Subject: Re: [lojban] Sapir-Whorf Hypothesis

From: Nathaniel Smith <[email-masked]>
Date: Tue, 12 Jun 2001 17:04:41 -0700

On Tue, Jun 12, 2001 at 03:22:14PM -0700, Robin Lee Powell wrote:
> On Tue, Jun 12, 2001 at 12:50:20PM -0700, Edward Cherlin wrote:
> > The best recent example is non-standard arithmetic, which comes in two
> > forms, one from Robinson's model theory, and the other from Conway's
> > advances in game theory. Both provide consistent but significantly
> > different arithmetics with actual infinitesimals, and both can be
> > extended to analysis. Without the appropriate definitions of terms and
> > proofs of theorems, there is no way anybody outside the field can
> > understand what either form is talking about, since mathematicians had
> > previously "proved" that arithmetic with infinitesimals was
> > impossible, and in particular Peano thought that he had proved the
> > impossibility of any non-standard models of the natural numbers.
> 
> As someone who almost has a math degree <grin>, I'm intrigued.
> 
> What are infinitesimals, and where do I find out about game-theory based
> arithmetic?

Infinitesimals are numbers which are larger than zero, but smaller
than any positive real number. Conway's game theoretic version is
purportedly described best in "Winning Ways" by Berlekamp and Conway,
but that's a two-volume several thousand page tome in which I have yet
to find them. The book I read is Knuth's "Surreal Numbers: How Two
Ex-Students Turned on to Pure Mathematics and Found Total Happiness".
It's a fun book, but not as detailed mathematically as one might want
-- great if you want to explore them on your own, though. "On Numbers
and Games" by Conway might also be good.

I don't know much about Robinson's infinitesmals; they're interesting
partly because he (and others) went through and redeveloped a whole
theory of calculus and analysis where dx/dy really does mean dividing
an infinitesmal by another infinitesmal... the book at hand mentions:

Robinson, Abraham. (1966). Non-Standard Analysis. Amsterdam:
North-Holland.

Robinson, Abraham. (1979). Selected Papers of Abraham Robinson.
Vol. 2: Nonstandard Analysis and Philosophy. New Haven: Yale
University Press.

Incidentally, I don't see how mathematical discourse is relevant to
S-W. Mathematical discourse has a very different character, notation
even more so, and refusing to accept ideas for philosophical reasons
is different from being unable to understand them because of a
restrictive language.

-- Nathaniel

-- 
But in Middle-earth, the distinct accusative case disappeared from the
speech of the Noldor (such things happen when you are busy fighting
Orcs, Balrogs, and Dragons).