One of the signal developments in semantic theory during my
lifetime has been an increased
appreciation for the role of prototypes
in organizing our knowledge. To
introduce or elucidate a concept, don’t try to define -- to delimit -- the
whole domain, which can get fuzzy around the periphery: give a central, typical, illuminating
example. Thus
sparrows, and animals like that
clams and the like
are not-half-bad characterizations of the idea of ‘bird’ or
‘mollusc’. You might need to learn
a bit more become earning your doctorate in zoology, but for ordinary folks,
for everyday purposes, that is pretty much what we mean by bird and mollusc.
There are tons of such structures in mathematics, where the
prototype literally came first and then was generalized. Thus, the integers are the
prototype for ring theory; although, as a matter of idiom,
mathematicians usually do not say “prototype”, but motivating example.
(For further examples, and a vigorous defense in principle,
from some leading Russian mathematicians, of the wisdom of motivating-examples
for mathematical pedagogy -- and indeed, even for mature cognition -- try
this: Abstraction and Generality.)
Now, this procedure works only if the prototype is
antecedently known. Thus, to
revert to zoology, suppose you run across the term Chelicerata and ask your
naturalist friend what that means.
Casually he replies, “Oh, that’s like harvestmen and solifugae, and all
that sort of thing.” You’d
probably be pretty much where you started out.
Such was my experience, while trying to learn what is meant by "vertex operator algebra", upon meeting this helpful hint:
“Chiral algebras are the
prototypical examples of a vertex operator algebra.”
So profound is my ignorance of anything touching on the
subject, so stygian is the blackness of my nescience, that it got me no further forward.
~
The status of “What is …?” questions differs pragmatically -- subtly but
importantly -- depending upon circumstances, in particular what field fills in
the blank. Thus, suppose you
ask, “What is exobiology?”, already knowing the sense of exo- and biology but
being unclear as to what these parts mean in conjunction (sort of like the
opposite of “internal medicine”, perhaps?
Like dermatology, or the study of scales and fur?). Upon being told that, no, it means the
study of extraterrestrial life (should any such exist), you have been told all
you need to know.
Take it a step further: “What is etymology,
as opposed to entomology?” (a question which, as a licensed
etymologist, I have indeed many times met). In this case
you’ll be enlightened merely by “Word-origins versus insects”. For here your semantic Wissbegierde is probably minimal, and
may be actually zero as regards one or other of the paronyms: you simply want to be set straight
about an easily confusable pair.
In this case, clarification happened via verbal signs; but it need not. The case shades into those in which no intension is involved at all, but merely
extension -- reference. As, “Which of the Wilson
twins is Bobby?” A
satisfactory answer might be simply pointing; the questioner is not asking, “Who is Bobby Wilson really -- as a person?”
We may say that, in this case, we have definition not as description, but as delimitation. Such a style of defining is called ostensive definition.
(Note: Such a
duality between interior/intension and boundary/delimitation may remind adepts of that jewel in the crown of the higher
calculus, the generalized Stokes Theorem, displayed in all its refulgence here. However, the analogy is superficial.)
A similar case from the vocabulary of mathematics. I ask, “What is a semigroup, as opposed to a quasigroup?” Without the second phrase, I might be asking for a concise
but contentful thumbnail, such as “like a group but lacking inverses”. But perhaps I simply stumbled across
both terms on the back of a cereal box. Now my friend admonishes: “Semigroups are the things we met briefly back in
undergraduate algebra, remember?
Quasigroups are fancy new items which, if you can’t even recall the
definition of a semigroup, you really oughtn’t to go bothering your head
about.”
O-oh, so-o-o … Do we feel patronized by that? Okay, here you go, an actual
definition, quoted from Wikipedia:
A quasigroup
(Q, *, \, /) is a type (2,2,2)
algebra satisfying the identities:
- y = x * (x \ y) ;
- y = x \ (x * y) ;
- y = (y / x) * x ;
- y = (y * x) / x .
Nun, alles klar?
~
Sometimes to be met with in the practice of math (the
praxis, rather than the pre-existing body of truth), and scarcely outside it,
is the dialectical definition: Posit, analyze, refute, sublate; repeat until golden brown. Lakatos’ Proofs and
Refutations illustrates the process.
(For an example of such a “working, preliminary definition”, click here.)
~
Further illustration of the vocabulary, with concrete
examples:
Just as Euclidean space can be
thought of as the model Riemannian
manifold, Minkowski space with the
flat Minkowski metric is the model Lorentzian manifold.
-- http://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold
This is not “model” in the sense of mathematical “model
theory”; it is an informal term
equivalent to the linguists’ prototypical. The latter is occasionally also
used; as here, by algebraic
topologist Raoul Bott, speaking pedagogically in the introduction to a text:
We use the de Rham theory as the
prototype of all cohomology.
~
For further considerations, focusing mainly on math, try
this:
For the lexicography of mathematics proper, this:
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