[deletion]
> Actually, only continuity is needed for proof; nevertheless the
> Weierstrasssche Nullstellensatz is also a fundamental theorem of
> analysis, but it is the calculus which figures the quintessence
> of analysis und not continuity while continuity figures the
> quintessence of topology. See the access to topology via the
> concept of limit points in [HocYou 1961] (John G. Hocking, Gail
> S. Young: Topology, Reading, ... 1961). These authors write in
> the introduction: "The very definition of a continuous function
> is an example of this dependence (on the properties of limit
> points). To exaggerate, one might view topology as the complement
> of modern algebra in that together they cover the two fundamental
> types of operations found in mathematics." This corresponds to
> what van der Waerden said in the above quotation. To emphasize his
> point of view a bit more, again van der Waerden will help (vol I,
> chapter 9, reelle Koerper): "Beim Studium der algebraischen
> Zahlkoerper spielen ausser den algebraischen Eigenschaften ihrer
> Zahlen gewisse unalgebraische Eigenschaften: absolute Betraege |*|,
> Realitaet, Positivsein, eine Rolle. Dass diese Eigenschaften sich
> nicht mit Hilfe der algebraischen Operationen + und * eindeutig
> definieren lassen, zeigt sich an folgendem Beispiel. ..." So there
> are arguments contra et pro. Ta/l Tura/n's proposal to call
> Funktionalalgebra that part of algebra dealing with FTA (note that
> Gauss' first proof needs the implicit function theorem for proof)
> etc. because of the significant use of functions there, this
> finally shows that in the matter of names Professor Faust may be
> right as Sanford Segel emphasized: Name ist Schall und Rauch
> (Names mean nothing) (Goethe, Faust, part I, Gretchenszene) as to
> whether the unavoidable "non-algebraic" portion of the usual FTA
> over the complex numbers is called analysis or topology seems
> irrelevant.
>
To give some more literal translations, names may be "sound and smoke",
"sound and haze"? or maybe "noise and haze?", which may suggest that
they "mean nothing", or at any rate that they are just labels which
give no insight into what is named. Still, it's hard to communicate
in natural languages without using names, especially when we are
separated by long distances and can't point. And it appears to me
that names in fact trigger meanings, which can differ from individual
to individual, but must remain in some sort of circumscribed region
of meaning for communication to be possible.
Now, historically speaking, topology both in name and in fact was
established after analysis, and to some extent as a generalization
of analysis (although one might except certain combinatorial
investigations). There are various species of topology, but
practitioners of densely populated species have devoted themselves
to study of continuous transformations and properties invariant
under such transformations. Due allowance must be made for the
fact that studying continuous transformations involves a certain
amount of study of discontinuities, but the center of mass, so to
speak, of this species of topology has been continuity. If
differentiability is added, we shade off into what most of us have
named differential geometry. As I see it, if we get too far away
from continuity in topological studies, we shade off into what
most of us call set theory.
So as far as the Fundamental Theorem of Algebra is concerned, I think
the distinction should be between algebra, on the one hand, and
"analysis and topology" (or "analysis or topology", if you prefer).
By the way, what does the name "algebra" stand for in the present
context?
Gordon Fisher gfisher@shentel.net