Like most people, I suspect, my dreams come in different kinds. Some of them are like short stories, and if I remember enough details these can be quite interesting. Many are like a confused movie, except that the confusion is usually not noticed until I wake up. In yet other dreams the main character really does seem to be much more identifiable as "me", partly because the people with whom the character interacts are often family or friends, or because the places the character visits are places I have been, and partly because he thinks much the same way my waking self thinks.
The fact that this dream self thinks so much like me makes the differences all the more interesting.
One thing that stands out is that at some level -- something like the subconscious of my dreaming self -- I know that I am dreaming. I do not have lucid dreams, but in some cases I end up debating with myself whether I can disregard an unpleasant event because it is just a dream or whether I am just in denial. This happened twice over the past week: in one dream I had accidentally backed my truck into a car behind me, and in another I was hunting with a friend (not anyone my waking self recognizes) and got into trouble both for shooting a moose on private property without the landowner's permission and for discharging a firearm too close to the road. The fact that I have never hunted nor lived in a place with moose makes that second dream particularly strange. I tend to wake up shortly after beginning to suspect that I am dreaming.
Geography is very confused in my dreams. I know the layout of the US states pretty well -- the main mistake I am likely to make is confusing Vermont and New Hampshire. Maps in my dreams are horribly confused. The distances are much reduced, with the cities that are landmarks on my drives each lying about one hour or less from its neighbor, and the roads are much more north/south and east/west than in reality. In my dreams, a good day's road trip would be from Tuscaloosa up into Indiana, then over to New Jersey and down to Philadelphia. In the real world this would be a trip of about 1300 miles; in my dreams it is more like 250 miles, and I seem to have made many such trips just of the heck of it. (Update 1/19/15: Last night I dreamed my dad had driven to Detroit and back as part of a long day's work, and the round trip took him 8 hours. In the real world it would have taken 32 hours.)
When I was a student I used to dream that I was studying for the test that was (in the real world) coming up the next day. I would try to read from my textbook, but it was impossible to focus on it, either visually or mentally. This usually happened just as it was time to get up, and eventually I realized I needed to wake up because the fuzziness of the textbook was a dead giveaway.
Something similar happens when I try to do math in my dreams. For instance, just last night I dreamed I was talking to someone about the manganese nodules on the seafloor (why we would be talking about this, I have no idea), and he suggested that there were only a few thousand of them. I knew that was much too small a number, but all my efforts to make an order-of-magnitude estimate were frustrated, even though I used dream paper to try writing it out. I made all kinds of mistakes: I mixed up the formula for the volume with the formula for the area,
even though I knew I was doing something wrong; I estimated the oceans to cover half the earth, rather than 70%; and I used 4000 km as the radius of the earth, when it is 4000 miles. I also estimated one nodule per square kilometer of ocean, but I knew that would be a generous lower bound. I can do a calculation like this in my head in just a few seconds while awake, but it was quite beyond me while asleep.
The weird thing is that once, back in high school, I went to bed knowing there was a serious problem with the proof for a theorem I was trying to prove. At about 3 am, I woke up knowing what the solution was to my difficulty, so I got up and wrote it down. Somehow I had solved the problem in my sleep. Sadly, this seems to have been a once-in-a-lifetime event.
