(Free Will is that thing we all have which makes us do the things we do. Yet science attempts to prove its non-existence ...)
The traditional argument about Free Will is "Free Will versus Determinism".
In other words, do we have free will, or is all our behaviour and thinking determined by something?
But Free Will is a subjective psychological concept, and also a moral concept, whereas Determinism is a physical, mathematical concept.
If we accept that Free Will and Determinism are opposites, then we are accepting that physics and mathematics have something to say about psychology and morality.
In particular:
If the laws of physics are deterministic, then Free Will doesn't exist at all.
Oddly enough, the physical interpretation of Free Will versus Determinism does not come from philosophical efforts of physicists; rather it comes from the efforts of theologians to understand the nature of God and God's relationship to human morality.
The idea is that God knows everything that can be known, so if our actions are predictable, even in principle, and God punishes us for our actions, then God is punishing us for something that He knew we were going to do anyway. Which seems a bit unfair.
"Free Will" is the thing that God supposedly gave us so that not even He could predict our behaviour.
We could investigate how physics may allow for non-determinism, but this leads to silliness such as quantum psychology.
A better approach is to realise that Free Will is a mystery, so we don't really know what it is, and we shouldn't assume that it is contrary to some particular physically defined concept. Such as Determinism.
We should investigate "Free Will" on its own terms, without narrowly defining it as being the opposite of something else.
Our personal experience of Free Will is that we observe an agency within us, which is responsible for our actions, and which may act against various emotional forces, such as desire or fear.
And from a scientific point of view, the primary fact about the human brain/mind is that it is an information processing system. The specific physics of the system is secondary.
The real "problem" of Free Will is not to prove or disprove its existence, but to equate our subjective and moral notion of Free Will with a corresponding concept which is part of a theory of the human brain as an information processing system.
