>...people begin with a compelling interest
>in some phenomenon, and they keep their eyes on prizing out some
>explanation...people will naturally turn to formal and
>mathematical models of the theoretical constructs of
>interest....they will have been talking about sets and functions
>and relations (oh my), or categories and morphisms and functors
>(OMG)...none of this very modern model of mediate reprsentation is
>intended
>to pose a terminal distraction from the intitial phenomenon of interest. (01)
Absolutely! (02)
Formal models enable engineering to take place. They allow people to
communicate precisely about a phenomenon in which they have a
compelling interest. They enable other people to understand exactly
what is being said about the phenomenon of interest. They enable
automated reasoning engines to process descriptions of the phenomenon
of interest to derive logical consequences of the descriptions that
were not immediately obvious to those who formulated the
descriptions. Sometimes this identifies weaknesses in the
descriptions that can be repaired to yield improved descriptions.
Other times, the consequences of the descriptions elicit great
surprise and consternation, but when checked against the world, they
turn out to be correct! Historically, this has been one of the most
powerful features of the scienfific method. One constructs a precise
formal theory of a phenomenon. The formal theory explains empirical
observations that have been obtained, but also entails some
highly-counter-intuitive consequeces that -- amazingly -- turn out
upon investigation to be correct! (03)
There are people who care more about formal models per se than they
do about the phenomena being modeled. Such people tend to become
pure mathematicians or logicians. (04)
There are people who care about the phenomenon and dislike
mathematics and logic. Because they find mathematics difficult and
unpleasant, they tend to balk at the intellectual effort required to
apply math and logic to the phenomena in which they have an intrinsic
interest. However, when they encounter a truly gifted teacher who
can explain the relevance, they usually do come to understand why
formal models are necessary and important. (05)
There also are people who have an affinity both for the formal models
and for some particular phenomenon to which formal models can be
applied. Such people are the ones who make the breakthrough advances. (06)
None of this is intended to disparage in any way other modes of
inquiry and understanding, such as analogy, informal case studies,
experiential learning, and the like. Formal models are not the ONLY
way to understand a phenomenon. They do not replace other ways of
thinking about a phenomenon. Nevertheless, they can be extremely
useful. And they are essential for engineering and information
technology. (07)
Kathy (08)
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