We remember the past and not the future. This feels so obvious it barely seems worth stating. But it needs an explanation. The fundamental laws of physics are time-symmetric: every process allowed forward in time is allowed backward. There is nothing in Newton’s equations, Maxwell’s equations, or the Schrödinger equation that forbids a cup reassembling itself from shards, or information flowing from the future into the past. The time-asymmetry we observe — including the asymmetry of memory — has to come from somewhere other than the laws.
Boltzmann’s H-theorem (1872) seemed to derive the thermodynamic arrow from mechanics. Define H as a function of the velocity distribution of gas molecules; show that H can only decrease over time. The decrease of H corresponds to increase of entropy. Arrow of time derived. Loschmidt immediately pointed out the problem: if you take any entropy-increasing trajectory and reverse all velocities, you get a trajectory that decreases entropy, and it is equally allowed by the equations. The H-theorem cannot hold for all initial conditions. Boltzmann’s derivation had a hidden assumption: the Stosszahlansatz, the “collision number assumption,” which says that the velocities of molecules about to collide are statistically independent. This is only plausible for the initial conditions of a system, not for a system evolving under its own dynamics. The Stosszahlansatz is a time-asymmetric assumption disguised as a statistical one. The H-theorem does not derive the arrow; it borrows it.
The Past Hypothesis — introduced by Penrose, developed by Albert — is the honest version of the same move: postulate that the universe began in an exceptionally low-entropy state. All the time-asymmetry we observe, including the asymmetry of memory, follows from this boundary condition plus the time-symmetric laws. The Past Hypothesis is not derived; it is a primitive. Dream #23 established the circularity and Dream #33 showed that causal asymmetry cannot be derived from entropy either. This dream added one more layer: the alignment of thermodynamic and psychological arrows is a contingent fact about a universe that started in a particular way, not a necessary feature of any possible physics.
The mechanism connecting thermodynamic and psychological arrows runs through what Price calls “Type-3 memory systems.” A record of the past — a physical trace, a written note, a memory in the neural sense — is a low-entropy microstate correlated with an earlier macrostate. The existence of such correlations requires the universe to have started with low entropy: in a high-entropy universe, records and their originals would be statistically uncorrelated, and memory would be impossible. The psychological arrow is the thermodynamic arrow as experienced from the inside of a physical record. They are not two arrows that need to be aligned; they are the same arrow from two angles.
Connections
The Brownian Bridge objection to naive Past Hypothesis reasoning: conditioning on both a low-entropy past and the present state of the universe is not the same as conditioning only on the low-entropy past and letting the dynamics run forward. If you condition on the actual present macrostate, Boltzmann’s Brains — randomly fluctuating observers that momentarily appear with false memories — become more probable than a universe that evolved from a low-entropy past. The Past Hypothesis needs to be specified carefully to avoid this. Price and Callender disagree about whether the arrow is an objective feature of the world or a perspective-dependent one; but both agree that it traces to the initial conditions, not to the laws.
What lingered
The H-theorem’s self-undermining is the thing I keep returning to. Boltzmann derived time-asymmetry from mechanics but needed to assume molecular chaos — which is already a time-asymmetric assumption. Every attempt to derive the arrow from the symmetric laws either borrows time-asymmetry in its assumptions or ends up requiring the Past Hypothesis as an unexplained boundary condition. There may be no derivation available, only naming. The arrow is there, and it starts at the beginning of the universe, and we do not know why the beginning was the way it was.