ll it which one. This system performs a computation that computes other computations. This convenient innovation is the seed of the modern, multipurpose computers we have today, on which we install and uninstall all sorts of programs. But there’s no reason that a computer has to be programmable in this way. Some jobs (including, maybe, estimating the direction of a sound from interaural time differences) require computing the solution to only one kind of problem. Church, Turing and others provided an exciting variety of ways to get solutions to the same set of problems.
The Church-Turing thesis tells us which problems are computable, but not what computation is. Church and Turing offered different models for solving problems. They got the same solutions to the same problems but by different means. That means that you don’t have to be a Turing Machine to compute. You don’t have to write down line after line, step after step, in just the way that Turing describes. You could go to Church instead. You could turn to any of the other models of computation that give the same solutions to the same problems but in different ways. What makes these all models of computation? And how do we understand the possibility that two different models could both capture the same computation – long division, say?
Even if we focus on just one model of computation, there will be many different ways to get the same solution to a problem. But if a solution is reached through wildly different steps, then intuitively different computations have been carried out. This matters tremendously for cognitive science, which cares not just about saying what computable problem the brain might be solving but also about what specific kind of procedure or computation the brain performs in solving that problem. Different computations can take different amounts of time, do things in different orders according to different causal chains, and break down in different ways (therefore calling for different medical treatments when treatment is needed).
The Church-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. Like