# Currying by using zippers and an allusion to the Cartesian Theater

Here is a recipe for understanding currying with the help of zippers.  (Done in graphic lambda calculus.)

We have a graph $A \in GRAPH$ which has one output and several inputs. We want to curry it. For this we have to artificially give names to the inputs, i.e. to number them (notice that such a thing is not needed in graphic lambda).

The next step is to use a $n$-zipper in order to clip the inputs, by using $n$ graphic beta moves, until we get this:

This graph is, in fact, the following one (we replace the $n$-zipper, which is just a notation, or a macro, with its expression).

The graph inside the green dotted rectangle is the currying of $A$, let’s call him $Curry(A)$.  This graph has only one output and no inputs.  (The procedure of currying can be made itself into a graph which is applied to  the output of $A$, but we stop at this level for this post.) The graph inside the red dotted rectangle is almost a list. We shall transform it into a list by using again a zipper and one graphic beta move.

Now we’re done!

As you see, the currying creates the list, or the list creates the currying, or both form a pair, like the homunculus $Curry(A)$ and the scenic space $List(1,2, ... , n)$, an allusion to my post on the Cartesian Theater.

# The Cartesian Theater: philosophy of mind versus aerography

Looks to me there is something wrong with the Cartesian Theater term.

Short presentation of the Cartesian Theater, according to wikipedia (see previous link):

The Cartesian theater is a derisive term coined by philosopher Daniel Dennett to pointedly refer to a defining aspect of what he calls Cartesian materialism, which he considers to be the often unacknowledged remnants of Cartesian dualism in modern materialistic theories of the mind.

Descartes originally claimed that consciousness requires an immaterial soul, which interacts with the body via the pineal gland of the brain. Dennett says that, when the dualism is removed, what remains of Descartes’ original model amounts to imagining a tiny theater in the brain where a homunculus (small person), now physical, performs the task of observing all the sensory data projected on a screen at a particular instant, making the decisions and sending out commands.

Needless to say, any theory of mind which can be reduced to the Cartesian Theater is wrong because it leads to the homunculus fallacy: the homunculus has a smaller homunculus inside which is observing the sensory data, which has a smaller homunculus inside which …

According to Dennett, the problem with the Cartesian theater point of view is that it introduces an artificial boundary (from Consciousness Explained (1991), p. 107)

“…there is a crucial finish line or boundary somewhere in the brain, marking a place where the order of arrival equals the order of “presentation” in experience because what happens there is what you are conscious of.”

As far as I understand, this boundary creates a duality: on one side is the homunculus, on the other side is the stage where the sensory data are presented. In particular this boundary acts as a distinction, like in the calculus of indications of Spencer-Brown’ Laws of Form.

This distinction creates the homunculus, hence the homunculus fallacy. Neat!

Why I think there is something wrong with this line of thought? Because of the “theater” term. Let me explain.

The following is based on the article of Kenneth R Olwig

but keep in mind that what is written further represents my interpretation of some parts of the article, according to my understanding, and not the author point of view.

There has been a revolution in theater, started by

“…the early-17th-century court masques (a predecessor of opera) produced by the author Ben Jonson (the leading author of the day after Shakespeare) together with the pioneering scenographer and architect Inigo Jones.
The first of these masques, the 1605 Masque of Blackness (henceforth Blackness ), has a preface by Jonson containing an early use of landscape to mean scenery and a very early identification of landscape with nature (Olwig, 2002, page 80), and Jones’s scenography is thought to represent the first theatrical use of linear perspective in Britain (Kernodle, 1944, page 212; Orgel, 1975).” (p. 521)Ben Johnson,

So? Look!

From the time of the ancient Greeks, theater had largely taken place outside in plazas and market places, where people could circle around, or, as with the ancient Greco-Roman theater or Shakespeare’s Globe, in an open roofed arena. Jones’s masques, by contrast, were largely performed inside a fully enclosed rectangular space, giving him control over both the linear-focused geometrical perspectival organization of the performance space and the aerial perspective engendered by the lighting (Gurr, 1992; Orrell, 1985).” (p. 522, my emphasis)

“Jonson’s landscape image is both enframed by, and expressive of, the force of the lines of perspective that shoot forth from “the eye” – notably the eye of the head of state who was positioned centrally for the best perspectival gaze.” (p. 523, my emphasis)

“Whereas theater from the time of the ancient Greeks to Shakespeare’s Globe was performed in settings where the actor’s shadow could be cast by the light of the sun, Jones’s theater created an interiorized landscape in which the use of light and the structuring of space created an illusion of three dimensional space that shot from the black hole of the individual’s pupil penetrating through to a point ending ultimately in ethereal cosmic infinity. It was this space that, as has been seen, and to use Eddington’s words, has the effect of “something like a turning inside out of our familiar picture of the world” (Eddington, 1935, page 40). It was this form of theater that went on to become the traditional theater in a box’ viewed as a separate imagined world through a proscenium arch.” (p. 526, my emphasis)

I am coming to the last part of my argument: Dennett’ Cartesian Theater is a “theater in a box”. In this type of theater there is a boundary,

“… scenic space separated by a limen (or threshold) from the space of the spectators – today’s traditional’ performance space [on liminality see Turner (1974)]” (p. 522)

a distinction, as in Dennett argument. We may also identify the homunculus side of the distinction with the head of state.

But this is not all.

Compared with the ancient Greeks theater, the “theater in a box” takes into account the role of the spectator as the one which perceives what is played on stage.

Secondly, the scenic space is not “what happens there”, as Dennett writes, but a construction already, a controlled space, a map of the territory and not the territory itself.

Conclusion: in my view (contradict me please!) the existence of the distinction (limen) in the “Cartesian theater”, which creates the homunculus problem, is superficial. More important is the fact that “Cartesian theater”, as “theater in a box”, is already a representation of perception, having on one side of the limen a homunculus and on the other side a scenic space which is not the “real space” (as for example the collection of electric sparks sent by the sensory organs to the brain) but instead is as artificial as the homunculus, being a space created and controlled by the scenographer.

Litmus test: repeat the reasoning of Dennett after replacing the “theater in a box” preconception of the “theater” by the older theater from the time of ancient Greeks. Can you do it?

On the beautiful idea of “aerography”, later.