I'll have to read the book, but in my mind, the (emprical) study of humans and their brains doesn't shed light on the metaphysical question of the nature of mathematics. What they find is how humans have developed to do mathematics. We could have evolved to be the way we are with or without mathematics being "out there". Survival in the physical world would lead us to "throw away the metaphors that don't work well". At any point in time a concrete human being would still be able to consider only a limited set of mathematical ideas i.e. for humans "mathematics bottoms out at "what goes on in human noggins"".
I'd say the patterns you mentioned in an earlier comment are a way for math (or parts of it e.g. some integers) to be "out there". If humans embody mathematics, then analogously so do those patterns.
> No, it's actually, "ta-da, we're not conscious! but here's why we think we are!"
And of course the "think" has a quality to it that the hard problem is about. It's interesting how illusionists and eliminativists explain away aspects of SE by invoking (other) aspects of SE. "You merely have an illusion of being conscious" - that illusion is the hard problem, so now explain that illusion. I could be having an illusion of an illusion of consciousness.
Imagine something that doesn't exist in the usual physical sense e.g. a dinner table on the Moon. Does that table exist? Not in the usual physical sense. Your thought or imagination of it does, though. What is that thought or image in your mind's eye "made of"? Sure, you might be able to correlate it precisely with certain neurons and yet you've not answered the question. You might call the mind's eye table an illusion, but you're not gonna deny that the picture of it exists in some sense. Three things exist: the physical table, your neurons and, separately, although not entirely independently from the neurons, (the picture of) the mind's eye table. Hence, the latter is part of the universe and the fundamental substrate of the universe must support if somehow, in a way that's different from the usual physical matter tables and neurons. Is your visual brain circuitry involved in the imagination, perhaps even generating the image in your mind's eye? Perhaps, but this doesn't answer the question. If we're nothing but our perceptions, then what the heck is that imaginary table that I'm visualizing quite well while there's no perception of an actual table? What are the physical laws characterizing such mind's eye objects, somehow coupled to ordinary physical matter and yet not of the same "stuff"?
Models like the one linked don't explain why SE exists in the universe. They posit certain physical/mathematical strucutures and claim that if this or that structure is present, then ta-da there is SE (or the illusion of it, which is the same thing). People in the stone age had a model of that kind: "this piece of matter, structured with two arms and legs - it's conscious". At some point we developed language and the model got a bit more precise by demanding the piece of matter emit certain sounds from a specific location on their body. What we have today is no different in kind. We've just become more precise at locating the pieces of human matter to verify the presence of conciousness (or illusions). None of that says why that configuration of neurons experiences or has illusions, only that it does. Science tells us that experience is in the nature of certain pieces of matter and we just have to accept that without further explanation, like the fact that electric charge exists and follows certain rules. Deeper "why" answers are out of the scope of current science.
It can't be "literally everything". It is experience that enables us to correct those misconceptions in the first place. It is by experiencing that we discovered and experimentally confirmed quantum theory i.e. that "the world is not classical". It is by reflecting on his experiences that Dennett came to his conclusions.
Build the thing on a sea platform. Launch the counterweight into water. Make it sharp-nosed like the payload, so it could potentially survive impact with water and be recovered.