Notes – On Understanding


A practical description of understanding has four parts; a descriptive part, a derivation part, a usage part, and an insight part.  Each part is important to the understanding of a concept and somehow related to at least one of the other three parts.


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.





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