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Solomonoff induction can be computed in the limit. During round i, simulate your chosen UTM on each valid input string (note: assuming that the valid input stri
by drdeca 8y ago
Solomonoff induction can be computed in the limit.
During round i, simulate your chosen UTM on each valid input string (note: assuming that the valid input strings are prefix free, which iirc is the standard for solomonoff induction) of length at most i, until you have simulated it for i steps total, or until it halts (whichever comes first).
When doing so, for any string, the proportion of the inputs (with respect to the coin flip measure thing) to the UTM that eventually halt with whatever particular output, which will have been found to halt , will approach "all of them".
Now, you can't quite say "run as long as you need to in order to get answer within this epsilon of correct answer", because you don't have a way of determining whether something hasn't halted yet but will, or if it never will, but, it is still computable in the limit. If you use absurdly (possibly quite absurdly) large but finite amounts of computational resources, it should give you a good enough approximation.
Humans make mental errors often enough. I don't see why one would think that, because solomonoff induction is only approximable, that that suggests human intelligence is uncomputable. Why not suppose that humans just have a very close approximation to the uncomputable thing, rather than the noncomputable thing itself?
Why believe that human intelligence is computable? Well, for me it is more of a "choosing not to believe that it is uncomputable". If I believed it uncomputable, I might hang too much of my philosophy/worldview on that assumption, and then if I turned out to be incorrect on that, too much might come crashing down.
Edit: note: I mean "not believing", not "disbelieving". Not(believe(p)) rather than believe(not(p)) .
- yters 8y agoAh, right. That's true. Since the elegant program is guaranteed to halt, then we'll get to it eventually. My mistake. That being said, even with infinite resources we will never know when that actually happens, so we still will never known when Solomonoff induction has succeeded. I don't see what will come "crashing down." Science is about being able to quantify and empirically test our hypotheses. Nothing about uncomputable intelligence seems to undermine this idea, but perhaps I am missing something. Instead, it broadens the range of hypotheses we can use to explain the world, which seems to be a good thing.
- drdeca 8y agoWhen I said crashing down, I meant in my own beliefs personally, not anything about the impact on science as a whole. Like, for example, if I built a justification for my believing in the existence of souls on top of the belief that the human mind is uncomputable, then if I turned out to be incorrect about the human mind, it might lead me to believe that souls don't exist (though that wouldn't really be a logical result, but I am not always rational). So, in order to have better foundations for my personal beliefs, I prefer not to rely on the assumption of uncomputability of the mind (nor on its negation). Basically, I want to avoid being too optimistic and believing too much because of it being convenient for my other beliefs.
- yters 8y agoIf your beliefs are consistent with a computable mind, then what stops you from also investigating the uncomputable mind? If true, then your beliefs are broadened, and if false nothing is threatened.
- drdeca 8y agoAh, I don't mean to reject the possibility that minds are uncomputable. Perhaps they are! It does seem likely that ideal minds would be (for reasons like those you describe). I just think I should require a rather strong argument to conclude that they are, instead of remaining non-commital.