(mathematical) sets of beliefs “It seems to me that they can if beliefs or the forming of beliefs take the form of brain states or changes in the structure of the brain, but I’m not sure.”
First, let me recommend a book that presents the conceptual difficulties in developing this notion that beliefs correspond to brain states. The book is “Memory and Mind” by Norman Malcolm.
I will try to summarize these difficulties as follows:
1. First, we seek a natural correlation between the elements of the two domains, and not a stipulated correlation. For example, like the relation between tides and phases of the moon vs the relation between the english language and morse code. So, the isomorphism between experiences and brain states is one by nature not by convention. To determine if there is a natural isomorphism, we propose a hypothesis and thru observation see if it holds in the world.
2. Problem of defining the elements in the “Experience” domain. - Take the desire “wanting to catch a bus” and designate it as an element, call it “E” . The problems start to arise when one begins to think of all the different circumstance one would call “wanting to catch a bus”. We find there is not one common factor among all the cases. So, what brain state element is to connect to what “Experience” element?
3. Problem of duration - where mental states and brain states employ different concepts of duration. For example, Wittgenstein put forward the following example, “Indeed one scarcely ever says that one has believed, understood or intended something “uninterrupted” since yesterday. An interruption of belief would be a period of unbelief, not for example the withdrawing of attention from what one believes - e.g. as in sleep.” This would be unlike the duration of many physical events, say the motion of a ball across the floor could be observed and clocked. Thus, how could we ever determine if one element was simultaneous with other element when the one kind of duration is specific and the other duration is quite vague.