From: Hal Finney (hal@rain.org)
Date: Wed Nov 13 1996 - 22:20:33 MST
From: hanson@hss.caltech.edu (Robin Hanson)
> Whether smarter beings can create still smarter beings at a faster
> rate than lessor beings can improve on themselves depends on how the
> difficulty of the task of getting smarter scales with intelligence.
> It could be that the task gets harder so fast that smarter beings take
> *longer* to get still smarter. We know far too little to say that
> this isn't the case.
Yes, this is a good point. The same thing may happen in general with
technological improvement. The law of diminishing returns can be
expected to apply to super-intelligent beings as well as to us. Whether
the greater capabilities in the future will be enough to outmatch the
greater problems they face, we can't say.
I read a book many years ago, Infinity_and_the_Mind, by Rudy Rucker.
I didn't really like it very much; it seemed to approach the mathematical
and philosophical concept of infinity from a rather mystical direction.
But he did make some points that have stuck with me.
Mathematically, there are many orders or classes of infinity. In fact,
there are an infinite number of infinities, because of the general rule
that the size of the set of all subsets of a set S is always bigger
than the size of set S. It also makes a different whether you are
talking about ordinal numbers, which might include infinity plus 1,
infinity times 2, and such, versus the more familiar cardinal numbers,
which are the sizes of things.
The interesting thing is that as you move to the higher cardinal numbers,
there is no general rule which can be used to understand the progression.
Each class of infinity represents a new direction, a new problem, which
must be understood on its own grounds. From the counting numbers, we
go to the real numbers (which are a lot more complex than most people
realize); then to the set of all functions. I don't think anybody
knows where we go from there.
This kind of transition is inherent to the problem; anything which can
be mapped onto an earlier level is no bigger than that value. It is
only when there is a change in the nature of what is counted that we
go to a new level of infinity.
This suggests to me that super-intelligences may be faced with similar
problems as they advance. There will be levels, or plateaus, where they
advance to the limit that is possible at that level. Then some new
fundamental breakthrough is needed to advance to a new level. Being
smart is not just a matter of being fast (in my opinion), but involves
qualitative changes.
Hal
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