...
> Thinking some more about it, it seems that new forms of math are as likely
> a candidate as any for ideas that cannot be expressed symbolically. But,
> I've never heard of a mathematical system who's rules exist, but cannot be
> described in terms of things already know to other mathematicians.
> Obviously symbols can be invented on the fly, so that's not a problem...its
> more that one could imagine a set of rules so far removed from present
> systems that there is no mapping. But, I know of no instances of someone
> with a real track record coming up with systems he/she cannot describe at
> all to any other mathematician.
>
> Dan M.
All of known mathematics can be coded into Set Theory, for
instance, which is pretty simple. (Two undefined terms: 'set' and
'is an element of', and around 10 axioms.) After a while, the
encoded forms are nothing like how anybody actually THINKS about
the ideas, but the encoding can certainly be done.
So in the worst case, one could use set theory to
unambiguously state what one's ideas were, and then do a lot
of hand-waving to get the "flavor" of it across.
If pushed, I would be prepared to say that something
which could not be coded into set theory was not mathematics
at all.
---David
0 = {}, 1 = {0} = {{}}, 2 = {0,1} = {{},{{}}}, 3 = {0,1,2},...
is the standard encoding of the natural numbers, which I believe
is due to Henkin.
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