>I cannot really engage with your question. First, because I am not an expert in philosophy

I feel the entire field of philosophy is mostly useless and full of categorization errors. I don't buy into it, I just use the word for speculative stuff. A lot of the things that the actual field of philosophy chooses to speculate on (Animism for example) is just random garbage. They also pair religion and logic together as two peered pillars of study, as if religion is just as fundamental as logic. I have little respect for the entire field from a formal perspective. In my reply, I was using the word informally.

>You see, in the pragmatist view of a statistician (and in the math that goes with it), randomness - for example in the form of statistical independence

No we're talking about randomness as a fundamental phenomena in a procedural and physical universe. Meaning that what is a definition of a pure stateless function that takes no input and will always output a completely random number? The probabilistic definition of a random variable does not actually produce such a function and sort of avoids the concept of randomness all together.

In physics, quantum physics specifically, such a function is not defined but is taken as an axiom. The actual location of a atom after decoherence is assumed to be axiomatically random.

>Instead, we need the "switch" variable to be uncorrelated with things we do not observe, or can not adequately model.

Arguably everything in the universe is correlated and shares a causal connection. A butterfly flapping it's wings in China causes a hurricane on the other side of the world. On a grander scale: A perturbation of an atom in one corner of the universe eventually changes the parameters of every atom in the universe.

According to certain theories in physics information can only travel as fast as the speed of light. Therefore according to this logic all things can be causally connected but only up to a point. If you imagine multitudes of infinite spheres continuously growing out of an atom at the speed of light, then everything outside of a sphere that started growing at t0 is guaranteed to not have a causal connection to any events that happened to the atom at t0 or after.

However in math we create our own playground devoid of all the messiness of the universe and it's atoms and limits to causal transmission across large distances. Within this world there seems to me, to be an issue with your notion of a switch to be "uncorrelated."

For something to share a causal connection with another thing, both have to be correlated. You cannot establish causality if the switch is uncorrelated with the observation. The result will always be a failure in establishing causation if you are selecting this type of switch.

There is a paradoxical problem with the definition here. I get your intuition and you understand mine as well but this definition seems to be incomplete or not an accurate illustration of both of our intuitions of this concept.

Using "randomness" as a concept and leaving "correlation" out of the definition seems to better fit our intuition of what's going on when *I* or another human flips the switch. After all, you cannot have a causal relationship without a correlative one.

>Statisticians know that the idea of causality is by default not a philosophically "true" concept.

I would say the just "don't go there." They stop at the definition of a random variable and don't dig any deeper. https://en.wikipedia.org/wiki/Random_variable <--- on wikipedia a random variable is defined informally in terms of "random phenomena." Additionally the formal definition is defined in terms of "possible outcomes" so they're really side stepping the problem here. But that's ok, it's not really needed to play the mathematical game of probability.

>Related to this, let me offer you the perspective of social scientists.

A social scientists opinion is not any more valid than an average persons opinion. A social scientists views human behavior as a black box. They don't just put one human brain in a black box, they actually put a population of human brains in a black box and they try to analyze everything from that perspective. It simplifies things but inevitably there's a lot of missing information. One piece of this information is whether the output of the black box is random. If you can't peer into the black box you can never know if it's random or not random. Thus the social scientists viewpoint doesn't mean any more than someone who isn't a social scientists because most people view the brain as a black box as well.

That being said if there was a field that could closest determine "free will" it would be neuroscience and quantum physics. When I am flicking that switch and attempting to randomize the timing of it all what neurons are firing and what chemical reactions are occurring in my brain? What is the exact mechanism producing this action? Does the randomness of the quantum world leak out of a chemical reaction to influence the macro output of my timing on flipping that switch?

>That is, they would reject the notion that free will is indeed (or at least always) random.

Again they can't reject anything or accept anything related to this. To them, this part of the universe is underneath a black box.

>Sorry again that I have to rely on the statistical view - it is the only one I know a little bit about.

Who says you have to restrict yourself to viewpoints from certain groups? When you restrict yourself to "the statisticians" viewpoint or "the social scientists" viewpoint you are also restricting your intelligence.