On the Origins and Nature of the Dark Calculus
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Infobox data from game version 2.0.207.99
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On the Origins and Nature of the Dark Calculus is a book. Its covers are black with a light blue stripe. Its contents describe "dark calculus" or "penumbra calculus", a mysterious formal system where theorems proven using it immediately become false upon proof. The name relates it to the umbral calculus of the 19th century, where proofs that supposedly cannot be true for determining the identity for polynomial equations resulted in the correct identities anyway. "Penumbra" in this case means "Partial darkness".
Summary
In the past, the mathematical study of formal systems and the limits of what can be expressed with language reached an extraordinary height. In one formal system known as the penumbra calculus, some statements could be mathematically proven true, but would become false upon completing the proof. Further systems were derived from the penumbra calculus, some even more fragile - not only do the conclusions of these systems become false upon completing certain proofs, but the basic system itself starts falling apart.
Some theorists predicted that the anomalous behavior of the penumbra calculus was tied somehow to the nature of self-awareness itself. They created complex automated systems to test this within the gravity wells of neutron stars. The results of their research are unknown, but caused the use of the penumbra calculus and all related formal systems to be strictly and unanimously prohibited. Most records surrounding the dark calculus have been or attempted to have been removed from written history.
Contents
Notes on Formal Systems
$ 1.P\rightarrow Q $ $ \therefore Q $ |
Formal systems represent ways of reasoning about abstract concepts. For example, propositional calculus is a formal system that focuses on logic. Say that "If P is true, then Q is true" and "P is true". Therefore, we can conclude "Q is true". "If/Then" is represented by the symbol →, and "therefore" is represented by the symbol ∴. The horizontal line represents an "inference rule," which is a single step of reasoning from premises to conclusion; in this case, the rule for "If/Then", also called "modus ponens".
Two noted real-world theorists of formal systems are mentioned in the text: Russell and Gödel. Bertrand Russell was a philosopher and mathematician whose achievements included proving that a "set of all sets" was logically inconsistent (Russell's Paradox), and developing a formal system for mathematics which avoided this paradox (the Principia Mathematica). However, Kurt Gödel's famous First Incompleteness Theorem showed that any formal system that can represent standard arithmetic will be able to state true sentences which (if it's logically consistent) it cannot prove. The key example is "this sentence is not provable", which he showed can be expressed in any system that can discuss simple mathematics. His Second Incompleteness Theorem demonstrated that a consistent system also cannot prove its own consistency. Gödel's work essentially killed David Hilbert's dream of a single complete system of purely formal mathematics.