Reality Statis of Mathematical Entities for Plato
One of the aspects in the intellectual world is reason. At the top of the ladder is the Good, followed by Form then Reason. The reason or thought is based on axioms. The reality statis of mathematical entities for Plato is that mathematical principles fall under Reason because they are too perfect to be considered part of the visible world.
Take for instance the mathematical advances of the Pythagoreans. Mathematics has a definitiveness that goes beyond the finest knowledge derived from experience. The construction of the Pythagorean Theorem, for example, cannot be found in the crude dimensions of space and time alone. At the same time it exhibits an amazing insight and certainty. The Pythagorean Theorem can only be explained by assuming that it is a concept formed brought about by the impact of a perfect world of geometrical forms. It is the association of our soul to some extent in that world of Forms that explains the fact that we are capable of grasping something that is exact and amazing even though we cannot realize the construction except by the use of crude methods in the sensible world.
In geometry and arithmetic, we make use of specific figures to show our ideas and make examples clear. In these sciences, certain postulates are made and conclusions are drawn from the postulates. The intelligible, on the other hand, is “that which the reason itself,” rather than image-assisted imagination,
lays hold of by the power of dialectic, treating its assumptions not as absolute beginnings but literally as hypotheses, underpinnings, footings, and springboards so to speak, to enable it to rise to that which requires no assumption and is the starting point of all, and after attaining to that again taking hold of the first dependencies from it, so to proceed downward to the conclusion, making no use whatever of any object of sense but only of pure ideas moving on through ideas to ideas and ending with ideas. (511b-c)
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