[FRIAM] Fractals/Chaos/Manifolds

[Robert J. Cordingley]

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, lrudolph at 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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