From: Wei Dai (email@example.com)
Date: Sun Sep 11 2005 - 09:29:21 MDT
> Wei Dai, you are correct. Mathematical reasoning is not usually regarded
> as the domain of probability theory. There are efforts to extend standard
> decision theory to remove the assumption of logical omniscience, under the
> heading of "impossible possible worlds". Figuring out exactly how to
> integrate this is, in fact, one of the things I'm currently doing some
> work on myself.
I'm aware of "impossible possible worlds", but as far as I can tell, there
hasn't been much work done in that direction beyond setting up some
formalism. It doesn't seem to offer any insight into how informal
mathematical reasoning actually works (i.e., how we come to believe some
unproved mathematical conjectures but not others).
Another direction, taken by Marcus Hutter's AIXI-tl and Schmidhuber's Godel
Machine, is to essentially hard wire difficult-to-prove/unprovable
mathematical conjectures into the AI as unchangeable assumptions. This may
not be obvious at first glance so I'll explain. The way those AIs work is to
start with an initial policy (handcrafted program for making decisions) and
then systematically search for provable improvements to the policy. The
initial policy necessarily makes implicit or explicit assumptions about the
truth value of mathematical conjectures. Such an assumption can't be changed
unless it can be proved to be false, otherwise it's impossible to prove that
changing it won't lower expected utility.
I'm curious what others think about how important this problem is. I know
that in my own field of cryptography, heavy use is made of informal
mathematical reasoning, since it has not be possible to invent ciphers that
can be proven to be computationally hard to break (this would require P!=NP
at a minimum). Likewise in cryptanalysis, many important algorithms can not
be proven to be correct and/or only have heuristic running times.
One might argue that the problem isn't important, because an AI with much
greater computational resources than we do will be able to easily prove or
disprove any mathematical conjectures that might be practically relevant.
But it's not obvious this will be the case. Gregory Chaitin has argued that
many interesting or natural mathematical questions may be independent of the
usual axioms (see http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm2.html). He
advocates that mathematics should be "quasi-empirical" and done more like
physics. I think this is already the case to a great extent, especially in
applied math, but we don't seem to know how to formalize the process so an
AI can also do this.
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