Gödel's Incompleteness Theorem

• Mathematical theorems prove power of mathematics has limits
• Gödel's theorem: if consistent axioms sufficiently complex, can formulate undecidable statements
• Mathematics incomplete - can't know if certain statements right/wrong
• Irrelevant for scientific practice
  • Can extend axioms with additional axiom declaring undecidable statement true/false
  • Physics theories = mathematical axioms + prescription for identifying structures with measurable quantities
  • Undecidable statements resolved by measurement, then add axiom agreeing with outcome
  • If no observable consequences, can ignore statement

Uncomputable Functions

• Mathematically irrelevant for different reason
• Problem: always involves something infinite
• Nothing real is infinite - theorems don't apply to nature
• Turing's halting problem example:
  • Algorithm either halts at finite time or doesn't
  • No meta-algorithm exists to determine if another algorithm halts
  • Works for infinitely large class of algorithms
  • Reality never needs algorithm answering infinitely many questions
• Wang tiling problem:
  • Squares with different colored sides
  • Question: can tile infinite plane without gaps, adjacent colors match?
  • Undecidable for arbitrary sets
  • Never need to tile infinite planes in practice
• Most real numbers uncomputable
  • No algorithm approximates them to finite precision in finite time
  • Science deals with finitely many digits + error bars
  • Mathematical curiosity, no scientific relevance

Unpredictability in Science

• Quantum mechanics: unpredictable element by postulate (uninteresting)
• Chaotic systems more interesting:
  • Even knowing initial conditions arbitrarily precisely
  • Can only predict for finite time
  • Unclear if happens in real world systems
• Navier-Stokes equation:
  • Used for weather forecasting
  • Whether leads to unpredictable situations = Clay Institute millennium problem
  • Even if proven unpredictable, tells little about nature
  • Only approximation - fluids actually particles described by quantum mechanics
  • Quantum mechanics lacks chaotic unpredictability
  • Maybe quantum mechanics itself not ultimate theory

General Problem with Impossibility Theorems

• Can never know if mathematical assumptions in physics truly correct
• May be replaced by better theories
• Physics = science, not mathematics
• Use mathematics because it works, not because nature is mathematics

Practical Relevance for Science

• Theorems relevant for science but not because they reveal nature's deep truths
• Relevant because we use theories with these properties in practice
• Tell us what we can do with our theories
• Navier-Stokes example:
  • Not fundamentally right for weather prediction
  • Currently used equation
  • Knowing when unpredictable situation coming matters
  • Want to know when unpredictability approaching to avoid it
• Nuclear fusion plasma application:
  • Plasma develops instabilities damaging containment vessel
  • Must shut down fusion process quickly
  • Makes fusion inefficient
  • Better understanding unpredictable situations could prevent them
• Financial system applications possible

Essay Contest Information

• Essay submitted to Foundational Questions Institute contest
• Topic: unpredictability, undecidability, uncomputability
• All essays available at FQI website
• Submission deadline: March 16th