Aaronson is talking about Graham‘s number when he writes:
“Recondite as it seems, the Ackermann sequence does have some applications. A problem in an area called Ramsey theory asks for the minimum dimension of a hypercube satisfying a certain property. The true dimension is thought to be 6, but the lowest dimension anyone’s been able is prove is so huge that it can only be expressed using the same ‘weird arithmetic’ that underlies the Ackermann sequence. Indeed, the Guinness Book of World Records once listed this dimension as the biggest number ever used in a mathematical proof.”
That‘s about a third of the way through the article. The numbers get a lot bigger after that.
As a footnote, the true dimension is no longer thought to be six — Aaronson’s essay was written in 1999 — and is now known to be at least 11. The Wikipedia article linked above has a reference for this improved lower bound.
That‘s about a third of the way through the article. The numbers get a lot bigger after that.
As a footnote, the true dimension is no longer thought to be six — Aaronson’s essay was written in 1999 — and is now known to be at least 11. The Wikipedia article linked above has a reference for this improved lower bound.