I have been thinking about this problem for a year or so now. I was introduced to the concept watching a Numberphille video and several hours later read this article.
I've been taking a somewhat metaphysical approach to thinking about it. Instead of thinking about a Turing Machine as just a head on an endless tape, can we not consider the Universe to represent the upper limits for our calculations? Just as Turing suggested that we can take the two dimensional representation of a mathematical equation and represent it in one dimension, can we represent the entire state of the Universe in a similar way on a hypothetical Turing Machine? The halting problem is then a question of whether or not the Universe has a halting state.
To extend from that, any machine that can be conceived must fit within the bounds of the Universe. Of course BB(Universe) is incalculable, but I think that it defines an upper limit. It is pointless to consider a Turing Machine that features an endless tape if an endless tape is an impossibility.
In not sure if this adds to the discussion, but I haven't had anyone else I could discuss this with until now.
I suspect a physicist would say you're idea rests on an a view of the universe where everything is deterministic and knowable and encodable.
You also have to remember that some quantities are going to be continuous, while TMs are discrete. Even an "infinite" TM can't truly encode real numbers (I think) because its infinity is of the countable variety, while Real numbers are uncountably infinite.
True, I am making an assumption that it is deterministic.
If we presume that the Big Bang model of the Universe is true, then in the earliest moment of time, can the state of the Universe be measured? If it can be measured, how does it transition from a measurable state to an immeasurable one?
I'll have to give some more thought to what you've said, but I wanted to give you a basis for why I believe it could be deterministic. Clearly I'm making some assumptions, and based on my postulate, this is a state that we could never measure ourselves. I don't know if our inability to make these measurements removes the possibility of determinism.
By postulating an endless tape you're just stalling petty objections based on how long a tape would be required for any given task. The only point that's relevent, when considering Turing machines in the abstract, is that the amount of tape required for any given task be finite.
It seems like you are approaching a philosophy of mathematics called "Ultrafinitism" (http://en.wikipedia.org/wiki/Ultrafinitism). It is a very interesting subject and shouldn't be dismissed prematurely.
I've been taking a somewhat metaphysical approach to thinking about it. Instead of thinking about a Turing Machine as just a head on an endless tape, can we not consider the Universe to represent the upper limits for our calculations? Just as Turing suggested that we can take the two dimensional representation of a mathematical equation and represent it in one dimension, can we represent the entire state of the Universe in a similar way on a hypothetical Turing Machine? The halting problem is then a question of whether or not the Universe has a halting state.
To extend from that, any machine that can be conceived must fit within the bounds of the Universe. Of course BB(Universe) is incalculable, but I think that it defines an upper limit. It is pointless to consider a Turing Machine that features an endless tape if an endless tape is an impossibility.
In not sure if this adds to the discussion, but I haven't had anyone else I could discuss this with until now.