ain computes. Some have dug in their heels, confident that cognitive scientists do know just what they are saying when they say that brains are computers: the claim is in fact a concrete, well-understood scientific hypothesis.
Unlike metaphors, whose value is independent of their literal falsity, scientific hypotheses are valuable insofar as they are (likely close to being) true and offer concrete, insightful explanations of phenomena. And, unlike metaphors, ideally one should be able to state precisely and with some completeness what a scientific hypothesis says about the world, and what experimental observations it predicts (give or take some background assumptions).
But the suggestion that we have on hand a precise and well-understood claim when we say that the brain is a computer feels forced. There really isn’t much clarity or agreement at all on what it means to say that the brain is a computer, even among those who agree that the claim is meant literally.
There is a whole mathematical field dedicated to understanding computation. Yet you might be surprised about how little clarity there is about what computation is. Foundational to the field is the Church-Turing thesis, named after the American philosopher-logician Alonzo Church (1903-95) and the English logician-mathematician Alan Turing (1912-54). Both Church and Turing were working to figure out what sorts of mathematical problems can and cannot be solved by blindly following rules of the sort you learn in order to do long division, long multiplication, or to solve quadratic formulas. In other words: what mathematical problems can be solved by rule-based calculation or, as it was called back then, computation? To answer this question, both Church and Turing developed, around the same time in the 1930s, mathematical models to capture what happens when one calculates the solution to a problem.
Church’s model of computation, the ‘lambda calculus’, emphasises giving explicit definitions of functions in terms of a small stock of very basic functions. To give a rough idea, the function 3 + (14 – x) can be defined as a sum, 3 + y, where y itself is defined as a subtraction, 14 – x. To determine what 3 + (14 – x) equals, when x = 6, one first solves for 14 – 6, and then adds that solution to 3, giving 11. The idea is to break up the problem into its smallest chunks, solving it chunk by chunk. Church proposed that a mathematical problem is computable just in case it can be expressed within his sparse language for defining functions in terms of more basic functions. For him, problemsolving involves giving an explicit definition of the problem, and then following that definition to its conclusion.
Turing’s model of computation emphasises rote rule-following. Remember how, when doing long division or solving for x, you would write one line and then, following a rule you were taught, you would write down another line underneath it that was based on the previous line, and so on until the last line you wrote gave the solution? Sometimes we felt entitled to skip a step or two, but the teacher would insist that you must ‘Show your work!’ Turing took this to the extreme. He defined a system of rules or constraints for writing one line based on a previous line, in which you could see all the work down to the last, minute detail, so that your teacher could not credibly ask you to show any more.
Suppose you start with a line of symbols expressing the problem to be solved:
… 21 x 3 = ? …
and you need to get to the final line expressing the solution:
… 63 …
Imagine you are writing everything down on graph paper, with no more than one item or symbol per square. You have an instruction manual telling you what to do at each step of the computation. You always start at a particular line, in a particular square on your notebook, and you handle only one square per step. Steps in the instruction manual say that, if the current square with which you are dealing contains such-and-such, then either erase that item from the square, leaving a blank spot in its place:
… 21 x 3 = [?] …
… 21 x 3 = 