On 10/12/2010 6:01 PM, Darren New wrote:
> Kevin Wampler wrote:
>> inputs I'm actually having trouble thinking of a case where this is true
>
> I don't know what "this" you're talking about. Your comments don't make
> sense in the context of the bit you quoted.
>

Yeah, the question I was attempting to ask was only tangentially related 
to the bit I quoted (and not very well thought out anyway).

To rephrase, I was wondering if you had any examples of an unbounded 
mathematical structure and an associated finite restriction where the 
kinds of questions it's "natural" to ask about the finite version are 
easier to answer than in the unbounded version?  In retrospect, 
finite-resource computation is an obvious example you already mentioned, 
but I'm having trouble thinking of others.


On a only partially related note, I was reading a blog entry on 
mathematical techniques for simplifying problems and ran across the 
relevant snipped:

"Make the Objects Really Big: One tool to better understand a problem 
that mathematicians use and we rarely seem to is: replace “finite” by 
various notions of “infinite.” The study of finite groups is extremely 
difficult: one reason is that the interplay between small primes and 
each other can be very complex and subtle. Studying instead infinite 
groups is not easy, but there are some cases where the theory does 
become much easier. For example, studying Lie groups is not easy, but it 
does avoid much of the “accidental” number issues that arise in finite 
groups. One way to see this is their classification occurred first and 
is simpler than the classification of finite simple groups."

I'm not really trying to prove a point with that, but it seemed 
interesting and relevant, and is one of the examples I had in mind of a 
case where infinities made things easier (hence my curiosity for more 
examples of the opposite).

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