Paimana

Notes on individuation, from the Shifāʾ to the singlet state

Horseness, in itself, is nothing but horseness.
— Avicenna, al-Shifāʾ

I

The question of individuation entered physics as an accounting error. Asked for the entropy of a gas, classical statistical mechanics counts arrangements: this molecule here with this velocity, that one there with that. The counting presumes that the molecules are individuals in the strong sense, such that a configuration in which molecule a occupies the left half of the box while b occupies the right differs, as a state of the world, from the configuration in which the two have traded places. Grant the presumption and a difficulty follows: if the partition between two samples of the same gas at the same temperature and pressure is removed, the theory reports that entropy has increased, though no instrument, and no sequence of instruments, will ever detect that anything happened. Gibbs repaired the books by dividing the count of arrangements by the number of possible permutations, in effect ruling that the exchange of two identical molecules is not a change. The repair was presented as bookkeeping. It was in fact a metaphysical concession, entered quietly and in the currency of a factorial: swapping a and b yields one and the same state of affairs, from which it follows that the labels were never attached to anything.

Quantum mechanics converts the concession into a principle. The states available to identical particles are, by construction, indifferent to permutation; exchanging the labels is not a physical operation the world can notice, and where Gibbs’s division had the look of convenience, this indifference is a structural fact about the state space itself. To see what the principle costs, consider the two electrons of helium in its ground state. Their spins are locked in the singlet configuration, which one writes, in a notation that will turn out to be the whole problem, as a superposition in which “the first” is up where “the second” is down, minus the reverse. Neither electron has a spin direction of its own. Ask after either one individually and the theory returns the identical answer for both, a perfectly mixed condition, up and down in equal measure. Their spatial situation is likewise common property: the same orbital, the same cloud of position. Every intrinsic property is shared by definition, since sharing mass, charge, and magnitude of spin is what makes them electrons; every state-dependent property is shared by the mathematics; and the relations each bears to the rest of the universe are the same relations.

The classical principles of individuation may be tested against this pair in turn. Trajectory first: there are no continuous paths, and when two wave packets overlap and then separate there is no fact of the matter about which particle emerged where; the electron detected on the left is not the one that entered from the left, nor the other one, because “the one” has quietly ceased to refer. Location next: the electrons of the singlet occupy the same spatial state. Impenetrability, the old assurance that two bodies cannot be in one place, was the ground on which the schoolmen rested individuation by matter under dimensions; it fails spectacularly, since bosons can be piled into a single state in any number whatever, and are, in every condensate a laboratory has ever cooled. Properties last: the bundle of accidents by which one might have hoped to tell the two apart is, in these states, a single bundle, held in common. What remains, once trajectory, place, and property have all been surrendered, is the question the tradition had been asking before any of these answers was tried.

II

The question has a name, mā bihi al-tashakhkhuṣ: that by which a thing is this thing rather than merely a thing of its kind. The question owes its sharpest formulation to Avicenna. The quiddity taken in itself, māhiyya min ḥaythu hiya hiya, is neither universal nor particular, neither one nor many; horseness is only horseness. In other words, nothing in what a horse is determines that there should be this horse, or that one, or any at all. Universality is something that happens to horseness in minds, particularity is something that happens to it in the world, and both happenings arrive from outside. The analysis carries an immediate consequence: if the quiddity is indifferent to being one or many, then no inventory of the quiddity, however exhaustive, will ever yield thisness, and individuation must be purchased elsewhere. Avicenna canvasses the familiar candidates: matter, as that which receives the form and can receive it twice; position, since two bodies exclude one another from a place; the hour of a thing’s origination; and the retinue of accidents (this color, this size, this scar), whose full aggregate no second individual, it is hoped, could exactly share.

An accident, though, inheres in a substance; for the scar to be this scar it must already be the scar of this horse; the accidents presuppose the very individuation they were hired to explain. Avicenna, to his credit, saw the circle. Pressed, he demotes them from causes to signs, ʿalāmāt, marks by which we tell individuals apart rather than that by which they are apart. The demotion is honest, and it is a surrender: the question concerned what makes individuals individual; how we discern them is a different question, and to answer the first with instruments fit only for the second is to change the subject. Suhrawardī, pressing the Peripatetics a century and a half later, made the objection a refrain: whatever individuates cannot itself be one more universal notion appended to universal notions, since a heap of the general is general still; the concrete is not assembled out of kinds. The Latins, working the same Aristotelian inheritance toward the same impasse, produced what is recognizably the same shortlist: matter under designated dimensions, Scotus’s haecceitas, Leibniz’s complete concept. The options are few; they were discovered early; each is circular, or brute, or both. The shortness of the list measures the hardness of the question.

What the tradition lacked was a case: its disputes were conducted over horses, coins, and souls, individuals so thickly dressed in accidents that the bundle always sufficed in practice. The decisive experiment, in which two individuals would share every property down to the last, monadic and relational alike, could only be imagined; Max Black would later imagine it as two iron spheres alone in a symmetrical universe. Nature, it turns out, had been running the experiment all along, in every atom past hydrogen.

III

The full accounting was carried out only in 1988, by Steven French and Michael Redhead. In the symmetrized states that quantum mechanics assigns to identical particles, the two share every monadic property, as the singlet already showed; but they also share every relational property, since whatever relation the first bears to anything, the second bears as well; the symmetry of the state guarantees it. The Principle of the Identity of Indiscernibles, Leibniz’s wager that two things must differ somewhere, therefore fails in every one of its versions: no intrinsic feature tells the electrons apart, and no relation to any third thing does either. If an individual is nothing over and above its properties, then the singlet contains one electron and it contains two, and the number two is doing work that no property funds. That is the end of the bundle theory of individuation.

A single possibility remained, and Simon Saunders identified it. There is a relation each electron bears to the other and neither bears to itself: having opposite spin to. The relation discerns nothing, in the sense that it does not sort the pair into the up one and the down one, there being no up one; but it is irreflexive, and an irreflexive relation cannot be instantiated in a domain of one. Borrowing Quine’s grades of discriminability, Saunders called the pair weakly discernible: not absolutely, by some predicate true of one and false of the other; not relatively, by some relation holding of them in one order and failing in the reverse; but weakly, by a relation that each bears to the other and neither could bear to itself. This is individuation of a kind the old shortlist never contained. The scholastic candidates had all been monadic (a designated matter, a bundle, a bare thisness) or at furthest relational to something third, as position is a relation to place and origination a relation to an hour. Weak discernibility is dyadic and mutual, and it stops deliberately short of the very thing the tradition assumed individuation must deliver: no fact settles which electron is which; the facts run out at the number two. (The rescue holds for fermions; Muller and Saunders later secured it for every state fermions can occupy. For bosons it remains contested, since the candidate relations are accused of smuggling in the distinctness they were meant to establish. I keep to electrons.)

What, then, of haecceity, the Scotist’s bare thisness? It survives, in the way a currency survives that nothing can be bought with. One may posit that each electron possesses a primitive identity, what French and Redhead called transcendental individuality, and the physics will leave the posit untouched and unemployed. Quantum statistics counts unlabeled states, and the haecceitist must therefore explain why nature’s counting never once notices the labels. The explanation can only take the form of a restriction: none but permutation-invariant states occur. The restriction’s sole function is to conceal the haecceities from every consequence they might have had. The wheel is retained; it turns nothing. Each of the modern positions is, on inspection, a scholastic position handed a laboratory: the bundle theorist refuted, the Scotist kept on without duties, the only new appointment a relationalist of a kind the tradition never thought to interview.

IV

A further step remains, and Paul Teller took it. Reading quantum field theory, he proposed that the deep description trades particles for quanta, entities that can be counted but not labeled. A field mode carries an occupation number (there are three quanta of such-and-such a momentum), and the number is a cardinality without a roster; one can say how many without there being any fact as to which. Aggregation without enumeration. In the field picture the electron is an excitation of the one electron field, a way the field locally is, and the innumerable electrons of the world are so many modes of a single bearer. Wheeler once put the limit case as a joke, telephoning Feynman to announce that all electrons have the same mass and charge because they are all the same electron, threading forward and backward through time. The mechanism was a joke; the content survives it. Perfect identicality is no mystery at all if “they” were never many things to begin with.

A reader of Mullā Ṣadrā will experience, at this point, a suspicious warmth. Ṣadrā’s doctrine, whose germ he credits to al-Fārābī, is that individuation is by existence, tashakhkhuṣ bi-l-wujūd. The argument begins exactly where Avicenna began, with the horseness that is only horseness, and turns the principle against its author. If the quiddity is indifferent to one and many, then nothing on the side of quiddity can ever close the distance to thisness: not matter, which is a quiddity; not the accidents, which are quiddities; not position or the hour, which are quiddities of relation. Adding indifference to indifference yields indifference. What individuates is the act of being itself; it was never one more ingredient in the recipe. For a thing to be individuated and for it to exist are one and the same; wujūd is the principle of individuation rather than its possessor. Quiddity is what the mind skims off the concrete act after the fact, which is, in outline, what the primacy of existence, aṣālat al-wujūd, amounts to. And since existence, for Ṣadrā, is a single reality admitting of degrees — this is tashkīk al-wujūd, gradation by intensity and weakness and by priority and posteriority, whose way-station was Suhrawardī’s lights differing by perfection and deficiency alone — the world comes out as one reality variously intensified, a single act in many grades, where the warehouse of substances, each carrying its own papers, used to stand. One field, many modes; one wujūd, many grades. The ease of the formula is itself grounds for suspicion.

V

By this I do not mean that quantum field theory confirms Mullā Ṣadrā; the distance between a rhyme and an argument remains to be measured.

The first part of that distance lies within the physics itself, which does not even confirm the field picture, if by confirmation one means selection. Bohmian mechanics, which is empirically equivalent to the standard theory in the non-relativistic regime, retains particles in the fullest classical sense: definite positions at all times, continuous trajectories that never intersect in configuration space, so that each electron is reidentifiable from moment to moment and individuated as securely as any billiard ball. The statistics come out identical. This is French and Krause’s own sober verdict after four hundred pages: the formalism underdetermines the metaphysics; a package with individuals and a package without both fit the phenomena, and the choice between them is made with philosophical instruments; no laboratory settles it. What physics has discovered is that electrons need not be individuals, a different and smaller thing than discovering that they are not.

What, then, has actually happened to the shortlist? One position has been bankrupted within its jurisdiction: individuation by the bundle of accidents fails for electrons, exactly and demonstrably, and a principle proposed as universal does not survive a domain in which it is false. The bundle may go on sorting horses; it is no longer that by which things as such are individual. A second position, haecceity, has been rendered idle. A third, weak discernibility, has been invented outright. And existence has not been crowned, because existence was never on the ballot; nothing in the formalism quantifies being by intensity. The difficulty extends beyond the missing measure. Ṣadrā’s wujūd is intensive, hierarchical, and oriented: its grades are ranks, ordered from the Necessary downward; its individuals are degrees of a reality that admits more and less; and its physics, the doctrine of substantial motion, is a physics of ascent. The quantum field is none of these things. Its modes differ in energy and in number; a state of three photons is not more intensely existent than a state of one; nothing in Fock space ranks its occupants by ontological dignity. To read tashkīk into the occupation numbers would be concordism, the endemic vice of the genre this essay is written in, whereby every equation is discovered to have been a verse all along. A piece built on this adjacency owes its reader the disclaimer at full volume. What the physics supplies is a constraint: any account of what makes a thing itself must henceforth cohere with permutation invariance, with the failure of the bundle, with the idleness of labels. Ṣadrā’s account satisfies the constraint; so do others. The physics offers nothing more than the constraint, and nothing less.

VI

None of this was, for the tradition, a question about electrons. Its urgent case was the middle one, ourselves. Avicenna lets the human soul be individuated at its origination by the body prepared for it; since the soul survives the body, he then owes an account of what keeps disembodied souls distinct once the matter that distinguished them is gone. Before it ever concerned electrons, the question of individuation was a question about the afterlife of thisness. But the tradition also knew a limit case at the top. The separate intellects, the angels of the philosophers, have no matter, and matter was the principle by which a form could be received twice; each intelligence is therefore the sole member of its species, a kind exhausted in a single instance, individuated, if the word still applies, by having nothing to be distinguished from. The electron is the same failure inverted. Its species has members beyond all counting and nothing whatever to distinguish them: no matter of its own, no accident unshared, no label the world will honor. The angel cannot be many; the electron cannot properly be several. Individuation fails at the summit for want of anything to multiply the form, and at the floor for want of anything to divide it. Between the two failures lies the middle kingdom of horses, scars, and souls, where the question was first asked. It still stands roughly as Avicenna left it: outside the quiddity, beyond the accidents, and now, on the best evidence, beyond the properties altogether. Ṣadrā wagered that the genitive in that by which is empty, the concrete never having been assembled from ingredients in the first place. The physics cannot confirm the wager. It has only made the older answers harder to give.