Abstract Objects and Universals

You read three apples, three stars, three beats. The threeness registered across all three without effort. You did not count each in turn.

Naming made the invariant readable. It did not create it. Three apples unseen still share threeness. What changes when a reader appears is that it is registered.

Try to deny that universals operate. The denial is a sequence of distinctions, each carrying the same kind of fact across instances. You are already inside.

An abstract object is not an object in a second realm. It is a stable relational structure readable across records without being identical to any one of them.

A universal is one kind of abstract object. An invariant readable across many records under an operation.

A property is a universal whose invariant appears through a stable coupling or response between a substrate and an operation that probes it.

A proposition is a recordable constraint on what configurations or trajectories obtain at a specified site.

The chapter earns these.

Plato: the many red things share Redness, a Form in a realm of its own, and they participate in it.

Aristotle saw the hole. How does a spatial apple participate in a non-spatial Form? His answer: universals are real, but only in their instances. Redness is in the red things.

The medieval nominalists denied universals altogether. Only particulars exist. The word red is applied to things that resemble each other. The universal is the name.

The conceptualists put universals in the mind — abstractions formed by beings meeting particulars.

Russell revived a limited Platonism. Armstrong defended universals that exist only as instantiated. Quine and Goodman reduced them to similar particulars and the predicates that group them.

The positions did not reduce to each other. The question did not close. Location is still where the disagreement lives.

Platonism captures that universals do not depend on any one instance. Destroy the three apples and threeness survives. Plato reached, rightly, for something more than the particular. He put it in the wrong place. The last chapter’s ship, wake and ocean handle this. Every ship through deep water makes the same wedge. The universality is in the structure of the wake, read across many ships. Not in a realm.

Aristotle captures that universals are not somewhere else. Redness is here, where the red things are. He erred in putting them inside the instances, like seeds in the apple. Universals are not in instances. They are between them.

Nominalism captures that nothing extra needs adding to the world. It erred in making the similarity conventional. Two red apples are alike as a fact about the apples and the architecture that registers them. The naming follows the similarity.

Conceptualism captures that universals become visible through recognition. It erred in locating them in the mind alone. Threeness obtains across three apples whether or not anyone counts. It is read only when counting is performed.

Four separations. Realm from world. Universal inside instance. Fact from convention. Mind from world. The structure contains none of them.

What is threeness?

A relation. The invariant that any system of distinguishable records carries under the counting operation.

Count the apples. Three. Count the stars. Three. Count the beats. Three. The invariant is what the operation produces. It is real because the operation is real. It is not in any one apple because relations are not in things. They are between things.

A universal is a structural invariant that any system of distinguishable records carries under a given operation.

Threeness under counting. Sphericity under rotation — the shape preserved under every turn about its centre. Primeness under division. Redness under a particular coupling of light, pigment and eye.

Particulars. An operation. An invariant the operation produces across them. The invariant is the universal. None of the three is in a separate realm. None is in any single particular.

In 1915 Emmy Noether proved, and in 1918 published, that every continuous symmetry of a physical system corresponds to a conserved quantity. The laws are the same here as a metre to the left: momentum is conserved. The same now as yesterday: energy is conserved. The same under rotation: angular momentum.

The symmetry is the operation. The conserved quantity is the invariant. It is real — measured, calculated, used in every prediction about a closed system. And it is not somewhere. Energy is not at a site. It is what the symmetry, applied to the system, produces.

Not a metaphor. A proven physical instance of the same form. The chapter’s claim is that universals share the structure of what Noether proved.

Not every predicate names a clean universal.

Red names a stable coupling invariant. Red or square names a union of two invariants under a policy the speaker chose. Not red names exclusion from an invariant. Taller than names a relation between particulars. Unicorn names a fictional record-pattern — real within a specified record-set, not as a biological universal.

Where the predicate is a policy, a disjunction or an exclusion, the structural reading still applies. The policy is what is being read.

Wittgenstein’s game is the hard case. Chess, football, solitaire, ring-a-ring-o’-roses share no single feature, only overlapping resemblances. Game is not one invariant. It is a policy that gathers several — competition, rule-bound piece-manipulation, group physical activity — because the network of overlaps is itself a real feature of the instances. Each overlap is a clean invariant. The network is real.

The distinction shows the location question was malformed from the start. It presupposed that universals are uniform single things with places. They are invariants, networks, policies, exclusions. None has a place.

Ask where threeness lives and you have asked where between lives.

Between is not anywhere. It is the relation between two things — everywhere they are, and nowhere else. Grounded in the relata. Not floating free. Not inside either one.

Threeness is the relation that obtains across three distinguishable records when they are counted. Everywhere they are counted. Nowhere else.

Located things are records. Universals are not records. They are relations between records, and relations do not sit at sites. They hold between them. Not unreal. Non-site-local.

Two and a half thousand years of debate, structured around a question with no well-formed answer. The right question is not where but what. Answer the what and the where dissolves.

Redness is not something the apple has alone.

In darkness it is not visible. Before an architecture with no eyes it is not registered. Before one that reads only ultraviolet it is not the property read.

Three things are involved. Wavelengths reaching the surface. Pigments that absorb some and reflect others. An architecture that responds to the reflected light with a signal.

One precision. The apple’s reflectance does not wait for an eye. That disposition is in the surface whether anyone looks. Redness as a property is not the surface alone. It is the stable coupling between disposition, illumination and a registering architecture. Not dependent on minds. Not collapsed into reflectance. The coupling pattern.

Two classes. Response properties — hardness under force, solubility in a solvent, mass under inertia — are read when an operation probes a disposition; the apparatus is the registering architecture. Registration properties — colour, smell, taste, pitch — are read by perceptual architectures through specific channels. Both are coupling patterns. The couplings differ.

Intrinsic properties are not exceptions. An electron’s mass travels with the electron. It is not hiding inside it. Intrinsic means structure that travels, not structure at a site.

This is why properties feel stubborn. The couplings are real. The materialist grabs one pole — the wavelength, the pigment, the firing — and calls it the property. The idealist grabs the other. The property is the pattern across them.

A proposition is what a sentence expresses. The apple is on the table. Seven is prime. Different sentences in different languages express the same one. It is not the sentence and not the speaker’s mental state. The tradition put it in a separate realm. Same dispatch.

One piece of vocabulary. A trajectory is a sequence of records linked by R. A trajectory-space at a site is the set of trajectories the substrate could carry, given what C permits. Trajectory-weights are which of those are still open and which are foreclosed.

A proposition is a recordable constraint on a space.

The apple is on the table records that, in the apple’s trajectory-space at that time, only trajectories with the apple on the table are open. True if the actual weights match. False if they diverge.

Seven is prime records that, in the operation-space of division, only one and seven return. True because the operation returns exactly those, at any site.

The apple could have been on the chair records that some trajectory, under a specified counterfactual variation, is open. Modal content. A later chapter.

Sherlock Holmes lives at 221B Baker Street records, inside the Holmes corpus, what that corpus assigns. True within the fiction. No referent outside it.

English, German and Italian sentences for the apple on the table are three records of one constraint. One proposition, many carriers, no realm.

Truth is correspondence between the proposition’s record and the actual structure at the relevant evaluation-space. Trajectory-weights for empirical claims. Stability under the operation for mathematical ones. Coherence inside the record-set for fiction.

The redness of red is the invariant across every instance of the coupling. Real. Not located. Read.

The primeness of primes is the invariant under division. Seven bears it. Eight does not. Not in seven. In the relation seven bears under the operation.

The truth-value of the apple is on the table is the relation between the recorded constraint and the actual weights. Not in the proposition, the apple, the table or the speaker. In the relation.

Modal propositions need an account of counterfactual variation. Chapter Eight.

Fiction-internal truth and truth about fictions nest differently. Not worked through here.

Higher mathematics extends the same machinery to richer operation-spaces. The formal papers.

Normative propositions — what ought to be — are the most open. Whether they reduce to constraints on trajectory-space is taken up in the closing chapter, on suffering and evil. This chapter does not commit either way.