I thought I made it pretty clear they propose a whole bunch of possibilities in the gaps of our current understanding. The problem is the proponents make out these possibilities are much more probable than any evidence suggests. The reason it is woo, is because they start reasoning about things based on no evidence and present it as "knowledge". Like most woo, if you accept the axioms without evidence, then you can create a world of knowledge that seems logically coherent based on those axioms. That's sort of the nature of woo. It's not attacking the proponents, it's just my observations of the situation.
You're right, I was a bit too harsh. You indeed mentioned that they make strong claims without solid evidence, which is a valid criticism.
However, from what I recall, they don't present the ideas in the book as facts or established theories like your critique seems to imply. They make it very clear that this is uncharted territory and that there is a lot of science to be done. In fact I think one of the main purposes of the book was to convince the reader that there is in fact tangible research to be done in the subject, and questions worth asking.
A large proportion of the book is indeed spent convincing the reader that panpsychism is a worthy theory, yes. And despite their lack of empirical data (which has proven hard to collect in this area), they make logically sound arguments that the theory is at least worth further consideration.
This is how many scientific theories start, as hypotheses that seem logically sound but lack specific empirical evidence. These hypotheses then guide how we design our experiments. The book argues that panpsychism is a worthy hypothesis, and it does not do so by referencing any woo or pseudoscience.
Mathematics is a bit different than science, science doesn't claim truths, it says what the best description of things are based on the facts, the facts are evidence based. When you skip the facts and evidence part and just claim things, then you are in the land of woo. At the basis of science is essentially the laws of thought which, like good axioms should be, claim the smallest possible thing to reason from ( https://en.wikipedia.org/wiki/Law_of_thought ). Woo tends to claim large things partially based on existing knowledge and partially based on things we don't know but may be possible.
Axioms are only useful if they are widely accepted and they create an useful system. The modern axiomatic system of mathematics was created to support an already useful system, and they are (mostly) based on actual indisputable realities of our world.
The main axiom of panpsychism, if I understand correctly, is that all or most objects of reality have a mind. Now, as an axiom in itself it is pretty imprecise and not self-evident at all, as it depends on the definition of "mind". Also, it doesn't seem to create testable or useful theories out of that axiom.
The axioms of mathematics were _chosen_, because they have properties that are immensely useful. You can decide to use other axioms, and get different results, which may also be useful.
