SUO: Re: "Abstract" and "dimensionality" - Re: SUMO axiomatization
Graham,
Your note raises many interrelated issues, which have to be sorted
out in any kind of standards document.
> "Horn, Graham" wrote:
> I have had considerable problems with trying
> to tie down the nature of pure structure, theory, etc. I tried to
> apply the term "concept" to this, but was ruled out of order on the
> basis that a concept had to have been conceived by someone, and by
> implication, at a given point in time. The latest draft of
> ISO/IEC 111793 accordingly defines "Concept" as
>
> "a unit of thought constituted through abstraction based on
> characteristics common to a set of objects".
This was a hot topic in the late 19th century, when people like
Peirce and Frege were trying to get rid of "psychologism" in logic.
Frege was only half successful, since he continued to use words
like "Gedanke" (thought) and "Urteil" (judgment). Peirce completely
eliminated that terminology by using the terminology of the
medieval scholastics, who were much more modern. He adopted the
word "proposition" (from the Latin "propositio"), and drew a clear
distinction between the proposition as content and its use in
a thought, a judgment, an assertion, etc.
> I fear the same attitude may be taken to "Abstract",
> since someone had to perform the (mental) task of "drawing away" the
> pattern or structure from the original observation or thought.
Finding good terminology that is both precise, easy to remember, and
free of misleading connotations is always difficult. Whitehead used
the term "eternal object" for the entities in my category Abstract
because they are all independent of time. However, that led many
people to think that they were somehow located in a kind of Platonic
heaven and had some sort of spiritual connotations.
I would like to identify all abstract entities with mathematical
structures, since they are of the same nature. (That doesn't solve
the problem, because then you have to face the question about what
mathematical entities actually *are*. Working mathematicians don't
worry about that issue, since they all accept a kind of Platonic
view that such things really exist in a mathematical heaven. The
mathematician Erdos was fond of saying that God had all of them
written down in a big book.)
My solution was to adopt the term "abstract", which is less frightening
to most people than "mathematical structure". However, I insisted that
the choice of terminology is irrelevant to the formalism. I said that
you should think of that category as having the name "A". The word
"Abstract" consists of 7 meaningless letters concatenated to "A",
which make it easier to remember.
> Personally, I feel there is something in the nature of
> pure structure or pattern that can exist independently of whether it
> exists in physical reality, and also independently of whether any
> intelligent being has conceived of or observed it.
Yes, indeed. The usual term for it is mathematics.
> By way of explanation, I would suggest that both
> Newton's law of gravity and Kepler's laws of planetary motion are not
> precisely accurate, and so do not exist in physical reality, but are
> rather mere approximations of physical reality. Yet, I would suggest
> that they are both timeless and without location or mass
> ("zerodimensional" in the physical sense), and hence existed
> infinitely before either Newton or Kepler were born, and will continue
> to do so after all records and other traces of them have been
> obliterated.
And that is exactly what mathematics is.
> The problem is I don't know what term I can use to
> apply to such pure structures or patterns that one can abstract or
> conceive.
I agree that there is no word that is immediately recognizable to
everybody and which is free from all misleading connotations. That
is why the "official" name of my category, which I usually write as
"Abstract", is really "A".
> Neither does "information" cover this. One can observe
> and document the spread of information, whether be it physical, such
> as through the spread of genes, etc., or purely conceptual, such as
> with nemes, by means of, say, semaphore communication.
I once considered the word "Information" for category A, but it also
had too many misleading connotations.
> The reason I ask this question is that there are times
> when I wish to consider these "zerodimensional" patterns
> structures.
Certainly.
> Incidentally, t have a related problem about the
> nature of "dimensionality". I wish to apply the term beyond the
> traditional physics scope of mass, length and time, into far more
> abstract areas. This is because it can be extremely useful in
> information management and information based sciences to apply to a
> far broader range of concepts. Statisticians psychologists and
> pharmacists do this when they are looking to ascertain which factors
> are relevant, and which irrelevant, for particular avenues of research
> and endeavour. The concept of "dimensionality", along with others like
> "mutual exclusivity" or "orthogonality" also come into play here, and
> can greatly facilitate conceptual design and analysis.
In my previous note, I used the term "dimension" for what the letter t
happened to represent. That was another misleading use. I should have
just used the word "parameter" for t.
> Can anyone suggest a rigorous definition that covers
> this broader scope?
As I said, mathematics is the subject that characterizes all possible
entities in category A, and every entity in category A is of the same
nature as the mathematical structures.
That is why Plato posted the sign above his Academy:
ageometretos medeis eisito
(Let no one ignorant of geometry enter here.)
In the _Republic_, Plato elaborated on that idea: "The knowledge
at which geometry aims is knowledge of the eternal, and not of
anything perishing and transient." That is the justification
for Whitehead's term "eternal object". To avoid all those
confusing connotations, I would prefer to use the term "mathematical
structure", but that just adds further confusion for those people
who are not mathematicians. So my only other suggestion is to
use the term "Category A". That is to be distinguished from
"Category P", where the letter P can be concatenated with the
7 meaningless letters "hysical".
John Sowa