Re: THEORY: OT Syntax (Was: Re: THEORY: phonemes and Optimality Theory tutorial)

On Sat, Nov 18, 2000 at 05:50:24PM -0500, John Cowan wrote:
[snip]
> What's bizarre is that that whole structure of infinitesimals which
> Cauchy & Co. so rightly discarded for epsilon-delta arguments can
> be restored to intellectual respectability by employing nonstandard
> numbers.  ("Nonstandard number" is a technical term here, folks, like
> "imaginary number" -- don't run away with it.)
Uh oh... you're wandering dangerously close to my home-brew, probably
nonsensical, theory of magnitudes, which features the "magnitude numbers":
an extension of the real numbers into a trans-continuum cardinality. The
frightening thing about these magnitude numbers is that there are
infinitely many levels of infinitesimals and infinite numbers.

For example, between every two arbitrarily close "polynomial magnitudes",
which correspond with real numbers, there are sub-polynomial magnitudes
("infinitesimals") that arise from logarithms. And between every two
arbitrarily close logarithmic magnitudes, there are sub-logarithmic
magnitudes ("sub-infinitesimals"?) that arise from iterated logarithms,
and so on, ad infinitum.

The same holds for the "large" magnitudes (which correspond with
"infinite" real numbers, if such things exist). Entire hierarchies of
these things arise from exponentials, tetration functions, etc..

> Wouldn't it be cool if there was a finite proof for G?  Nobody actually
> knows if it's true -- but if it were, nonstandard numbers would be
> *hard-wired* into number theory, willy-nilly.
Hmm, I'd like to find out more about these "non-standard" numbers -- any
web resources on this? :-)

[snip]
> "Mathematics is perhaps the only science in which foundational work can
> be replaced at will."  --I forget who
LOL!! Yeah, I saw this quote before. I think it was by a famous
mathematician... perhaps Hilbert? I forget too...

> > "Proof" in math can be pretty darn ephemeral sometimes!
>
> "Almost all proofs have bugs, but almost all theorems are true."
>         --A math/CS friend of mine
!!! This is such a good quote, I'm gonna "steal" it for my signatures file
:-)

> > To my knowledge calculus stayed around 'cause it worked, and because
> > later mathematicians were able to find a much more solid theoretical
> > foundation for it.
>
> Just so.
[snip]

Hmm... perhaps my magnitudes theory has hope.... *ducks* :-P


T

--
Computers shouldn't beep through the keyhole.