The Oxford Companion to Philosophy, 1995

ABSTRACT ENTITIES. The dichotomy between the abstract and the concrete is supposed to effect a mutually exclusive and jointly exhaustive ontological classification. The dichotomy is, however, too naïve to be of theoretical use. There are many different ways, themselves vague, to mark the distinction: abstract entities are not perceptible, cannot be pointed to, have no causes or effects, have no spatio-temporal location, are necessarily existent. Nor is
there agreement about whether there are any abstract entities, and, if so, which sorts of entity are abstract. Abstract entities, conceived as having no causal powers, are thought problematic for epistemological reasons: how can we refer to or know anything about entities with which we have no causal commerce? Hence the existence of nominalists, who try to do without abstract entities.

The AI says:

Abstract entities are things that do not exist in space or time and have no physical properties, but are said to exist in some non-physical sense. Standard examples are numbers, mathematical objects, sets, propositions, properties, relations, and meanings. The number 2, for instance, is not located anywhere, does not cause physical effects, and does not change, yet it is treated as something we can refer to and reason about. Abstract entities are usually contrasted with concrete entities, which exist in space and time and can causally interact (people, trees, rocks). Philosophical disagreement concerns whether abstract entities really exist (Platonism), exist only as human constructions (nominalism, fictionalism), or exist as dependent on minds or language (conceptualism).

Wikipedia

In philosophy, a fundamental distinction exists between abstract and concrete entities. While there is no universally accepted definition, common examples illustrate the difference: numbers, sets, and ideas are typically classified as abstract objects, whereas plants, dogs, and planets are considered concrete objects.

Philosophers have proposed several criteria to define this distinction:

1. Spatiotemporal existence – Abstract objects exist outside space-time, while concrete objects exist within space-time.
2. Causal influence – Concrete objects can cause and be affected by other entities (e.g., a rock breaking a window), whereas abstract objects (e.g., the number 2) lack causal powers and do not cause anything to happen in the physical world.
3. Metaphysical relation – In metaphysics, concrete objects are specific, individual things (particulars), while abstract objects represent general concepts or categories (universals).
4. Ontological domain – Concrete objects belong to the physical realm (or both the physical and mental realms), whereas abstract objects belong to neither.

Another view is that it is the distinction between contingent existence versus necessary existence; however, philosophers differ on which type of existence here defines abstractness, as opposed to concreteness. Despite this diversity of views, there is broad agreement concerning most objects as to whether they are abstract or concrete,such that most interpretations agree, for example, that rocks are concrete objects while numbers are abstract objects.

Abstract objects are most commonly used in philosophy, particularly metaphysics, and semantics. They are sometimes called abstracta in contrast to concreta. The term abstract object is said to have been coined by Willard Van Orman Quine. Abstract object theory is a discipline that studies the nature and role of abstract objects. It holds that properties can be related to objects in two ways: through exemplification and through encoding. Concrete objects exemplify their properties while abstract objects merely encode them. This approach is also known as the dual copula strategy.

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