Rob and Pat, (01)
I think we are beginning to converge. I mostly agree with
Pat's explanations, but I'd like to add a few comments. (02)
RF>> I would still like to isolate _what_ it is about these theories
>> which is good. It's not the reals themselves. What is it? (03)
PH> The reals are simply a way to quantify continuous variations
> which admit of arbitrarily small changes, and functions on them
> which can be differentiated, so it is meaningful to speak of
> rates of change. The basic "_what_" is probably continuity and
> differentiability. These fields all study phenomena which arise
> only in such continuous/differentiable domains. (04)
I agree with Pat, but I'd like to add a few points: (05)
1. What is important is not the real numbers themselves, but the
much richer mathematical structures that can be built with
multidimensional spaces with real number coordinates instead
of just the combinations of Boolean {0,1}. (06)
2. Holograms and catastrophe theory are just two examples of
powerful techniques that can be supported with real numbers.
Blum, Smale, et al., discuss the advantages of the wider range,
but they admit that there are still vastly more kinds of
structures that have yet to be explored  many of which may
be much better suited to modeling cognitive mechanisms than
the ones explored so far. (07)
3. Although Blum, et al., did not mention AI specifically, I agree
with Pat that there is no sharp dividing line between the methods
used in AI and comp. sci. In fact, many of the techniques that
are now standard practice in languages such as Java were pioneered
in LISP for AI applications  among them are recursive functions,
list processing, garbage collection, and even the ifthenelse
statement. In fact, the Semantic Web uses just a subset of the
technologies developed in the AI systems of the '70s and '80s. (08)
RF> Any references for the possible relevance of all this (catastrophe
> theory, chaos, manybody systems, and now holograms) to AI, interests
> me greatly. I don't care whether it is presented in terms of reals
> or computational theory. (09)
All the theory has been developed with real numbers during the past
four centuries. Computationally, the real numbers are approximated
by floating point arithmetic (which is rarely used in AI systems). (010)
Blum et al. talk about both the computational and theoretical issues.
For example, many algorithms that take polynomial time with exponent
N on Boolean algebras can be approximated by polynomial algorithms
with exponent N1 in floating point arithmetic. An Nsquared algorithm
that cannot be scaled to the size of the WWW could be replaced by a
linear algorithm that can scale to terabytes, petabyes, and beyond. (011)
RF>> But I reserve the right to use words more generally if I feel
>> it will communicate my meaning more effectively (or if no such
>> community of broadly agreeing peers yet exists.) (012)
PH> But in all these cases, such a community does exist... (013)
In principle, Humpty Dumpty was right that you can use any word
in any way you wish. But in practice, you'll create confusion
and misunderstanding unless you take the audience's background
into account when choosing words to express yourself. (014)
If you have a radically new idea, it's better to coin a new term
than to confuse people by using the old words in unusual ways. (015)
John (016)
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