Platonism

Platonism is the view that mathematical objects exist independently of human thought and language. It treats mathematical truths as discovered rather than invented.

Mathematical object: An abstract entity, such as a number, function, or set, studied in mathematics. Platonism makes the existence of these abstract entities its central claim.
Epistemology of mathematics: The study of how mathematical knowledge is possible and justified. It asks how Platonists can know truths about objects outside space and time.
Mathematical formalism: The philosophy that mathematics concerns formal symbol systems and their rule-governed transformations. Formalism explains mathematical practice through symbols and rules rather than an independent realm.
Plato: An ancient Greek philosopher whose dialogues explore Forms as realities distinct from sensible things. The modern position borrows its name from Plato's account of a reality beyond perception.
Quine–Putnam indispensability argument: The argument that indispensable mathematical commitments in science support belief in mathematical entities. It is a leading argument for Platonism based on mathematics' role in science.
Abstract object: An entity without a location in space or time, such as a number or proposition. Mathematical objects are commonly understood as a distinctive kind of abstract object.
Benacerraf's identification problem: The problem that mathematical structures can be represented by different set-theoretic constructions without a unique identification. It challenges how Platonism identifies particular abstract objects such as numbers.
Mathematical intuitionism: The philosophy that mathematical objects and proofs are grounded in mental constructions. Intuitionism makes mathematical existence depend on construction, unlike Platonism.
Theory of Forms: Plato's account of eternal, intelligible Forms as the realities imperfectly reflected by sensible things. It provides the historical model behind the label “Platonism,” though mathematical versions differ.
Mathematical universe hypothesis: The proposal that physical reality itself is a mathematical structure. It extends mathematical realism by identifying the physical world with mathematical structure.

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