Dear Del,
this was a very exciting essay for me to read---thank you for submitting it to this contest! You provide a highly original perspective, and argue it well. I like the starting point: Einstein and Hilbert, each poised in there respective quest for certainty, set up to be foiled by incompleteness and quantum unpredictability. It's perhaps no accident their names are, together, enshrined in the centerpiece or general relativity, the Einstein-Hilbert action---with general relativity itself, the 'marble of geometry', reflecting their shared convictions.
I was also taken by your observation that "mathematically modelling the physical world without deep understanding can be compared to machine translation without comprehension." I think this is a highly insightful comparison. An algorithm that produces words merely based on some probability distribution as abstracted from massive volumes of text is not much different from a human being predicting measurement outcomes from a probability distribution abstracted from massive numbers of experiments---successful, perhaps spectacularly so, but ultimately without even any real appreciation as to the reason of this success.
I wonder---what would a Turing test for such understanding look like?
The path you plot is a daring one---well, beautiful mathematics, with austerely certain foundations, hasn't fulfilled its promise, so let's look to (ugly) randomness, to propositions 'true for no reason', as Chaitin put it elsewhere. This is close to my own view---I, likewise, try to find a comprehensible foundation for quantum mechanics, analogous to Einstein's explanation of the Lorentz transformation, and likewise, I've been steered towards the notions of incompleteness and randomness, including Chaitin's specific take (which is to say, here's the obligatory advertisement for my own essay; you might also be interested in the paper I first worked on these ideas).
You suggest an intriguing concept regarding Lorentz dilation as a compression of time. I will have to mull this over a litte; at the outset, the intuition instilled by special relativity bristles a little at the singling-out of time (from spacetime) this seems to imply. But it also makes me think of two possibly related notions. One is the recent proposal by Dragan and Ekert that quantum mechanics could derive from the usually discarded 'superluminal' solutions to the defining equations for the Lorentz transformations---perhaps this is a way your 'unordered time' could enter into the picture, leading to indefinite causal orders.
The other is Seth Lloyd's discussion regarding the ultimate physical limits of computation. (Well, I thought it was in that article, but it might have been another one---I can't quickly find it there.) Anyway, the idea is for there to be an analogue to the Bekenstein bound in time---related not to the entropy, but rather, to the action, giving the number of state transitions that can be implemented in a given time frame. Perhaps this could yield a universal time scale---sort of the 'blocks' of your unordered time, which come either in the 'usual' direction or in the direction deriving from the 'superluminal' Lorentz sector.
Anyway, as you can see, your essay lots of---perhaps overly speculative and rash---ideas for me. I'm sure I will come back to it many times. I'm glad to have discovered it before the end of the voting period.
Thanks, again, and good luck in the contest!
Cheers
Jochen
