MEH.
http://plato.stanford.edu/entries/platonism-mathematics/ said:
1. What is Mathematical Platonism?
Mathematical platonism can be defined as the conjunction of the following three theses:
Existence.
There are mathematical objects.
Abstractness.
Mathematical objects are abstract.
Independence.
Mathematical objects are independent of intelligent agents and their language, thought, and practices.
First of all Plato does not at all seem to have supported this, so why term it as platonism? One of his most famous passages, the allegory of the cave, even has Socrates argue that math objects are on the lower part of the triad of phenomena outside the cave: math, abstract eide (termed 'ideas' usually, but the term is Eidos and means Category/type) and then the over-archetype of 'benevolence', which is paralleled to the Sun (one cannot see it without ruining his eyes, etc).
Of course Plato founded the Academy, where most of the courses were math-related, including geometry and number sets (eg powers of numbers and their trigonometric presentation in spirals). But neither PlatoSocrates nor any other of the people before claimed that math is the ladder to actual Gnosis (Knowledge), apart from Pythagoras who indeed seems to have been of this view.
Socrates and Plato often argue that one cannot have any perfect knowledge of any phenomenon, cause he can only know it up to some level of detail, whereas the phenomenon always has parts below that detail as well (unless axiomatically those final parts are not further examinable, such as the notion of "a single point" in geometry).
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Concurrent to Socrates, the main positions are by the Eleans (Parmenides/Zeno etc) and the Democritians (Democritos(?)/Anaxagoras/Protagoras). The former argue that man is either entirely locked out of 'reality', or he is bounded by his own finite ability for thought while some kind of 'shadow of a shadow' of reality may reach him in some of his reasoning. The latter argue that no phenomenon is independent of the observer/mind. 'Man is the meter of all things' is a subset of that view.
Edit: The term 'mathematical platonism' seems to refer to the view that math objects may be seen as Eide themselves (ie things which exist by themselves, regardless of observers). I think this is not really about math, though, due to aforementioned views of Plato in his own works, and the special position of the discussion of Eide at the time.
Keep in mind that a core of their debate in the 5th BC was about more defined areas of thought rising from (problematically in their view) axioms, such as in math.In general the term rises with the dialogue between Socrates and Parmenides, and is meant as some theorising on what 'reality' may exist outside of human perception,
but that we may have dialectic reason to notice actual shadows of it in our own thought.
Useful to note that Parmenides reaches no conclusion.
