Godel's shadow: what mathematics says computation can never do
Roger Penrose builds on Godel and Turing to argue that understanding goes beyond algorithms, Edward Frenkel sees paradox and limits at the heart of mathematics, and Stephen Wolfram describes computational irreducibility. Each identifies a different hard limit on what calculation can predict or prove.
What three limits on calculation come up in these conversations? Show answer
Unprovable truths (Gödel), undecidable questions (Turing) and outcomes that cannot be predicted without running every step (Wolfram).
What does Gödel's theorem say? Show answer
Any consistent system of rules rich enough to include arithmetic contains true statements it cannot prove.
What did Roger Penrose first fear about the theorem? Show answer
That it meant there were things in mathematics you could never prove. He came to see it as showing what understanding can do.
What is Penrose's argument in three moves? Show answer
A proof can be checked by a computer. You trust the rules because you understand them. That understanding sees a truth the rules cannot prove.
What conclusion does Penrose draw about the mind? Show answer
Understanding is not a computational procedure. He admits the argument is disputed and tied to physics that does not yet exist.
What hope did Gödel end, in Edward Frenkel's history? Show answer
The belief around 1900 that all of mathematics could be derived mechanically from the right axioms.
Why does Frenkel call incompleteness life-affirming? Show answer
It leaves room for new discovery. Mathematics cannot be finished by a mechanical process.
What is computational irreducibility? Show answer
Wolfram's idea that even with exact rules, often the only way to find out what happens is to run every step.
What are pockets of reducibility? Show answer
Places where a shortcut does exist and we can jump ahead. Wolfram says science and invention consist of finding them.
How do Penrose and Wolfram differ on minds? Show answer
Penrose says understanding goes beyond computation. Wolfram says minds are a limited kind of computation bound by the same limits.
What does irreducibility imply for AI and software? Show answer
Building a machine does not mean you can predict or control it, and no sandbox is ever perfect.
What do the limits not mean? Show answer
That computation is weak, that mathematics stops, or that everything is unpredictable.