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Hmm, I think its the same problem, you are somewhat arbitrarily drawing the line between "math" and the universe (or maybe I'm agreeing with you and don't realize it). A halting/decide-able problem is one that is precisely about tractability - does it end, and can you decide that. The maths is about whether there is a tractable solution, in theory ignoring any energy/mass requirements. The maths of the maths of the tractability issue is also one of tractability - if you provide an infinite solution to an infinite problem then is it a solution or an admission of defeat? You can't even decide if you can decide that you can decide. If there is no maths for the maths for the decidability question of is it tractable (not even energy computeable), then in my mind there is no difference between the two, i.e. you have no higher order reduction with which to express the lower order problem. Without a constant time solution, there may not be enough math to express the solution to the problem, i.e. this may just be another set of infinities, as you mentioned the integers, irrational numbers, precision, etc. (which are lower order entities for which we have tractable mathematical solutions). And the lower energy bound of the universe to the set of problems to me remains interesting, in that maths itself is an invented set of relationships, in the sense that the set is invented and less in that the relationships are invented (they exist but we value certain relationships over others, so the set is invented), so any maths that cannot survive and exist within the information space that is the universe is to some degree irrelevant - if there is no way to relate to it, does it exist? I think so, we simply lack the ability to process or observe it. Consider, for instance, a constant time solution to the decideability problem, however the maths to prove it require a set of equations and theorems larger than the processing time of the universe. Then a solution exists, perhaps, but we will never discover it. It doesn't mean it doesn't exist, but it does not exist for us, and we can neither prove nor disprove that one exists. |