Earlier quoted context omitted.
That's precisely the problem - we can't yet formalize the concept we understand as "consciousness", and so any attempts to find a physical/mathematical definition are almost doomed to fail, as they can't really guarantee they are defining the actual concept we have in our minds when we say "consciousness".
My argument is a bit more subtle. I'm saying asking what "consciousness" is like asking "what does it mean to be American." Does it make sense to attempt to find a mathematical definition of "American." No because being American is largely a made up concept. It's a complex category no doubt but it is not fundamental to the structure of the universe. I'm saying "Consciousness" is the same thing. It's an ingrained conc…
Regardless of what you think the mind is (if you believe, like me, that it is fundamentally a physical computer, or if you believe it is something else entirely), our picture of the universe is never going to be satisfactory until we can account for how we integrate with it. So yes, for every made up human notion, there needs to be some way to physically define that human notion, if we had a physical model of the human mind (even if that definition is as simple as "an illusion caused by such and such series of events").
It may well be that consciousness is not a fundamental concept in the sense that it is a uniquely human illusion, not something that could not be properly extended to non-human agents. But even if that is true, it doesn't mean that there can't be a precise definition of how the feeling of consciousness arises in humans, what properties of our minds make us think that we have this property and that it is important.