Arrow system

June Kerby
Mon, 26 Apr 1999 20:15:50 -0500 (CDT)

Sorry for not reading your whole arrow paper first (I read the Introduction
and the first part of the Proposal), but ... I have one question as
to the fundamental nature of arrows as you define them: 

Is an arrow given only its head and tail, or does it also have "color" so to
speak? Otherwise phrased, if two arrows have the same head and tail, do they
necessarily express the same concept? 

I am going to assume a "colored" arrow system,
in which an arrow of color "c" from "a" to "b" implies the fact that "a"
and "b" are related by the relation "c". That system would make sense to me
and would be quite useful. On the other hand, I don't see an uncolored arrow
system as particularly useful (although I'd be happy for you to enlighten me,
if uncolored arrows are what you've been talking about); what meaning would be
implied by an arrow from "a" to "b"? that "a" is related to "b"? If you can
say that things are related but cannot tell how, then you're rather
restricted. Or were you planning on picking one universal relation, so that
for all arrows (with head "a" and tail "b"), it is implied that "a" is
related to "b" in a certain way? If so, what is that "certain way"? Or is it
left up to a metasystem to decide what the "certain way" is? If so, I'd like
to see an example, because it's not obvious to me that a good "certain way"
will even exist for systems. Or perhaps you've been talking about a colored
arrow system the whole time...

> lets' assume that it takes N graphs of arrows to describe all of the aspects
> of an equation embodied by a written statement.  these N graphs wouldn't
> apply to just that equation, though.

Hmm, what do you mean by "N graphs of arrows"? Do you just mean "N arrows",
or are you talking about sets of arrows?

> example:
> 		n=m+1
> there would be an arrow between the atoms for 'n' and 'm+1' that would be
> part of the graph linked to '=' (think of '=' as a relation, and as a set of
> arrows).

I don't think I understand what you said... It would make sense for "n" and 
"m+1" to be joined by an "="-colored arrow... but it sounds like you're
talking about uncolored arrows now.

Actually, now I think about it, it seems you are talking about colored arrows,
with the restriction that no two arrows can have the same color (perhaps
"location" would be a better metaphor now than "color"). You are expressing
the relationship between "n" and "m+1" using an arrow, and then using other
arrows to express the relationship of the arrow with "=". This seems futile
to me, because only the same meaningless arrows can be used, and the relation
between this arrow with "=" will never be able to be established. Why not
just used colored arrows? Although it may seem more complex at first, it
actually seems simpler to me, with less restrictions, especially from a
textual syntax (rather than graphical) point of view. I'll used the syntax 
"a r b" to say that "r" relates "a" and "b" (there it exists an arrow of
color "r" from "a" to "b"). The original equation could be expressed like 

n = x
1 + y
m y x

I'm using "+" as a relation between a number and a relation that adds that
number (this is sort of like currying, only with relations instead of
functions), for instance if "3 + x", then x relates numbers in which the
second is 3 higher than the first (i.e. x would relate 1 and 4, x would relate
2 and 5, etc.). Also, note that "1", "x", "n", and "m" are not themselves 
relations (they are numbers). Forcing all terms to be relations would be a
limit to the system that would severely cut down on its usefulness, because it
is often useful to relate things other than relations. 

Actually, it's beginning to seem as
if you are actually using arrows only as low-level representations,
and that an arrow could stand for anything, and so an arrow could stand for
a number. That technique seems bad to me because it corrupts the semantics
of the system. If you are only interested in a good low-level way to
represent concepts, then why not just go back to streams of bits? It would
be much simpler to communicate concepts in this way, but much harder to
interpret them. Similarly, using arrow-patterns to represent numbers adds
an unnecessary level of symbolism...

> there would be an atom for 'm+1' would be an arrow in the '+' graph linking
> the atoms for 'm' and '1'.
> 'n', 'm', and '1' would be arrows (selectors) from sets of atoms: 'n' and
> 'm' would be part of a particular user context vocabulary (called an
> ontology), and '1' would be an arrow from the set of natural or integer or
> whatever kinds of numbers (again, in a graph).

This has pretty much been covered... Using arrows to represent concepts
other than relations seems like a bad idea to me. However, the use of terms
other than relations does not imply the need to artificially distinguish 
between "arrows" and "objects". One can just say that "r" relates "a" and "b";
there is no need to say whether "a" and "b" are also arrows or what they are.

Any comments ... ?

- Brent Kerby ("Iepos")