Notes – On Understanding

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The first part is
descriptive. Descriptive understanding
needs the ability to precisely define an object or concept. A pencil is a graphite-based writing
instrument. In logic, negating a
concept twice is logically equivalent to the concept. The description may be stated in a symbolic language and remain a
description.

ØØp º p

The second part is
derivation, and requires the ability to derive an object or concept from some combination
of primitive and non-primitive objects or concepts. The derivation part is similar to the descriptive part with a
very subtle difference. A description
explains what an object is or what it does.
A derivation explains how an object becomes what it is. The derivation of a pencil might be: a
pencil has a graphite rod wrapped in wood;
the wood protects the brittle graphite; graphite has the property that
it leaves a mark on a surface and tends to remain on the surface. The double negative theorem derivation
relies on reflexive property of logical equivalence and the distribution of
negation over logical equivalence.

The third part is
usage, and describes how an object affects other objects. One may write a journal with a pencil by
listing observations made each day. The
reflexive property of logical equivalence and distribution of negation over
logical equivalence are used to prove the double negation theorem. Unlike the descriptive part and the
derivation part, usage adds purpose to the object. Using a pencil makes the pencil a writing instrument or a drawing
instrument. One could use the double
negative theorem to prove deMorgan’s laws for example.

The fourth part of
understanding is insight, and it is the most elusive. Having insight about the subtleties of a concept can be the sum
of experience and analysis based on the other three parts of understanding or
can simply become. The way that insight
works is not well known.

The four parts of
understanding are related. The
derivation part contains elements that have a usage part. A theorem uses axioms and other
theorems. An axiom contains a simple
descriptive part, but is also used to prove other theorems. A theorem contains two parts; description
and derivation. Better understanding
comes when the theorem is used, because usage adds a third, different but equal
dimension to understanding. Insight is
a skill that creates things out of nothing, and is needed to some extent to
fuel the other three parts. Insight and
the other three parts form an interesting dependent relation. Every good description is based on some
insight, and every insight grows out of any one of the other three parts.

Understanding has
many dimensions, and each dimension is both dependent and independent of each
other. Understanding grows only when it
grows in all parts together.

References:

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Gries, David and Schneider,
Fred B., 1993, *A Logical Approach to Discrete Math*, New York, NY:
Springer-Verlag