On 8/19/2012 2:43 PM, Stephen P. King wrote:
On 8/19/2012 4:30 PM, meekerdb wrote:
On 8/19/2012 12:51 PM, Stephen P. King wrote:
I understand that 2+2 = 4.
I still cannot explain how and why I understand "2+2 = 4".
"2+2=4" is easy.
"I understand 2+2=4" is quasi infinitely more complex.


Dear Bruno,

As I see it, the quasi-infiitely more complex aspect of "I understand that 2+2=4" follows, at least, from the requirement that many entities capable of making such statements can point to examples of 2+2=4 and communicate about such statements with each other however far away in space and time they are from each other. We can ignore the fact that there is a collection of entities to whom the statement "I understand that 2+2=4" has a meaning. You need to get a grip on the nature of meaningfulness. Searle has tried to do this with his Chinese Room idea but failed to communicate the concept. :_(

Maybe Bruno will introduce a new modality to his logic Up="Understands p".  :-)

Brent
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Hi Brent,

That would be wonderful if possible. AFAIK, understanding is contingent on demonstrability, e.g. I understand p if and only if I can demonstrate that p implies q and q is not trivial and q is true in the same context as p. I think that Bruno's idea of "interviewing a machine" is a form of demonstration as I am trying to define it here. In my thesis, demonstrability requires that the model to be demonstrated is actually implemented in at least one possible physical world (i.e. satisfies thermodynamic laws and Shannon information theory) otherwise it could be used to implement a Maxwell Demon.

BTW, it was an analysis of Maxwell's Demon that lead me to my current ideas, that abstract computation requires that at least one physical system actually can implement it. This is not ultrafinitism since I am allowing for an uncountable infinity of physical worlds, but almost none of them are accessible to each other (there exist event horizons, etc.). Consider the case where a computation X is generating an exact simulation of the behavior of molecules in a two compartment tank with a valve and there exists a computer Y that can use the output of X to control the valve. We can easily see that X could be a subroutine of Y. If the control of Y leads to an exact partition of the fast (hot) and slow (cold) molecules and this difference can be used to run Y then some might argue that we would have a computation for free situation. The problem is that for the hot/cold difference to be exploited to do work the entire apparatus would have to be coupled to a heat reservoir that would absorb the waste energy generated by the work. Heat Reservoirs are interesting beasts....

If your computer simulation is acting as Maxwell's demon then you don't need a heat reservoir. The demon makes one tank hot an the other cold so a heat engine runs on the difference. Unfortunately this is impossible because such a simulation would require defining the initial state of the particle's position and momentum in the two tanks. This is not available for free. To determine it by measurement takes at least as much free energy as can be recovered after implementing Maxwell's demon.


See http://www.nature.com/news/the-unavoidable-cost-of-computation-revealed-1.10186 for more on this.

But if you're doing a calculation once on a given machine it's not necessary to erase the result. In Feynman's paper on quantum computing he note this gets around Landauer's limit. So long as the evolution of the computation is unitary no energy need be dissipated. So I don't see how the result is relevant to Bruno's UD.

Brent


We ignore the role played by our physical world in our philosophical/mathematical/logical discussions to our peril!

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Onward!

Stephen

"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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