Hi Jason!
Oh, that's a great point about N vs NP, thank you for pointing that out! I will correct that!
And these are all extremely good points, all of which are crucial for this topic. After thinking about these for quite some time now, I feel more comfortable moving away from the idea of abstract vs. physical and going to the "other side" of the Church-Turing thesis. Turing machines are a very useful abstract tool, but it is only one way to understand a system. But rather than focusing on lambda calculus, I'm now more concerned about where these spaces for abstractions "come from."
Here's what I mean. I can think of a Turing machine in two ways. One is the configuration of the abstract machine (the lookup table) and the other is the states that it produces by running it. As it runs, it marches from one state to another state and given all possible initial conditions, one could map out the entire state space and the "legal" transitions between states. Because of the Halting Problem, it is impossible to look at an abstract Turing Machine and decide whether or not it will eventually halt. It is like looking at a lookup table and asking if it is possible to know some properties of the resulting state-space map. I think the reason this is the case is the entire Turing machine system is described in two ways: A lookup table and a state-space map. In that sense, these two "languages" to describe the same system are encoded in two different spaces of their own. There's the state-space map space, and the lookup-table space.
But are these two spaces the best way to represent a dynamical system? The Church-Turing thesis says this representation is equivalent to a world where programs and data are really the same things. But are there other possibilities?
This gets extremely difficult because it's essentially asking if there exists an "optimal" description for a system, which suggests some kind of objective reality. But I assert that for at least trying to understand biological systems, "optimality" makes no sense without the context of an observer.
So now I'm thinking that state spaces only exist within an interaction between something and something, like an observer and a dynamical system (possibly another observer or even itself). The state space of all possible interactions between me and my cat depends entirely on our current physical configurations at that exact time. Without ears, I couldn't hear him meow, although I could see him do it if he was in my view. Without feet, we would need to invent a new way to play.
Going back to the stock market, I think your points are absolutely correct. For an individual broker, their goal isn't to create a perfect model of the financial world, but rather to make a slightly better prediction than everyone else. I am mostly comfortable thinking that the human factor, just a human by themself, is entirely physical, but then the abstract is only defined by the human's interaction with other "stuff."
I like the old philosophical question: "Does Coulomb's law exist in absence of charged particles?" I don't know enough field theory to answer this for physics, but for biology, I think the answer is a resounding "no." Conservative/liberal politics do not exist without humans, species cannot exist without niches, and fish cannot exist without water. I think the notion of abstract exists within interactions.