On 1/3/2013 12:19 AM, Hassan Aït-Kaci wrote:
> The notation "2^S" for a set S denotes the set of all subsets of S -
> i.e., it powerset (also written P(S) sometimes). It is because its
> cardinality |S| is equal to 2^|S| that this notation has been used. (01)
I agree. (02)
But that is no excuse for writing statements like: (03)
> "An intensional relation (or conceptual relation) ρ^n of arity n
> on <D,W> is a total function ρn : W → 2^D^n from the set W into
> the set of all n-ary (extensional) relations on D" (04)
Mathematicians don't think like that. They only use such language
when they are deliberately trying to frighten the unwashed. (05)
Following is a quotation by Paul Halmos, whose books were used to
teach the mathematicians who talk that way. But thankfully, he
never wrote that way. If he had, nobody would have ever read
his books. (06)
As Halmos said, the intuition is fundamental. After you understand
the fundamental ideas, writing the formalism is trivial: "it is more
the draftsman’s work not the architect’s." (07)
Teaching ontology by burying the fundamental insights under
the trivial notation is pedagogical malpractice. (08)
John
________________________________________________________________________ (09)
Paul Halmos: (010)
“Mathematics — this may surprise or shock some — is never deductive
in its creation. The mathematician at work makes vague guesses,
visualizes broad generalizations, and jumps to unwarranted conclusions.
He arranges and rearranges his ideas, and becomes convinced of their
truth long before he can write down a logical proof... the deductive
stage, writing the results down, and writing its rigorous proof are
relatively trivial once the real insight arrives; it is more the
draftsman’s work not the architect’s.” (011)
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