In one German dictionary I found /mannigfaltigkeit/ translates to
/variousness/ which seems pretty obtuse but indicates it may have less
to do with the original entymology of /manifold/
(https://en.wiktionary.org/wiki/manifold Entymology 1). Per Dean's pdf,
perhaps it's a made up usage inspired by Gauss/Riemann who had a concept
about topological space but needed a word for it? That is to say
'manifold' (in English) was a neologism in its time based on an
appearance of the German word?
Robert C
On 3/1/17 2:21 PM, lrudo...@meganet.net wrote:
The word, as a term of Mathematical English (which is of course quite a
distinct dialect of
English) is a calque of the Mathematical German word "Mannigfaltigkeit".
Franklin Becher, in
the first paragraph of the lead article in the October, 1896, issue of the
American
Mathematical Monthly, "MATHEMATICAL INFINITY AND THE DIFFERENTIAL", doesn't
quite use the word
yet, but makes its origin clear enough.
---begin---
Mathematics, as defined by the great mathematician, Benjamin Pierce, is the
science which
draws necessary conclusions. In its broadest sense, it deals with conceptions
from which
necessary conclusions are drawn. A mathematical conception is any conception
which, by means
of a finite number of specified elements, is precisely and completely defined
and determined.
To denote the dependence of a mathematical conception on its elements, the word
"manifoldness," introduced by Riemann, has been recently adopted.
--end--
In his article on the foundations of geometry, available at
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/Geom.html ,
Riemann distinguished two types of "Mannigfaltigkeit", the discrete and the
continuous:
---begin---
cat
Grössenbegriffe sind nur da möglich, wo sich ein allgemeiner Begriff vorfindet,
der
verschiedene Bestimmungsweisen zulässt. Je nachdem unter diesen
Bestimmungsweisen von einer zu
einer andern ein stetiger Uebergang stattfindet oder nicht, bilden sie eine
stetige oder
discrete Mannigfaltigkeit;
| Google Translate >
Size terms are only possible where there is a general concept, which allows
different modes of
determination. According as, according to these modes of determination from one
to another, a
continuous transition takes place or not, they form a continuous or discrete
manifoldness;
---end---
In Riemann's (eventual) context, those sentences would be understood now (at
least by
topologists of my sort, which is to say, geometric topologists, cf.
http://front.math.ucdavis.edu/math.GT) as sketching the modern concept of a
(topological or
differentiable) manifold as a "mathematical conception" that can "precisely and
completely
defined and determined" by a collection [called an "atlas"] of "modes of
determination"
[called "charts"] among (some pairs of) which there are also given "continuous"
(i.e.,
topological) or perhaps *smooth* (i.e., differentiable) coordinate changes.
I dispute, incidentally, the claim that 3-manifolds are too hard to understand;
they're *just*
at the edge of that, but not over it (whereas 4- and higher dimensional
manifolds are
DEFINITELY over that edge, in various well-defined mathematical ways; e.g., the
problem of
determining whether two explicitly-given n-manifolds, n greater than 3, has
been known for a
long time to be computationally intractable [you can embed the word problem for
groups into
the manifold classification problem for n greater than 3], and much more
recently has been
shown to be doable in dimension 3).
The French word for (something a little more general than a) manifold is
"varieté", by
the way; same sort of reason, I assume.
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