The question “What is time?” has been an eternal problem for philosophers. However, you can define a physical time by making the second law of thermodynamics analytic. How can we define the time in our mind, then?
1. Time as the increase in entropy
What characterizes time is its irreversibility, while the entropy of the universe as an isolated system also increases irreversibly. So, some physicists regard the second law of thermodynamics not as a law of entropy but as the definition of time. That is to say, they conceive that time is the increase of entropy rather than entropy increases as time goes by.
However, what philosophers have tackled is not such physical time but time in mind. Does the semantic entropy in the ideal world as well as the thermal entropy in the real world increase as time passes? An information system maintains itself by distinguishing a sense from nonsense and truth from falsehood. As time elapses, our experience and thinking grow larger. Accordingly the semantic entropy, namely other possibilities of interpretation also go on increasing. So, semantic entropy also seems to increase irreversibly.
However, as the excess of the meaning is reduced by oblivion, the semantic entropy in the ideal world does not always increase. It is natural that the principle of maximum entropy should not apply to information systems as open systems, because it can originally apply only to isolated systems.
2. Our mind resists the increase in entropy
Is the concept of entropy unnecessary when we analyze the time in mind, then? No. I would rather find significance in the fact that the principle of maximum entropy does not apply to consciousness. That is to say, consciousness of time is resistance to the increase in entropy, while time itself is the increase in entropy.
Time is often compared to the flow of a river, maybe because it flows irreversibly like time. It is because the frog is against the flow of the river that it can feel the flow. If it gives up the effort of clinging to the pier and is carried away at the mercy of flow, it will no longer feel the flow, since water flows with the frog.
The frog feeling the flow is analogous to us having the consciousness of time. We, as life systems and information systems, go on making an effort to resist to the increase in entropy in the environment. Giving up this effort means our death and, once we die, we will not have the consciousness of time any longer.
But not all systems have consciousness of time, although all systems are negentropy that resists increase of entropy. The system that does not waver in selection has no consciousness, much less consciousness of time. To use the metaphor of the river again, it corresponds to the pier that resists the flow determinately. Unlike the frog, the pier does not feel the flow of the river.
3. Requisites for time consciousness
What conditions are necessary and sufficient for a system to have the consciousness of time? The system whose selection is programmed by genes or a momentary being at the mercy of indeterminacy need not have the consciousness of time. It is only those who can predict the future or decide their behavior on the basis of past experience, that is, can learn that can and must have the consciousness of time.
When the learning beings fail in something, they trace the present effect to the past cause and no more repeat the same failure in future. This tracing back to the past is the resistance that the learning beings put up to the irreversibility of entropy. The reversibility of the ideal world enables the universal and trans-temporal science, though actually science has been intra-temporal and subject to historical relativity.
In the actual world, you can observe that red ink, dropped on water in a tank, spreads all over until the whole water turns pink, but you never observe the reverse of it. You can virtually observe the phenomena against the principle of maximum entropy, when you run the movie the opposite way. Of course, since the movie projector consumes electricity, it actually increases entropy. So, the principle of maximum entropy is never broken. Even if the frog resists the flow of the river, it does not reverse the flow.
Although we are temporally limited beings, we try to transcend the limited moment toward the past and the future. It is neither the completely trans-temporal nor the completely intra-temporal but the intermediary transcendental beings that can have the consciousness of time.