gnosis92

http://en.wikipedia.org/wiki/Philoso...tics#Platonism

Platonism

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 naive 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", an unchanging ultimate reality that the everyday world can only imperfectly approximate. The two ideas have a meaningful, not just a superficial connection, because Plato probably derived his understanding from the 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, which 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.

Yahzi

There are two kinds of people: those who think the world derives from truth, and those who think truth derives from the world.

One of them is wrong.

Tuvar Ane Ingolenen

There are two kinds of people: those that think there are two kinds of people, and those that do not.

They are both right.

JohannGoodflag

I think this phenomenon of apparent Platonism (with a belated retreat to formalism) is largely due to the way we talk about mathematical objects --- or rather, that we talk about mathematical objects. We talk about mathematics using noun-phrases, even though in terms of how mathematics is used, everything which a mathematician uses as a noun describes what someone else would use as an adjective, or at best, the name of a marking on a ruler or other measurement device.

Mathematics takes adjective-phrases and turns them into noun-phrases. That this is linguistically expedient, in that it makes it much easier to talk about what we talk about, is beyond question. But it makes us prone to thinking that things such as the number three have an independent existence of sorts. Doing this is a bit of intellectual short-hand which may make it easier to effectively do mathematics, in the same way that it's easier to have discussions about other rule-based-systems (e.g. video games and table-top RPGs) if you speak as though the things in those systems actually exist. Someone who isn't a Platonist can take up that mode of thinking temporarily as a convenience, and can get caught up in it for a while like a Star Trek fan arguing over when warp drive will actually be invented.

I would claim that mathematical Platonism is likely to be the result of mistaking the rules of the system for the rules of the universe (as was originally done with Euclid's Elements and its' predecessors) or making the old mistake of assuming that things which can be concieved of necessarily exist, or correspond at least partially to things which exist.