is in Book X of the Nicomachean ethics, he also brings in Eudoxus at the end of Book I. The context here is Aristotle’s discussion of whether that which is ultimately good is also praiseworthy.

Why is this an interesting discussion? Let me try to explain.

It might seem, at first glance, that if something is so good that it is, in fact, the ultimate good, then surely it must (at the very least) be something praiseworthy. This was the view held of Speusippus, an older contemporary of both Eudoxus and Aristotle who became Plato’s successor as the leader of the Academy. Later on, this has later come to be known as a Stoic and a Kantian view.

Let’s say, then, that we accept that what is ultimately good is also praiseworthy. Then we might ask: How does this fit with the view, held by both Eudoxus and Aristotle, that what is ultimately good is that which ultimately constitutes a benefit to a person? It’s not a good fit. Why is that? Because it seems strange to praise people for being benefited and, quite possibly, it seems even stranger to blame someone for being harmed.

Let’s say, for the sake of the example, that what is ultimately good is happiness. Then imagine that a poor woman, Penelope, wins the lottery. If she does, and this brings her both short-term and long-term happiness, then we might be glad on her behalf, but it would be misplaced to praise her for this. Or imagine that Anaxis gets robbed, and the robbers beat him so badly that he is barely able to survive. If he is, and this brings him less short-term and long-term happiness, then we might feel sorry for him, but it would be misplaced to blame him for this.

If the ultimate good is happiness, therefore, it seems plain that the ultimate good is not something that, in and of itself, is praiseworthy.

According to Aristotle, however, we should not expect that which is ultimately good to be praiseworthy. Why? Aristotle explains:

Anything that is praised seems to be praised for its being of a certain kind and its standing in a certain relation to something else: the just person, the brave person, and the good person and virtue in general we praise for their actions and what they bring about. And we praise the strong person, the fast runner, and each of the others, because he is naturally of a certain kind and stands in some sort of relation to something good and excellent. This is clear also from praise of the gods. For it seems absurd that they should be judged by reference to us, but this happens because, as we have said, praise involves reference to something else. But if praise applies only to things standing in relations, clearly it is not praise that applies to the best things, but something greater and better. This is in fact obvious, since the gods and the most godlike of people we call blessed and eudaimon. The same goes for things that are good, since we never praise happiness as we might justice, but rather call it blessed, as something better and more divine.

And Eudoxus seems to have been right in pressing the claims of pleasure to supremacy. He believed that the fact that it is not praised despite its being a good indicates that it is better than things that are praised; and he thought that god and the good are like this, because it is by reference to these that other goods are praised. For praise is indeed appropriate to virtue, since it makes us the kind of people to perform noble actions; eulogies, however, are bestowed on what is achieved in the spheres of the body and of the soul alike. But perhaps clarity here is more the job of those who have gone into the subject of encomiums [high praises in public speeches]. For us, anyway, it is clear from what has been said that happiness is something honourable and complete.

And that it is so seems to follow as well from its being a first principle. It is for the sake of this that we all do all the rest of our actions, and the first principle and cause of goods we take to be something honourable and divine. (EN.I.12, 1101b10-1102a4)

So according to Eudoxus — and here Aristotle agrees with him — we should not expect that which is ultimately good to be praiseworthy. Praise belongs to that which contributes to what is ultimately good. What is ultimately good, however, is beyond praise.

The hedonist

Aristotle tells us, in Nicomachean Ethics (EN), that “Eudoxus thought pleasure was the good.” (X.2, 1172b8) He then proceeds to give an account of Eudoxus’ four arguments for hedonism:

Argument #1: All seek pleasure

Aristotle explains:

Eudoxus thought pleasure was the good because he saw that all things, whether rational or not, aim at it. And in everything, he says, what is worthy of choice is good, and what is most worthy of choice is best (aristos); thus the fact that everything is borne towards the same thing shows that this is what is best for all, since each thing finds its own good, as it finds its own food; and that which is good for all things and at which all aim is the good. (EN X.2, 1172b10-14)

This argument starts with the premise that there is a tendency among all beings to seek pleasure. In support of this, it might be said that we see this tendency to seek pleasure in ourselves, in other people, and (or so it seems) in animals. If we accept this, then unless we are willing to say that all such beings are fundamentally misguided here, this then indicates that pleasure is something good. Insofar as pleasure is not just something that some beings seek sometimes, but that which most beings seek most consistently, this indicates that it is the chief good.

Argument #2: All shun pain

The second argument (in the order of Aristotle’s presentation) is as follows:

[Eudoxus] thought that the matter was just as clear if one considered it from the opposite point of view. Pain, he thought, is in itself something to be avoided by all, and therefore, similarly, its contrary is something to be chosen by all. (X.2, 1172b19-21)

At a first reading, it might seem that while Eudoxus’ first argument is an argument for the goodness of pleasure, his second argument is an argument for the badness of pain. But that’s just the first half of the argument. Sure, Eudoxus says that pain “is in itself something to be avoided by all,” and that does count in favor of its badness. But the argument is also — or, I will suggest: primarily — an argument for the goodness of pleasure.

Notice that the beginning of the argument is about pleasure (“the matter” refers back to pleasure’s goodness in the previous argument) and that the end of the argument is also about pleasure (“something to be chosen by all.”) Eudoxus point is that pleasure’s goodness can be demonstrated, not just the way he did in the first argument, but also by an alternative route: it can be inferred from pain’s badness.

This is an argument from contraries, which is an argument that seeks to establish a conclusion about something by means of an inference from what we know about its opposite.

Why this roundabout argument for the goodness of pleasure? One reason this might be a useful argument is that even if someone, despite Argument #1, still does not accept that pleasure is good, it is more difficult to deny pain’s badness than to deny pleasure’s goodness. If someone denies it, one can ask if one may stretch or pinch the skin on their forearms — or something else that is painful, yet harmless) If pain is not bad, that should not be a problem. (If they say that the mere time that it takes should be compensated, then $10 for letting a strong man strech/pinch their forearms for 5 minutes should be a bargain. That is, after all, a §120 per hour compensation.)

Argument #3: The goodness of pleasure is not instrumental

Argument #1 and Argument #2 have shown that pleasure is good. What Eudoxus wishes to argue, however, is not just that pleasure is good, but that pleasure is good in and of itself. For admittedly, something could be good without being good in and of itself: It could be good just instrumentally, e.g. in virtue of contributing to something further (such as money). To show that the goodness of pleasure is not just instrumental, Eudoxus launches his third argument:

What is the most worth choosing is what we choose neither because of nor for the sake of something else. And everyone agrees that pleasure is like this, since no one asks for justification of anyone’s being pleased, on the assumption that pleasure is worthy of choice in itself. (X.2, 1172b21-24)

What does this mean?

To get at the point that Eudoxus is making, imagine that I see that you are heading toward the marketplace. When I do, I might ask you: “What is going to the marketplace good for?” This can be a reasonable question since, presumably, there is some reason why you are going there. Let’s say that you: “To get wine.” This answer makes sense, for wine can be nice and you can get it at he marketplace. I might then continue to inquire by asking you: “But what is getting the wine good for?” This might perhaps also be a reasonable question, because it need not be obvious what the occasion is. You might answer: ‘‘Well, I want it for the pleasure of drinking it.’’ Now, if I proceed to ask “But what is the pleasure of drinking the wine good for?”, our discussion is likely to reach an awkward end. The reason for that, it seems, is that the pleasure is not good for anything further; rather, it is simply that for which going to the marketplace and getting wine is good.

Eudoxus point, then, seems to be that we regard the fact that something is pleasant as a reason for seeking it, and we do not ask what seeking pleasure, in turn, is good for. For this to be sensible, pleasure will have to be good in and of itself.

Argument #4: Good things are made even better by adding pleasure

Aristotle writes that Eudoxus

said too that pleasure, when it is added to any other good, such as acting in a just or temperate manner, makes the other good more worthy of choice; and that the good is increased by the addition of itself. (X.2, 1172b24-26)

Eudoxus’ point is that if we have something that is good, and we then add pleasure to it, that something is thereby made even better. But if adding pleasure (while keeping all else equal) is sufficient to make something better, then pleasure must be good in and of itself. If pleasure were not good in and of itself, how could adding it (while keeping all else equal) make anything better?

So these are Eudoxus’ arguments for hedonism. Or, at least, these are Eudoxus’ arguments they way Aristotle presented them in Book X of the Nicomachean ethics.

Importantly, Aristotle does not just present Eudoxus’ arguments; he also assesses them. In this post, however, I will not go into his assessments. I will deal with his assessments in my coming post on Aristotle.

This is not, however, the only place in the Nicomachean Ethics where Aristotle brings in Eudoxus. He also does so at the end of Book I.

The metaphysics of goodness

Let me now try to relate Eudoxus’ hedonism to his metaphysical views.

One of the central questions under discussion in Plato’s academy was this: What do all of the particular things that are good have in common, in virtue of which they are good?

The answer given by the mature Plato is that all of the particular things that are good are good in virtue of being manifestations of goodness as such, which he calls the Form of the Good. Goodness, then, is something that we, in contemporary philosophy, would call a non-natural fact. It is non-natural because goodness is not identical to anything particular around us. Although particular things around us might be good, goodness as such, while manifested in things that we can perceive in the world around us, is not itself something that we can directly perceive.

Aristotle is critical of Plato’s theory of Forms, including the most important of forms, which is the Form of the Good. In Metaphysics, Aristotle argues that, because of their transcendence, Plato’s Forms contribute nothing either to the being of sensible things or to our knowledge of them:

Above all one might discuss the question what on earth the Forms contribute to sensible things, either to those that are eternal or to those that come into being and cease to be. For they cause neither movement nor any change in them. But again they help in no way towards the knowledge of the other things (for they are not even the substance of these, else they would have been in them), nor towards their being, if they are not in the particulars which share in them; though if they were, they might be thought to be causes, as white causes whiteness in that with which it is mixed. [This is an] argument which first Anaxagoras and later Eudoxus and certain others used. (Met I.9, 991a = Met XIII.5, 1079b)

So here we learn that Eudoxus held that forms, rather than being transcendent, are immanent and directly available to sense-experience. This fits very well with hedonism, since if hedonism is true, then goodness is pleasure, and in that case, goodness is directly available to sense-experience.

Eudoxus and Philebus

Plato’s dialogues are typically named after one of the most prominent participants in the discussion. As readers of previous post might recall, moreover, Plato’s main attack on hedonism is found in Philebus. But who is Philebus?

The dialogue opens like this:

SOCRATES: Well, then, Protarchus, consider just what the thesis is that you are now taking over from Philebus—and what our thesis is that you are going to argue against, if you find that you do not agree with it. Shall we summarize them both?” (11a-b)
PROTARCUS: Yes, let’s do that.
SOCRATES: Philebus holds that what is good for all creatures is to enjoy themselves, to be pleased and delighted …
PHILEBUS: Absolutely, Socrates.
SOCRATES: Do you agree, Protarchus, to take over this thesis that’s now offered you?
PROTARCHUS: I am afraid I have to. Fair Philebus has given up on us.
(Philebus 11a-c)

So the view that pleasure is the good (i.e. hedonism) is presented as Philebus’ thesis, and Protarchus has come to also accept this thesis. Philebus, however, gives up on the discussion, and for this reason, the task of defending hedonism is left to Protarchus.

Although the task is left to Protarchus, however, Philebus does not disappear entirely from the dialogue. He reappears briefly to reaffirm that he is a hedonist (12a) and to ask how Socrates’ points relate to the overall discussion (18a; 18e-f; 22c). Then, finally, he appears in the middle of the dialogue to affirm that he thinks that pleasure and pain are thinks that admit of more or less (at 27e-28b).

From the latter half of the dialogue onwards, however, Philebus makes no more appearances. That, moreover, is convenient for Plato (or, in the dialogue, for Socrates), for Protarchus soon becomes convinced of everything Socrates has to say.

At the end of the dialogue, Protarchus has become Socrates’ obedient yay-sayer. On the last page of the dialogue (67a-b), Protarchus’ responses to Socrates’ claims are “You did say that” (67a1), “Very true” (67a4), “Exactly” (67a7), “Undeniably” (67a9), “Apparently” (67b1), and “We are all agreed now that what you said as as true as possible, Socrates.” (67b6)

Would Philebus have submitted so completely to Socrates? Somehow I doubt it.

The last time we hear from Philebus is at 28b - not just in the Philebus, but it any of Plato’s work. Indeed, it is the last we hear of him in any surviving text from antiquity. John M. Cooper, the editor of Plato: Complete Works, writes that “we know nothing of Philebus, apart from this dialogue” and, in fact, that “he may be purely fictional.”²

Why am I bringing this up?

Because Philebus might have been Eudoxus.³

How come? Here are three reasons for suspecting this:

First, it is strange that the name Eudoxus is not mentioned anywhere in Plato’s dialogues. Why? Because, at the time, Eudoxus was the most prominent defender of hedonism, and as we have seen, Plato discussed hedonism extensively in his work. It is very unlikely, moreover, that Plato was not aware of Eudoxus and his hedonistic views. Eudoxus was a very prominent figure in ancient Athens, ancient sources list him as one of Plato’s students (DL 8.86-88), and he even seems to have had a prominent role in the running of Plato’s academy: In Life of Aristotle, written by the Neo-Platonist Ptolemy-el-Garib circa 300 AD, Aristotle joined the Academy during the “time of Eudoxus” (tempore Eudoxi). This was prior to when Plato wrote the dialogue Philebus.⁴

If Philebus is Eudoxus, this is no longer strange. Then it is also no longer strange who this otherwise completely unknown Philebus character might have been.

Second, “Philebus” and “Eudoxus” are names (or nick names) with similar meanings. Philebus is a combination of phíl- (beloved), hḗbē (youth), and -us (masculine ending), so it means friendly young man. Eudoxus is a combination of eu- (well), dox (opinion), and -us (masculine ending), so it means a well-regarded man.

Third, all of the philosophical views that Plato attributes to Philebus are compatible with the views of Eudoxus.

So might Philebus have been Eudoxus? Most likely, we will never know for sure. It is, however, an appealing hypothesis.

1. When I refer to Aristotle’s writings, I use Bekker numbers. Unlike page numbers, Bekker numbers are the same across all scholarly editions of Aristotle’s works. My quotes from Aristotle’s Nicomachean Ethics are from Roger Crisps’s 2014 translation. My quotes from other works by Aristotle are from Jonathan Barnes’ (ed.) The Complete Works of Aristotle. Princeton University Press, 1985.

2. John M. Cooper, “Philebus” (introduction to the dialogue) in Plato: Complete Works. Hackett, 1997, p. 398.

3. I was made aware of this hypothesis in the recently (December 2023) published mini-book on Eudoxus in the Cambridge Elements in Ancient Philosophy Series. See Richard Davies, The Hedonism of Eudoxus of Cnidus (Cambridge University Press, 2023), p. 13.

4. Ptolemy -el-Garib, Life of Aristotle. Available at www.attalus.org/translate/aristotle.html