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Why do philosophers debate whether abstract objects such as numbers exist?

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Marble bust of Aristotle with a beard and draped robe, showing detailed facial features and wavy hair, set against a dark background. — metaphysics Detail of a Roman copy (2nd century bce) of a Greek alabaster portrait bust of Aristotle (c. 325 bce); in the collection of the Museo Nazionale Romano, Rome.

Philosophers debate the existence of abstract objects—such as numbers, sets, and functions—primarily because these entities appear to be essential to mathematical and scientific theories, yet they defy standard physical and spatiotemporal definitions.123

The debate centers on several core metaphysical and epistemological challenges:

Ontological Commitment: Realists, often called Mathematical Platonists, argue that because mathematical theorems provide true descriptions of the world, we must accept that the objects they describe—numbers and sets—actually exist as real, objective, and nonphysical entities. Conversely, nominalists reject the existence of abstracta, often invoking Ockham’s razor to argue that we should not multiply entities beyond necessity.124
Epistemological Problems: A major hurdle for realism is explaining how humans, who exist in space and time, can acquire knowledge of objects that are, by definition, nonspatiotemporal and noncausal.12
Indispensability: Some philosophers argue that because scientific theories rely on mathematics, we are committed to the existence of abstract objects to maintain the validity of those theories.2

Ultimately, the disagreement persists because there is no consensus on whether mathematical truth requires the existence of abstract objects or if such theories can be reinterpreted without them.24