Artificial Minds - Spring 2009 Class notes for Monday, February 2 (week 3) =============================================================================== DISCLAIMER: These are my own working lecture notes, which I'm posting here in case they're useful. I'll try to post my notes on a more or less regular basis (assuming that I have some to post), but I make no guarantees as to their completeness or comprehensibility. They're also not too pretty, unless you're a fan of plain text and ASCII art, so I'd recommend using them as a supplement rather than a replacement for your own note-taking. In other words, you're still responsible for the material we cover in class, whether or not that material is included here. =============================================================================== Mechanizing computation and thought processes (continued) --------------------------------------------------------- Can we capture primality as a formal set of rules? Axioms: xyDNDx where x and y are hyphen-strings ("bigger numbers can't divide smaller numbers") Examples: 2 does not divide 1 --DND- 3 does not divide 1 ---DND- 4 does not divide 3 ----DND--- 5 does not divide 2 -----DND-- 6 does not divide 1 ------DND- 6 does not divide 3 ------DND--- Rule: If xDNDy is a theorem, then so is xDNDxy ("if x does not divide y, then x does not divide x+y") Examples: 2 does not divide 1 --DND- (axiom) 2 does not divide 3 --DND-- - 2 does not divide 5 --DND-- -- - 2 does not divide 7 --DND-- -- -- - ... 3 does not divide 1 ---DND- (axiom) 3 does not divide 4 ---DND--- - 3 does not divide 7 ---DND--- --- - 3 does not divide 10 ---DND--- --- --- - ... 4 does not divide 3 ----DND--- (axiom) 4 does not divide 7 ----DND---- --- ... 5 does not divide 2 -----DND-- (axiom) 5 does not divide 7 -----DND----- -- ... 6 does not divide 1 ------DND- (axiom) 6 does not divide 7 ------DND------ - 6 does not divide 13 ------DND------ ------ - ... Strings of the form --DND---- ("2 does not divide 4") cannot be theorems because there is no way to generate them from the axioms and the above rule. To do so, we would have to start out with --DND, which is not an axiom. Our formal system now exactly captures the mathematical idea of "does not divide": every such fact is represented by some theorem or axiom, and every theorem or axiom represents a true statement. Let's now throw in more theorems to represent the idea of a number having no divisors from 2 up to a certain point... Examples: 25 is divisor free from 2 up to 4 49 is divisor free from 2 up to 6 11 is divisor free from 2 up to 10 Rule: If --DNDn is a theorem, so is nDF-- ("if 2 does not divide n, n is divisor-free up to 2") Examples: 2 does not divide 7 --DND------- (derived before) 7 is divisor free up to 2 -------DF-- 2 does not divide 3 --DND--- (derived before) 3 is divisor free up to 2 ---DF-- Rule: If nDFx and x-DNDn are both theorems, then so is nDFx- ("if n is divisor-free up to x, and x+1 does not divide n, then n is divisor-free up to x+1") Examples: 7 is divisor free up to 2 -------DF-- (derived before) 3 does not divide 7 ---DND------- (derived before) 7 is divisor free up to 3 -------DF--- 4 does not divide 7 ----DND------- (derived before) 7 is divisor free up to 4 -------DF---- 5 does not divide 7 -----DND------- (derived before) 7 is divisor free up to 5 -------DF----- 6 does not divide 7 ------DND------- (derived before) 7 is divisor free up to 6 -------DF------ Rule: If x-DFx is a theorem, then Px- is a theorem ("if n is divisor free up to n-1, then n is prime") Examples: 3 is divisor free up to 2 (derived before) 3 is prime P--- 7 is divisor free up to 6 -------DF------ (derived before) 7 is prime P------- We can't derive P-- since we would first have to derive 2 is divisor free up to 1 --DF- but there is no way to derive a theorem of the form xDF- Axiom: P-- ("2 is prime") ----------------------------------------------------------------------------- Can all of mathematics be captured as a formal system? Can the mathematical thought processes of people be mechanized? This was the dream of early 20-century mathematicians, e.g. David Hilbert, Bertrand Russell, Alfred North Whitehead. Hilbert's dream: - start with any mathematical question at all - encode question as a string of formal symbols - mechanically follow the rules - if string is derivable, the answer is yes - if string's negation is derivable, the answer is no - any question at all about mathematics could be answered this way Kurt Go"del showed this dream to be an illusion in 1931 Go"del's Incompleteness Theorem (1931): For any formal mathematical system capable of representing the natural numbers, there exist true mathematical statements that can never be derived by the system. Provability is a weaker notion than truth. Godel devised an encoding of the formal system's symbols as numbers. For example: & <---> 1 => <---> 4 P <---> 7 | <---> 2 ( <---> 5 Q <---> 8 ~ <---> 3 ) <---> 6 R <---> 9 First insight: Manipulating symbols is equivalent to transforming numbers "P & Q" = 718 "Q & P" = 817 "P & Q => Q & P" = "from the number 100a + 10 + b, we can make 100b + 10 + a" Next insight: symbol-strings can be constructed that refer to specific numbers Next insight: a symbol-string can refer to its OWN number Final insight: a symbol-string can be constructed that means the following: "This symbol-string cannot be derived using the system's rules" What happens if it is derivable by the system? - we would then have just derived a false statement! What happens if it cannot be derived by the system? - we could see that it would clearly be true - which would mean there is at least one truth the system cannot derive - this truth is forever inaccessible to the system Human reasoning can recognize the truth of this statement, but the formal system cannot. Does this mean that human reasoning is more powerful than formal systems? "Godel's Theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines." ---J.R. Lucas, "Minds, Machines, and Godel" (1961) But wait! We are *outside* of the formal system S, so we can perceive S's "Godel statement" as being true, even though S itself can never do so from within. However, our own mind could have its *own* Godel statement, which it could never perceive as being "true".