Could there be two things that share every qualitative feature, with the same shape, the same mass and the same relations to everything else, yet still remain two? The question sounds idle, but it probes the deepest commitments of any theory of individuals. If the answer is no, then what a thing is like settles which thing it is. If the answer is yes, numerical distinctness floats free of qualitative character altogether.

Leibniz answered no. His identity of indiscernibles holds that qualitative indiscernibility entails numerical identity. On this view there cannot be two substances that differ solo numero, by number alone. Three centuries of debate have tested the principle against thought experiments and, more surprisingly, against physics.

This article examines the principle in three stages: its precise formulation, Max Black's celebrated counterexample of two spheres, and the challenge posed by quantum particles. At each stage the underlying issue is the same. Is individuality grounded in qualities and relations, or does it require something further, such as a bare particular or a primitive thisness, that no description can capture?

Begin with a principle almost no one disputes. The indiscernibility of identicals says that if x and y are identical, they share all properties: x = y → ∀F(Fx ↔ Fy). This is close to analytic. If Cicero is Tully, nothing true of Cicero can be false of Tully. Apparent exceptions arise only in opaque contexts involving belief or description, and these do not express genuine properties of the object.

The identity of indiscernibles is the converse: ∀F(Fx ↔ Fy) → x = y. Everything turns on what the quantifier ranges over. If F includes identity-involving properties such as being identical to a, the principle is trivially true, since only a has that property. The interesting versions restrict F to qualitative properties, whose specification makes no reference to particular individuals. Even then a choice remains. A strong version admits only intrinsic qualities. A weaker version also admits relational ones, such as being three metres from a red cube.

Leibniz intended something substantive. Each substance, on his view, has a complete concept containing everything true of it, so distinct substances must have distinct concepts. He reportedly challenged a courtier to find two leaves exactly alike in a palace garden. More consequentially, the principle underwrites his relationalism about space in the correspondence with Clarke. A universe shifted wholesale five metres east would be indiscernible from ours. By Leibniz's principle it would therefore be the same universe, and absolute space, which distinguishes the two, posits a difference that makes no difference.

The principle is also bound up with Leibniz's principle of sufficient reason. God could have no reason to place one of two indiscernibles here and the other there, so no such choice could arise. The identity of indiscernibles therefore functions less as an isolated axiom than as the keystone of a metaphysics in which every fact about individuals is ultimately a qualitative fact.

Takeaway

The identity of indiscernibles is only as interesting as its restriction on properties. Admit identity-involving properties and it becomes trivial; restrict it to intrinsic qualities and it becomes easy to doubt.

In his 1952 dialogue, Max Black asks us to conceive a universe containing nothing but two iron spheres. They have the same diameter, temperature, colour and chemical composition, and they lie some fixed distance apart. Every intrinsic property is shared. Every relational property is shared too: each sphere is the same distance from an iron sphere of exactly that kind. No observer exists to name one sphere or point to it. If this world is possible, then two things can be indiscernible even when relations are included, and the principle fails as a necessary truth.

Defenders have several replies. Ian Hacking argued that the scenario is underdescribed. The same qualitative facts could be described as a single sphere in a curved, non-Euclidean space, where travelling far enough in one direction returns you to the sphere you left. Nothing in the qualitative description decides between the one-sphere and two-sphere readings. On this view, Black's insistence that there are two quietly assumes what is in dispute.

A second reply, drawing on Quine, introduces weak discernibility. The spheres are not absolutely discernible, since no formula holds of one and fails of the other. But they stand in an irreflexive relation, being at non-zero distance from each other, that neither bears to itself. Reformulate the principle in these terms and Black's world satisfies it. Critics answer that the relation seems to presuppose the numerical distinctness it claims to ground, which makes the defence look circular.

Robert Adams pressed the other way. Imagine a near-variant of Black's world in which one sphere carries a tiny blemish, then remove the blemish. The unblemished sphere seems to persist as the same individual. This suggests that distinctness is grounded in a primitive thisness rather than in qualities. The case is therefore less a refutation than a choice point between two metaphysics of individuation.

Takeaway

Black's spheres do not so much refute the principle as force a choice: either numerical distinctness is primitive, or conceivability is a poor guide to how many things a world contains.

In classical physics, particles of the same kind can always be told apart by their trajectories, since two particles cannot occupy the same place at the same time. Quantum mechanics removes this resource. Particles of the same type are 'identical' in the physicist's sense: their joint state must be symmetric or antisymmetric when the particles are swapped. Consider two electrons in the ground state of helium, with spins forming an antisymmetric singlet.

Steven French and Michael Redhead argued in 1988 that such particles violate even weak qualitative versions of the principle. Each electron's reduced state is the same, so every one-particle observable has the same expectation value for both. Their relational properties also match, since each relates to the other exactly as the other relates to it. Unless one posits something beyond the formalism, a transcendental individuality, the electrons appear to be two yet wholly indiscernible.

Simon Saunders replied with weak discernibility. In the singlet state, the electrons stand in the symmetric, irreflexive relation has spin opposite to. Along any axis, each has spin opposite to the other's, but nothing has spin opposite to itself. Fermions, on this view, are weakly discernible, and a suitably weakened principle survives. Later work by Muller and Seevinck argued that relations defined through the formalism extend weak discernibility even to bosons, though this remains contested.

The alternative is more radical. Perhaps quantum particles are not individuals at all, and identity simply does not apply to them in the usual way. French and Krause develop this idea using quasi-set theory, and ontic structural realists treat the relations themselves as fundamental. Either way, the original principle cannot be saved intact. It must either be weakened or confined to entities that quantum theory does not describe.

Takeaway

Saving Leibniz's principle may mean accepting objects that are individuated only relationally: genuinely distinct, yet with nothing intrinsic to tell them apart.

The first lesson is that formulation decides almost everything. The trivial version, which admits identity properties, holds. The strong version, restricted to intrinsic qualities, is widely abandoned. The contested ground lies between them, where relations are admitted and the kind of discernibility matters.

Behind the technical options sit two pictures of individuals. On one, a complete qualitative description fixes how many things there are. On the other, distinctness is primitive and can outrun any description. Black's spheres and helium's electrons both press on this boundary.

Weak discernibility offers a middle path that grounds distinctness in irreflexive relations. Whether a relation can ground the very plurality it seems to presuppose is the question that remains.