Re: Langauge Constets (was Natural Semantic Metalanguage)

On Fri, 23 Nov 2007 08:19:02 EST, MorphemeAddict@WMCONNECT.COM wrote:

>In a message dated 11/22/2007 9:16:12 PM Central Standard Time,
>mike@STARMAN.CO.UK writes:
>
>> That's a different thing entirely. How can you derive (for instance)
>> graphing, topology or set theory from just 0 and 1?
>> Mike
>
>I was wondering the same thing after my last post.
>The objects of those disciplines could be treated as abstract meanings of
>various numbers, then relevant rules applied to them.
>Maybe.
You could do that, but you'd be using the numbers for little more than their
value as labels; I at least wouldn't find that very satisfying.  Sure, Jot
can be written in 0s and 1s, but that's merely a notational convention.  You
don't need to increment or add or multiply them to evaluate a Jot program,
so they may as well have been white boxes and black ones.

It's a program with a long and illustrious history to lay down the
foundations of mathematics, and in particular derive everything from a small
select set of initial axioms and objects and such.  In foundations, rather
than taking numbers as fundamental, a typical approach is to begin by
developing first-order logic and then to move on to axiomatic set theory, in
terms of which one does everything else.  This is the way Russell and
Whitehead went about it in _Principia Mathematica_ (1910--1913).  Numbers,
just like any other sort of object, are in this approach realised as
particular sets.

You can, of course, look at systems for arithmetic in which numbers are
fundamental; this is the context in which Goedel set his incompleteness
theorems.  Goedel's proof of the first incompleteness theorem relies
crucially on Goedel numbering, which is very much like your "abstract
meanings of various numbers".  So doing an encoding into the domain of
numbers can certainly be useful in foundations; it's just less useful as a
place to start, in that fewer other areas of math are "naturally downstream"
from it.

Pushing the minimalism as far as possible, you could also ask just how much
these axiom systems can be slimmed down.  This is a question fitting right
in with Stephen Wolfram's New Kind of Science program -- _exactly_ how
simple can something be and still yield universal ~ arbitrarily complicated
behaviour?  He addresses it to a degree, for instance at
  http://www.wolframscience.com/nksonline/page-773
So, in the usual setup, it takes seventeen axioms and not a few symbols to
get arithmetic.  You can shrink this a fair bit, in the case of basic logic
from seven axioms and three operators (plus equals) to one and one, but the
results are extremely unintuitive and far too much of a pain to do anything
substantial with (343 steps with lemmas to reproduce the Sheffer system
there; see page 810).  For other domains the shortest axiom systems aren't
known, and we might never know them without exhaustive search, and they'll
likely be even less intuitive and user-friendly.
So if you're building a _conlang_ on these principles, well, you'll probably
fail for essentially Mungojelly's reasons, that semantics of the real world
just don't reduce usefully like that.  But if it _weren't_ for that, what
you'd have to look forward to is a conlang where it takes a couple hundred
uses of your ten-odd words to say "mouse" or "have", and nobody wants
something that extreme.

Alex