Re: Contexts (was Classes vs. Instances)

["John F. Sowa" <sowa@BESTWEB.NET>]

Paul,

I look at the full range of semiotics, which includes
sign processing by every organism from bacteria to humans
(and perhaps beyond).

 > Induction is not the sameAs deduction even if we are
 > considering the category of reasoning "processes".

They are certainly different.  Peirce divided semiotics
in three parts, which are now called syntax, semantics,
and pragmatics.  He classified the reasoning processes as
part of semantics, and the most basic reasoning process
of all is analogy.  Abduction, induction, and deduction
are more specialized and disciplined versions of analogy.
Humans (and other mammals) naturally reason by analogy,
and they have to go to school to learn the more specialized
and disciplined kinds of reasoning.

 > "Deduction" to computer science (and "some" mathematicians)
 > is sameAs formal inference.  Some mathematicians see a far
 > more complex situation regarding "what is a proof" and what
 > is inductive and deductive argument.  Abduction does not
 > exist in pure mathematics.

On the contrary, analogy is the primary method used by
mathematicians to make their discoveries.  Abduction is
the formal term used for the crucial insight.  Deduction is
merely the housekeeping part at the end when mathematicians
tidy up the results for publication.

People were using mathematics for centuries before the
Greeks formalized the methods of proof.  In geometry, which
is the branch they developed the furthest, the real insights
come in drawing the proper diagram that enables mathematicians
to see the relationships.  Once the correct diagram has been
drawn, the proof is trivial.

As an example, suppose I asked you to prove the theorem of
Pythagoras.  Unless you remember your high-school geometry
very well, you'd probably have to think a bit before writing
down the proof.  However, if somebody showed you the attached
diagram (pythago.gif), it's not necessary to know any algebra
to see that the two large squares are the same size, the four
triangles inside each square are the same size, and therefore
the inner square on the left must have the same area as the
sum of the two inner squares on the right.

If you must do algebra, you can write the equation directly
from the diagram on the right:

    (a + b)**2 = c**2 + 4(ab/2)

Expanding,

    a**2 + 2ab + b**2 = c**2 + 2ab

Subtracting 2ab from each side,

    a**2 + b**2 = c**2

As this proof illustrates, the real insight is the act of
abduction (i.e., guessing what diagram to draw).  After
you do that, the algebra (i.e., deduction) is trivial.

John