Mathematical Platonism is the form ofrealism that suggests thatmathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers.
The termPlatonism is used because such a view is seen to parallelPlato'sTheory of Forms and a "World of Ideas" (Greek:eidos (εἶδος)) described in Plato'sallegory of the cave: the everyday world can only imperfectly approximate an unchanging, ultimate reality. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popularPythagoreans of ancient Greece, who believed that the world was, quite literally, generated bynumbers.
A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is theUltimate Ensemble, a theory that postulates that all structures that exist mathematically also exist physically in their own universe.
Kurt Gödel's Platonism[1] postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many thingsEdmund Husserl said about mathematics, and supportsImmanuel Kant's idea that mathematics issynthetica priori.)Philip J. Davis andReuben Hersh have suggested in their 1999 bookThe Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat toformalism.
Full-blooded Platonism is a modern variation of Platonism, which is in reaction to the fact that different sets of mathematical entities can be proven to exist depending on the axioms and inference rules employed (for instance, the law of theexcluded middle, and theaxiom of choice). It holds that all mathematical entities exist. They may be provable, even if they cannot all be derived from a single consistent set of axioms.[2]
Set-theoretic realism (alsoset-theoretic Platonism)[3] a position defended byPenelope Maddy, is the view thatset theory is about a single universe of sets.[4] This position (which is also known asnaturalized Platonism because it is anaturalized version of mathematical Platonism) has been criticized by Mark Balaguer on the basis ofPaul Benacerraf'sepistemological problem.[5]
A similar view, termedPlatonized naturalism, was later defended by theStanford–Edmonton School: according to this view, a more traditional kind of Platonism is consistent withnaturalism; the more traditional kind of Platonism they defend is distinguished by general principles that assert the existence ofabstract objects.[6]