(Or, Excerpt from a Marginal Outburst to a Passage by Ian Stewart Concerning “Impossibly Complicated” Initial Conditions)
Yes, but why are they impossibly complicated? Specified, of course (smash the bottle a million times under precisely controlled conditions, you won’t get the same precise disorder twice), but then how is that so crippling?
What if entropy increases by the conversion of the simple to the complex, the less specified to the more specified and the expenditure of heat can do nothing but specify? No, that feels silly as stated.
Nothing can be replicated, that’s why. Time exists because nothing exists twice, not in any dimension, because of the absence of perfect duplicates. No bit of the cosmos matches any other bit absolutely exactly. (There’s a possibly falsifiable hypothesis there, I suppose, but not for a mere poet.) All replication is approximation, and each approximation involves irreparable loss. Nothing exists twice because no two slices of spacetime, however close, however fine, are truly identical or forever the same.
Time and space, the spacetime continuum if you like, are neither one an emergent property of the other (pace Smolin) but are together equally emergent properties of incomplete differences at all scales in any (including mathematically “higher”) dimensions.
Both the absolute pervasiveness of difference and its inevitable, simultaneous incompleteness in all phenomena, however large or small, through to infinity, is the ground ontological condition, the ontological foundation of all known, experienced, prosthetically observed, and measured existence. (And it doesn’t functionally matter, for us, if difference is “in here,” that is, a product of us as observers, or “out there” as a part of reality beyond our observations. It wholly pervades all our available means of observation—wholly—so, if it is only in us, it doesn’t matter except as idle speculation to guess as much. Us or out there, endless incomplete difference is all we’ve got, all we’ve ever spotted, from our bodies to our languages, from our geometry drawn in the dust to our subatomic particle colliders and gravity-wave detectors.)
From that fact, all the rest, all the other conditions—entropy, spacetime, entanglement, and all their apparent paradoxes—derive inevitably. This absolute perfusion of difference that is somehow never complete difference, the difference of the infinitely, variably similar, the perfect pervasiveness of incomplete difference within any and all phenomena, however defined, measured, confined, sliced, expanded or bounded, always then and there, whatever we ever find of there or then—this is everything. All bits only somewhat different, nothing exactly the same.
You can’t go back because nothing can match. You can’t easily get from the smashed bottle to the former whole because you can’t ever get two smashes (nor two whole bottles) the same, not in any dimension you care to enumerate or name. There is no perfect symmetry. Even reversible equations cull change.