As you might see, this cannot be a worse title. But recently I was inspired by Zizek’s interpretation on Kant’s transcendental objects and Lacan’s objet petit a, in his book Less Than Nothing, to make these many things connected in a gradual sense.
To begin with, one will assume a counting from zero. With ‘set representation’, one can write down:
{0, 1, 2, ...}
You might think it’s the set of natural number, the aleph-naught, or something similar. It’s not yet informative to that extent. Symbolically, one can argue that ... is too fuzzy. It nontheless means the “constant possibility of +1“, as if the magical infinity “spits out” one but itself still remain intact.
{...}
{0, ...}
{0, 1, ...}
{0, 1, 2, ...}
It’s fascinating to think there is a “point of impossibility”, an inaccessable X beyond all finite ones, lying at the ‘end’ of the series:
{0, 1, 2, ..., X}
The impossible/inaccessable X within the set can be viewed as a ‘transcendental one perceived empirically’.
Now we have a pair of ideas opposite to each other:
XWhat if there is a short-circuit between them? What if the impossible X itself is the constant possibility of adding one, represented as {...} (and any other representations with ... inside like {0, 1, 2, ...})?
With one more step forward, one can write this down carefully:
{0, 1, 2, ..., {0, 1, 2, ...}}
It’s exactly a counting from zero to one. One can continue adding the symbol X in the nested set, and unfold it further:
{0, 1, 2, ..., {0, 1, 2, ..., X}}
{0, 1, 2, ..., {0, 1, 2, ..., {0, 1, 2, ...}}}
The “constant possibility of adding one” is now inscribed into the level where the unfolding goes on. To make it more concise, if one takes {0, 1, 2, ..., X} as a totality of “constant possibility of adding one”, and = as the short-circuit operation:
X = {0, 1, 2, ..., X}
It is the transfinity.
A simple variant of the above result would be like:
X = (1, X)
X = (1, (1, X))
X = (1, (1, (1, X)))
It’s a stream of 1s. In Racket, with delay or lambda, one can define such a stream:
(define 1s (cons 1 (delay 1s)))
(define 1s (cons 1 (lambda () 1s)))
Stream is no more than transfinity. It’s a “constant possibility of unfolding”, a stream of empirical objects, by definition. But one may not easily relate an arbitrary stream to the impossibility of reaching a ultimate element (transcendental object) within itself. One can regard the stream itself as an ‘external framework’, or computationally, a ‘generator’ of all positive and empirical objects.
Another more general variant of the above result is something called ‘fixed point’ or ‘fixpoint’ (recall ‘point of impossibility’):
X = F(X)
where F is some mathematical or computational function. Here the transfinity lies in, at the most obvious level, the constant possibility of applying the function F:
X = F(X)
X = F(F(X))
X = F(F(F(X)))
One can say it’s too general and has nothing informative. There is nontheless one tiny tweak to be done to make it powerful - what if X is also indexed by another variable, that is, X is some g(x)?
g(x) = F(g(x))
This is a recursion without ‘termination conditions’, with F as the ‘wrapper’ of the recursive call to g. To make it practical, one needs to specify the very data type of x and create at least one termination condition, for example:
g(0) = 0
g(x+1) = F(g(x))
Racket version:
(define (g x)
(cond
[(zero? x) 0]
[else (F (g (sub1 x)))]))
A recursion is a ‘quasi-transifinity’ with an empirical ‘ultimate object’. It always returns half-way during the impossible journey to the transfinite.
Question:
Gaze and Silence are two symbolic mechanisms familiar to continental schools. To show them properly, I have to refer to Lacanian theory (the Big Other and the objet petit a). There are two ways to illustrate them, one in set-theoretical style, another in categorical way (you can have Yoneda at hands).
Suppose we have a (finite) collection/set, say {0, 1, 2, 3}. The critical idea here is: identity is not free - i.e. one cannot justify 0‘s identity 0 = 0 simply by its own presence. Instead, it’s very identity should be obtained through it’s absence and difference from other elements - i.e. one can say 0 = {1, 2, 3}, 1 = {0, 2, 3}…
Note the italic other here. The Otherness seems to be more explicit and tangible in category theory. But let me first introduce the Big Other under set theory - A is defined as:
A = {0, 1, 2, 3, A}
In other words, A is something like the set itself, while it’s also part of the set. It has the ability to unfold itself unboundedly (repetition). Compared to any “usual others” like 0, A‘s identity is indeed a self-referential tautology (the Master’s Signifier). One can easily imagine its political counterpart - people with various identities “issued” by a special agency, who pretends to be just a “usual element”. One should not miss a dialectical tension here - only by degrading itself into the commonplace can it impose an effective rule:
A = {0, 1, 2, 3}
This is an imaginary-idiot tyrant which is clearly not the case for modernity. But rather, it’s a trick played everywhere - call someone a tyrant, and others gaze it out, then they fight within the constitutive-illusion.
Now, one cannot help but ask the question: how does the Big Other successfully “hide” itself? The answer is - by objet petit a - a is defined as:
a = {a, A}
It says: A has its own master, which is a. Just like A can repeatedly unfold itself, so does a here. As a result, A is as common as the usual others. But is this true? Unfolding of a only generates a dull sequence of A which is suspicious enough.
Here the proper catch is: where does this mysterious a come from? Is it really something behind A? The answer is: from the commonplace elements - i.e. one element from {0, 1, 2, 3} is “sacrificed” to play the role as seemingly transcendental a. One should recall Hegel’s famous line “essence is the appearance qua appearance”, here the essence is the illusionary signal that there is an a beyond A.
To sum up, Gaze is the = operation in 0 = {1, 2, 3}, 1 = {0, 1, 2}…(Probably a good question to think about: what is the gaze from the Big Other?)
In category theory, the Yoneda Embedding shows the gaze proper:
X ↦ Hom(_,X)
This says an object “itself” is no more than an empty letter. Its determination is by the set of all arrows (gazes) coming from other objects.
Questions: