Mathematics and Computability

The library keeps circling a single knot: Gödel showed formal systems can't fully account for their own truths, and Turing, arriving from a completely different direction — "what would a machine do?" — landed on the same impossibility, but deeper. Gleick captures the moment the distinction between data and instructions collapses: every Turing machine can be encoded as a number, fed to the universal machine, and the whole edifice of "computation" becomes self-swallowing. Penrose then takes this result and does something the others won't: he treats it as evidence about minds, arguing that because we can *see* the truth of a Gödel sentence our own algorithm can't reach, mathematical understanding can't be algorithmic. The counterarguments are serious — you'd need to know your own algorithm to construct its Gödel sentence, and you don't — but what's striking is that Penrose and his critics are fighting over the same gap Turing opened: the space between a system running and a system comprehending what it's doing. Computability turned out to be a Platonic object discovered simultaneously by Church, Turing, Post, and Herbrand-Gödel through completely different formalisms, which means the boundary it draws — between what's computable and what isn't — isn't an artifact of any particular notation. It's a wall.