One should bear in mind that Searle writes a great deal about issues such as the relation between mind and brain, and the limits of artificial intelligence, and what these limits might tell us about that relation. He is not so much concerned with what is computable in completely abstract terms as with whether we can think of the brain as a computer, and if so, whether this has consequences for our understanding of minds. The issue of interpreting the significance of the Church-Turing thesis in this context is considerably more subtle than most accounts allow.
Firstly, Turing's use of 'computable' really answers to the concept of 'effectively computable'. The distinction may seem a fine one, but it potentially masks a misinterpretation of the Turing thesis. Copeland has a useful discussion of this point in section 2 of the SEP article on the Church-Turing thesis, and there are a number of references there to follow up.
Secondly, it is appropriate to impose a physical restriction on what counts as a computer. If we do not do so, there are any number of ways of specifying a hypercomputer, i.e. a computer that transcends the capabilities of a Turing machine. An analog neural network, for example, is capable of being a hypercomputer, but it would need to operate with arbitrarily fine precision. In our quantum universe, no such computer exists. So the definition of a computer has to be about physics and not just mathematics. Again, there are a number of useful references in the SEP article and in the Wikipedia article about hypercomputation.
Thirdly, it is problematic to explain how the Turing thesis applies to an interactive computer. A Turing machine is conventionally envisaged as a black box device, with an indefinitely long tape and a machine head that moves up and down reading and writing numbers onto the tape. Input to the machine takes the form of the initial state of the tape and its output is the final state. If an external agent writes onto the tape while a computation is in progress, then this violates the specification of a Turing machine. But this is in effect exactly what happens when a computer interacts with its surroundings. This has led some theorists, such as Peter Wegner and Dina Goldin, to claim that what they refer to as the strong Church-Turing thesis is false: it does not apply to interactive computers. (For example, see Goldin D., Wegner P. (2005) The Church-Turing Thesis: Breaking the Myth. In: Cooper S.B., Löwe B., Torenvliet L. (eds) New Computational Paradigms. CiE 2005. Lecture Notes in Computer Science, vol 3526. Springer, Berlin, Heidelberg.) Wegner and Goldin are not entirely 'mainstream' in their views, and for my part, I would be unwilling to say that the thesis is false, but we must be careful about interpreting its significance. At any given instant in time we could take all the interactive input that a computer has received, encode it in some way and include it on the input tape of some Turing machine. But an instant later, when the computer has received some more interactive input, we would have a different specification. The interactive computer is not a single Turing machine but a long sequence of Turing machines that evolves over time in accordance with the input it receives from whatever it interacts with. Without a complete specification of the portion of the universe that the computer is capable of interacting with, we cannot specify what future Turing machine it will evolve into. This suggests that the concept of computable functions has relatively little to tell us about the capabilities and limitations of artificial intelligences, or for that matter human beings.
