Abstract and Concrete

The mug is somewhere. The number seven is not. That difference feels obvious, and almost every dispute in the rest of this subject leans on it: whether mathematical objects exist, whether properties are the kind of thing that can be seen, whether minds are physical, whether God could be located. It is therefore uncomfortable that no definition of the difference has ever commanded agreement.

The paradigms

Both definitions are evasive, and deliberately so. They say what sort of thing each is by pointing at examples, which is enough to be going on with and is not an analysis. David Lewis called this the way of example, and observed that it works by hoping the sense of the distinction will emerge from the list. It usually does emerge. The trouble comes when a case arrives that the list does not settle.

The standard analysis

The most-used attempt defines the abstract by what it lacks, an approach Lewis called the way of negation. Two lacks do the work.

Put together, the account says an object is abstract if and only if it is non-spatial and causally inefficacious. The Stanford Encyclopedia entry on abstract objects calls this the standard account, and it earns that status by getting the paradigms right. Seven is nowhere and does nothing. The mug is somewhere and does a great deal.

Two features of the account are worth noticing before it is criticised. The definition is negative all the way through, so it tells you nothing about what abstract objects are like, only about what they are not. And it makes the distinction exhaustive and exclusive by fiat, which means any hard case has to be forced into one side or the other rather than being allowed to sit between them.

Where it breaks

The counterexamples are well known and none of them is exotic.

Consider a novel. Middlemarch is not identical to any particular printed copy, since the copies can be destroyed without the novel ceasing to exist, and it is not located anywhere in the way a copy is. So it counts as abstract. But it was created at a particular time by a particular person, and it has affected a great many readers, so it is neither timeless nor causally inert. Either the account is wrong or novels are not abstract, and neither option is comfortable.

Consider impure sets. The set containing exactly your two hands has concrete members, and sets are the paradigm abstract objects. Is that set located where your hands are? The account has to say no, which makes the set's relation to its members obscure.

Consider spacetime itself. It is not located in space and time, since it is what locations are locations in. Calling it abstract on that ground is plainly wrong.

Nearby distinctions it is not

A good deal of confusion comes from running the abstract and concrete distinction together with others that cut across it. Lewis called this the way of conflation, and it is worth separating the two most common cases.

The first is the distinction between the repeatable and the unrepeatable, which the ontology ladder treats under universals. These are not the same distinction. A property might be abstract and repeatable, but so might a set be abstract and unrepeatable, since there is only one set of your two hands. Being unrepeatable does not make something concrete.

The second is the following distinction, which is genuinely useful and is genuinely not the same one.

Types are often abstract and tokens are often concrete, which is why the two distinctions get run together. The correlation is not perfect. A performance of the Ninth is a token and is concrete. A printed copy of Middlemarch is a token and is concrete. But the type they are tokens of is created, dated, and causally consequential, which is exactly the case the standard account handles badly. Keeping the two distinctions apart is what lets that problem be seen rather than hidden.

Why it matters downstream

Three arguments in this tier and the next depend on the distinction, and each depends on a different part of it.

The epistemological objection to mathematical objects depends on causal inefficacy specifically. If numbers can cause nothing, then nothing about them reaches us, and the account of how we know about them has to work without causal contact.

The argument against properties as entities depends on the spatial condition. If properties are non-spatial, then admitting them expands the world beyond the physical, and philosophers who want a purely physical inventory have a reason to look for a paraphrase.

The debate about fictional characters depends on the hard case directly. If created artefacts can be abstract, then Holmes can be something without being a person, and the previous article's third option becomes available.

None of these arguments is settled by the distinction. What the distinction supplies is the place where each of them has to do its work.

The next article turns from the divisions among things to what happens when a single thing alters.