Re: [STDS-1788]: Another philosophy concerning unbounded intervals
> Date: Sun, 14 Jun 2009 05:37:33 -0500
> From: Ralph Baker Kearfott <rbk@xxxxxxxxxxxxx>
> To: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
> Subject: [STDS-1788]: Another philosophy concerning unbounded intervals
>
>
> P-1788 participants,
>
> I attach a paper that espouses a different philosophy concerning
> dealing with unbounded intervals. I do not propose it as a
> particular stand we should take, but supply it mainly as background
> material. However, it does supply a specification for how
> 0 * \infty should be defined, along with a reasoned argument.
> (Also, the concept of \infty is redefined here, as "grossone.")
> I hope this isn't too much of a red herring. In any case, I found
> it entertaining reading while over the North Pole last week.
>
> Sincerely,
>
> Baker
Baker,
You're right. It IS entertaining.
If I may be permitted a critique based on a quick read of the
paper, it appears to be a number system based on (finite)
polynomials, P(omega), with coefficients in the reals & an
indefinite that is allowed to take a formal value that is
considered to be roughly equivalent to aleph0.
Near the end he seems to reject some uses of the axiom of
choice but that doesn't matter much & it helps him to consider
elements such as grossone/2 (but not grossone/pi).
Such things have been considered in the past. Even on computers.
Indeed, in 1992 Prof Sussman at MIT & I briefly toyed with a
system of the form r + r*eps (where eps was an infinitesimal)
for the purpose of natural computation of derivatives in Scheme
along the lines of Leibnitz. It worked a bit but had problems
with l'Hospital for obvious reasons. Then Gerry figured out how
to create the D operator as a functional that takes any function
& returns the function which computes its derivative. He & Jack
Wisdom went on to create their classical mechanics system from
that.
Such things can work but I should point out some difficulties
in realizing them on computers.
Roundoff error can be a bitch. If I have two floating point
numbers f >> g such that f + g = f then there is a danger of
making an infinite error in that:
g*omega + 4 = (g*omega + 1) + ((f*omega + 5) - (f*omega + 2))
!= ((g*omega + 1) + (f*omega + 5)) - (f*omega + 2) = 4
While this approach to infinites is pretty much independent of
intervals, I can see that one might want to consider polynomials
with interval coefficients & an infinite indefinite.
In such a case the above problem would be replaced with result
intervals of formally infinite width.
I don't know if that's good or bad. You guys can answer that
question better than I can.
But as it is something that could be built upon the edifice of
WHATEVER interval model we choose I suspect it could be done
without much consideration on our part.
As usual, your mileage may vary.
In this case, by a possibly infinite amount. :-)
Enjoy,
Dan