Ontological Positions: A Primer

Before the map, the meta-question. When a mathematician says "there exists a prime greater than a million," what kind of existence is asserted?

Mathematical Platonism holds that mathematical objects exist independently of minds, language, and physical reality — they are discovered, not invented. The integers were there before humans counted them. This is the working assumption of most practicing mathematicians.

Formalism (Hilbert's program) holds that mathematics is a game of symbol manipulation under rules. Existence is proof-relative: "there exists" means "the string 'there exists...' is derivable from the axioms." The formalist does not ask what numbers are, only whether theorems about them are derivable.

Structuralism holds that mathematical objects are positions in structures. The number 2 is the second position in the natural-number structure; it has no intrinsic properties beyond its structural relations. Two different implementations of the natural numbers (as sets, as Peano-axiom models, as cardinal abstractions) are equally legitimate.

Intuitionism (Brouwer, Bishop) holds that mathematical objects are mental constructions. Existence requires an explicit construction, not merely a proof that non-existence leads to contradiction. The law of excluded middle — every proposition is either true or false — is rejected for statements about infinite collections.

Empiricism (Mill, Kitcher) holds that mathematical truths are highly general empirical facts. This view is now a minority position, largely because it struggles to account for the necessity that mathematical proofs appear to carry.

This document uses the language of structural Platonism — objects are described as existing, but what is meant is that they occupy determinate positions in structures that are coherent under their axioms. Whether those structures exist beyond human minds is left open.

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