Comment by nightowl_games | Hacker News Reader

Comment by nightowl_games | Hacker News Reader

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Yes, I believe it is the same as Wolfram's "Computational Irreducability". I'll copy paste the question I emailed him about it:

> In computers, floating point numbers have a limited precision, based on how many bits are used. Does the innate 'error' in floating point numbers have implications on your theory of 'Computational Irreducability'? Are the 'errors' introduced with each floating point operation is fundamentally computationally irreducable? Does this have a domino effect that causes basic classical mechanics to become computationally irreducible? Like the integration of velocity into position, it is a O(1) operation to compute the future position given a velocity and a starting point, but if a computer is rendering each individual frame, it will accumulate error in each frame, and the final frame may end up in a different position. Does this concept relate to the real world through things like the Planck length? Is space perhaps not purely continuous?

I have a partial answer for that. ;-) For at least some functions, iterative solutions exist which converge to correct solutions. For at least some iterative solutions the rate of convergence (the estimated error bounds) is a continuous function: it can be seen that convergence proceeds normally (I should be more careful with that word) as long as the estimated error bounds for the iteration is larger than the computationally induced error. I'll hazard a guess that with constraints some formulations which model a three body problem are solvable to the point where the estimated error bounds approaches the computationally induced error due to floating point imprecision.

The thing about the three body problem, same as here, is that there are regions in the initial position and velocity space where very small changes produce extremely large differences in function output.

There has been some work / recognition of the value of topological understanding to orbital mechanics, although I can't find whatever I read a few months ago. Best I could find was https://en.wikipedia.org/wiki/Symplectic_geometry and this DDG search looks promising: https://duckduckgo.com/?q=symplectic+integrators+solar+syste...

Those are some interesting open questions about physics, but the happenstance limitations of modern computer hardware have no bearing on them.

>Is space perhaps not purely continuous?

Pretty sure that's the entire basis of quantum mechanics

space? or energy?

No - space and time are continuous in quantum mechanics, despite the fact that other quantities such as angular momentum and energy are discrete (quantized). Relevant comment from Luboš Motl, a string theorist who produced a proof that, under some assumptions, quantum mechanics on discrete space and time is inconsistent, echoed by Peter Shor: https://physics.stackexchange.com/a/9077

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