The Problem of the Many may also be a consequence of the significant use of set theory in analytic philosophy along with the view that inanimate "composite objects" are nothing but "simples arranged object-wise," as Peter van Inwagen has maintained.

Van Inwagen criticized the tendency of metaphysicians to pick out selected "parts" or even just some properties of an object and claim to see another individual, as the Stoic Chrysippus did in his so-called "Growing Argument."

Recall that the Skeptics accused the Stoics of putting two entities at the same place and the same time, making us all double. Now this was only because the Stoics distinguished the substance (οὐσίας) or substrate (ὑποκείμενον) from the "peculiarly qualified individual" (ἰδίος ποιὸν), much as Aristotle saw a man as a combination of matter and form, body and mind.

Plutarch says if the Stoics add two individual qualifications to one and the same substance, there could also be three or four or more...

(1) One can hear them [the Stoics], and find them in many works, disagreeing with the Academics and crying that they confuse everything by their 'indiscernibilities' and force a single qualified individual to occupy two substances. (2) And yet there is nobody who does not think this and consider that on the contrary it is extraordinary and paradoxical if one dove has not, in the whole of time, been indiscernible from another dove, and bee from bee, wheat-grain from wheat-grain, or fig from proverbial fig.

Adding a second individual to the same substance may refer to the puzzle of Dion and Theon?
And here Plutarch anticipates the modern problem of the many

(3) What really is contrary to our conception is these people's assertions and pretences to the effect that two peculiarly qualified individuals occupy one substance, and that the same substance which houses one peculiarly qualified individual, on the arrival of a second, receives and keeps both alike.
For, if two, there will be three, four, five, and untold numbers, belonging to a single substance; and I do not mean in different parts, but all the infinite number of them belonging alike to the whole.

(Plutarch, Moralia, Against the Stoics on Common Conceptions 1077c—E, in The Hellenistic Philosophers, A. Long and D. Sedley, p.171)

(Essay Concerning Human Understanding, Of Identity and Diversity, Book II, ch xxvii)

The Problem of the Many is mostly associated with the modern metaphysician Peter Unger. who named it in 1980, and Peter Geach, who the same year showed how his metaphysical cat Tibbles could be reimagined as 1,001 numerically distinct cats by plucking each of 1,000 cat hairs.

Losing hairs reminds us of a variation of the Sorites puzzle of the heap of grains of wheat. It asks for the exact moment that a man becomes bald as his last few hairs fall out.

David Wiggins tells us that Geach's first version of Tibbles was as a cat that loses just one part, his tail, in an update of the " body-minus" problem of Dion and Theon,

If we remove something inessential (say one water molecule from a cloud, one hair from the second Tibbles, a foot from Dion, a tail from the first Tibbles, a leg from Descartes, or replace one plank in the Ship of Theseus), do we have a new entity, as the Academic Skeptics first argued?

Can we discover a criterion of parthood that makes some "proper parts" mereologically essential to the identity of the whole?

If we could, that would stop merely dialectical claims about different sets of the simplest components of a material object that are picked out by a metaphysician to start an argument. Van Inwagen attacks this as the "Doctrine of Arbitrary Undetached Parts."

Unger and van Inwagen independently came up with the extreme opposite position from the Problem of the Many, which became known as "mereological universalism," the belief in the existence of arbitrary "mereological sums." Give a set with a large number N of simple members, the Problem of the Many suggests that the N! different combinations of those members composes a new object.

In 1980 Peter Unger formulated what he called "The Problem of the Many." It led Unger to propose that nothing exists and that even he did not exist, a position known as nihilism.

Today this is the metaphysical problem of material composition and of vagueness.

let us start by considering certain cases of ordinary clouds, clouds like those we sometimes seem to see in the sky.

As often viewed by us from here on the ground, sometimes puffy ‘‘picture-postcard’’ clouds give the appearance of having a nice enough boundary, each white entity sharply surrounded by blue sky. (In marked contrast, there are other times when it’s a wonder that we don’t simply speak singularly of ‘‘the cloud in the sky’’, where each visible cloudy region runs so messily together with many other cloudy ‘‘parts of the sky’’.) But upon closer scrutiny, as may happen sometimes when you’re in an airplane, even the puffiest, cleanest clouds don’t seem to be so nicely bounded. And this closer look seems a more revealing one. For, as science seems clearly to say, our clouds are almost wholly composed of tiny water droplets, and the dispersion of these droplets, in the sky or the atmosphere, is always, in fact, a gradual matter. With pretty much any route out of even a comparatively clean cloud’s center, there is no stark stopping place to be encountered. Rather, anywhere near anything presumed a boundary, there’s only a gradual decrease in the density of droplets fit, more or less, to be constituents of a cloud that’s there.

With that being so, we might see that there are enormously many complexes of droplets, each as fit as any other for being a constituted cloud. Each of the many will be a cloud, we must suppose, if there are even as many as just one constituted cloud where, at first, it surely seemed there was exactly one. For example, consider the two candidates I’ll now describe. Except for two ‘‘widely opposing’’ droplets, one on one side of two overlapping cloudy complexes, way over on the left, say, and another way over on the right, two candidate clouds may wholly overlap each other, so far as droplets goes. The cited droplet that’s on the left is a constituent of just one of the two candidates, not a component of the other; and the one on the right is a component of the other candidate, not the one first mentioned. So each of these two candidate clouds has exactly the same number of constituent droplets. And each might have exactly the same mass, and volume, as the other.

(Mental Problems of the Many. Oxford Studies in Metaphysics, 23, Chapter 8. p.197.

In his 1990 book Material Beings, Peter van Inwagen said Unger's original insight that there are many ways to compose a cloud from innumerable water droplets should be called"mereological universalism.

Van Inwagen denies there is any way for simples to compose anything other than themselves, which van Inwagen calls "mereological nihilism.

Geach worked on problems of identity and some time in the early 1960's reformulated Chrysippus's ancient problem of Dion and Theon as "Tibbles, the Cat."

In 1968, David Wiggins described Geach's first version of Tibbles. Where Theon is identical to Dion except he is missing a foot, we now have a cat named Tibbles and a second cat named Tib who lacks a tail.

In 1980, Geach repurposed his metaphysical cat Tibbles. Geach's second version of Tibbles is widely cited as a discussion of the problem of vagueness or what Peter Unger in the same year called the Problem of the Many.

If a few of Tibbles' hairs are pulled out, do we still have Tibbles, the Cat? Obviously we do. Have we created other cats, now multiple things in the same place at the same time? Obviously not.

Geach argues that removing one of a thousand hairs from Tibbles shows that there are actually 1,001 cats on the mat. He writes:

The fat cat sat on the mat. There was just one cat on the mat. The cat's name was "Tibbles": "Tibbles" is moreover a name for a cat.—This simple story leads us into difficulties if we assume that Tibbles is a normal cat. For a normal cat has at least 1,000 hairs. Like many empirical concepts, the concept (single) hair is fuzzy at the edges; but it is reasonable to assume that we can identify in Tibbles at least 1,000 of his parts each of which definitely is a single hair. I shall refer to these hairs as h₁, h₂, h₃, . . . up to h_1,000.

Now let c be the largest continuous mass of feline tissue on the mat. Then for any of our 1,000 cat-hairs, say h_n, there is a proper part c_n of c which contains precisely all of c except the hair h_n; and every such part cn differs in a describable way both from any other such part, say c_m, and from c as a whole. Moreover, fuzzy as the concept cat may be, it is clear that not only is c a cat, but also any part c_n is a cat: c_n would clearly be a cat were the hair h_n plucked out, and we cannot reasonably suppose that plucking out a hair generates a cat, so c_n must already have been a cat. So, contrary to our story, there was not just one cat called 'Tibbles' sitting on the mat; there were at least 1,001 sitting there!

All the same, this result is absurd. We simply do not speak of cats, or use names of cats, in this way; nor is our ordinary practice open to logical censure. I am indeed far from thinking that ordinary practice never is open to logical censure; but I do not believe our ordinary use of proper names and count nouns is so radically at fault as this conclusion would imply.

Everything falls into place if we realize that the number of cats on the mat is the number of different cats on the mat; and c₁₃, c₂₇₉, and c are not three different cats, they are one and the same cat. Though none of these 1,001 lumps of feline tissue is the same lump of feline tissue as another, each is the same cat as any other: each of them, then, is a cat, but there is only one cat on the mat, and our original story stands.

Thus each one of the names "c₁ ; c₂, . . . c_1.000 or again the name "c", is a name of a cat; but none of these 1,001 names is a name for a cat, as "Tibbles" is. By virtue of its sense "Tibbles" is a name, not for one and the same thing (in fact, to say that would really be to say nothing at all), but for one and the same cat. This name for a cat has reference, and it names the one and only cat on the mat; but just on that account "Tibbles" names, as a shared name, both c itself and any of the smaller masses of feline tissue like c₁₂ and c₂₇₉; for all of these are one and the same cat, though not one and the same mass of feline tissue. "Tibbles" is not a name for a mass of feline tissue.

As David Wiggins has argued, we only have relative identity between any two objects

So we recover the truth of the simple story we began with. The price to pay is that we must regard " is the same cat as " as expressing only a certain equivalence relation, not an absolute identity restricted to cats; but this price, I have elsewhere argued, must be paid anyhow, for there is no such absolute identity as logicians have assumed.

(Reference and Generality, 3rd edition, p.215.)

The Problem of the Many, Stanford Encyclopedia of Philosophy