From: Paul Fidika (Fidika@new.rr.com)
Date: Fri Aug 13 2004 - 00:13:35 MDT
>I just got the message from Simon Gordon which said
>[ "Mathematics" in this case should be read as "the
>platonic reality" or "the domain of all abstract entities.].
>Is it ok at SL4 to make up your own language? I took
>math at university and I heard nothing about "domain of
>abstract entities". Maybe I missed those classes?
Simon isn't just making that up--"platonic reality" plays a very important role in the philosophy of mathematics. For example, after Gödel had just proved that the Continuum Hypothesis could not be disproven within Zermelo-Fraenkel Set-Theory, he went on to give some arguments for why he believed that the Continuum Hypothesis was FALSE! Now, some people might say that Gödel should not be allowed to get away with this; they believe that the truth or falsity of a statement should only be considered relative to some formal system of axioms, and Gödel cannot assert the falsity of a statement without making reference to some system within which it is false. They believe that statements of Mathematics are just character-strings which we manipulate according to some fixed-rules (axiomatic systems) and categorize as true, false, or not-well-formed, and refuse to recognize any further semantics within the statements than this. Sure, you can take this philosophical stance, but you're really missing out on a lot...
A Platonist is someone who can assert the truth or falsity of a statement of Mathematics without making reference to any particular formal system within which it is true or false; mathematical statements are entities in and of themselves which exist in a "Platonic realm" in the sense that they are either as objectively true or false as anything in our universe can be objective. For example, I, for one, believe that the Riemann Hypothesis is either true or false, even if no one is ever able to prove it either way, or even if it is proven that it is impossible to prove the Riemann Hypothesis within our universe because our universe is simply too small to contain its proof.
When the people on this list (as far as I can determine) say that "All is number", they mean that a piece of Mathematics exists, in the strongest sense of the word "exists", regardless of whether or not someone has discovered it. To me, this is simply an "obvious" truth, but the only real argument for the existence of the Platonic realm is that one can never disprove its existence; to disprove the statement "Platonia does not exist" one would have to first presume that it does exist, i.e., that statements can have objective truth-values independent of a formal system.
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