The reason we don't remember the future has to do with an arrow that gets ignored. That of information flow between the brain and the outer world. One could call it the sensory arrow of time. With our senses, we absorb information only in the future direction i.e. along the thermodynamic arrow of time. We never absorb information along the opposite direction of time. In other words we never emit it along the normal direction. We actually do emit information (speaking, touching) but not in the "future-memory' way. I will explain how emitting information is the key to the definition of future memories.
Imagine there were beings who somehow experienced time in reverse. They call the future past and the past future. Now from their point of view everything happens in reverse. Dissipated heat collects into light and goes back into stars, smell molecules waft back into roses, etc. However one should note that they can't observe these reverse-dissipation events.
For them it is us who are the future-remembering beings. How they "see" us can give us a clue to what future memory can look like to us. Now from our point of view, information from objects enters our bodies, gets stored as memories and finally decays / gets corrupted. But as per them, who live reverse lives, our memories arise randomly, reverse themselves and then the information from those memories (lights, fragrances, sounds) is sent out to the external world where it gets absorbed by objects.
Thus it follows that when we, the past remembering beings, want to talk of future memories, then it must be about forming memories, reversing and converting them into information and then sending that information out.
So, if we wanted to have a memory of the future, we could, with some physiological modifications, do something like this. We could imagine an object like a blue square and remember this imagination. Now we reverse this memory and emit blue photons from our body in such a way that this light forms a blue square on a white screen. This is a valid memory because it has a causal connection with the square and is also a memory of the future because the square came after the memory. Imagined in reverse the photons collect and fall on the white sheet, get reflected back in a blue stream, enter our body, get converted to nerve signals and are finally imprinted on our brains as a memory. This series of these events is not improbable because entropy decreases in the past here.
So just like past memories are 'decodable' impressions of past information, future memories must be decodable sources of future information.
This definition might seem problematic at first because this way you seemingly can't remember "complex" events like looking at a frog. But then even past memories are also always inferences. This has to do with the 2nd law of thermodynamics. It is very likely that information comes from objects than just randomly flowing into us in the shape of objects. So it is improbable that we will see frogs just because light rays collected in such a way by chance and impinged on our eyes.
Future-remembering beings would have a hard time keeping track of the world around them because generally information spreads out from objects. However if perhaps there were some chance events where local entropy was reversed like yellow painting which absorbed yellow and heat gave off white light, then their memories could make sense.
We also need to remember that in the end memories point to information transfer from the body to the brain and not from the outside world to the body. Touching something cold for example makes information from the body flow outward but it still sends impulses to the brain.
Remembering the future is not about time travel or knowing a future event without having any connection to it. And if there was indeed a way of knowing the future, it would still not mean having a memory of the future unless causality was involved.
This definition also avoids any paradoxes. You can't use this memory to change the future. If you do then the memory was simply false. Also, claiming to change the future by destroying a future memory would be akin to claiming to change the past by implanting a false past memory.
We don't remember the future because we don't / can't convert our memories into outside information in a way that is consistent with how we decode that memory in our heads. It makes zero evolutionary sense to align the sensory arrow against the thermodynamic arrow of time.
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My rather simple answer, which almost no one has paid heed to or understood, is that we can remember the future; it's just not worth it nor is it really spectacular. And there is nothing mystical about it. Hear me out; no pseudoscience here but quite a simple answer to an often asked question. Please ask me if you don't understand something about it.
So here it goes. You imagined a blue circle in your head and stored it as a memory. You then reversed (forgot) that imagination and that forgetting was the exact reversal of the process of remembering so that it led to emission of blue light from your body. That light produced an image of a blue circle on a white screen for an instant. Thus, that imagination was a memory of the future because you remembered it before it happened. If you reverse this whole process you can see how the blue photons from the fleeting circle flew into you and became your memory.
On the other hand if you imagined a blue circle and simply saw one the next day it would be chance and not a future memory because the blue circle was not formed by your memory.
Remembering the future doesn't mean knowing something you have no connection to like the winning number of the next lottery. lt is simply the reversal of the process of remembering (memorization reversal and sensory perception reversal).
Adding some points for further clarity -
Any valid memory must always have a causal connection with the corresponding information outside the memory housing body. And a mechanism should exist in that body to decode that memory because on their own memories don't represent anything. The definition of future memory here meets both these criteria.
In summary, just like past memories are 'decodable' impressions of past information, future memories must be decodable sources of future information.
So, we don't remember the future because we don't / can't convert our memories into outside information in a way that is consistent with how we decode that memory in our heads.
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