This post is more of a psychological or philosophical musing, only loosely connected to mathematics. Last night I had a peculiar dream that I remember quite precisely. I was sitting in my point-set topology class, and the teacher introduced ultrafilters. (For some reason I knew in the dream that they would play a crucial role in proving Tychonoff’s theorem.) Then she gave us the following (slightly weird, but easy) assignment:
Let be an infinite set and let be a non-principal ultrafilter on . Then show that is a topological space.
I am quite sure that I was never given this assignment in real life. In hindsight, I find the following points remarkable:
— is indeed a topological space;
— is a topological space for every filter , you don’t need the filter to be maximal or non-principal, so in the real world such a problem would not be stated in terms of non-principal ultrafilters;
— I’m pretty certain I haven’t looked at (ultra)filters as a topology in their own right.
In the dream I feverishly tried to show that distinct elements could be separated by disjoint members of thereby committing at least two mistakes on different levels:
(1) in order to show that a collection of subsets of a set you don’t need to prove the separation property (Hausdorff-ness) and
(2) no two members of any filter are disjoint.
I realised both points immediately upon my awakening (in a double sense), and in real life it would never have occurred to me to try the weird and fruitless approach that I took in my dream.
The reason why I found this dream interesting is the following. Many of us experience (and remember for some time afterwards) very awkward dreams dealing with situation that are physically impossible (e.g. allowing you to hover above the ground without further aid) or otherwise highly implausible, or that memory plays tricks on us within the dream and we even notice it within the dream. Dreams are a radically different experience compared to our waking hours; but for the first time I noticed that also there are some interesting distortions of formal thinking that may accompany dreams. I have had quite implausible and buggy thinking in other dreams, but never before have I been able to pin it down in a formal and precise setting.
These were the thoughts that passed my mind when I was laying awake this morning. Also, I was reminded of a fellow student that I met in my first year calculus class. At some point he claimed that he was sure he had found a bijection between and . This alone wasn’t noteworthy — such mistakes can happen in the beginning of your studies. I pointed him to Cantor’s diagonal argument showing that the reals are uncountable. But he told me I was only cunningly tricked into believing was uncountable. In truth, he claimed, there was an intricate and deep bijection between the natural numbers and the reals, and this knowledge was deliberately hidden by our calculus professor (and the math books). This claim struck me as really odd, but I shrugged it off and got on with my studies. (Later, my fellow student was diagnosed with schizophrenia and had to discontinue his studies).
Interestingly, schizophrenia is characterised by distorted perception, disruptive formal thinking, memory distortions and generally holding severely implausible beliefs. All of those symptoms can be a part of dreams as well. When reflecting on my dream, I was so strongly reminded of the episode of my fellow student that I felt there must be some underlying mechanism or phenomenon connecting dreams and schizophrenia. But since I am not medically trained, I should be careful with such hypotheses.