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It is a commonplace in philosophy that every thing or object is either abstract or concrete. Abstract objects are sometimes called abstracta (sing. abstractum) and concrete objects are sometimes called concreta (sing. concretum). The abstract-concrete distinction is often introduced and initially understood in terms of paradigmatic examples of objects of each kind. (See Fig. 1)

Philosophers disagree over just what makes the items in the first column abstract. One well-known proposal is that an object is abstract if and only if it lacks a location in space. Thus justice is abstract because it seems impossible to say where it is. One potential problem for this proposal is that certain typically abstract objects, like the game of tennis, arguably do have a sort of spatial location (e.g. "Tennis is alive and well in New York City"). Another problem is that some arguably concrete mental objects (e.g. Tim’s pang of concern for his eldest daughter) do not have spatial location.

Another popular proposal for drawing the abstract-concrete distinction has it that an object is abstract just in case it lacks any causal powers. A causal power is an ability to affect something causally. Thus the empty set is abstract because it cannot act on other objects. One problem for this view is that it is not clear exactly what it is to have a causal power. When my six dogs chew up my copy of The Economist, presumably the dogs are exercising their causal powers. Why shouldn’t we say in this case that the number six is also exercising its causal powers? There is no widely accepted answer to questions such as this. ^[1]

Abstract objects have often garnered the interest of philosophers because they are taken to raise problems for popular theories. In ontology, abstract objects are considered problematic for physicalism and naturalism. Historically, the most important ontological dispute about abstract objects has been the problem of universals. In epistemology, abstract objects are considered problematic for empiricism. If abstracta lack causal powers or spatial location, how do we know about them? It is hard to say how they can affect our sensory experiences, and yet we seem to agree on a wide range of claims about them. Some, such as Edward Zalta and arguably Plato (in his Theory of Forms), have held that abstract objects constitute the defining subject matter of metaphysics or philosophical inquiry more broadly. To the extent that philosophy is independent of empirical research, and to the extent that empirical questions do not inform questions about abstracta, philosophy would seem specially suited to answering these latter questions.

• Abstraction
• Nominalism
• Object (philosophy)
• Philosophy of mathematics
• Platonic form
• Problem of universals
• Universal (metaphysics)

• Abstract Objects at the Stanford Encyclopedia of Philosophy
• Nominalism, Realism, Conceptualism, from The Catholic Encyclopedia