lbThingrb

Thanks! I find this approach more intuitive than the proof of Sperner's lemma that I found in Wikipedia. Along with nshepperd's comment, it also inspired me to work out an interesting extension that requires only minor modifications to your proof:

d-spheres are orientable manifolds, hence so is a decomposition of a d-sphere into a complex K of d-simplices. So we may arbitrarily choose one of the two possible orientations for K (e.g. by choosing a particular simplex P in K, ordering its vertices from 1 to d + 1, and declaring it to be the prototypical positively oriented simplex; when d = 2, P could be a triangle with the vertices going around... (read 1063 more words →)

Thanks, this is a very clear framework for understanding your objection. Here's the first counterargument that comes to mind: Minimax search is a theoretically optimal algorithm for playing chess, but is too computationally costly to implement in practice. One could therefore argue that all that matters is computationally feasible heuristics, and modeling an ideal chess player as executing a minimax search adds nothing to our knowledge of chess. OTOH, doing a minimax search of the game tree for some bounded number of moves, then applying a simple board-evaluation heuristic at the leaf nodes, is a pretty decent algorithm in practice.

Furthermore, it seems like there's a pattern where, the more general the algorithmic... (read more)

I didn't mean to suggest that the possibility of hypercomputers should be taken seriously as a physical hypothesis, or at least, any more seriously than time machines, perpetual motion machines, faster-than-light, etc. And I think it's similarly irrelevant to the study of intelligence, machine or human. But in my thought experiment, the way I imagined it working was that, whenever the device's universal-Turing-machine emulator halted, you could then examine its internal state as thoroughly as you liked, to make sure everything was consistent with the hypothesis that it worked as specified (and the non-halting case could be ascertained by the presence of pixie dust 🙂). But since its memory contents upon halting... (read 500 more words →)

This can’t be right ... Turing machines are assumed to be able to operate for unbounded time, using unbounded memory, without breaking down or making errors. Even finite automata can have any number of states and operate on inputs of unbounded size. By your logic, human minds shouldn’t be modeling physical systems using such automata, since they exceed the capabilities of our brains.

It’s not that hard to imagine hypothetical experimental evidence that would make it reasonable to believe that hypercomputers could exist. For example, suppose someone demonstrated a physical system that somehow emulated a universal Turing machine with infinite tape, using only finite matter and energy, and that this system could somehow... (read more)