What Is a Manifold?
&quot;Quick&quot; etymological fact about what mathematicians in English call Manifolds:<p>If one would translate Riemann&#x27;s original German word for them, &quot;Mannigfaltigkeit&quot;, it would translate to &quot;manifoldyhead&quot;, or, more understandable to the speaker of Modern English:<p>Manyfoldyhood.<p>(Think of -head as in &quot;Godhead&quot;, not as in &quot;Brotherhood&quot;, in the same sense that the &quot;ring&quot; in &quot;algebraic ring&quot; refers to &quot;ring&quot; in the sense of &quot;smuggler ring&quot;, not in the sense of &quot;gold ring&quot;. The two suffixes merged in English, and have very different and rather complex etymological origins.)<p>While this word sounds somewhat ridiculous and perhaps a bit infantile in English, it, in my opinion, conveys what we mean by them significantly better.<p>Clifford tried to accommodate Riemann&#x27;s highly specific word choice in his translation, as Jost notes:<p>&quot;The English of Clifford may appear somewhat old-fashioned for a modern reader. For instance, he writes “manifoldness” instead of the simpler modern translation “manifold” of Riemann’s term “Mannigfaltigkeit”. But Riemann’s German sounds likewise somewhat old-fashioned, and for that matter, “manifoldness” is the more accurate translation of Riemann’s term. In any case, for historical reasons, I have selected that translation here.&quot;<p>Unfortunately, neither Jost (a native German speaker!) nor Clifford realized that English can and does accommodate Riemann&#x27;s exact meaning directly.<p>To make a comparison which might require quite a bit of German knowledge beyond high school education to describe in exact linguistic terms, but which native speakers should hopefully find intuitive to distinguish (I at least, do):<p>Translating &quot;Mannigfaltigkeit&quot; as &quot;Manifoldness&quot; seems equal to mistranslating &quot;Geheimnis&quot; as &quot;Secrethood&quot; and &quot;Geheimheit&quot; as &quot;Secretness&quot;, whereas the opposite pairing would yield an accurate (albeit not necessarily immediately apparent—in terms of the differences between the two—to the English native speaker, unless explicitly pointed out) translation. So much for the last suffix, but that still leaves insertion of the one before it (&#x27; <i></i>* foldy <i></i>* &#x27; instead of &#x27; <i></i>* fold[( <i></i>* )]&#x27; unclear.)<p>One may observe the difference involved there by a converse example, also involving &quot;Geheim&quot;, as well as the root word of it, &quot;Heim&quot;, by dragging forth a rather rare and archaic—but none the less highly likely intuitive to the native speaker—word:&quot;Geheimig&quot;. Whose meaning starkly differs from both &quot;Geheimnis&quot; and &quot;Geheimheit&quot;.<p>For further language related hijinks related to &#x2F;Manyfoldyhoods&#x2F;, see:<p>A) The Dutch word for them, which would back translate to &quot;Variety&quot;; and<p>B) This quote by Poincaré:<p>&quot;I prefer the translation of Mannigfaltigkeit by multiplicity, because the two words have the same etymological meaning. The word set is more adapted to the Mannigfaltigkeiten considered by Mr. Cantor and which are discrete. It would be less adapted to those which I consider and which are discontinuous.&quot;<p>(As I don&#x27;t speak French, I can&#x27;t make much of any statements about the accuracy about the etymological claim by Poincaré, so I&#x27;ll close with another Poincaré quote instead:<p>&quot; Mathematics is the art of giving the same name to different things.&quot;<p><pre><code>  —Henri Poincaré)</code></pre>