+1. 

Mathematical notations are used to communicate with people who understand them, and in so doing to convey their intentions precisely.  Those notations serve as a lingua franca among mathematicians and often engineers and scientists, in much the same way that Latin served as a lingua franca among European universities for 4 centuries.  Formal language has nothing to do with "frightening away the unwashed".  It is about using a language suitable for the intended audience.  The object is to be understood.

I agree that there is a lot of pseudo-mathematics in published papers in certain fields, notably many applications of computer science, but that is a consequence of the ill-considered view of some journal editors and reviewers that a professional paper should have mathematical notations in it.

Halmos's point is that you have to develop the proposition by inspiration and induction, come to believe the proposition, devise an approach to a proof, formalize what you have conceived, and then validate the details.  Writing mathematics is in that sense knowledge engineering.  But it is exactly the same way in which any engineering task is performed -- you see a need, you devise an approach to accomplishing a function that fills it, then you formalize that approach as a design (or a software program), and then you test and validate it.  To communicate it to others, you send them the formal design or the program itself, in either case using a formal language that you expect your peers to understand.

-Ed

P.S. That Paul Halmos described the development of a formal design as "a draftsman's task" is an insult to engineering that is typical of a mathematician.

-- 
Edward J. Barkmeyer Email: edbark@xxxxxxxx
National Institute of Standards & Technology
Systems Integration Division, Engineering Laboratory
100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
Gaithersburg, MD 20899-8263 Cel: +1 240-672-5800
"The opinions expressed above do not reflect consensus of NIST, 
 and have not been reviewed by any Government authority."

[I disagree, John. The point of using mathematical (or logical) notation is to make your statement precise and unambiguous. You can't have it both ways: 1) avoiding terminology wars by using precise logical statements (witness the recent discussion here on "individual"), and 2) avoiding logical and mathematical notation by using supposedly clearer natural language. One reason for the success of ontological engineering in general, as for formal ontology in philosophy, is that one can write logical axioms, thereby curtailing somewhat the endless argumentation in English and other natural languages that has gone on in philosophy for generations (and often goes on here). Logic and mathematics are formal tools that ontologists and computer scientists need to use, so they can express their ontological analysis in logical expressions that they can then share/compare. Ontological analysis does not reduce to logical analysis, but uses logic as a way to formalize what it wants to say. I do think that you need to use simple but precise English for neophytes, but at some point ontologists (and ontological engineers) need to learn logic and at least some mathematics. One can use good English and formal definition together. Thanks, Leo

-----Original Message-----
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-
bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F Sowa
Sent: Thursday, January 03, 2013 6:36 AM
To: ontolog-forum@xxxxxxxxxxxxxxxx
Subject: Re: [ontolog-forum] Intensional relation
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.

I agree.
But that is no excuse for writing statements like:

"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"

Mathematicians don't think like that. They only use such language
when they are deliberately trying to frighten the unwashed.
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.
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."
Teaching ontology by burying the fundamental insights under
the trivial notation is pedagogical malpractice.
John
___________________________________________________________
_____________
Paul Halmos:
“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.”
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