Where does the feeling that "time moves forward" come from?
It's interesting to look at this view from two sides -- from the reductionist approach, in terms of the grounding of minds in physical systems; and also the phenomenological approach, in which one takes subjective experience as primary.
Putting together these two perspectives, one arrives at the conclusion that the directionality of time, as perceived by a mind, has to do with: entropy increase in the mind's environment, and entropy decrease in the mind's "theater of decisive consciousness."
A Reductionist View of the Origin of the Directionality of Time
Microphysics, as we currently understand it, doesn't seem to have this. In both classical and quantum physics, there is no special difference between the forward and backward direction in time.
Julian Barbour, in his excellent book The End of Time, argues that the directionality of time is an artifact of psychology -- something added by the experiencing mind.
It's commonly observed that thermodynamics adds an arrow of time to physics. The increase of entropy described by the Second Law of Thermodynamics implies a directionality to time. And the Second Law has an intriguing observer-dependence to it. If one assumes a conservative dynamical system evolving according to classical mechanics, there is no entropy increase -- until one assumes a coarse-graining of the system's state space, in which case the underlying complex dynamics of the system will cause an information loss relative to that coarse-graining. The coarse-graining is a simple sort of "observer-dependence." For a detailed but nontechnical exposition of this view of entropy, see Michel Baranger's essay "Chaos, Complexity and Entropy."
In this view, an argument for the origin of the directionality of time is as follows: The mind divides the world into categories -- i.e. "coarse-graining" the set of possible states of the world -- and then, with respect to these categories, there emerges an information loss corresponding to one temporal direction, but not the other.
A Psychological View of the Origin of the Direction of Time
Next, what can we say about the origin of the directionality of time from the psychological, subjectivist, phenomenological perspective?
Subjectively, it seems that our perception of the directionality of time is largely rooted in our perception of causality. Confronted with a pool of semi-organized sensations, we perceive some as causally related to others, and then assign temporal precedence to the cause rather than the effect.
Now, grounding temporal direction in causation may seem to introduce more confusion than clarification, since there is no consensus understanding of causality. However, there are certain approaches to understanding causality in the philosophical literature, that happen to tie in fairly naturally with the reductionist approach to grounding the directionality of time given above, and bear particular consideration here for that reason. I'm thinking especially of the view of causality as "information transmission across mechanistic hierarchies," summarized nicely by Phyllis Illari in this paper.
If causality is viewed as the transmission of information from cause to effect via channels defined by "mechanistic hierarchies", then we may see the direction of time as having to do somehow with information flow. This is loosely similar to how Baranger sees entropy emerging from the dynamics of complex systems as perceived relative to the coarse-graining of state space. In both cases, we see the flow of time as associated with the dynamics of information. However, to see exactly what's going on here, we need to dig a bit.
(I don't necessarily buy the reduction of causality to information transmission. But I do think this captures an important, interesting, relevant aspect of causality.)
Another point made by Illari in the above-linked article is the relation between causality and production. However, I find it more compelling to link causality and action.
It seems to me that the paradigm case of causality, from a subjective, psychological point of view, is when one of our own actions results in some observable effect. Then we feel, intuitively, that our action caused the effect.
We then interpret other phenomena we observe as analogous to instances of our own enaction. So, when we see an ape push a rock off a cliff, we can imagine ourselves in the position of the ape pushing the rock, so we can feel that the ape caused the rock to fall. And the same thing when it's not an ape but, say, another rock that's rolling into the first rock and knocking it off the cliff.
In this hypothesis, then, the root of temporal directionality is cause, and the root of causation is our interpretation of our own actions -- specifically, the assumption that the relation between an action and its preconditions, is fundamentally conceptually different than the relation between an action and its results.
Another way to say this is: the carrying-out of an action is viewed as a paring-down of possibilities, via a choosing of one action among many possibilities. Thus, carrying-out of an action is viewed as a decrease of entropy.
So, psychologically: The directionality of time ensues from the decrease of entropy perceived as associated with enaction -- via means of analogical reasoning which propagates this perceived entropy decrease to various perceptions that are not direct enact ions, causing them to be labeled as causative.
Putting the (Reductionist and Subjectivist) Pieces Together
On the face of it, we seem to have a paradox here: physically, the directionality of time comes from entropy increase; but psychologically, it comes from entropy decrease.
However, there's not really any paradox at all. This is merely a relative of the observation that living systems habitually decrease their entropy, at the cost of increasing the entropy of their environments.
The directionality of time, from the perspective of a given mind, appears to ensue from a combination of
- entropy decrease in the foreground (the "acutely conscious", explicitly deciding mind -- the "global workspace")
- entropy increase in the background (the environment of the mind)