SUO: Re: Repairs Of Things Renumbered
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Matthew,
I extract the relevant passages from last time,
make a few clarifications and corrections, and
continue the discussion from where I left off.
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So let me suggest a certain analogy with arithmetic,
the one that may easily leap to mind whenever words
like "irreducibility" rise up or roil up to becloud
the air, and so let us now resort to the relatively
clear and untroubled view that we may have of those
so-called "natural numbers": N = {0, 1, 2, 3, ...}.
I always forget if we are supposed to count zero --
but we might as well throw it in for a good measure.
Let us think about N, at least, as it appears with
respect to the operation of multiplication and the
corresponding relation of divisibility. I am sure
that you can recite as well as I all of the rather
familiar homilies, so let us all boldly jump right
into the middle of it, and if we need to backtrack
later on to pick up something prior that we forgot,
well, nothing says that we can't do that on a case
by case and a "need be" basis.
So what's on first? -- in that "relatively first" sort of way?
I guess we would need a definition of the "local irreducibles",
to wit, the "prime mumbers", defined, if I can remember all of
the ins and outs of the official definition, as something like:
"A number that is divisible solely by 1 and by itself, no more."
So, for instance, if you find yourself thinking of this, and you will,
on some future occasion, and you are telling somebody else about what
all we revealed to ourselves in our talks about the nature of numbers,
and especially about these primary, indecomposable, irreducible ones --
and they interrupret you to say "What the heck are you talking about --
are not all whole numbers utterly decomposable and reducible like so:
n = O, or n = 1, or else n = 1 + ... + 1, (listing the special cases
at the start and then indicating the pattern of the general break-up,
according to the custom that forms the common convenient convention)?"
And so you have to back up to explain that you are talking about
a different mode of analyzing and synthesizing numbers than that
which arises from what is apparently their own present view of N
as it appears with respect to the operation of addition and that
associated relation of "precedence", of "subordination", or just
plain being "less than", along with that entire host of problems
that revolve about one's contemplation of so-called "partitions".
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One of those "familiar homilies" that I glossed over
just a bit too quickly was the definition of a prime.
For sundry reasons, like the convenience of ensuring
the "unique factorization" of naturals into products
of primes, it is conventional to exclude the numbers
0 and 1 from being considered as primes. Let us say:
| A prime is a natural number that has
| exactly two divisors, 1 and itself.
I know that you have frequently suggested that we ought to think about
the relations between different ontologies by way of the mappings that
it makes sense to ply between their various "domains of being" (DOB's).
That is just the brand of thing category theory is all about, which is
merely to say that many other folks have found it useful to do so, too.
And that is more or less the sort of thing that we are doing here when
we think about the possible analogy between a set of relations of many
distinct arities and the set of natural numbers N. In this particular
case, however, where the "source" domain M is a sufficiently inclusive
set of relations to be interesting, and where the "target" domain N is
the natural numbers, the potential analogies are not likely to be very
useful, but merely something to contemplate as ad hoc mental exercises.
Let me try to take stock of the points
that I am trying to make with all this:
1. The interesting situations, from the POV of my aesthetic, anyway,
are those where one has a big collection various-arity relations,
each k-ary relation of which collection, if viewed in respect to
its extension, is a set of k-tuples from its specific relational
domains, say, X<1>, ..., X<k>.
2. The various attributions of "reducibility" or "irreducibility"
that we might make in respect to a given relation in this set
will have no meaning at all except in relation to a specified
definition of the kinds of "composition" and "reduction" that
one has in mind.
This is just how a very combinatorial-, concrete-, discrete-minded
sort of person thinks about this array of issues. It is a type of
reasoning and representation that turns out to be just what I need
and just what I am always eventually driven to whenever I actually
come down to the brass syntax of programming parsers, provers, and
data-base pursers.
To be continued, ...
Jon Awbrey
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> Matthew West wrote:
> >
> > Dear John,
> >
> > As one of those mystified by "triadic irreducibility"
> > please see some comments below.
> >
> > > Pat,
> > >
> > > Peirce and Whitehead were both familiar
> > > with the point you keep repeating endlessly:
> > >
> > > > Had Peirce or Whitehead lived a little longer maybe they would have
> > > > become aware of the fact that any n-ary relation can be defined in
> > > > terms of binary relations, with the aid of the existential quantifier.
> > > > The translation, as I know you know, John, is this:
> > > >
> > > > R(t1,...,tn) --->
> > > >
> > > > (exists e)(R(e) & first(e, t1) & second(e, t2) & ... & nth(e, tn))
> > > >
> > > > where 'first', 'second', etc., are some fixed set
> > > > of binary relations. (In case grammar these correspond
> > > > to cases such as 'agent', 'subject' and so on, and the 'e'
> > > > is something like an event or a situation, of type R,
> > > > corresponding to the verb of the simple sentence, as in:
> > >
> > > Now let me use your example to explain the point which both
> > > CSP and ANW were trying to make, which and you keep missing:
> > >
> > > > Gave(John, Book, Mary, yesterday)
> > > >
> > > > --->
> > > >
> > > > ( exists e )
> > > > ( Giving (e) &
> > > > agent (e, John) &
> > > > subject (e, book) &
> > > > recipient (e, Mary) &
> > > > time (e, yesterday)
> > > > ).
> > >
> > > This is the transformation that Ernst Schroeder used in his 1890 book
> > > for replacing triadic and higher relations with dyadics. Both Peirce
> > > and Whitehead were very familiar with it, and they both rejected it
> > > because it is *not* a decomposition into dyads. It is merely
> > > a relabeling of the arguments 1, 2, 3, 4 with
> > > labels "agent", "subject", "recipient", and "time".
> >
> > MW: Well I disagree. What it is really doing is recognising
> > an object that was hidden by the initial incomplete analysis,
> > i.e. the giving activity in which the other objects are
> > participants (except the time which gives the temporal
> > location of the activity).
> >
> > > What CSP and ANW were trying to explain is that the verb "give"
> > > and many other irreducible triads *cannot* be reduced to
> > > conjunctions of dyads of the following form:
> > >
> > > give(x,y,z) = part1(x,y) & part2(y,z) & part3(y,z).
> >
> > MW: Doing this would be just silly, but it isn't what Pat has suggested.
> >
> > > In this decomposition, part1, part2, and part3 are
> > > pure dyads that only relate two arguments at a time.
> > > In Schroeder's decomposition, which you keep repeating,
> > > you have to introduce a new argument e, which links the
> > > monad named "giving" to each of the other arguments:
> > > agent(e, John), subject(e, book), recipient(e, Mary),
> > > and time(e, yesterday).
> >
> > MW: Yes, but the new argument represents a real object.
> > My experience is that most of the time when there is
> > a triadic or higher relation there is a hidden object --
> > often an activity (or process or event if you prefer)
> > and sometimes another relation.
> >
> > > Notice that you have *not* performed a reduction
> > > of a triad (or tetrad in this case) to a conjunction
> > > of dyads because your new argument e links "giving" to
> > > every one of the other arguments. All thetriadicity (or
> > > tetradicity) is still buried in the monad named "giving".
> >
> > MW: Well if we follow that line of argument
> > there is only one enormous relation of all objects.
> >
> ...
> >
> > Regards
> >
> > Matthew
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