Does panpsychism claim that even mathematical entities, like numbers and functions and sets, are conscious entities? Or is it restricted to physical objects?
This is a great question in that what you have asked uncovers two logical fallacies, the slippery slope argument and circular reasoning, that is located at the root of panpsychism. It does not have a clear notion of what a "thing" is. Let me clarify my point by dissecting this chain of reasoning:
- Everything has a mind
- Everything should include abstractions and fictions, otherwise it cannot be "every" thing
- Then abstractions and fictions too shall have consciousness.
Panpsychism proposes a ubiquitous mind, as an atomic physical/ideal particle or structure, in the natural world. If we accept the proposition that abstract concepts and fictions do not have consciousness, they become non-existent. This is obviously paradoxical as abstract concepts do exist, proven by the fact that I am writing this line.
On the other hand, if we accept that they are conscious entities -as they are part of "everything", they should be by the panpsychist perspective- we also imply they should have a physical or structural correlate of consciousness. Where in the natural world do these particles sit? Every time you make use of an abstract concept, you should also be creating "physical" or "structural" (depending on your stance between idealism or materialism) entities. This argument could be countered by proposing the entity that is created is not atomic but rather is emergent, which then implies consciousness does not have an atomic physical (or structural) correlate and falsifies the first assumption (that everything has a physical correlate of consciousness) altogether.
As you can see, if we accept that they are conscious, circular reasoning is unavoidable and as such, they should not.
I learned and heard of panpsychism from Eastern religions and Leibnitz, who as a universalist was well aware of and very interested in eastern mythology and religions. In Leibnitz later years he developed Monadology as a final summary to depict his panpsychism view using his famous monad, which I suspect is derived from the old indo-european word "Manas", very similar in meaning to English's word "Mine".
In his view, certain higher level monad like animal soul and human rational mind with reasoning/abstracting capabilities can have consciousness. Only monad is real substance in this world and monads exist everywhere, but one person can only have one distinctive soul-monad which is self-conscious. Plants also have monads but they're of bare type without consciousness. A math entity is only a very abstract concept only found to be in human mind level monad, thus the math concept itself has no consciousness, but the containing mind has.
A computer's memory also can have math entities, but computers are material composites which can be infinitely divisible without a unifying consciousness, thus according to him, this computer can not have consciousness. it can manipulate and compute these math entities faster than any human mind, but it simply does not "understand" any meaning of its computation. And I totally agree with him on this question.
There are many different approaches to pansychism.
In Buddhist Yogacara philosophy, subjectivity is seen as fundamental, with all phenomena occur in the intersubjective space of shareable experiences. That is, no pure objective reality, because no one can experience that, but no pure subjectivity or isolated thinker either because that arises interactively - as illustrated in the metaphor Indra's Net. In this framework, relations between minds would be fundamental, with material experiences like geometry occuring within that, ie within the alaya vijnana, or space of mentalising/narrative possibility.
In physics, symmetries and conservation laws have been shown to be equivalent. If the set of fundamental constants has been determined by the Strong Anthropic Principle, then the symmetries and arising of minds are intimately related, and perhaps equivalent in our patch of all possible universes. That is not all mathematics being a kind of entity, but maybe E8.