I realized something about information density and compressibility. I will try to break the following down into something simpler, but:
So think about it.
We know that optimal polynomial time algorithms exist for classes with finite obstruction sets. Because the elements in those obstruction sets have a finite number.
But the number of elements in that finite obstruction set, (the constant used in the algorithm) when it is within a certain range of numbers, has a probability of being expressible (compressible) that is close to zero.
So even if the polynomial-time algorithm exists for such a class, the proof for it cannot be compressed in polynomial time when the constant the algorithm uses is one of those inexpressible/incompressible numbers.
Which would mean that… If an optimal polynomial-time algorithm exists for compressing and enumerating the proofs for all the algorithms that do have expressible constants, then its own constant can either be expressible or inexpressible. For the same reasons, it is almost certainly inexpressible.
So:
The optimal polynomial-time algorithms MUST exist for those classes, but the constants used in them are likely to be incompressible/inexpressible.
The optimal polynomial-time algorithm for compressing and enumerating the proofs of those algorithms cannot itself be compressed in polynomial time if the constant it uses also is incompressible/inexpressible, which it has a 99.99999 percent chance of being. [b]But like I said, there is a very small probability that the constant this meta or sorting algorithm uses,- the one that compresses and enumerates the proofs of the others,- happens to be one of the sporadically expressible numbers in this range. Very small. But the chance exists.
That is why nobody is ever going to solve the P/NP problem. Assuming the constant in that meta-algorithm is expressible, (like I said, 99 percent chance it isn’t) you would have to find it somehow. Find it when it is sandwiched (if it is even expressible: it’s not) between trillions upon trillions of numbers on either side of it that can’t even be written down in our universe if we stuck a digit on every Planck unit. The expression needle in a hay stack? Yeah, this is much worse.
But maybe it just randomly happens to be one of the sporadically expressible/compressible numbers. Call it Parodites’ constant. If that number just happened to be expressible, that means you could run the meta-algorithm and enumerate the proofs for all the otherwise un-findable algorithms whose expressible constants are scattered in between unending series of inexpressible numbers. It would be like a magical crystal ball that let you overcome the physical limitations on computers.[/b]
Basically, we know optimal algorithms exist for certain problems that we cannot prove. (Graphs with finite obstruction sets, but the specifics are not important.)
But the number, the actual value of the constant such algorithms use, because it is in a certain range, is almost 100 percent likely to be inexpressible/incompressible.
That means even if the algorithm exists, the proof still can’t be obtained when the constant in the algorithm is one of these inexpressible/incompressible numbers.
An optimal algorithm also exists to sort and enumerate the proofs of all these other algorithms,- these algorithms for problems we can’t prove despite knowing the algorithms to optimally solve them exist.
The constant used in this meta-algorithm has a non-zero chance of being expressible. It is small. Just as small a chance as it is for the others, that its constant happens to be expressible/compressible.
In sum: The number of algorithms with inexpressible constants in them is infinite; the number of algorithms with expressible constants in them is finite, therefor an algorithm to sort the finite ones exists. That is this meta-algorithm. But its own constant can be expressible or inexpressible. If the constant in that meta-algorithm happens to be, against all odds, expressible/compressible, then that means whoever had that number, whoever had that constant, could run a program that enumerates proofs for all the other algorithms with expressible constants,- algorithms which we otherwise have no possible hope of ever finding because their constants are scattered like needles in a haystack, surrounded on either side by incomprehensible sequences of un-writeable numbers. So the value of this constant, if it is indeed expressible, is beyond the value of any other piece of information.
The subject that exists the universe knows the algorithms or the universe couldn’t exist right? Or… would the algorithm still fail to capture subjectivity/quality?
A question/reflection I am left with is, Perhaps we do not find perfect shapes in nature because there is more to nature and reality than order, which allows for change, variation, and the strikingly new–for active, vital existence? Perhaps the perfect order holds all this together just enough so we do not spiral off into chaos or nothingness? Perhaps we need a balance of order and movement in order to exist, to be real? So we find the Golden Ratio in roses and snail shells, we find fractals in trees and shorelines, but not perfectly. That is to say, not without movement. Perhaps you cannot approach perfection outside a medium that would cease to exist if it stopped moving? Perhaps when we seek for harmonic teleology underlying all projects, we find the answer to our question, to all of our questions, is a perfection that defies the containment of mathematization? Perhaps we discern it before we even know what to ask? Perhaps we know it by all our empirically subjective movements towards or away from perfect will, all apparent oppositions and parallels? Perhaps not just mathematization, but our very words, are like dream catchers, catching what we cannot even imagine, but only hypothesize, like “infinite sets”, like “self=other”, like “the last sunset before you never saw her again”, like “reality”? How do we know if words like those contact or communicate (with) reality, or vice versa? Do we need certainty before we move? Are we not moving? Just enough?