The Significance of Alan Turing

All physical systems can be modeled and simulated by the actions of a logical structure known as the universal Turing Machine.  If we can demonstrate any functions and activities of the mind which are beyond the capabilities of a Turing Machine then we have produced strong evidence that the mind is a non-physical entity. 
 
The challenge of materialism
The main intellectual challenge to Buddhism nowadays is materialism. There are other challenges, but they could hardly be described as intellectual.   Materialism comes in various flavors, but its bottom line is the claim that the mind is nothing more than the physical activities of the mechanism of the brain. This, of course, invalidates Buddhism and indeed all other spiritual paths. In this view, spirituality, ethics, art etc are just by-products of purely mechanistic processes.

So Buddhism, being a philosophically justified system rather than one based on blind faith, needs to refute materialism by logical argument. This is where Alan Turing comes in.  Although Turing is nowadays best known as a computer pioneer, code-cracker and victim of the British establishment's vicious homophobia, he first attained prominence as a philosopher of mathematics.

Negating the mechanistic model of mind
One of the methods used in Buddhist philosophy to refute erroneous views is to identify the 'Object of Negation'  This approach consists of  obtaining a precise definition of the assertion which is to be refuted, and then demolishing it by analyzing its contradictions and inadequacies. Such an object of negation is provided by the Turing Machine as a model for the mind.

The Turing Machine (TM) is a logical/mathematical structure, a kind of thought-experiment, which doesn't necessarily need to implemented as an actual physical machine to be of use for philosophy.

The first advantage of using the Turing Machine for philosophical discussion is its great precision and clear definition. We find it difficult to refute 'materialism' and its more modern variant 'physicalism', not because of the strengths of the arguments for them, but because of their fuzziness and incoherence. The definitions of matter and physics are, when examined in detail, surprisingly vague and imprecise.  On the other hand, the Turing Machine gives a precise definition of 'mechanism', in the philosophical sense of the fundamental basis of physical actions.

The second advantage of using the TM model is the all-encompassing generality of the Turing machine as the model for all physical systems.   It is the mother of all mechanisms, the archetypal computer and the basic method of implementing all algorithms. All the apps on your phone and tablet are Turing Machines. All physical systems may be simulated by appropriately programmed Turing Machines.

However, when we examine the components of the TM in detail, we find that none of them...

(i) are capable of holding meaning. They do not possess any semantic capabilities, or 'intentionality' to use the technical term.

(ii) are capable of registering qualitative states ('qualia') such as sensations of greenness,  pleasure, pain etc.

And when we check the basic repertoire of operations that the TM can carry out (which corresponds to the instruction-set of a general purpose computer) we fail to find any combination of these operations that can operate on the components to produce semantic or qualitative phenomena.  In Buddhist terminology the Turing Machine is all rupa and no nama.   It cannot produce two of the main aspects of sentience.

Conclusion
The archetypal physical mechanism of the Turing Machine cannot of itself produce mental phenomena.  If we want to completely explore the mind, we need to look elsewhere in addition to the physical, mechanistic structure of the brain.