Does the fact that physics has been shown that at the most basic level to be indeterministic support free will? Or does the fact the we cannot predict whether a neuron fires or not, in the process of making a decision, also support free will?
These are two versions of indeterminism but they are quite different. For our purposes here, let us call the first physical indeterminism and the second we can call neuronal indeterminism. Now by free will we mean something works above and beyond the natural processes of the natural world, it is often called contra-causal free will - as going against or trumping the natural causal processes - or libertarian free will (LFW).
I started a series on free will a while ago where I had already questioned that such a libertarian free will is likely an irrelevant or redundant concept for responsibility, that is it is just not needed, hence my descriptions of how responsibility could possibly work made no reference to such a free will. I indicated then but have not yet developed the idea that not only LFW is not needed but it actually contradicts responsibility. As an intermediate step, here I intend to briefly address two arguments for how LFW could work given what we know of the world.
There are two ways of looking at this. The first is over disputes over different interpretations of quantum physics and the second is over randomness.
In the the most popular interpretation was the Copenhagen interpretation, which sought to show how from a entirely deterministic Schrodinger equation (the wave function) one could obtain only a statistical description of quantum events - we could not deterministically predict a single event but could (incredibly) reliably describe a multitude of quantum events via statistical mechanics. This interpretation required the mysterious "collapse of the wave function" - an indeterministic event, since there is no deterministic equation to describe this collapse itself, only that it does. The interpretation was rather basic just saying the maths works and that is really all that counts. Of course some us inquisitive humans were not satisfied with this and sought better understanding of the collapse of the wave function.
This was partly because it led to a view that the universe was indeterministic and some had issues with this. Still was this view just the result of our epistemological limitation of the ontology of the quantum world? That is due to our limited knowledge of what is really going on? Well one can use different interpretations to help understand the same facts differently.
Yes there are different interpretations of the quantum facts that do not imply, unlike some, that the universe is indeterministic. To name but two the de Broglie-Bohm hidden variables interpretation and the growing in popularity (arguably now the most popular amongst professional physicists) Many Worlds plus Decoherence interpretation (which says there is no collapse of the wave function - that is just the way it looks to us). These, and others, one way or another, argue that the universe is not indeterministic but deterministic.
Still, one of the issues in quantum physics is there is no way to experimentally differentiate between these interpretations. If one could, then these would no longer be just interpretations, since they could then make empirical different claims that could be tested and used to select one over another. That is these interpretations are entirely subjective.
Clearly supporters of LFW are likely to chose an indeterministic interpretation of the universe, on the presumption that this lets in the possibility of a LFW. Regardless of this still, being an entirely subjective choice, an example of the tail wagging the dog, does it help them? In two ways, the answer is no.
First, since the different interpretations make no difference and make entirely the same predictions of all the current known quantum facts, there is no room here for LFW. Instead there is nothing that shows that we are not a part of one unified natural universe, describable by a single wave equation.
Still what if they are correct and whether we can get beyond our current epistemological limitations or not, that the universe really is ontological indeterministic, does this help? Again the answer is no, since such indeterminism would be the purest randomness possible and nothing more. There can be no hidden information caused by some LFW or anything else, the quantum theories are the most accurate predictive models by a huge margin over any other physical theory and if there were any form of hidden information - that could somehow translate into influencing brain states or for any other purpose - it would have been empirically detected (even if not at all understood) and it has not - ever. Quantum mechanics theories (not the interpretations) are the most reliable, accurate and robust we have ever developed and have been tested in millions of experiments and have not failed in any way that would provide any evidence to allow in the possibility of LFW.
A LFW supporter has no evidence from quantum physics to support any claim for LFW however you look at it. A LFW supporter would have to provide a theory which makes claims that can be disconfirmed and then test it to see that it has not been disconfirmed. To date they have done no such thing. Merely pointing to indeterminism here is empty rhetoric.
This is far simpler but one can also draw on a vague parallel over the collapse of the wave function here. A brain can be considered as a network of neurons - indeed we have developed a large variety of artificial neural networks (ANN) designs inspired by how we think the brain works and to better understand how the brain works.
Now many of these ANNs are not attempting emulate real brains but whether they do or not, still they are pretty much all statistical models, hence there is some parallels with quantum models and statistical mechanics. That is these work by not being wholly depending what a specific individual neuron does in a network, which is quite the opposite of logical and other models (where if one logical inference is changed there is a whole cascade of affects that can be attributed to that rule change). Similar to quantum mechanics, the challenge is not to predict what an individual neuron does but what an ensemble of neurons do. This is how neural nets are robust and reliable because they have a different type of sensitivity to "noise" and any individual neuron might be redundant in a ANN reaching a "decision". One could simplistically say they are analogue models versus the digital models of logic networks and artificial intelligence. Whilst this whole paragraph is an obvious over simplification, this is sufficient for our purposes here.
Could one say that a real brain, a real neuronal network is indeterministic, indeterministic because one does not know or be able to predict that for any repeatable decision whether a specific neuron will fire or not? No, simply this is how these networks work, they work by being robust to noisy inputs so that similar but not identical inputs can generate the relevantly appropriate outputs and part of that ability is due to not relying on specific neurons always firing for specific inputs. So, far from any neuronal evidence showing some neuron firing or not as evidence for LFW, it shows no such thing, rather this is how they are meant to work.
Secondly, whether these real and artificial networks work as some form statistical pattern recognition models or otherwise, there is nothing in them that relies in the slightest on any quantum indeterminacy (Penrose aside but and most regard his work as both controversial and unsubstantiated by any evidence). Any argued for neuronal indeterminacy is entirely consistent with and indeed relies upon the known descriptive laws of nature - physics, chemistry or electromagnetism and so on.
A LFW supporter has no evidence from "neuronal indeterminacy" to support any claim for LFW however you look at it. A LFW supporter would have to provide a theory which makes claims that can be disconfirmed and then test it to see that it has not been disconfirmed. To date they have do such thing. Merely pointing to indeterminism here is empty rhetoric.
So anyone who wants to argue for a libertarian free will does not have any evidence to back such arguments from either the world of physics or the world of brains, any claims of indeterminacy in either area today provides zero evidence for such a libertarian free will.