In a

previous thread , Ken Aizawa suggests that I'm insufficiently pluralistic about computation in cognitive science and to substantiate his criticism he points to his forthcoming article "Computation in Cognitive Systems; It's not al about Turing-Equivalent Computation" (available on

his website ).

Having read Ken's nice paper, I only have time for a few quick comments.

1. Ken correctly points out that there are several notions of computation. (I make the same point in a

paper that he refers to.)

2. Ken correctly points out that many people including myself think there is something special and theoretically deep about what he calls Turing-equivalent computation, by which he seems to mean the kind of computation that can be performed by Turing machines (computation of Turing-computable functions). They're right, because in fact this is the core theoretical notion of computation, with lots of deep results about it.

3. Ken correctly points out that the notion of Turing-computable functions can be generalized to study functions of natural numbers (or equivalently of strings of letters from a finite alphabet) that are not computable by Turing machines. This enterprise was started by Turing himself and is a large branch of computability theory. (Anyone who takes a nontrivial course in computability theory knows this.) But contrary to what Ken seems to suggest, the study of functions that are uncomputable by Turing machines is not based on a different notion of computation from that of Turing machines--it's the very same notion; in fact, the whole subject matter is defined in terms of functions that are like those computable by Turing machines but cannot be computed by Turing machines.

4. Ken persuasively argues that Turing machines and the related notion of computabiltiy probably played only a minor role in McCulloch's thinking at the time he wrote his 1943 paper with Pitts. But Ken seems to underestimate the theoretical significance of computation-theoretic results in characterizing the power of McCulloch-Pitts nets and other neural networks. (The latter obviously is not discussed in the 1943 paper.) Ken also seems to underestimate the important role that the connection between McCulloch-Pitts nets and Turing-computability played in the history of cybernetics and cognitive science. For the beginning of an account of that history, based on extensive archival research, see Chapters 5 and 6 of

my Ph.D. dissertation .

5. Ken asserts that the notion of "computed vs. uncomputed cortical maps" deployed by some neuroscientists "is not a Turing--equivalent form of computation" (p. 17). But I didn't notice anything in Ken's paper that determines what relationship there is or isn't between the notion of computation deployed in this area of neuroscience and Turing-computabilitiy, so I don't know why Ken is so confident in his assertion.

6. Ken also points out that often cognitive scientists talk about computation as symbol manipulation or digital symbol manipulation, without mentioning the kind of "finitude constraints" that are important to Turing-computability. This is true but doesn't mean that the finiteness constraints are not implicitly assumed to be in place (except by people like Jack Copeland); after all the brain has only a finite number of neurons etc.

7. Ken's pluralism seems to be based on something like the following argument: if scientist A uses "computation" in pursuit of goal X and scientist B uses "computation" in pursuit of goal Y and X is different than Y, than scientists A and B use two different notions of computation. This is a fallacy. Maybe there are two different notions of computation, maybe they aren't. It takes a lot more than this to show that two notions of computation are the same or different. More generally, it takes a lot of theoretical work to compare and contrast different notions of computation and see how they relate to one another. That's why, contrary to what

Ken suggests , it's very helpful to have an umbrella notion of computation, of which other notions (including all those mentioned by Ken) are species.

8. In conclusion, reading Ken's paper convinced me that I have just

just the right kind of pluralism for sorting out competing claims about computation in cognitive science.

Having read Ken's nice paper, I only have time for a few quick comments.

1. Ken correctly points out that there are several notions of computation. (I make the same point in a paper that he refers to.)

2. Ken correctly points out that many people including myself think there is something special and theoretically deep about what he calls Turing-equivalent computation, by which he seems to mean the kind of computation that can be performed by Turing machines (computation of Turing-computable functions). They're right, because in fact this is the core theoretical notion of computation, with lots of deep results about it.

3. Ken correctly points out that the notion of Turing-computable functions can be generalized to study functions of natural numbers (or equivalently of strings of letters from a finite alphabet) that are not computable by Turing machines. This enterprise was started by Turing himself and is a large branch of computability theory. (Anyone who takes a nontrivial course in computability theory knows this.) But contrary to what Ken seems to suggest, the study of functions that are uncomputable by Turing machines is not based on a different notion of computation from that of Turing machines--it's the very same notion; in fact, the whole subject matter is defined in terms of functions that are like those computable by Turing machines but cannot be computed by Turing machines.

4. Ken persuasively argues that Turing machines and the related notion of computabiltiy probably played only a minor role in McCulloch's thinking at the time he wrote his 1943 paper with Pitts. But Ken seems to underestimate the theoretical significance of computation-theoretic results in characterizing the power of McCulloch-Pitts nets and other neural networks. (The latter obviously is not discussed in the 1943 paper.) Ken also seems to underestimate the important role that the connection between McCulloch-Pitts nets and Turing-computability played in the history of cybernetics and cognitive science. For the beginning of an account of that history, based on extensive archival research, see Chapters 5 and 6 of my Ph.D. dissertation .

5. Ken asserts that the notion of "computed vs. uncomputed cortical maps" deployed by some neuroscientists "is not a Turing--equivalent form of computation" (p. 17). But I didn't notice anything in Ken's paper that determines what relationship there is or isn't between the notion of computation deployed in this area of neuroscience and Turing-computabilitiy, so I don't know why Ken is so confident in his assertion.

6. Ken also points out that often cognitive scientists talk about computation as symbol manipulation or digital symbol manipulation, without mentioning the kind of "finitude constraints" that are important to Turing-computability. This is true but doesn't mean that the finiteness constraints are not implicitly assumed to be in place (except by people like Jack Copeland); after all the brain has only a finite number of neurons etc.

7. Ken's pluralism seems to be based on something like the following argument: if scientist A uses "computation" in pursuit of goal X and scientist B uses "computation" in pursuit of goal Y and X is different than Y, than scientists A and B use two different notions of computation. This is a fallacy. Maybe there are two different notions of computation, maybe they aren't. It takes a lot more than this to show that two notions of computation are the same or different. More generally, it takes a lot of theoretical work to compare and contrast different notions of computation and see how they relate to one another. That's why, contrary to what Ken suggests , it's very helpful to have an umbrella notion of computation, of which other notions (including all those mentioned by Ken) are species.

8. In conclusion, reading Ken's paper convinced me that I have just just the right kind of pluralism for sorting out competing claims about computation in cognitive science.