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The Computational Theory of Mind. computation. Functions. 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 y. Functions.
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Examples of Functions • 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 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 • 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.
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.
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
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 Q
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) 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 →I
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.
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!
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.
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 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: • 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.
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.
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.
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 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.
