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===platonismo====
Platonism is the form of realism that suggests that mathematical 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 term Platonism is used because such a view is seen to parallel Plato's belief in a "World of Ideas" (typified by Plato's 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 popular Pythagoreans of ancient Greece, who believed that the world was, quite literally, generated by numbers.
The major problem of mathematical platonism is this: 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 answer might be Ultimate ensemble, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe.
Gödel's platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things Husserl said about mathematics, and supports Kant's idea that mathematics is synthetic a priori.) Davis and Hersh have suggested in their book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism (see below).
Some mathematicians hold opinions that amount to more nuanced versions of Platonism. These ideas are sometimes described as Neo-Platonism.
===Filosofia da Matemática===
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