The Computational Theory of Mind - PowerPoint Presentation

Slide1

The Computational Theory of MindSlide2

computationSlide3

FunctionsSlide4

Examples of Functions

f(x) = x

2

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) = x

2

+ y – 4

y

’s password for website x

x

and ySlide5

Functions

A function is any relation between inputs and outputs where: for each distinct input there is only one output.Slide6

AlgorithmsSlide7

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.Slide8

Change-Giving Algorithm

Take

the largest coin of

n

cents where

n

≤ the amount owed.

Reduce

the amount owed by

n

cents.

If

the amount owed is 0 cents, return all coins taken and stop.

Go

back to State (line) 1.Slide9
Slide10

Computation

Computation is the concrete use of an algorithm (program) to find the output of a function given its inputs. It requires:

A representation of the inputs.

Basic means of manipulating its representations.

A set of instructions that use the basic means of manipulating to run the algorithm.Slide11

Abacus ComputerSlide12

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=GcDshWmhF4A

Slide13

Logical computationSlide14

Truth Functions

A special subset of functions is the truth-functions. 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 QSlide15

Truth FunctionsSlide16

Arguments

An argument (in the philosophical sense) is a pair: a set of propositions, called the “premises,” and another proposition, called the “conclusion.”Slide17

Proofs

In logic, we prove conclusions from their premises using basic rules of inference like modus ponens (“

E”).Slide18

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. Slide19

(P → Q), (Q → R)├

(P

→ R)

1 1. (P

→

Q) A

2

2. (Q

→

R)

A

3

3.

P

A (for →I)

1,3 4. Q 1,3 →E

1,2,3 5. R 2,4 →E

1,2 6. (P

→

R) 3,5 →ISlide20

Proofs

A proof is a type of program that computes conclusions from their premises:

A representation of the premises.

Basic means of manipulating its representations.

A set of instructions that leads one to a representation of the conclusion.Slide21

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.Slide22

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.Slide23

Automatic Reasoner

But can we use the laws of physics to build an automatic

reasoner

, as we did with the marbles and addition? Yes!Slide24

Logic GatesSlide25

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.Slide26

Automatic Reasoners

This is important!

We’ve created things that can use logic and reason on their own.Slide27

Universal computersSlide28

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.Slide29

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.Slide30

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.Slide31

Universal Computers

Nowadays, many people carry around universal computers in their pockets.Slide32

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++.Slide33

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.Slide34

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.Slide35

The computational theory of mindSlide36

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.Slide37

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.Slide38

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.Slide39

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).Slide40

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.Slide41

Logical Relations

From

:

If Joe fails the final exam, he will fail the course.

If Joe fails the course, he will not graduate.

It follows logically that

:

3. If Joe fails the final exam, he will not graduate.Slide42

Logical Relations

If you believe

:

If Joe fails the final exam, he will fail the course.

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.Slide43

Mental Processes are Rational

Computers are the only things (besides minds) that we have so far discovered that are reason-respecting in this way.

This gives us some reason to think that maybe minds are in fact computers.Slide44

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.Slide45

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”).Slide46

The Necker CubeSlide47

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 CATSSlide48

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.