FOM: Thomae and non-archimedian domains
John Case
case at eecis.udel.edu
Fri Nov 28 22:25:29 EST 1997
On Nov 26, 15:33, Walter Felscher wrote:
Bill Tait, on Nov.12th, wrote in connection with Goedel and
infinitesimals that
Cantor is quoted by Dauben as saying that Johannes Thomae
(who had an office down the hall from Frege) was the
first to ``infect mathematics with the Cholera-Bacillus
of infinitesimals''.
While I do not know what Cantor actually may have referred
to, Thomae (1840-1921 , since 1872 Professor in Halle, since
1879 in Jena) does have a documentable connection, not with
infinitemals, but with non-archimedian extensions of the
reals: in his
Abrisz einer Theorie der complexen Funktionen, Halle 1870
and his
Elementare Theorie der analytischen Functionen einer
complexen Ver=84nderlichen., 2te Aufl., Halle 1898
he represented the orders of growth of real functions by
lexicographically ordered semigroups of sequences of
integers. It then was Paul du Bois-Reymond 1882 who
expressed the idea that the totality of orders of growth
(which he called the infinitaere Pantachie) should be viewed
as an expansion of the continuum; for a more modern
presentation of his work cf. G.H.Hardy, Orders of Infinity,
Cambridge 1924.
It may not be superfluous to point out that this achievement
of Thomae's is NOT connected with his opinions on the
foundation of numbers, analyzed so masterfully in Frege's
"Ueber die Zahlen des Herr Thomae".
W.F.
}-- End of excerpt from Walter Felscher
That was interesting. It's nice to have non-standard analysis now to assuage
worries re positive mathematical ontological status for infinitesimals. (-8
Four items occur to me.
1. The Weirstrauss delta-epsilon approach to calculus leads more naturally to
thinking about error analysis in numerical analysis and the infinitesimal
approach fits better many situations for physicists. Re the latter: for
example, in physical optics when one wants to set up equations for predicting
the shape of fringe patterns resulting from the passage of light through some
partial obstruction, it's a lot simpler to proceed by summing up contributions
from judiciously chosen infinitesimal volumes. Cognitive efficacy should be a
criterion for choosing among e.g., extensionally equivalent, definitions too.
2. It is a separate problem to worry about the _physical_ ontological status of
infinitesimals. If the universe is discrete, then physically there aren't any,
and calculus is just a brilliant tool for getting approximations by smoothing
out unknown, fluctuating, and/or messy discretizations. For example, already
since electrical charges are discrete, Maxwell's equations are such an
approximation.
3. If the universe (including space) is discrete, then it may not be possible
_physically_ to get arbitrarily close to or be right on a singularity.
Mathematically, the point of singularity may make good sense, but it may
correspond to no acutally physically reachable point. I'd like to see more
physics worked out with this possibility strongly in mind. A hard part is
which of many incomparable workings out to use.
4. It's too bad aleph_0-mind calculations likely can't be realized physically.
Then one could decide first-order (but not second order) arithmetic with a
physical device. While we likely can't do infinitely many steps in finite
time with a physical device and Turing computability still models (idealized)
physical computability, the new possibilites in quantum computing show we
need to revise our definitions (at least intensionally) of (idealized)
_feasible_ physical computation. It's still open whether the definition of
feasibly computable is thereby changed extensionally.
(-8 John
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