Free choice
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What does it mean that something "causes" something else? What does it mean for something to be "possible" even if it does not happen? There are many concepts that have been around for centuries or millennia and for which we have always "kind of known what they mean" intuitively, but without a precise, formal account.
Fortunately, with the progress of scientific research, there comes a time at which we are able to bring a formal account to a concept. This is how philosophy was the incubator to many of the modern scientific fields, usually when there was such a turning point: mathematics, physics, biology to name a few examples.
As it turns out, the revolutions brought about by Albert Einstein's relativity theory as well as quantum physics in the 20th century have given us (candidate) formal accounts for what a "cause" is and what a "possibility" is.
In special relativity, time is not absolute and the order in which two events happen might depend on who is looking. However, in some cases, this order is the same no matter who watches: A occurs, absolutely, before B. This happens when it is possible for a light beam to be sent from A and received by B. This allows us to use this as the definition of A causing B: A causes B if it occurs before B no matter who the observer is. Of course, it doesn't mean that only A causes B: B is jointly caused by everything that precedes it no matter who watches, this is called B's past light cone.
This is not the only definition of causality on the table. There is another very (if not more) popular approach taken for example by Judea Pearl or David Deutsch and based on counterfactual statements resembling (although this is more complex than it seems) "A causes B if it is true that, if A didn't happen, B wouldn't either". This sort of definition is very pragmatic and useful, because it is meant to be used in situation in which somebody can act on A, and thus indirectly on B, something of paramount importance in Data Science.
Is it a problem to have competing definitions? I don't think so. Names are only a matter of convention. When I have conversations with colleagues or friends and we realize we have two different definitions for the same term, I often suggest to pursue the conversation by duplicating the terms, for example here type-1 causality and type-2 causality. That way, we do not lose our energy on terminology and can continue arguing on the actual structure of the reasoning. Then each one of us is free, when back home, to reformulate our insights in their own terms, and we both learned something from each other.
But let us come back to type-2 causality, which uses a subjunctive conditional: "If A didn't happen, B wouldn't either." This sort of statements has been extensively covered in literature, with a seeding book by David Lewis in 1972. Such a statement is called a counterfactual implication, and it is not a truth functional, in the sense that it does not only depend on the truth assignment of A and of B, as "A implies B" or "A and B" would.
Rather, its semantics depends on some possible worlds semantics. Lewis suggested that a sentence like "If A didn't happen, B wouldn't either" formally means that, in the closest possible world in which A doesn't happen, B doesn't either.
We just mentioned possible worlds. Possible worlds semantics are very popular in modal logic, because they allow us to define many terms we use everyday, but that are otherwise difficult to define:
Something is necessary if it happens in all possible worlds (must).
Something is possible if it happens in at least one possible world (could).
Something is impossible if it does not happen in any possible world (cannot).
Something is contingent if is both possible and not necessary (may or may not).
We are in one of these possible worlds, the actual world. A popular terminology, especially in science fiction, is to call the other possible worlds parallel worlds. In fact, quantum physics gave us the fundamental socle on which we can actually talk about parallel worlds without being considered crazy. Whether or not parallel worlds are real is the subject of intense debate, culminating in the many-worlds interpretation of quantum mechanics, suggested by Hugh Everett III.
But even if they are not real, parallel worlds are nevertheless a core part of the reasoning in quantum physics: indeed, as soon as we start using probabilities, we assume that there is an underlying Omega space with all possibilities; a probability is then just an assignment of numbers between 0 and 1 on these possibilities (called elementary events), in such a way that it all sums up to 1.
Likewise, when we talk about entangled particles, and assume that Alice measures it up and Bob down; and that "if Alice had measured it down, Bob would have measured it up" -- again, we have a counterfactual statement, and are reasoning on another possible world, a parallel world, in which things are different.
So, the interest for counterfactuals is largely motivated by quantum physics. In fact, what many refer to as a "spooky action at a distance" in the context of entangled particles is just a counterfactual statement. It is not an action, and it is also not spooky. But then, why is it nevertheless referred to in that way in mainstream literature? Because of...
In quantum foundations literature, the free choice of the physicist to perform whichever experiments they see fit is a fundamental and almost undisputed assumption -- so undisputed that it is often only implicit, even though I am delighted that, especially for a few years, it has been appearing increasingly more explicitly.
Free choice, in the context of impossibility proofs based on Bell inequalities, is defined in terms of conditional probabilities. A decision made by a physicist is modelled with a random variable that we shall call A (for measurement Axis) and that is located somewhere (SV, for Spacetime Variable to use Renato Renner's and Roger Colbeck's wording). It is deemed a free choice if
P(A=a) = P(A=a | B=b)
for any random variable B located in the past. In other words, to use Renato Renner's and Roger Colbeck's wording, it is "independent of anything it couldn't have caused".
A (non-trivial) implication of the above non-correlation is the counterfactual statement: "If I had made decision a' instead, then B would still have been the same: b" or more generally: "If I had made decision a' instead, then the past would be the exact same". For future reference, let us call this type-1 free choice.
Now, if we switch to Lewis's semantics for such statements, something very interesting comes out: in the closest possible world in which my decision is different, the past is exactly the same as in the actual world. This presupposes the existence of a possible world in which the past is exactly the same, but in which my decision is different. This is exactly where it itches, and why I believe that the type-1 free choice assumption is too strong and unrealistic. Indeed, a decision is a process happening in the brain, it is a succession, a cascade of events. Thus, if a decision had been different, then the succession of events in the brain just before that would have been different. In fact, Libet showed a few years ago that the implementation of a decision in the brain precedes the awareness of the decision; the consciousness only has a "veto right" to proceed or not.
Is this bad news? Do we have no free choice at all? Does it mean we are not responsible of our actions?
On the exact contrary. In fact, I believe that the above definition of free choice would make us less responsible, not more: how can we be accountable for actions that are indistinguishable from random choices?
Rather, Jean-Pierre Dupuy suggested, almost 30 years ago, a different approach. He suggested to define free choice (let us call it type-2 free choice) as follows: I have free choice if I could have acted otherwise.
So, type-2 free choice means that in possible worlds wording, it means that there exists a possible world in which I act otherwise. But the closest possible world in which I act otherwise might have a slightly different past -- this is the difference. In fact, it is even compatible with being perfectly predictable.
Why do I spend so much time studying this direction of research? Because I believe:
As it turns out, it is possible to re-bootstrap game theory and adapt it to this alternate form of free choice, in a way that is meaningful, and that even seems to increase social utility. While investigating the topic, I found out that, as early as 1983, Douglas Hofstadter (author of the best-selling book "Gödel, Escher, Bach" known to many computer scientists) pioneered this non-Nashian reasoning with Superrationality on symmetric games. In 2004, we formalized the reasoning for games in extensive form (trees) and a few years ago, I extended it to non-symmetric games in normal form and later to games with imperfect information (generalized decisions made in Minkowski spacetime). If you want to learn more on non-Nashian game theory, you can read , which also includes a lot of background knowledge to understand the topic.
That type-2 free choice is .
That it is the key to potentially extending quantum theory to , based on models in which we solve for the equilibrium of a gigantic game played between us and the universe with a non-Nashian forward induction (which is less computationally expensive than Nash and Subgame Perfect Equilibria).
That it can lead to a greater good in society with both individual freedom and high social utility, as the non-Nashian game theory approach acts as , some sort of Adam Smith 2.0.