Tuesday, January 21, 2014

Definition, Description, Delimitation

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:

    What is Mathematics?”

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