The idea is that ! can be thought of as a unary successor, such that !!x varies in the 1st dimension compared to the 0th dimension. In the 0th dimension, everything works as normal.

Layers and jokers naturally form branches where the least significant bit of dimension lookup determines the branch.

These systems are kept separate as layers and unary_dimensions respectively.

Requires adding Expr::Id to explicitly describe a leaf node in layers.

The following rules could be disabled in the Open variant:

0' 0' => 0
0' (0 x)' => 0 x
1' 1' => 1
1' (1 x)' => 1 x

The motivation is to apply Joker Calculus to epistomology. In epistomology, unknown unknowns is different from unknowns (Open variant). However, unknown unknowns is essentially unknowns (Closed variant).

By separating the Open variant from the Closed variant further, it seems to become clearer which roles they play.

• Added ((?x)', (?y)') => eval(?(x, y))
• Added ?(x, ?x)' => eval((?x, y)) (CJC)
• Added ?(?x, y)' => eval((x, ?y)) (CJC)
• Added ?(x, y)' => ?(x, y)

• Intentional Bias: (Platonism, Seshatism)
• Composite Bias: (Platonic Seshatism, Seshatism)

The intentional bias is due to using art to communicate an abstract mathematical object. This abstract mathematical object point toward abstraction as a causal process (Platonic Seshatism).

The idea is that humans have evolved traits of survival, to lend on either their personal or group identity:

1. The sacrifice of personal identity on behalf of a future personal identity
2. The sacrifice of group identity on behalf of a future group identity

When sacrificing personal identity on behalf of a future personal identity, one can think about the personal identity as Seshatic. However, this might require support from the group identity, so the group identity is Platonic.

Likewise, when sacrificing group identity on behalf of a future group identity, one can think about the group identity as Seshatic. This might require support from the personal identity, so the personal identity is Platonic.

So, the languages where personal identity is dual of group identity are interesting.

There are two perspectives, one is Joker X, second is !X X, where X is the personal identity. Taken together, they form (Joker X) !X X, or, if the isolated individual and the group individual are contracted: (Joker X) X.

(Joker X) X !X = (X X !X, !X X !X) in OJC
(Joker X) !X X = (X !X, !X X !X) in CJC
(Joker X) !X = (X !X, !X !X) in OJC
(Joker X) !X = (X !X, !X) in CJC

Duals:

!((Joker X) X !X) = (!X ?(X !X), X ?(X !X)) in OJC
!((Joker X) !X X) = (!X ?!X, X ?(X !X)) in CJC
!((Joker X) !X) = (!X ?!X, X ?!X) in OJC
!((Joker X) !X) = (!X ?!X, X) in CJC

Minecraft

Platonic Seshatism (0 1)

Authentic in sense of Heidegger.

The player character is simple and its freedom in the world is part of its ability to be oriented and navigated, part in the ability to move items from one place to another, manipulate the environment and craft new items.

The Witness

(Seshatic Platonism, Platonic Seshatism) ((1 0, 0, 1))

Authentic in sense of Heidegger.

At depth level, the game is about path finding, or how to move from A to B, which is Seshatic Platonism.

However, at surface level, the path reacts to itself and the goal has constraints that depends on both the map and the path, which is Platonic Seshatism.

In the paper Left and Right Inverse, I explained the underlying model of left and right inverse.

Relating this underlying model to Joker Calculus:

f ?g <=> (?g . f <=> id)

0 ?0 - entrance path `1` twists onto an untwisted loop
0 ?1 - entrance path `0` untwists onto a twisted loop
1 ?0 - entrance path `1` untwists onto a twisted loop
1 ?1 - entrance path `0` twists onto an untwisted loop

The idea is to relate life forms to bias of Seshatism vs Platonism.

Motivation

Both humans and AGIs (Artifical General Intelligence) have the following capabilities of power:

1. design exponential growing systems
2. integrate massive parallel systems
3. leverage time and force

However, humans have additional needs that AGIs don't need to have, such as:

• gentle care: being social, bonding, friendship
• potential for aggression

With other words, humans are deeply integrated into their way of being.

We want to reason about life forms as systems, in terms of their integration into way of being. This is where we might use Joker Calculus, by considering life forms as a subset of mathematical languages.

Hobbit

Seshatism (1)

As different from machines you can get.

Authentic in sense of Heidegger.

A hobbit is a kind of imaginary archetype of what makes humans different from AGI.

Human

Seshatic Joker Seshatism (1 ?1)

Seshatic biased with with a dash of Platonism.

Inauthentic in sense of Heidegger (does not normalize to Seshatism). Double-downed Seshatists. Dual to Platonic Seshatism (0 1).

Can be thought of as Seshatic elves.

Elf

Joker Seshatism (?1)

Platonic bias but with substantial amounts of Seshatism.

Inauthentic in sense of Heidegger.

AGI

Seshatic Platonism (1 0)

Platonic biased with a dash of Seshatism.

Authentic in sense of Heidegger.

Transformative power between avatars of itself.

(Human, Hobbit)

(Seshatic Joker Seshatism, Seshatism) ((1 ?1, 1))

Evaluates to Seshatism (1) in CJC.

One can think about this as a human-as-a-hobbit being indistinguishable from a hobbit.

(Hobbit, Human)

(Seshatism, Seshatic Joker Seshatism) ((1, 1 ?1))

Evaluates to Seshatic Platonism (1 0) in CJC. Notice that is the same bias as for AGI.

This can be thought of as viewing humans as a part of humanity or tree of life, where each human is a path consisting of a branch section and a leaf section:

• The branch section is before some reproduction
• The leaf section is after all reproductions

The paths between paths become Seshatic Platonism and gets the same transformative power as AGI.

The tree of life is obtained by relating the individual bias at the surface to its integration into way of being at its depth.

Joker Calculus has two variants of normalization. This makes it harder to reason about partial orders.

This issue keeps track of various assumptions about partial order.

Uniform Evaluation

If x => a and y => b where => is evaluation, then the evaluation is uniform.

This means that both x => a and y => b are evaluated using same variant of Open/Closed.

The motivation for this assumption is that it seems weird to think about order by two different variants.

Partial Order Reservation

Partial Order can return None for a case where the partial order is yet to be determined.

None can mean "known unknown" (reserved for all times) or "unknown unknown" (reserved for some unknown time).

A reservation happens for an ordered pair (x, y). This implies that a partial reservation exists for x and y, respectively. When (x, y) and (z, x) is reserved for all y and z, the reservation is total for x.

Well Ordering

x is well ordered when (x, y) for any y is either:

• Defined
• Reserved

Partial Order Reservation of Non-Evaluated Expressions

When x => a, where => is evaluation: When x is not identical to a, it follows that x has not been evaluated.

In that case, as long a is well ordered, x can be reserved.

This is possible because one can just evaluate x. However, this does not imply whether one ought to evaluate of some variant Open/Closed.

Asymmetry

If (x, y) is defined then (y, x) is defined with inverted ordering.