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:
http://worldofdrjustice.blogspot.com/search/label/Realism ]
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