The limits of knowledge/rev X

Tommy Parker for Quanta Magazine

Recursion, feedback loops and the journey to forever resonates when viewing this terrific illustration showing just how wondrous reality truly is. At the same time, while looking at the image, the limits of knowledge comes to the fore as one can never know everything about any given event thanks to Heisenberg's uncertainty principle, Godel's incompleteness theorems and Quantum Mechanics because there is no certitude, ever. 

This sentence is false.

Proof of the limits of knowledge writ large applies because if the sentence is true, how can it be false at the same time?

Uncertainty also applies in classic space as one cannot ascertain the momentum of a cue ball and its position at the same time. You can get very very close but not perfect.

As for certitude, go to a parking lot that's full. You know a space will open but not when.

David Hilbert (left) wanted to put mathematics on firmer foundations. Kurt Gödel (center) and Alan Turing later showed that Hilbert’s dream was impossible.

University of Göttingen (left); Kurt Gödel Papers, the Shelby White and Leon Levy Archives Center, Institute for Advanced Study; GL Archive/Alamy Stock Photo

Researchers worried about the limits of formal mathematical reasoning long before the P versus NP problem was first articulated — indeed, long before the beginning of modern computer science. In 1921, struggling with the same question that would grab Carmosino’s attention nearly a century later, the mathematician David Hilbert proposed a research program for grounding mathematics in absolute certainty. He hoped to start from a few simple assumptions, called axioms, and derive a unified mathematical theory that met three key criteria.

Hilbert’s first condition, consistency, was the essential requirement that mathematics be free of contradictions: If two contradictory statements could be proved starting from the same axioms, the whole theory would be unsalvageable. But a theory could be free of contradiction and still limited in its reach. That was the motivation for Hilbert’s second condition, completeness: the requirement that all mathematical statements be either provably true or provably false. His third criterion, decidability, demanded an unambiguous mechanical procedure for determining whether any mathematical statement was true or false. Addressing a conference in 1930, Hilbert declared: “Our slogan shall be ‘We must know, we will know.’”

Just a year later, Gödel delivered the first blow to Hilbert’s dream. He proved that a self-defeating statement like “this statement is unprovable” could be derived from any appropriate set of axioms. If such a statement is indeed unprovable, the theory is incomplete, but if it’s provable, the theory is inconsistent — an even worse outcome. In the same paper, Gödel also proved that no mathematical theory could ever prove its own consistency.

The limits to knowledge rules ... yet again.

Then in 1936, a 23-year-old graduate student named Alan Turing rephrased Hilbert’s decidability condition in the then-unfamiliar language of computation and dealt it a fatal blow. Turing formulated a mathematical model — now known as the Turing machine — that could represent all possible algorithms, and showed that if Hilbert’s procedure existed, it would fit within this model. He then used self-referential methods like Gödel’s to prove the existence of undecidable statements — or, equivalently, uncomputable problems that no algorithm could solve.

How to negotiate reality ...

Diagram of the OODA loop

Situational awareness applies as does Destruction and Creation.