h-Turing thesis is just silent about deeply important features of computations. It says what set of problems can be computed and it says this by reference to two models of computation. But the thesis doesn’t say what it is about the different models of computation that makes them all models of computation. It doesn’t say what makes two problemsolving methods ‘the same’ computation or ‘different’ computations. So it says nothing about what a computation is. (Turing did try to give an account of computation, but it is debatable how successful it is. The American mathematician Richard Shore worried that addressing these issues might be pie in the sky.) Many roads lead to Rome. The Church-Turing thesis told us the location of Rome, but not what makes something a road.
We haven’t even touched the main sources of difficulty. The mathematical theory of computation is just that: mathematics. But the brain, like your PC, is a concrete thing. It’s a piece of nature. The stuff of mathematics doesn’t break down or decay. Numbers don’t chip at the edges; 1 + 2 will never take on a different value because moss started growing over the number 2 or because the operation of addition got jammed. Even if we had a precise, adequate and agreed-upon mathematical definition of what computation is (and we don’t), this wouldn’t tell us what it means for a part of the natural world – the sort of thing that can rust or sneeze – to perform a computation. One problem is to understand the nature of computation. Another is to understand the nature of computation in nature.
In fact, we often say that something in nature performs a particular computation even when that thing doesn’t always give out the answers that the computation, by definition, is supposed to give out. My laptop runs the latest version of my word processor even though it will sometimes stall on account of the 732 tabs that I have open on my web browser. A Lego set can compute addition even if some of the pieces get gunk in them, preventing them from moving as they should. Likewise, your brain can compute the direction from which a sound arises according to a particular trigonometric rule, even if it sometimes makes errors that are incompatible with that rule. We expect these errors in anyone, and we find greater errors among those of us who are hearing-impaired. But why think that most people’s auditory systems are solving the same trigonometric problems in error-prone ways rather than solving their own personally tailored problems in an error-free way?
It isn’t as arbitrary as it seems. The reasonable thing to say is that something performs a computation if it would normally perform that computation. Abnormal and non-ideal circumstances – having a head cold while walking through an organ-grinding festival – can lead to errors. This causes surprisingly few difficulties in cognitive science. Cognitive scientists are able to identify, in testable and motivated ways, which behaviours are the result of abnormal or interfering factors. But even though we have a stable, usable empirical concept of what computation a system ‘would normally perform’, it is wildly unsettled what it actually is for something to ‘normally perform’ one computation rather than another.
That’s just the beginning of our problems. In saying which problems could be solved by rule-bound calculations, Church and Turing defined models in which a set of symbols expressing the problem (the line ‘21 x 3 = ?’ in your notebook) is transformed into a set of symbols expressing the solution (‘63’). But the very same models could be used to convert meaningless gibberish (‘j&^@’) into other meaningless gibberish (‘•¡ø∂’). Would those transformations then count as computations? Turning back to the natural world: does it really matter that the firing of a neuron means that a sound came from thataway, or does it just matter that when this neuron here starts firing (whatever news or information that neuron might have about the world), it triggers those neurons over there to start firing?
If our brains are computers, our mental lives m