Minds and Computers : An Introduction to the Philosophy of Artificial Intelligence

Exercise 9.2
The second example above codes the solution to Exercise 8.4.
Code up the solutions to the other exercises in Chapter 8.
Although we are leaving program codes in exponential form, they
resolve to natural numbers. Admittedly their resolutions are very
large natural numbers, but they are natural numbers nonetheless. As
such, we now have a unique and e
ffective procedure for coding any
register machine program of any length as a single natural number
which is uniquely and e
ffectively decodable. This means we now have
a mechanism for referring to programs within programs, since a reg-
ister can hold the code of a program.
Exercise 9.3
Give three examples of numbers which are, for di
fferent
reasons, not program codes.
Exercise 9.4 (Challenge)
What is the smallest natural number which is a program code?
9.3 A UNIVERSAL MACHINE
Armed with the method of Gödel coding, we are now ready to
describe the register machine program which can compute any algo-
rithmic function.
Let [UM] be the register machine whose program is described by
the following procedure.
First, decode the contents of R
1
. If R
1
does not contain the code
of a program then halt. If R
1
does contain the code of some program
P, then add 1 to all the registers referred to in P, then run P and on
termination copy the contents of R
2 
to R
1
.
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It is clear that this procedure is e
ffective, hence, by the Church/
Turing thesis, there is such a register machine program.
If we were to run [UM] with the input #P, a
1
, . . . , a
n
in the first n
+ 1 registers, it would deliver the same result as running a machine
with program P and the input a
1
, . . . , a
n
in the first n registers.
This is why the procedure for [UM] involves adding one to all the
registers in P (if R
1
decodes to #P) and why we copy R
2
to R
1
on ter-
mination of P – because the inputs of P will all be displaced along one
register in [UM].
Consequently [UM] can emulate any register machine program.
We simply put the code of that program in the first register and the n
inputs of the function in the subsequent n registers and run [UM] as
described.
The register machine [UM] is known as a universal machine – it can
compute any register machine computable function. More simply,
[UM] is a computer.
Modern digital computers, as we know them, are instantiated uni-
versal machines. Personal computers, mainframes and even super-
computers are no more powerful than [UM] – there is nothing that
they can compute that [UM] cannot. In fact, [UM] is more powerful
than any physical machine since it is a theoretical idealisation which
is unconstrained by physical manifestation. We will have more to say
about this in the following chapter.
We have now completed our survey of computational theory.
Equipped with our new understanding of what computers are, it is
time to return to philosophical material and discuss the theory of
mind which our central concern is to evaluate – computationalism.
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C H A P T E R 1 0
COMPUTATIONALISM
So far, we have considered the question of what minds might be and
in doing so, have examined a number of philosophical theories which
aim to provide an answer to this question. We suspended that discus-
sion after introducing functionalism and turned our e
fforts to devel-
oping a rigorous account of what computation is. It is now time to
pick up where we left o
ff.
In this chapter, we are going to use the understanding of compu-
tational theory we built up in the previous section to flesh out the
functionalist framework in a particular way.
There are various ways in which one can be a functionalist, depend-
ing on how one analyses the functions of mental states. Our aim is to
give a fair and precise treatment of the theory that these functions are
to be fleshed out in computational terms.
In doing so, the utility of having developed an account of compu-
tational theory will become apparent. In the first instance, we can now
speak of computation without simply engaging in loose talk – we
have a precise formal definition and some subtle distinctions at our
disposal. This allows us to see that certain objections sometimes
raised against computationalism do not actually target the theory –
they target straw men by virtue of an insu
fficiently sophisticated
understanding of what computers are.
To target a straw man – or to commit a straw man fallacy – is to
characterise an opposing position as being weaker than it actu-
ally is and to then argue against the weaker position. Arguing
against a weaker misconception does no work at all against the
actual opposing position. Hence, if one commits this fallacy, one
is said to build a straw man simply for the purposes of knocking it
down.
We begin with a clear characterisation of computationalism.
Following this, our first priority is to address, and clear up, some
possible misconceptions of the theory. We are then going to discuss
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precisely what a computationalist is committed to and consider some
immediate implications.
In this chapter, we will also discuss some of the merits of compu-
tationalism and deal with some prima facie objections to the theory.
Evaluating computationalism more fully will be the concern of the
remainder of the book. We are going to see how disparate material
from the cognitive disciplines bears importantly on the tenability of
the theory.
Let us now address the question of precisely what is involved in
claiming that mental states are computational states.
10.1 WHAT COMPUTATIONALISM ISN’T
Computationalism is the view that to have a mind is to instantiate a