ONT Re: Why Tri?
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TRI. Note 2
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So we take up the task, one more time, of recalling why
3-adicity is so indispensable to understanding Peirce's
theories of continuity, information, inference, inquiry,
logic, reality, signs, and so on, just to name them in
alphabetical order.
I imagine that it would probably be more interesting right off the bat
if I were able to begin with the matter of "synechism", or what Peirce
called his theory of continuity, but I'm afraid that I cannot really
do a responsible job of that without taking care of many discrete
and discretionary issues first.
However, I can leave this hint that Peirce's theory of continuity is
closely akin to what has been in recent times called "generativity",
as when we speak of the "generative capacity" of a language, and in
a related sense, of the "generative power" of a mathematical system.
All of these can be seen to depend on 3-adic relations at the core.
By way of trying to begin fresh, let me start
with this very apt quotation from recent days:
| But this is not a simple matter; the interpretant
| determines the meaning, (the quotation, for example,
| means nothing unless the interpreter has collateral
| knowledge about the object, which might confirm the
| meaning. As I say to my students: "Quotations do
| not 'speak for themselves'". The object needs be
| considered. (Gary Richmond).
In that brief statement we can already see a striking recognition
of the integral relationship between the character of a "symbol",
as an essentially interpretive and thus pluperfectly 3-adic type
of sign, the 3-adicity of all sign relations in general, and the
appreciation of human frailty in the process of fixing beliefs
through inquiry. I say this because it is the fact that our
signs are always properly-partially informative that forces
us to call to mind our independent acquaintance with the
object and the reconstitutive powers of the interpretant
in order to make any true sense of signs at all.
Less obvious, perhaps, is the connection with such issues
as continuity and generativity. For the moment I can but
approach them by way of the related issue of recursedness.
Here is a picture of every child's first recursively defined function,
the "factorial" f : N -> N, such that f(x) = x! = x*(x-1)*...*3*2*1,
where the star (*) is being used here as a sign of multiplication.
x f f(x) = m(x, f(p(x)))
o- - - - - - - - - >o
|\ ^
| \ |
p | o--------------->| m
| |
v |
o------------------>o
p(x) f f(p(x))
This diagram illustrates the recursive definition
of the factorial function as f(x) = m(x, f(p(x))),
where the pre-fabricated materials m and p of
the construction will be explained shortly.
The definition of the "factorial" as
the "multiplication by predecessors"
continued until it bottoms out at 1,
is already intuitively clear, and so
what we're about here is nothing but
the teasing out of a compact formula.
Let N be the non-negative integers, N = {0, 1, 2, ...}.
I use the term "prefunction" for a "partial function",
that is, a construct that is like a function, but not
defined on all of the arguments in its declared domain.
In the diagram, the "f" on the dashed line at the top
of the picture stands for the function f : N -> N that
we would like to define on the generic argument x in N.
The prefunction p : N ~> N is given as the predecessor function,
p(x) = (p - 1) except when x = 0, in which case it is undefined,
a fact that is often indicated by writing f(0) = _|_ = "Bottom".
Insert your own "Midsummer Night's Dream" joke here ___.
The "f" on the too, too solid line at the bottom
of the picture stands for the notion that we've
already been given the actual value f(p(x)) of
the function f on the predescent argument p(x).
The prefunction m : N x N ~> N is named "more or less multiplication".
It is defined to be just like the multiplication of two integers in N,
whenever its input arguments actually are valid elements of N, but if
one or the other is undefined, then a special set of exceptions apply.
In particular it is set that m(0, y) = 1, whether y is defined or not.
In computer science this is called a "regime of lazy evaluation" (ROLE),
and what it means in practical computational terms is that once you get
the news that the first argument to m is 0, you need not wait around to
see whether the second argument to m is going to turn out to be defined
or not, but that you may feel safe assured to report that m(0, ...) = 1.
In fine, the 2-adic relation f : N -> N is being defined in
terms of the 2-adic relation p : N ~> N in conjunction with
the 3-adic relation m : N x N ~> N. The triadic character
of the relation m can be described in the following wise:
| But this is not a simple matter; the interpretant
| determines the meaning, (the quotation, for example,
| means nothing unless the interpreter has collateral
| knowledge about the object, which might confirm the
| meaning. As I say to my students: "Quotations do
| not 'speak for themselves'". The object needs be
| considered. (Gary Richmond).
That is to say, it is very important that we keep the initial object x
in mind along with all of the other signs that we may have to evaluate
in the process of deciding some feature of that object.
Jon Awbrey
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