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#136 in Math

MIT/Apache

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Joker Calculus

An implementation of Joker Calculus in Rust

Based on paper Joker Calculus, by Daniel Fischer, William Alexander Morris and Sven Nilsen (2021).

Plato

Plato, an influential figure in Western philosophy. Source

Example: Hello Joker

use joker_calculus::*;

fn main() {
    let a = platonism();
    let b = not(a.clone());
    assert_eq!(b.eval_closed(), seshatism());
}

Introduction

Joker Calculus is used to describe language bias in philosophical positions.

You can think of Joker Calculus as extending bits with two modes of evaluation (Open and Closed):

0 = Platonism
1 = Seshatism

!0 = 1 (in both Open and Closed variant)
!1 = 0 (in both Open and Closed variant)

There is a "Joker" operator ?:

?0 = (0, 1) = Something that appears to be 1 but actually is 0
?1 = (1, 0) = Something that appears to be 0 but actually is 1

(0, 0) = 0
(1, 1) = 1

There is also a perspective operator:

0 1 = Something that is like 1 but seen from the perspective of 0
1 0 = Something that is like 0 but seen from the perspective of 1

These operators are used to build more complex expressions, such as:

0 ?1 = 0 (in Closed variant) = Something who stands for 1 but "embraces" 0
1 ?0 = 1 (in Closed variant) = Something who stands for 0 but "embraces" 1

This also allows expressing positions that are co-dependingly "stuck in inauthenticity":

!(1 0) = 0 ?0 = Something who stands for 0 but "can not perceive itself from" 1
!(0 1) = 1 ?1 = Something who stands for 1 but "can not perceive itself from" 0

Authenticity and inauthenticity in sense of Heidegger

Martin Heidegger thought a lot about "authenticity and "inauthenticity" in Being. This idea has been used in Joker Calculus from a Wittgensteinean perspective of the relationship between language and logic.

In Joker Calculus, one says a language bias is "inauthentic" if it contains a Joker after being evaluated in the Closed variant. This means, the language can not get rid of some internal boundary between two languages of different biases.

For example, when I see your position 0 from my position 1, I can express this as 1 0. This is considered an authentic position, even though I am biased.

However, when I pretend to have some position 0 while actually having a position 1, this is considered an inauthentic position, since it can be expressed as (1, 0) = ?1. This language bias is inauthentic because it contains a Joker.

A third example is when I "embrace" a position 0 while holding a position 1, which is expressed as 0 ?1 = 0 in the Closed variant.

  • I hold the position 1
  • I appear to hold the position 0, hence (1, 0) = ?1
  • My position is seen from the position 0 as embracing 0, hence 0 ?1 = 0 in the Closed variant

Joker Calculus can sometimes get rid of Jokers when evaluating in the Closed variant. These positions are considered authentic.

Identity as evaluation termination

The id operator preserves the identity of the argument.

For example, !id(0) does not evaluate to 1. However, Joker Calculus can still reason about it. In the Closed variant, !!id(0) becomes id(0).

In philosophy, this corresponds to opaque language bias. You can also think of it as treating involutions along an unknown language dimension.

Motivation

Joker Calculus is a strongly normalizing formal model of higher duality between Platonism and Seshatism.

In philosophy of metaphysics, Platonism is the view that there exists such things as abstract objects. Platonism had a profound effect on Western thought.

In the philosophy of mathematical language design, the core axiom of Path Semantics implies the existence of a dual version of Platonism called "Seshatism".

With other words, to the degree one can take the view of Platonism in philosophy, there is a corresponding but opposite position of Seshatism. Seshatism is just as deep and complex as Platonism, because there is a precise mathematical relation between the two ways of constructing mathematical languages.

Seshatism must not be confused with Nominalism, which is important in the debate about philosophy of metaphysics. Nominalism plays a less important role in the philosophy of mathematical language design. You can learn more about this in the essay What is Wrong With Platonism?.

Seshatism is currently being studied under the AdvancedResearch organisation.

"The Joker" as a mathematical universe was inspired by the philosophy of Alan Watts, who held a lecture about the topic.

A higher duality, in the sense of the Joker Calculus, means that languages can have different surface and depth levels. These configurations of surface and depth levels also have their duals.

Open vs Closed Joker Calculus

Joker Calculus comes in two variants, one called "open" and one called "closed".

  • In the closed variant, the higher dualities are involutions
  • In the open variant, the higher dualities are adjoints

Example: Open vs Closed

use joker_calculus::*;

fn main() {
    let a = seshatic(platonism());
    let b = not(a.clone());
    let b_open = b.eval_open();
    let b_closed = b.eval_closed();
    assert_eq!(b_open, platonic(joker(platonism())));
    assert_eq!(b_closed, platonic(joker(platonism())));

    let a_open = not(b_open).eval_open();
    let a_closed = not(b_closed).eval_closed();
    assert_eq!(a_open, seshatic(joker(joker(platonism()))));
    assert_eq!(a_closed, seshatic(platonism()));
}

No runtime deps