[FRIAM] Fractals/Chaos/Manifolds

[lrudolph at meganet.net]

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.