Transcript The Computational Theory of Mind

The Computational Theory of
Mind
COMPUTATION
Functions
Examples of Functions
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f(x) = x2
Mother of x
x’s definition in the Oxford English Dictionary
Your password for website x
It is not true that x
g(x, y) = x2 + y – 4
y’s password for website x
x and y
Functions
A function is any relation between inputs and
outputs where: for each distinct input there is
only one output.
Algorithms
Algorithms
An algorithm is an effective procedure for
calculating a function.
You can think of it as a list of steps where: if you
follow the steps correctly, you will always get
the right answer.
Change-Giving Algorithm
1. Take the largest coin of n cents where n ≤ the
amount owed.
2. Reduce the amount owed by n cents.
3. If the amount owed is 0 cents, return all
coins taken and stop.
4. Go back to State (line) 1.
Computation
Computation is the concrete use of an algorithm
(program) to find the output of a function given
its inputs. It requires:
1. A representation of the inputs.
2. Basic means of manipulating its
representations.
3. A set of instructions that use the basic means
of manipulating to run the algorithm.
Abacus Computer
Mechanical Computers
Abacuses are nice, but they’re prone to human
error. For computation to work, all the steps of
the algorithm need to be followed exactly. What
we want is a mechanical computer, one where
physics performs the computations.
https://www.youtube.com/watch?v=GcDshWm
hF4A
LOGICAL COMPUTATION
Truth Functions
A special subset of functions is the truthfunctions. These are functions whose input is
truth-values (true or false) and whose outputs
are truth-values:
• Not P
• P and Q
• P or Q
• If P, then Q
Truth Functions
Arguments
An argument (in the philosophical sense) is a
pair: a set of propositions, called the “premises,”
and another proposition, called the
“conclusion.”
Proofs
In logic, we prove conclusions from their
premises using basic rules of inference like
modus ponens (“E”).
Arrow Elimination: →E
The →E rule says that
if on one line we have a conditional (φ → ψ)
and on another line we have the antecedent of
the conditional φ
then on any future line, we may write down the
consequent of the conditional ψ
depending on everything (φ → ψ) and φ
depended on.
(P → Q), (Q → R)├ (P → R)
1
1. (P → Q)
2
2. (Q → R)
3
3. P
1,3 4. Q
1,2,3 5. R
1,2 6. (P → R)
A
A
A (for →I)
1,3 →E
2,4 →E
3,5 →I
Proofs
A proof is a type of program that computes
conclusions from their premises:
1. A representation of the premises.
2. Basic means of manipulating its
representations.
3. A set of instructions that leads one to a
representation of the conclusion.
Validity
An argument is valid := If the premises are all
true, then the conclusion must be true.
Equivalently: It is impossible for the premises to
be true and the conclusion to be false.
Soundness
Importantly, classical logic is provably sound.
This means that it is truth-preserving: no proof
leads from true premises to a false conclusion.
Every argument that can be proven is a valid
argument.
Automatic Reasoner
But can we use the laws of physics to build an
automatic reasoner, as we did with the marbles
and addition? Yes!
Logic Gates
Mechanical and Digital Computers
[WATCH VIDEO]
In modern day digital computers, the physics
isn’t gravity, but instead electromagnetism:
computer chips are built with transistors.
However, the basic principle is still the same.
Automatic Reasoners
This is important!
We’ve created things that can use logic and
reason on their own.
UNIVERSAL COMPUTERS
Programs as Data
The inputs and outputs of programs are the data
that it manipulates.
But programs themselves can be data too: I
could have a program that took as inputs two
other programs P1 and P2, and two numbers, n
and m, and then returned “P1” if P1(n, m) was
higher than P2(n, m) and “P2” otherwise.
Universal Computers
Here’s a question then:
Is there a program P that can take any other
program P*, plus the inputs to P*, and then tell
you what P* would do with those inputs?
P would be a “universal simulator,” able to run
any program you gave it as data.
Universal Computers
In 1936, Alan Turing
proved that there was
such a program, and that
you could in principle
build a computer that ran
it: a universal computer.
Universal Computers
Nowadays, many people
carry around universal
computers in their
pockets.
Writing Software
When you write code for a computer, you don’t
write 0’s and 1’s. That’s because it doesn’t run
your code: it simulates it. The programs it runs
are “machine language” programs that don’t
look anything like C++.
Non-Universal Computers
Most computers we use
aren’t universal
computers.
A cash register computes
the sums of the items
purchased. But you can’t
play Angry Birds on it.
Read-Write Memory
In order to be a universal computer, you must
have a read-write memory: a memory that
allows you to store a symbol and then to
retrieve it.
This isn’t all there is to a universal computer, but
it is a necessary condition for being one: no
finite state machine is a universal computer.
THE COMPUTATIONAL THEORY OF
MIND
The Computational Theory of Mind
The computational theory of mind says that the
brain is a universal computer and that the mind
is the program that it runs.
It is a version of functionalism, since what makes
something a computer is not what it’s made out
of (transistors, dominoes, Legos, brain cells) but
instead it’s the relations of its states.
Some Evidence
In his 1957 book Syntactic Structures, Noam
Chomsky proved that certain actual human
languages are unlearnable unless the human
mind has the architecture of a universal
computer.
Similar Evidence
The fact that I can work out what different
computer programs will do with different inputs
seems to suggest that I am a universal computer.
Mental States are Multiply Realizable
There’s already plenty of reason to believe in
functionalism, and CTM is just a type of
functionalism that is more detailed and explains
more things (e.g. rationality).
Mental Processes are Rational
processes are reason-respecting. Many of your
mental states cause other mental states, and do
so in a way that if the causing states represent
something that is true, then the caused state
represents something that is also true.
Logical Relations
From:
1. If Joe fails the final exam, he will fail the course.
2. If Joe fails the course, he will not graduate.
It follows logically that:
3. If Joe fails the final exam, he will not graduate.
Logical Relations
If you believe:
1. If Joe fails the final exam, he will fail the course.
2. If Joe fails the course, he will not graduate.
These beliefs can cause you to also believe:
3. If Joe fails the final exam, he will not graduate.
Mental Processes are Rational
Computers are the only things (besides minds)
that we have so far discovered that are reasonrespecting in this way.
This gives us some reason to think that maybe
minds are in fact computers.
No Computation without
Representation
As we’ve seen, to be a computer requires that
one be able to represent and manipulate
representations of the inputs and outputs to
functions.
This means that IF the brain is a computer, and
the mind is its software, THEN the mind has
representational states.
The Language of Thought
If the mind has representational states, then
there is some format the representations are in.
One idea is that the format is a language that is
a lot like a computer language for an electronic
computer or a natural, spoken human language:
the language of thought (sometimes:
“Mentalese”).
The Necker Cube
The Language of Thought
The idea would be that when you think “dogs
hate cats,” there are discrete ‘words’ of the
language of thought, DOGS, HATE, CATS. These
are your ideas. The thought is a ‘sentence’ that
is made out of those ideas:
DOGS HATE CATS
Systematicity
You can use those same ideas in different
combinations:
CATS HATE DOGS
The LOT hypothesis thus predicts mental
systematicity: that people who can think that
cats hate dogs can think that dogs hate cats.