The advent of quantum biology, light harvesting molecules, bird navigation, perhaps smell, suggests that sticking to classical physics in biology may turn out to be simply stubborn. Now Turing machines are discrete state, (0,1), discrete time T T+1, subsets of classical physics and define algorithmic. We all know they, like Shannon information, are merely syntactic. Wonderful mathematical results such as Chaitin’s Omega, the probability a program will halt which is totally non-computable and non-algorithmic tell us the human mind, as Penrose also argued, cannot be merely algorithmic.
Mathematics is creative. So is the human mind. We understand metaphors, "Tomorrow and tomorrow and tomorrow creep at this petty pace ..." but metaphors are not even true or false. All art is metaphoric, language started gestural or metaphoric, we live by these, not merely by true false propositions and the syllogisms they enable. No prestated set of propositions can exhaust the meanings of a metaphor and if mathematics requires propositions, no mathematics can prove that no prestated set of propositions can exhaust the meanings of a metaphor. Thus the human mind, in Pierce’s "abduction", not induction or deduction, is wildly creative in unprestatable ways.
The causal closure of classical physics precludes more than an epiphenomenal mind that cannot "act" on the world, be it a Turing machine or billiard balls, or classical physics neurons. The current state of the brain suffices to determine the next state of the brain (or computer) so there is nothing for mind to do and no way for mind to do it! We’ve be frozen in this stalemate since Newton defeated Descartes’ Res cogitans.
Ontologially, free choice requires that the present could have been different, a counterfactual claim impossible in classical physics, but easy if quantum measurement is real and indeterminate: the electron could have been measured to be spin up or measured to be spin down, so the present could have been different.
A quantum mind however, seems to obviate responsible free will. False, for given N entangled particles, the measurement of each alters the probabilities, by the Born rule, of the outcomes of the next measurements. In one extreme these may vary from 100% spin up on the first to 100% spin down on the second and so on for N measurements, entirely non random and free if measurement is ontologically indeterminate If probabilities of N entangled particles vary between less than 100% and 0% we get choice and an argument suggest we can get responsible choice in the "Strong Free Will Theorem" of Conway and Kochen.
We will never get to the subjective pole from third person descriptions. But a single rod can absorb a single photon so it is conceivable to test if human consciousness can be sufficient for quantum measurement. If we were so persuaded, and if the classical world is at base quantum then the easy hypothesis is that quantum variables consciously measure and choose, as Penrose and Hameroff in "Orch Or" theory and others suggest. We may live in a wildly participatory universe, consciousness and will may be part of its furniture, and Turing machines cannot, as subsets of classical physics and merely syntactic, make choices where the present could have been different.