Saturday, December 15, 2012

Language and Math

There are a couple of different ways in which that blandly even-handed conjunction, Language and Math, can be desymmetrized.  As:

(1)  The subject-matter is language, seen in a mathematical light.  This leads to ‘mathematical linguistics’, in which I have no interest.
(2) The subject-matter is mathematics, seen from a linguistic perspective. 

The latter endeavor again splits:

(2a)  We examine the way actual mathematicians talk in their subject, quite the way francophone linguistics must be based on the way actual Frenchmen talk.
(2b)  We consider the sheer subject-matter of mathematics, in complete independence from the quirks and foibles of present-day mathematicians, and examine the ways in which  either any ideal approach to this matter, or perhaps even the structure of the matter itself, can or must be scene in a perspective of ‘language’ in some relevantly extended sense.

(2a) is a lot of fun; I have been collecting examples over the years, of syntactic and semantic phenomena peculiar to this field of discourse, which perhaps can be shared someday.   There is more intellectual substance to this exercise than in the usual dialectological study  (that is to say, inventorying the predictable quirks of this or that regional patois;  that of a Geistesgemeinde is another matter entirely, and forms the backbone of my own corpus-based Dialect Notes, available on the high side), but it is of no importance either to linguistics or to mathematics  as theoretical disciplines.

(2b) has grown and grown in core importance, beginning peripherally with the obtention of clarity on non-Euclidean geometries, and becoming foundational with Russell-Whitehead and later Gödel.

Thus, a typical example, from the introductory paper in a symposium volume commemorating Gödel’s 60th birthday:

From the viewpoint of a realistic philosophy of mathematics, the imcompletability theorm can be regarded  not as calling into question  the independent reality of mathematical entities such as sets or numbers, but rather as indicating an essential limitation in the expressive power of symbolism:  the limitation being that no symbolism can fully succeed in characterizing a system of objects as rich as the natural numbers.
Stephen Barker, “Realism as a Philosophy of Mathematics”, in:  J. Bulloff et al, eds. Foundations of Mathematics (1969), p. 4

That paper is disappointingly brief and even shallow, and will not be considered further.   But it does raise a semiotic issue -- only, one not restricted to mathematics.   Our natural language is, after all, inadequate for discussing anything that really exists out there independently of ourselves:  such as (to take one plump example), a penguin.  Philosophically incurious beings that most of us are, we do not notice how tongue-tied we really are, when it comes to expressing anything beyond a few platitudes;  only in new technical areas do we become semantically self-conscious (“Is it a wave?  Is it a particle?  Is it neither?  Is it both?”)   But try as we might, we shall never manage to express the essence of The Quintessential Penguin.

[Note:  The paragraph quoted above is Platonist, in that it assumes that something can exist, even if we cannot name it -- here, to be sure, in a more sophisticated version of that truism.  For essays relating to mathematical Platonism, click here:  ]

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