Thoughts for MLK Day

Folks in our neck of the professions are beginning a three-day weekend.   The temperature outside is in the twenties, with a bit of a wind;  so I shall be celebrating the memory of the late Reverend Doctor  by remaining in my bear-cave with a strong pot of java (which is the liquid that fuels mathematicians;  and though I am not one of those, I like to partake of their imbibing), and dipping into a spot of differential geometry. 

Already I have stumbled upon a couple of passages  suitable for eventual insertion into the ongoing essay “Language and Math” (see below).

The first illustrates our point about mathematicians consciencely fussing over their own terminology.   (This situation is of course at utter variance with what obtains today in the public square, where ranters pay no attention to what their terminology might mean, or whether it is minimally self-consistent, and simply blurt out whatever pops into their heads, quickly forgetting what they themselves opined five minutes before:  the whole state of affairs being  in any event  epistemologically divorced from any actual evidence, raw emotion being processed into semi-articulate speech  the way high-fructose corn syrup is processed into just about everything these people eat.)  Even at its loosest or most intuitive, mathematical discourse is quite precise and well-founded, yet we often meet with a crise de conscience sémantique on the part of the writer, who then gives us passages like this one:

The most important type of topological vector space for us  is the Banachable space (a TVS which is complete, and whose topology can be defined by a norm).   We should say Banach space when we want to put the norm into the structure.  There are of course many norms which can be used to make a Banachable space into a Banach space, but in practice, one allows the abuse of language which consists in saying “Banach space” for “Banachable space”, unless it is absolutely necessary to keep the distinction.

-- Serge Lang, Introduction to Differentiable Manifolds (1962), p. 4

While an embryo in utero, young Stefan could not properly yet be called “Banach”, but only Banachable.

(Note:  Lang was originally French, and his phrase “abuse of language” is a calque of the much commoner French original, par abus de langage.)

This struck me as fussy at first:  I had never seen this mouthful of a word, “Banachable”, before;  it is a phonological monstrosity, and quite useless as a rhyme-word in verse.  But semantically and morphologically, it is in the same line of work as the much commoner word metrizable.  No doubt if we had first learned Banachable from our wet-nurse, we should find the word perfectly normal.

(“Can you get this thing to me by Tuesday?” -- “Sorry, that’s just not Banachable.”)

-- At this point, we have really left the field of semantics, and entered that of literary criticism.

~

The next is more substantive, and concerns the topic (2b) of Language and Math, (concerning consilience between expression and content), which is out of our range, but we may quote a leading mathematician:

In the above discussion of the path-dependence of parallelism for a connection, I have been expressing things usuing the physicist’s index notation.  In the mathematician’s notation, the direct analogues of these particular expressions are not so easily written down.  Instead, it becomes natural to follow a slightly different route. (It is remarkable how differences in notation  can sometimes drive a topic in conceptually different directions! )   This route involves another operation of differentiation, known as Lie bracket.

-- Roger Penrose,  The Road to Reality (2004), p. 309f.

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