Gödel’s Proof and Einstein’s Dice: Undecidability in Mathematics and Physics – Part I

by Jochen Szangolies

God’s dice? Image by S L on Unsplash

TWA Flight 702 left New York at 7 AM on Monday, Feb. 4 1974, to arrive in London at 7 PM—some 40 minutes early. We know this thanks to the meticulous note-taking habits of visionary physicist John Archibald Wheeler, coiner of such colorful terms as ‘quantum foam’, ‘wormhole’, ‘superspace’ and ‘black hole’.

Wheeler spent the flight occupied with what he is perhaps best remembered for: pondering his ‘Really Big Questions’ (RBQs), among which we find perennial mysteries such as ‘How come existence?’ or ‘What makes meaning?’. The RBQ that occupied Wheeler on this particular day, however, was one that in many ways lay at the nexus of his thought: ‘Why the quantum?’

Wheeler had been a student of Bohr and Einstein, and thus, had learned about quantum mechanics straight from the horse’s mouth. Yet, he would struggle with the implications of the theory for the rest of his life, referring to the fundamental indefiniteness of its phenomena as the ‘great smoky dragon’. He was searching for a way to dispel the smoke, and in the note composed on Flight 702, draws a surprising connection to another remote frontier of human understanding—the phenomenon of mathematical undecidability, as discovered in 1931 by the then-25 year old logician Kurt Gödel. (The note itself is available online at the John Archibald Wheeler Archive curated by Baruch Garcia.)

At first sight, one might suppose this connection to be little more than a kind of ‘parsimony of mystery’: in substituting one riddle for another, the total number of unknowns is reduced. Indeed, the idea of ripping Gödel’s result from the austere domain of mathematical logic and injecting it into physical theory is controversial—earlier, during the writing of his magisterial textbook on General Relativity, Gravitation, with Charles Misner and Kip Thorne, Wheeler had confronted Gödel himself with the idea, who did not react too enthusiastically. Read more »