This is the third column in this series on
"Virtual Logic". In the first column I promised
more about the self-replicating gremlin, but turned my
attention to the Robbins problem in Boolean algebra for
the second column. In this column, we return to the
gremlin.
Lets begin by recalling the gremlin in
typographical form. The gremlin, G, has the property that
Gx = (xx)
where (a) is our typographical
convention for an image of a "cap" over a. A
gremlin, when it encounters an object on its right (the
gremlin, being a denizen of the line
discriminates between right and left) duplicates that
object and puts the two copies inside the cap.
A harmless idea this duplication, is it
not? So it seems, unless we apply the gremlin to itself.
Then we find that Gx = (xx) becomes GG = (GG). For
the sake of abbreviation, let J = GG. Then J =
(J). J is an entity that can produce a cap (or
in the line, a parenthesis) around itself. So we have
J = (J) = ((J)) =
(((J))) = ((((J)))) = …
ad infinitum.
This leads us to consider the
"limiting form" omega:
W = (((((…))))).
If we allow the idea of an infinite
nest of parentheses, or an infinite nest of caps within
caps, then we have a spatial example of a form W that
literacy satisfies the equation
W = (W ).
The form W satisfies this
equation identically, since an exact copy (albeit
infinite) of it sits inside itself. The concatenation of
gremlins J = GG : J = (J). does not actually sit
inside itself. Rather, J is a machine that when
you operate it produces a shell about itself: J
----> (J). We have obscured this distinction by
our cavalier use of the equals sign!
Underneath this use of the equals sign
is an equally cavalier use of the notion of identitv.
It is through the notion of identity that this discussion
moves from pure mathematical considerations to matters of
personal identity, and to the identity of the observer in
any selfobserving system.
Can we ever assert of two entities A
and B that "A is identical to B"? This cannot
be done. If A and B are given as distinguished from each
other, then they are not identical. When we say that A is
identical to B we really mean "identical in certain
respects". Thus the two Omegas in this line, W and W , are
identical in form but not in position. They are however
congruent figures - one can be moved rigidly to coincide
with the other.
And what about the identity of A with
itself? Is this A identical with this A? This is even
more precarlous, since A must be seen as different from
itself in order for the question to arise. This is
exactly what we have meant by self-identity all along,
and it stems from the experience of the self. We (as
western speakers of Indo-European languages) take as
axiomatic that there is one self that is each of us. And
that self is a self capable of observing itself. In order
for me to observe myself, l must split myself into a self
that sees and a self that is seen. I SEE that the self
seen and the seeing self are identical. The self becomes
identical to it/her/him self. This axiomatic move (that
each of us performs) becomes the reference point for a
notion of identity. We find identity in a return to unity
after an act of separation. We project this same pattern
upon the objects of our experience.
As for ourselves, we take reference
points in space and in time. Thus I listen to my words
right now, comprehending them after they are spoken. In
this way I view a trace of myself in the past. It is
myself, and I identify that trace as my trace.
This reflectivity takes on a timeless aspect, much like
viewing in a hall of mirrors the successive images all at
once. Remarkably, we do not seem to create in our
cognition those infinite sequences so easily seen in
mirrors and in recursive constructions such as W =
(((((...))))). It is enough for us to point once to our
trace without going into an infinite regress of pointing.
The absence of this infinite regress in our mental and
perceptual experience is an important experiential
adjunct to the axiom that
The self is one. In a being prone to
falling into the infinite regress, the self would either
fragment into many selves, or it would fall endlessly
down the rabbit hole of perceiving perceiving.
Just so, the gremlin does not insist on
an infinite construction in order to obtain GG= (GG). We
are a little surprised to find GG inside those
parentheses after applying the operator G to itself, but
after all G just duplicates an X and puts the copies in a
box.
We are a little surprised to encounter
ourself upon examining the cleft between the world
outside and the world within. But after all, the operant
self on the boundary between inside and outside just
takes the form and its opposite (they inform a common
boundary) and puts them both in the box of comprehension.
The gremlin should have the last say in
this matter, and he does, telling us that his mode of
being can produce paradox as easily as pulling a rabbit
from a hat. For example, let us interpret the symbolism
AB to mean that "B is a member of A".
AB = "B is a
member of A".
Let ~A mean "not A":
~A = "not
A"
Now define R by the equation
RX = ~XX.
Reading this equation out we see that
it says "X is a member of R if and only if it is not
the case that X is a member of X." This is the
famous Russell set. Substituting R for X we have
RR = ~RR.
This is the Russell Paradox. The set R
is a member of itself if and only if it is not a member
of itself. Bertrand Russell devised this paradox around
the turn of the century. As the reader can see, the
Russell paradox is nothing more than a special case of
the action of the gremlin. The Russell Paradox produces a
"thing" (RR) that is identical with what it is
not. Is not the self identical with what it is not? This,
in itself, constitutes one kind of resolution of the
paradox. But we will leave this topic (more about the
Russell Paradox another time) and make a shift to matters
of reference.
The lndicative Shift
We have seen how self-reference arises
naturally from certain acts of reference. Here is a
variation on that theme. Let the formalism A----->B
denote a reference from A to B. For example, A can be the
name of B. Now define the SHIFT of the reference A
-----> B to be a new reference, namely from A# to AB:
A# -----> AB.
SHIFT(A -----> B)
= A# -----> AB.
Here we have a new name, denoted A#,
and a new referent, denoted by AB. The new name, A#, will
be called the "meta-name" of A. The new
referent, AB, will be called the "A-projection of
B."
An example of the movement from A
-----> B to A# -----> AB is the social situation of
meeting a person (B) and learning their name (A). After
learning the name, we use it in the meta-form A#. When I
see Roger I see a "person with name Roger" and
from my place of reference I call this person Roger, but
that "Roger" has the property that his name
occurs both in my mind and in association with the locus
of his physical body in space (the referent). In ordinary
language, the name in quotes is the equivalent of the
meta-name. In the indicative shift we notate Roger# to
indicate the new status of the learned name.
Now lets suppose that the meta-naming
operator, #, itself has a name. Let
M -----> #
denote the reference that names the
meta-naming operator. Apply the shift to this reference
and we obtain
SHIFT(M -----> #)
= M# -----> M#.
The meta-name of the name of the
meta-naming operator refers to itself.
More generally, suppose that A
-----> #B; then, shifting, we find
SHIFT(A -----> #B)
= A# -----> A#B.
You can think of A#B as a sentence that
discusses A#. Since A# is the name (referent) of this
sentence A#B, we see that A#B is discussing itself by
using its own name.
In this way, the indicative shift shows
how self-reference can arise in a very simple formal
language. The self-reference that arises in the
indicative shift, is a syntactic analogue of the famous
Quine sentence (Willard van Ornam Quine - Harvard
philosopher and logician). Here is a variant of the Quine
sentence:
"Creates a self-referential
sentence when appended to its own quotation" creates
a self-referential sentence when appended to its own
quotation.
The Quine sentence is indeed
self-referential and it contains within it the
instructions for determining when the self-reference
happens. We read the Quine sentence and see that it has
the structure of "Q"Q where Q tells us that the
result of appending Q to its own quotation will result in
a self-referential sentence.
In the indicative shift, we have
removed the internal speech of sentence fragment
Q = Creates a self-referential
sentence when appended its own quotation
and made an external grammar of the
shift.
The meta-operator # has a natural name,
namely its placement in quotes.
"Anything"-----> Anything and so
"#" -----> #.Then
SHIFT( "#" -----> #) =
"#"# -----> "#"#.
From the context of the indicative
shift, the operator # has the property that it creates a
self-reference when it is appended to own quotation.
lnstead of # being a sentence fragment, this potential
for creating self-reference is a way to describe how #
behaves in its context. The semantics of the sentence
fragment has been replaced by the syntax of the
shift.
It is amusing to go backwards and
devise a generalised Quine sentence fragment that will
correspond to #. Here is one phrasing of it:
# = Creates a reference from itself
appended to A, to B appended to A whenever A refers to B.
Then "#"# is the construction
"Creates a reference from
itself appended to A, to B appended to A whenever A
refers to B" Creates a reference from itself
appended to A, to B appended to A whenever A refers to B.
The construction "#"# is seen
to refer to itself through its own self-definition. This
sort of bootstrapping is at the heart of our apparently
stable notion of the self, and it is at the heart of the
structure of systems that embody their own descriptions.
An Exercise
Let us say that an expression of the
form #F is "active". Then a reference to #F
such as g ----> #F shifts to g# -----> g#F and
creates a self-reference. However g#F is no longer active
unless g is active. On the other hand, if #g ----> #F
then
SHIFT(#g ----> #F)
= #g# -----> #g#F,
The self-reference remains active when
the referent or name is active.
"What is in a
name?
A rose by any other
Name
Would smell as sweet".
I urge the reader to explore the
relationship between activity as a syntactic property and
activity as an analog of wakefulness (or consciousness).
The awake self is always ready to perform a metalevel
shift.
As a parting shot, we ask the reader to
consider the last sentence of this column. Who can verify
the truth of the last sentence of this column?
The Last Sentence of this Column:
No person who is awake while reading
the last sentence of this column can verify the truth of
that sentence.
