# Math's Fundamental Flaw

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## Game of Life

• John Conway, creator of the Game of Life, passed away in 2020 due to COVID-19.
• The Game of Life involves an infinite grid of cells which are either alive or dead.
• It's governed by two rules: a dead cell with three live neighbors becomes alive, a live cell with less than two or more than three neighbors dies.
• The game evolves automatically, generating a variety of patterns that may be stable, oscillatory, or even move across the grid.
• Some patterns grow indefinitely, but predicting the ultimate fate of any pattern is undecidable.
• No algorithm can determine in finite time whether a pattern will stabilize or grow without limit.
• Many systems besides the Game of Life are also undecidable, such as Wang tiles, quantum physics, and even the game Magic: The Gathering.

## Start Writing Down a New Real Number

• Georg Cantor's proof showed that natural numbers and real numbers between 0 and 1 are not the same in size; real numbers are more numerous.
• Cantor's diagonalization argument constructs a new number that cannot be on any list pairing naturals with real numbers, showing the existence of uncountable infinities.
• His work was part of a greater reevaluation of mathematics, challenging notions like the concept of a limit and the true nature of infinity.
• A divide among mathematicians arose, with intuitionists rejecting Cantor's work and formalists embracing it, aiming to establish a rigorous mathematical foundation through set theory.
• Cantor's set theory led to paradoxes highlighted by Bertrand Russell, such as the set of all sets that don't contain themselves, which prompted further scrutiny of mathematical foundations.

## Paradox of Self-Reference

• Russell discovered a paradox using a village barber analogy, which led to a contradiction about who shaves the barber.
• Intuitionists believed this paradox proved set theory was flawed.
• Mathematicians, including Zermelo, resolved this issue by redefining set concepts to remove problematic sets.
• Despite these resolutions, self-reference problems in math persisted, highlighted by Hao Wang's undecidable tiling problem, similar to Conway's Game of Life.

## Gödel's Incompleteness Theorem

• Gödel demonstrated that any sufficient mathematical system will contain true statements that lack proofs.
• His work proved Hilbert's belief in the absolute completeness, consistency, and decidability of mathematics was incorrect.
• Gödel's Incompleteness Theorem suggests that truth and provability are not synonymous.
• Gödel's second theorem shows that a consistent formal system cannot verify its consistency, leaving the possibility of future contradictions.
• The only remaining question from Hilbert is whether math is decidable, an issue addressed separately.

## Is Mathematics Decidable

• In 1936, Alan Turing invented the concept of the modern computer, originally imagined as a mechanical device to determine the decidability of mathematics.
• Turing's machine consists of an infinite tape with zeroes or ones, a read/write head, and a set of instructions.
• The Turing machine can perform any computable algorithm with its simple operations: overwrite, move left or right, and halt.
• Turing machines can potentially solve problems like the twin prime conjecture by halting when a solution is found or running infinitely if unsolvable.
• Turing proposed a hypothetical 'H machine' capable of determining whether any Turing machine would halt on a specific input.
• Upon modifying H to create H+, it was proven that such a machine could not consistently predict its own behavior, leading to a contradiction.
• Consequently, Turing concluded that mathematics is undecidable, implying some problems might be unsolvable, such as the twin prime conjecture.

## The Spectral Gap

• Gapless quantum systems can undergo phase transitions at low temperatures, whereas gapped systems cannot.
• Determining if a quantum system is gapped or gapless was shown to be an undecidable problem in 2015.
• Even complete knowledge of a system's micro-level interactions does not guarantee the ability to predict its macro-level properties.
• Turing's machines, representing the pinnacle of computational systems, still define the limits of what computation can achieve.