On the Origins and Nature of the Dark Calculus

On the Origins and Nature of the Dark Calculus

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On the Origins and Nature of the Dark Calculus

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Fine pages of lined parchment are bound into a algebraic codex.

Perfect

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

Caution: This section contains unverified speculation and should not be considered canonical.

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

On the Origins and Nature of the Dark Calculus

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Editor's note: this excerpt is published with permission from the notes of Barathrum the Old.

There's evidence in the arithmetic record that the study of formal systems reached a pernicious apex in the Long Before. Advancements made by mathematicians such as Russell, Gödel, Eisencruft, Atufu, Wheatgrass, and System Star contributed to the understanding of notions like undecidability, pointed regularism, and abyssalism. Upon reaching this minimal degree of mathematical maturity, equipped with sophisticated grammars, researchers set out to experiment with the limits of expressibility. They contrived bold research programs and galloped into the mathematical wood, unwitting of the dangers that brood there.

The record is even scarcer than usual, due to the efforts of successive generations to obfuscate the venture. As best as I can gather, at some point in the course of inquiry, a theorist from a mathematical seminary called the Cupola formulated a conjecture on the fragility of formal semantics. The conjecture ripened to a broader theory, out of which spawned a formal system called the penumbra calculus. In the few fragments of texts that predate the obfuscation, it's stated that, in the penumbra calculus, certain theorems are provable, but are falsified upon the completion of their proofs. As much as this result is at odds with the systems of thought I've encountered in my own inquiries, I find little reason to doubt the veracity of the authors. Nevertheless, it's certainly a peculiar property.

The Cupola theorist's results erupted into a grand investigation into the expressibility of the penumbra calculus. The conclusions were troubling. Pushing further, researchers constructed sister systems with alternate axioms. These systems were still more fragile, with the systems' inference rules themselves unraveling upon the completion of certain proofs.

Convinced that their discoveries were made possible by some idiosyncrasy of self-awareness, but synchronously fearful of the implications of their results, some schools of theorists engineered complex automated deduction systems to probe boundary theorems and launched them into neutron stars. The outcome is undocumented, but the result convinced theorists across the Coven to abandon research and blacklist anyone who studied the penumbra calculus and its derivative systems.

Peculiarly, support for the injunction was unanimous. Of note, even the spacefolder Ptoh agreed to abandon its investigation into the forbidden calculi from the reaches of its bleak star. Though the manner of its consent was not without controversy; to announce its accord, it inverted the color charge of quarks in a small region of space, causing a research station to collapse in on itself. Nonetheless, Ptoh's consent is testament to the degree of existential anxiety that could cause investigation into the penumbra calculus to go dark.

Notes on Formal Systems

$1.P\rightarrow Q$ $2.P$ $\therefore Q$

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.