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When we decide to move an arm, some argue that it is an example of a mental event causing a physical effect. But doesn’t recent science show that free will may be illusory and from a time perspective, the notion of “choosing” to move the arm happens after unconscious reactions have already willed it?
If so, does this technically mean we have never observed a non physical cause causing anything physical? Would the mental event be an effect rather than a cause?
To quote a cliche, the jury is still out on the question of the nature of the mind. There seems to be no doubt that thought and contemplation are somehow associated with physical processes in the brain, and certainly when you decide to lift your arm, the movement is triggered by electrochemical effects. The question is about what happens in between. Is there some kind of free-floating decision making that is not directly determined by the laws of physics acting on the particles that form your brain? Does the brain simply provide the working environment in which the mental causes operate? There is a challenge whichever way you look at it. If you decide that consciousness somehow lies above the physical processes, then you have to find a way to account for the fact that your conscious decisions are able to influence the movement of electrons and ions to trigger the nerve impulses that move your arm. On the other hand, if you suppose that thinking is all to do with the movement of charged particles in the brain, then you have to find a way to explain how those movements give rise to the sensations associated with consciousness. It is an insoluble pancake and a conundrum of the utmost impenetrability, as a certain policeman would have said.
The most convenient evidence (and as I'd soon argue, the most robust) for non-physical causation is introspection. I choose to raise my arm and the thing goes up. Period! To question that by unwarranted extrapolation from physical science is invalid specially when the hard problems about mind-brain relations are not yet resolved by mainstream science/philosophy and when there are even concepts from modern science itself that are consistent with mental causation such as quantum indeterminacy (ruling out a causally closed physical world), and brain plasticity (ruling out any fixed and solid correlation between mind and brain). So why extrapolate from modern science only when it helps a prior "physicalist" bias and ignore contrary evidence/theory?
Back to introspection, it has been dismissed as an unreliable means of knowledge in modern philosophy even though consciousness (the agent of introspection) is by far the most indubitable and vivid reality of life to the point that physicalists have really gone to crazy absurd lengths to deny its reality. (See this paper for example. Thanks to Conifold for sharing this the other day).
Moreover, if scientists are willing to look into more "anomalous" mental phenomena, like scientists at Division of Perceptual Studies, UVA have, they'll find clearer empirical evidence of mental causation, such as very peculiar and extreme forms of psycho-physiological influence as in stigmata, maternal impressions, hypnotic-induced shapes on skin, and even more radical cases of mind influencing someone else's body as in telepathy, that are either extremely difficult or outright impossible to explain under physicalism.
Scientists at DOPS have long made their case in their pioneer work Irreducible Mind, and are yet to receive any worthy rebuttal.
Agent causation is an established fact. We are able to perform voluntary actions that are not mere causal reactions to prior events. We can decide what we do.
Voluntary muscle actions seem to be caused by converting chemical energy to heat and kinetic energy and controlled by the mind who decides which muscles should contract and when.
The neural signals that trigger the muscles to contract are still physical events that need to be caused by something. The brain is a physical organ often compared to muscles using chemical energy brought by the bloodflow to do what it does, which is mainly processing information and sending muscle control signals.
The brain's capacity to process information is called the mind. The mental and physical processes in the brain are deeply interconnected, but still fundamentally different processes doing completely different things studied by different branches of science.
We still don't know enough about how exactly the mental and the physical processes interact, but it seems that the mind is capable of causing physical changes within the brain. Muscle actions are just causal consequences of the changes in the brain's motor cortex.
The mental event, the decision to move an arm, does seem to be the cause of the physical movement. The decision cannot be an effect, as it is not a physical event. Decisions are not caused by prior events, decisions are based on knowledge about prior events.
N.B., in this answer I equate "non-physical causes exist" with "free will exists" since OP seems to have picked the latter as indication of the first. I have nothing much to say about non-physical causes that are not free-will/mind related... and as Stewart pointed out in his answer, what is a "cause" anyways...
TLDR: nobody knows, but there are strong opinions to be had.
But doesn’t recent science show that free will may be illusory
Absolutely not. There is a lot of philosophy out there which argues about free will and whether it is an illusion or not. Some of this philosophy takes modern scientific findings about certain measurements of brain activity into account to further its arguments. Free Will itself is more a concept than something physical - it's more a question about definitions and beliefs than something you could ever measure.
There is no science out there about this whatsoever, simply because science, while it has made good progress in brain research (i.e. how the visible or microscopic bits and pieces, the electricity, chemistry of the brain work, and how the whole mess may be structured into components or not...), it is awfully far away from producing any findings about how the mind works. Including questions about whether the mind, conscience, self and so on are a pure physical phenomena (i.e., illusions) or whether there is a different, non-physical aspect out there.
While it is easy - and getting steadily easier - to enjoy the "it's all physical" view these days, with many good arguments, there really is no falsifiable scientific evidence whatsoever; nobody has yet come up with a scientific, physical, experiment to prove the one or the other alternative.
It's eerily like the question whether gods exist: it is a question outside of science; there is simply no experiment out there that would prove or disprove the proposition. Like with god, unless you already believe in some super-physical aspect of our minds from the get go, it is very hard to find arguments supporting that assumption. And if you do believe in such, then it is very hard to find arguments dissuading from it.
So to your original question:
Do non physical causes exist?
Nobody knows, there are no experiments to prove or disprove it; there is no reason to believe they do. On the other hand, while there is no particular reason to believe that they do not exist, either; science is encroaching ever more and more and explaining ever more phenomena that are somewhere on the border between brain and mind, making it ever more likely that they do not exist (and if they do, then not in our minds at least; or to such a minuscule amount that it plays no role in everyday life).
On a tangent, the closest you can get to answer the question about free will and how the mind works is, as far as I'm concerned, not scientific, but simply self-reflection (in the form of meditation, e.g. Vipassana or similar purely observing, "insight" based meditations). You can, if you wish, view this as a kind of science, if you view yourself as the scientist; your own mind as the field of study; and yourself as the complete audience; and with unfortunately no way to extrapolate from your findings to something with relevance to other minds (i.e., no matter how long you meditate, you still have no way to solve e.g. the brain-in-a-vat question, or the living-in-a-simulation question; and whatever you personally tell me about your findings while meditating is nothing I can take for a fact about my mind, nor do I have any way to prove or disprove what you tell me about yourself).
On a tangent's tangent: while I personally find some of the science (i.e. that which you quoted about a limb moving before the "spark" has been measured in the brain) fascinating, I personally judge some of the purely philosophical arguments about the (non-)existence of Free Will much much more powerful. For example: one way - maybe the only way - to prove the existence of Free Will would be to "rewind the universe" to a point in the past, and check whether a person decided something different this time around. If a phenomenon requires such an absurd "proof", I strongly favor Occam's Razor.
Heck, and even if we were able to perform that experiment, and a person did indeed decide something differently, how would we ever know that it was not just random chance (i.e., physical quantum-based effects), which would be a very poor form of "Free Will" indeed.
Secondly, from personal experience, both while meditating and just while being mindful of what I do myself, I very much have the feeling that I would have decided any decision I had ever made exactly the same, if the universe were indeed rolled back. I personally would not be asking for science to disprove Free Will, but would challenge it to prove it (with my assumption that there will not be such a thing, ever); the same as I would not expect science to disprove God, but would rather challenge believers to prove it.
Let's consider the mind as an object that we can decribe correctly (i.e. correct enough to satisfy a set standard) in two alternative ways: material and mental. To every mental state there are some material states that correspond to (not cause) the mental state. You can study the brain from either perspective or from both. Both perspectives are valid.
In physics, matter has a double-nature of being waves and particles at the same time. In some contexts it is more convenient (i.e. it is more easy to grasp, analyse and understand the system) from the particle perspective, in other contexts from the wave perspective. Likewise, with the brain.
Note that I am not intending by this allegory that particles=material and waves=mental. It's just that the double-nature of the brain and of matter are on par when it comes to understanding; they both seem to break the laws of logic and common sense: How can a particle also be a wave? How can a thought also be a material process? Well, that's just how it seems. We are the universe looking at itself. How could that not lead to paradoxes?
How do you know that conscience only happens after the movement (whether it be of an arm or even of a neuron) when you only ever measure the movement? Do you see the problem? "Scientific observation" can't determine that the mental event is an effect rather than the cause because by design it only investigates mind-independent events. Moreover, if the human mind has an immaterial cause behind its choices, you can't observe this cause through material investigation. Its immateriality can only be demonstrated rationally, if at all.
The papers here and here purport to demonstrate it, and the proof can be summarized as follows. Material processes are all indeterminate, and at least some mental processes are determinate. Therefore, the mind can't be reduced to the material, and must have some kind of immateriality behind its determinacy. For an example of my own, consider this emoji, which should be material enough: 🔺. Upon seeing it you can take it to be a triangle, or a strongly seasoned dorito, or just a bunch of dots on a screen, or an irregular polygon with way more sides than 3 if you scale it up enough that the individual pixels become visible squares, etc., thus showing the inherent indeterminacy of the emoji itself. But either one of these concepts will be determinate themselves. If you choose to think specifically about the concept of triangle then you'll definitely be thinking about triangularity, not about polygons with more than 3 sides or clusters of dots, since they are mutually exclusive. This thought can't be reduced to or be strictly caused by the image of the emoji because 1) the same image could just as well have given rise to the other concepts since they all also properly describe the image, 2) the same concept encapsulates an infinity of particular triangles besides the one in the emoji, and 3) the image and even the stimuli it causes in your body may change ever so slightly, e.g. by irregularities in neuronal activity and environmental lighting, while the concept does not. Among other reasons. So the cause of your choice must be somehow over and above the material phenomena surrounding it, which also means it can't be directly observed by outsiders which only have access to the material phenomena.
As I see things, everything in the universe is physical, because I am defining "physical" to have that meaning.
From my point of view, this even includes the possible existence of things that are typically labeled "supernatural".
There are the aspects of physics that we already know about, the aspects of physics that we are going to learn about in the future, and quite likely there are aspects of physics (as I define it) that we will never learn about. (The last category could be because the human mind never gets that far in our study of physics, or because some aspects of physics are simply unknowable.)
The nitty-gritty — the details — of how we humans go about making a decision is obscure to us, apparently hidden in our subconscious mind. The fact that we do not (and probably cannot) know such details is no grounds for assuming that we have free will.
As always, the mere lack of knowledge of a thing's existence is no grounds for concluding the lack of existence of the thing.
I take it that physicalism is true. The idea of psychologism being true leads me to think that all things can be reduced to being physical, thus making physicalism being true if somehow everything was of a mental substance ("mind of God").
I would answer this with a counter-question. Hear me out:
Do physical causes exist?
What does cause mean? If Ball A hits Ball B, and we say "A causes B to move", what we mean is, the impact of ball A happened first.
The phrasing of the question ("time perspective", etc) implies that sequence of events and a time, is important to this definition of cause.
But time is a component of the physical universe. Physicists measure it, and will tell you how gravity changes it.
So a non-physical cause must be something that excludes time, otherwise it would be physical.
If a non-physical cause excludes time, that implies it is something physics can't measure; physics being the study of the physical. It also implies this type of cause is of the eternal, abstract variety.
For me, this shifts it around. There are no physical causes, only physical effects. Ball B is the effect of ball A, which was the effect of something before, etc, etc ...
I think the fundamental issue with Cartesian dualism lies in defining mental events, the human mental life, as "non-physical". From that unnecessary assumption follows all sorts of difficulties, such as the one highlighted in the OP: how come a non-physical "ghost" have physical effects?
The human mind is evidently causal. I can ask my wife to do something for me, and if she is so disposed she will do it. The example shows that symbolic information, an exchange of sentence such as "Honey did you see my glasses? - They are on the kitchen table" can result in a physical outcome. Thoughts can be shared, people can be persuaded. Thoughts are causal. Otherwise, there's neither a need nor a possibility for philosophy.
Therefore I believe the mind is physical, in some yet unknown manner of "being physical". Or perhaps we already know. My money is on brainwaves.
Not in any useful sense. Suppose you observe some physical effect, and posit that it is caused by some non-physical cause. By nature of the cause being non-physical there is absolutely nothing you can measure or observe to indicate that the cause actually happened. I can also posit that the physical effect is due to some entirely different non-physical cause, and there is also no evidence I can produce to indicate my cause actually happened. We're each stuck claiming a different non-physical cause for the physical effect - it's impossible that we're both right, yet it is also impossible to favor one explanation over another since there is no (and indeed there can be no) evidence supporting either one.
Basically, if you claim that a physical effect has a particular non-physical cause, I can make an equally compelling claim that the effect was due to any other non-physical cause. If you claim your arm went up because you non-physically willed it, I can claim your arm went up because I non-physically willed it. Any argument that says that's not possible is going to come back to physicality of causes and effects.
Non-physical causes could conceivably be linked to any and all effects and can't be supported or disproved one way or another, so it seems like a rather useless concept. Observing physical event B and claiming "A caused B" while having no evidence whatsoever of A is rather pointless, and the fact that all other unsupported explanations X, Y, and Z are equally compelling only further shows that non-physical "causes" don't really explain anything at all.
