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who can tell me what a proof is any
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ideas and what is a proof anyway and
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thoughts yeah very good I like that
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that's that's very close to what we're
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going to do here yeah
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now that's a special kind of proof
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though that's a mathematical proof and
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I'm going to write a definition very
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close to that in a few minutes but
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proofs exist beyond mathematics can
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anybody think of a higher-level notion
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of what a proof is that's correct what
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you said but there's a higher meta level
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notion of what a proof is beyond that
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may have no logical deductions
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potentially they have no assumptions any
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thoughts better proof okay well I think
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generally a proof is considered across
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multiple fields as a method for
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ascertaining the truth and you described
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one method now by ascertaining I mean
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establishing truth verifying truth and
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there's lots of ways to ascertain truth
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in society and even within science what
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are some examples of ways that we
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ascertain truth in society yeah
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observation experiment and observation
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excellent that's that's the bedrock of
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physics I mean who really knows if
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there's gravity out there
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well we observe it and so we then
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conclude that's the truth there's
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gravity and we have laws about it that's
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one good way what's another way of
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sir taining truth across scientific
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disciplines or inside beyond science
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just in society how is truth established
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one of the ways yeah yes truth yeah
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that's great truth is the opposite of
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falsehood how do we establish falsehood
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what are the ways in doing that how do
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you decide something is not true and
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every day ya find counter examples yeah
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that's good so in fact even a step more
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general sampling counter examples are
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ways you know if you do something an
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experiment 10 times and every time it
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comes out one way huh that's truth maybe
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but there's fields where that becomes
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truth what about other ways how are we
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going to decide if Roger Clemens is
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guilty of perjury for lying about
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steroids to Congress he did it or he
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didn't how are we going to decide that
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what's the truth behind how is that true
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it's going to be ascertained yeah
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examining evidence and who makes the
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conclusion there the jury truth is
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established by juries or judges you know
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like oh I can never pronounce his name
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the Illinois Governor Blagojevich he's
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guilty that's the truth okay
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of you know not of conspiracy trying to
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sell Obama sent a seat but of lying to
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the authorities about campaign financing
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Oh Jay is guilty not of killing his wife
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but of breaking into an apartment to
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steal back some of his merchandise okay
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so judges and juries make decisions on
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truth what are other truths bigger
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truths even than judges and juries in
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society
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there's one really big one that causes a
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lot of issues yeah what is it
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religion the Word of God broadly
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construed here or religion you know now
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that one's really hard to argue about
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you know because you believe it
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you know and it's especially if you're
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not talking to God regularly and
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somebody is well it's hard to argue
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about you know the truth so you rely on
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others to interpret it for you often you
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know a priest or minister or rabbi okay
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now it gets complicated because you can
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end up with conflicting truths based on
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you know who you think you're talking to
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or who you represent translator is for
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you you know another one is the word of
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your boss whatever the boss says is
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right
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often in business the customer is always
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right that's the truth whatever the
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customer says all right you know what
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Donald Trump is your boss you better
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agree or you're fired
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right yeah
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often in classes the professor says it
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it is true because the authority said it
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that's not true here all right that will
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not hold and one of the nicest thing
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about math that I like a lot is that the
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youngest student can stand up against
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the most oldest most experienced
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professor and win an argument on
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mathematics Beck I am you know I really
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I do get pleasure when a student comes
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up and proves me wrong I loved it when
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that student came in and she showed me
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what I said wheelie wasn't right when
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you looked at it carefully or in a
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different light you know now sometimes
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if I do it on the board here and it's in
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class well you know that's fun I feel a
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little embarrassed afterwards but but
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it's a good thing about mathematics is
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that you can have that kind of dialogue
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okay another one which is related to the
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word of God sometimes is inner
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conviction very popular in computer
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science believe it or not you know with
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the mantra there are no bugs in my
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program can't tell you how many times
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you know you hear that closely-related
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is
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I don't see why not something is true
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and that's a good one because that
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transfers the burden of proof to anybody
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who disagrees with you
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you don't have to prove you say I don't
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see why it's not true all of a sudden
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the other person who's questioning you
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becomes their job to disprove you which
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is not not so good okay now in
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mathematics there's this higher level
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and someone stated it very clearly up
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there let me write it up here in
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mathematics we have a mathematical proof
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is a verification of a proposition by a
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chain of logical deductions from a set
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of axioms now that's a bit of a mouthful
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there's three important components here
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propositions logical deductions and
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axioms and we're going to spend the rest
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of the class today talking about each of
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these and then give an example of a
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proof
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we'll start with propositions a
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proposition is a statement
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that is either true or false you may not
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know which one but it's one of the other
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simple example two plus three equals
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five now that's a true proposition
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here's one that's a little more
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interesting
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for all n in the set of natural numbers
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N squared plus n plus 41 is a prime
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number
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all right now I've used some notation
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here this is the upside-down a is the
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for all symbol how many people have not
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seen that symbol before I had a bunch of
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you yo you're going to see a bunch of
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symbols here first week and that means
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for every possible choice of N and this
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is in the natural numbers which is the
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set 0 1 2 3 so forth it's the natural
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numbers it's basically the integers but
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not negative so we're saying for every
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natural number ie 4 0 4 1 4 2 of the 3
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and so forth this expression is a prime
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now a prime number is a number that is
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not divisible by any other number
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besides itself and 1 so 2 3 5 7 or prime
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9 is not because it's 3 times 3 now this
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part here is called the predicate and a
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predicate is a proposition
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whose truth depends on the value of a
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variable in this case n all right this
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is referred to as the universe of
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discourse it's the space of all things
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we're talking about we're only talking
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about natural numbers here this is
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called a quantifier will see more
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quantifiers later all right now to see
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if this proposition is true we need to
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make sure that this predicate is true
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for every natural number n so let's see
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if we can check that let's try some
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values so we'll try and is 1 2 3 and so
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forth and we'll compute N squared plus n
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plus 41 and then we'll check is it prime
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so for N equals 0 N squared plus n plus
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41 is 41 it's 41 prime yeah nothing
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divides 41 but itself and 1 alright
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let's try 1 1 squared plus 1 plus 41 is
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43 43 prime yes let's try 2 we get 4
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plus 2 is 6 plus 41 47 he's 47 Prime yes
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looking good 3 I got 9 12 53 53 prime
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yeah and I could keep on going here I
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could go down to 20
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I get 420 461 in fact that is a prime
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and I could just keep on going here go
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down to 39 I get 1601 you can show
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that is a prime the first 40 values of n
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the proposition is true the predicate is
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true it is prime now this is a great
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example because in a lot of fields
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physics for example statistics often you
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know if it's true you checked for T
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examples that's above and beyond the
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call of duty it's always true so yeah
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this must be true right no wrong often
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you'll see this in a lot of scientific
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fields
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it is not true can anybody give me an
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example of n for which N squared plus n
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plus 41 is not prime yeah 40 good let's
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see about 40 40 squared plus 40 plus 40
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1 is 16 81 what's that equal 41 squared
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so it is not prime somebody give me an
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obvious example where it's not prime 41
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yeah 41 square with that everything is
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divided by 41 but 40 is the first
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breakpoint so the first 40 examples work
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and then it fails so this proposition is
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false even though it was looking pretty
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good
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there's a reason I'm doing this if I'm
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going to do it some more here I'm going
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to beat you over the head with it here's
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a famous in mathematics statement a to
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the fourth plus B to the fourth plus C
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to the fourth equals D to the fourth has
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no positive integer solutions that is a
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proposition now
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this proposition was conjectured to be
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true by Euler in 1769 Euler's a big
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honcho in math you know we still talk
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about him a lot even though he's been
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dead for centuries it was unsolved for
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over two centuries
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Matheson's worked on it
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it was finally disproved by very clever
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fellow named no Mel Kees 218 years later
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after was conjectured he works at that
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other school down the street and he came
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up with this a equals ninety five
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thousand eight hundred b equals two
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hundred seventeen thousand five hundred
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nineteen c equals four hundred fourteen
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thousand five hundred sixty you don't
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have to remember these numbers we're not
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going to quiz you on that four hundred
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twenty two thousand four hundred eighty
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one now he claims I've never personally
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checked it but presumably people have
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you plug those in here and you have an
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equality so he says so in fact correct
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true proposition is there do exist a b c
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d in the positive natural numbers such
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that a to the fourth plus B to the
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fourth plus C to the fourth equals D to
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the four
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all right I used a new quantifier here
274
00:16:25,189 --> 00:16:31,410
called there exists instead of an upside
275
00:16:29,160 --> 00:16:37,769
down a it's a backwards e don't ask me
276
00:16:31,410 --> 00:16:40,049
why that's what it is the plus means you
277
00:16:37,769 --> 00:16:41,790
can't have zero or negative numbers so
278
00:16:40,049 --> 00:16:49,889
the these are the positive natural
279
00:16:41,790 --> 00:16:51,509
numbers and here's your predicate which
280
00:16:49,889 --> 00:16:55,529
of course the truth of this depends on
281
00:16:51,509 --> 00:16:57,059
the values of a b c and d alright take a
282
00:16:55,529 --> 00:16:59,459
long time to figure out that actually
283
00:16:57,059 --> 00:17:00,269
there was a solution here obviously
284
00:16:59,459 --> 00:17:04,100
everything they tried till that time
285
00:17:00,269 --> 00:17:04,100
failed let me give you another one
286
00:17:08,360 --> 00:17:26,809
313 x cubed plus y cubed equals z cubed
287
00:17:15,000 --> 00:17:26,809
has no positive integer solutions
288
00:17:28,660 --> 00:17:34,610
this turns out to be false but the
289
00:17:32,120 --> 00:17:38,540
shortest smallest counterexample has
290
00:17:34,610 --> 00:17:41,120
over a thousand digits this one was easy
291
00:17:38,540 --> 00:17:43,210
they only have six digits so there's no
292
00:17:41,120 --> 00:17:45,470
way ever you'd use a computer to
293
00:17:43,210 --> 00:17:49,490
exhaustively search thousand digit
294
00:17:45,470 --> 00:17:51,940
numbers here to show it's false now of
295
00:17:49,490 --> 00:17:57,500
course some of you are probably thinking
296
00:17:51,940 --> 00:18:00,340
why on earth would I care if 313 times s
297
00:17:57,500 --> 00:18:03,590
Q plus y cubed equals Z cubed has a
298
00:18:00,340 --> 00:18:05,150
solution now that probably won't be the
299
00:18:03,590 --> 00:18:08,390
last time that thought occurs to you
300
00:18:05,150 --> 00:18:10,760
during the term and why on earth would
301
00:18:08,390 --> 00:18:14,090
anybody ever try to even find a solution
302
00:18:10,760 --> 00:18:16,700
to that you know I mean mathematicians
303
00:18:14,090 --> 00:18:19,280
are sort of a rare breed now actually in
304
00:18:16,700 --> 00:18:23,570
this case that's really important in
305
00:18:19,280 --> 00:18:24,950
practice okay this equation is an
306
00:18:23,570 --> 00:18:26,870
example of what's called an elliptic
307
00:18:24,950 --> 00:18:29,120
curve elliptic curve and you study these
308
00:18:26,870 --> 00:18:31,670
in your really a specialist in
309
00:18:29,120 --> 00:18:33,200
mathematics and graduate school or if
310
00:18:31,670 --> 00:18:36,080
you work for certain three-letter
311
00:18:33,200 --> 00:18:37,880
agencies because it's central to the
312
00:18:36,080 --> 00:18:41,360
understanding of how to factor large
313
00:18:37,880 --> 00:18:45,410
integers that means you know factoring
314
00:18:41,360 --> 00:18:48,530
showing that what was it 16 81 is 41
315
00:18:45,410 --> 00:18:49,510
times 41 I say okay who cares about
316
00:18:48,530 --> 00:18:53,420
factoring
317
00:18:49,510 --> 00:18:57,770
well factoring is the way to break
318
00:18:53,420 --> 00:19:00,050
crypto systems like RSA which are used
319
00:18:57,770 --> 00:19:03,140
for everything that we do electronically
320
00:19:00,050 --> 00:19:05,060
today you do have a PayPal account you
321
00:19:03,140 --> 00:19:07,700
buy something on online you're using SSL
322
00:19:05,060 --> 00:19:09,110
they're all using crypto systems most of
323
00:19:07,700 --> 00:19:11,480
which are almost all which are based on
324
00:19:09,110 --> 00:19:13,820
number theory and in particular that
325
00:19:11,480 --> 00:19:16,640
based on factoring and if you can find
326
00:19:13,820 --> 00:19:18,950
good solutions to things like this or
327
00:19:16,640 --> 00:19:21,440
solutions to things like this all of a
328
00:19:18,950 --> 00:19:24,140
sudden you can get an angle and a wedge
329
00:19:21,440 --> 00:19:28,120
on factoring and it's because of that
330
00:19:24,140 --> 00:19:31,610
that now RSA uses a thousand digit
331
00:19:28,120 --> 00:19:33,560
modulus instead of hundred digit modulus
332
00:19:31,610 --> 00:19:34,730
is like they used to use because people
333
00:19:33,560 --> 00:19:37,340
figured out how to factor them how to
334
00:19:34,730 --> 00:19:40,190
break the crypto system if you can break
335
00:19:37,340 --> 00:19:41,390
those crypto systems I can't rule the
336
00:19:40,190 --> 00:19:43,740
world but it's close
337
00:19:41,390 --> 00:19:45,900
all right so you'll learn would talk
338
00:19:43,740 --> 00:19:48,210
more about this the week after next when
339
00:19:45,900 --> 00:19:50,130
we do number theory and we work up to
340
00:19:48,210 --> 00:19:52,680
RSA and how that crypto system works and
341
00:19:50,130 --> 00:19:54,840
why factoring is so important all right
342
00:19:52,680 --> 00:19:56,220
so yeah you all don't have to worry
343
00:19:54,840 --> 00:19:57,870
about we all have to worry about this
344
00:19:56,220 --> 00:20:00,720
but these things are important and the
345
00:19:57,870 --> 00:20:02,310
bigger message is that you don't just
346
00:20:00,720 --> 00:20:05,460
try a few cases and if it works you
347
00:20:02,310 --> 00:20:07,020
think it's done because not that's not
348
00:20:05,460 --> 00:20:09,300
how the game works in mathematics you
349
00:20:07,020 --> 00:20:13,440
can get an idea of maybe it's true but
350
00:20:09,300 --> 00:20:15,660
doesn't doesn't tell you the answer all
351
00:20:13,440 --> 00:20:17,070
right let me give it another one this is
352
00:20:15,660 --> 00:20:19,890
another very famous one that you
353
00:20:17,070 --> 00:20:30,440
probably most you have heard of the
354
00:20:19,890 --> 00:20:30,440
regions in any map can be colored in
355
00:20:30,980 --> 00:20:40,310
four colors so that adjacent regions
356
00:20:44,300 --> 00:20:49,710
have different colors you know like a
357
00:20:47,850 --> 00:20:51,870
map of the United States every state
358
00:20:49,710 --> 00:20:54,900
gets a color I have two states share a
359
00:20:51,870 --> 00:20:57,600
border they have different colors you
360
00:20:54,900 --> 00:20:59,760
know so you can distinguish them this is
361
00:20:57,600 --> 00:21:01,380
known as the four color theorem and it's
362
00:20:59,760 --> 00:21:03,360
it's very famous in the popular
363
00:21:01,380 --> 00:21:05,370
literature how many people have heard of
364
00:21:03,360 --> 00:21:07,740
this theorem before yeah
365
00:21:05,370 --> 00:21:09,990
okay so you've all heard of it as a long
366
00:21:07,740 --> 00:21:12,660
history was conjectured by somebody
367
00:21:09,990 --> 00:21:13,920
named Guthrie in 1853 he's the first
368
00:21:12,660 --> 00:21:18,170
person to say this ought to be possible
369
00:21:13,920 --> 00:21:21,150
and there were many false proofs over
370
00:21:18,170 --> 00:21:23,610
the ensuing century one of the most
371
00:21:21,150 --> 00:21:26,400
convincing was a proof using pictures by
372
00:21:23,610 --> 00:21:29,310
kemppi in 1879
373
00:21:26,400 --> 00:21:31,040
you know 26 years later and this proof
374
00:21:29,310 --> 00:21:32,700
was believed for over a decade
375
00:21:31,040 --> 00:21:35,190
mathematicians thought the proof was
376
00:21:32,700 --> 00:21:37,830
right until another mathematician named
377
00:21:35,190 --> 00:21:40,980
he would found a fatal flaw in the
378
00:21:37,830 --> 00:21:42,840
argument now this proof I can't be
379
00:21:40,980 --> 00:21:45,480
consistent of drawing pictures of what
380
00:21:42,840 --> 00:21:47,220
maps have to look like your summer
381
00:21:45,480 --> 00:21:48,930
saying a map has to look like one of
382
00:21:47,220 --> 00:21:51,810
these types and would draw pictures of
383
00:21:48,930 --> 00:21:55,000
them and then he argued that those types
384
00:21:51,810 --> 00:21:58,180
that you pictures of it worked for
385
00:21:55,000 --> 00:22:01,450
proofs by picture are often very
386
00:21:58,180 --> 00:22:02,980
convincing and very wrong and I'm going
387
00:22:01,450 --> 00:22:05,140
to give you one to start lecture next
388
00:22:02,980 --> 00:22:08,410
time be proof by PowerPoint which is
389
00:22:05,140 --> 00:22:11,920
even worse than proof by picture and it
390
00:22:08,410 --> 00:22:13,150
is compelling you know and the point
391
00:22:11,920 --> 00:22:15,550
will be to show you proofs by picture
392
00:22:13,150 --> 00:22:17,290
are generally not a good thing because
393
00:22:15,550 --> 00:22:19,720
your brain just locks in oh that's what
394
00:22:17,290 --> 00:22:21,580
it has to look like and you don't think
395
00:22:19,720 --> 00:22:24,190
about other ways that it might look like
396
00:22:21,580 --> 00:22:29,230
now the four-color theorem was finally
397
00:22:24,190 --> 00:22:31,330
proved by Appel and Haken in 1977 but
398
00:22:29,230 --> 00:22:36,070
they had to use the computer to check
399
00:22:31,330 --> 00:22:39,010
thousands of cases now this was a little
400
00:22:36,070 --> 00:22:40,510
disturbing to mathematicians because how
401
00:22:39,010 --> 00:22:43,900
do they know the computer did the right
402
00:22:40,510 --> 00:22:44,980
thing you know you can your colleague
403
00:22:43,900 --> 00:22:46,150
writes of proof on the board you can
404
00:22:44,980 --> 00:22:47,860
check it but how do you know the
405
00:22:46,150 --> 00:22:51,580
computer didn't mess up or not do some
406
00:22:47,860 --> 00:22:55,300
cases now everybody believes it's true
407
00:22:51,580 --> 00:22:59,520
now but it's unsatisfying a few years
408
00:22:55,300 --> 00:23:02,560
ago a 12-page human proof was discovered
409
00:22:59,520 --> 00:23:04,630
but it's not been verified and people
410
00:23:02,560 --> 00:23:07,440
are very suspicious of it because the
411
00:23:04,630 --> 00:23:10,270
proof of the main lemma says quote
412
00:23:07,440 --> 00:23:13,090
details of this lemma is left to the
413
00:23:10,270 --> 00:23:14,770
reader see figure 7 that's what the main
414
00:23:13,090 --> 00:23:16,240
lamb of the proof is but people think
415
00:23:14,770 --> 00:23:20,340
that maybe we're some good ideas there
416
00:23:16,240 --> 00:23:23,790
but very suspicious proof all right
417
00:23:20,340 --> 00:23:32,920
let's do another one another proposition
418
00:23:23,790 --> 00:23:37,080
also very famous every even integer but
419
00:23:32,920 --> 00:23:49,750
two actually positive integer but two is
420
00:23:37,080 --> 00:23:54,900
the sum of two primes for example 24 is
421
00:23:49,750 --> 00:23:54,900
the sum of 11 and 13 which are prime
422
00:23:56,370 --> 00:24:06,880
anybody know is this true or false this
423
00:24:00,100 --> 00:24:07,530
proposition yeah yeah that's right me
424
00:24:06,880 --> 00:24:10,360
too
425
00:24:07,530 --> 00:24:14,890
nobody knows if this is true or false
426
00:24:10,360 --> 00:24:18,600
this is called Goldbach's conjecture was
427
00:24:14,890 --> 00:24:21,840
conjectured by Kristian Goldbach in 1742
428
00:24:18,600 --> 00:24:24,580
this is a really simple proposition and
429
00:24:21,840 --> 00:24:27,160
it you know as it's amazing it's not
430
00:24:24,580 --> 00:24:29,050
known in fact I spend you know a couple
431
00:24:27,160 --> 00:24:30,250
years working on I thought oh well this
432
00:24:29,050 --> 00:24:34,630
has to be easy enough to prove when I
433
00:24:30,250 --> 00:24:36,670
was a younger and didn't get very far so
434
00:24:34,630 --> 00:24:40,570
people still don't know if it's true and
435
00:24:36,670 --> 00:24:42,490
in fact it was listed by the globe as
436
00:24:40,570 --> 00:24:44,160
one of the great unsolved mysteries so
437
00:24:42,490 --> 00:24:46,180
if you get out this globe article here
438
00:24:44,160 --> 00:24:49,270
on the handout everybody have this
439
00:24:46,180 --> 00:24:50,770
handout you don't we'll get it passed
440
00:24:49,270 --> 00:24:53,710
out somebody missing the handout up up
441
00:24:50,770 --> 00:24:57,760
over there and over here if we get those
442
00:24:53,710 --> 00:25:00,370
passed out now it lists the three
443
00:24:57,760 --> 00:25:05,560
conjectures do you see Goldbach's
444
00:25:00,370 --> 00:25:07,240
conjecture there now can anybody point
445
00:25:05,560 --> 00:25:09,940
out something that's a little disturbing
446
00:25:07,240 --> 00:25:16,180
about what the globe says about
447
00:25:09,940 --> 00:25:19,210
Goldbach's conjecture yeah it gives the
448
00:25:16,180 --> 00:25:25,630
example like set up 24 is 11 plus 13 it
449
00:25:19,210 --> 00:25:27,250
says 20 is the sum of nine and 11 now if
450
00:25:25,630 --> 00:25:28,870
we're allowed to use things like nine as
451
00:25:27,250 --> 00:25:33,310
primes Goldbach conjecture is pretty
452
00:25:28,870 --> 00:25:35,350
easy to prove is true all right
453
00:25:33,310 --> 00:25:36,580
this won't be the last time we get
454
00:25:35,350 --> 00:25:38,650
examples from the literature if I queer
455
00:25:36,580 --> 00:25:40,960
gonna do this a lot along this theme of
456
00:25:38,650 --> 00:25:42,970
you cannot believe everything you read
457
00:25:40,960 --> 00:25:43,930
and now the globe is easy pickings but
458
00:25:42,970 --> 00:25:46,690
we'll do some more interesting ones
459
00:25:43,930 --> 00:25:48,580
later now this article lists to other
460
00:25:46,690 --> 00:25:52,630
famous conjectures which most people
461
00:25:48,580 --> 00:25:57,730
believe to be true the Riemann
462
00:25:52,630 --> 00:26:01,330
hypothesis after a 1859 paper written by
463
00:25:57,730 --> 00:26:02,890
bernhard riemann suggests that zeroes in
464
00:26:01,330 --> 00:26:05,110
an infinite series of numbers known as a
465
00:26:02,890 --> 00:26:07,510
zeta function form along a straight line
466
00:26:05,110 --> 00:26:08,020
the complex plane the hypothesis has
467
00:26:07,510 --> 00:26:11,620
been proved
468
00:26:08,020 --> 00:26:14,740
to 1.5 billion zeros not far enough to
469
00:26:11,620 --> 00:26:16,390
prove it completely now if they did 1.5
470
00:26:14,740 --> 00:26:19,240
trillion zeros it wouldn't be far enough
471
00:26:16,390 --> 00:26:23,860
to prove it completely of course and
472
00:26:19,240 --> 00:26:26,080
then the pranker now actually the Rhema
473
00:26:23,860 --> 00:26:27,880
thesis a couple years ago somebody
474
00:26:26,080 --> 00:26:30,430
credible claim to approved it
475
00:26:27,880 --> 00:26:32,980
proof turned out not to be right then
476
00:26:30,430 --> 00:26:36,100
there's the plonker a conjecture now
477
00:26:32,980 --> 00:26:38,980
this one was finished off it was proved
478
00:26:36,100 --> 00:26:42,030
to be true in 2003 by a Russian named
479
00:26:38,980 --> 00:26:45,580
Gregor Perelman the conjecture says
480
00:26:42,030 --> 00:26:49,090
roughly speaking that 3d objects without
481
00:26:45,580 --> 00:26:50,740
holes like not a doughnut are equivalent
482
00:26:49,090 --> 00:26:53,980
to the sphere they can sort of be
483
00:26:50,740 --> 00:26:55,540
deformed into a sphere this is known to
484
00:26:53,980 --> 00:26:56,860
be true in four dimensions and higher
485
00:26:55,540 --> 00:26:59,560
but nobody could prove it for three
486
00:26:56,860 --> 00:27:01,150
dimensions until Perelman came along now
487
00:26:59,560 --> 00:27:04,270
there's a bit of a controversy around
488
00:27:01,150 --> 00:27:06,460
this guy he had an 80-page proof but
489
00:27:04,270 --> 00:27:08,020
didn't have all the details so then
490
00:27:06,460 --> 00:27:11,820
other teams and mathematicians got
491
00:27:08,020 --> 00:27:14,350
together and wrote 350 pages of details
492
00:27:11,820 --> 00:27:16,240
and then most people believe now that
493
00:27:14,350 --> 00:27:17,950
it's right and that his original proof
494
00:27:16,240 --> 00:27:19,750
might not have had all the details but
495
00:27:17,950 --> 00:27:23,170
he had the right structure of the proof
496
00:27:19,750 --> 00:27:24,820
so he won prizes for this you know he
497
00:27:23,170 --> 00:27:27,550
won the highest prize in mathematics the
498
00:27:24,820 --> 00:27:29,830
Fields Medal and just earlier this year
499
00:27:27,550 --> 00:27:32,320
he was awarded that $1,000,000
500
00:27:29,830 --> 00:27:34,180
millennium prize and there's about six
501
00:27:32,320 --> 00:27:36,130
problems or so that if you solve one of
502
00:27:34,180 --> 00:27:38,230
them the clay is 2 gives you a million
503
00:27:36,130 --> 00:27:40,360
dollars and he's the first one to win
504
00:27:38,230 --> 00:27:42,910
the million dollars now the guy's a
505
00:27:40,360 --> 00:27:44,740
little strange he rejected the Fields
506
00:27:42,910 --> 00:27:46,600
Medal and refused to go to the ceremony
507
00:27:44,740 --> 00:27:49,950
where he was being honored and he's
508
00:27:46,600 --> 00:27:53,290
recently rejected the Millennium prize
509
00:27:49,950 --> 00:27:56,410
in any way this area is murky and we
510
00:27:53,290 --> 00:27:59,760
have an expert to explain it all for us
511
00:27:56,410 --> 00:27:59,760
on video which I thought I'd show
512
00:28:05,430 --> 00:28:11,400
all right let's do a simpler one here
513
00:28:13,120 --> 00:28:22,810
for all n MZ and greater than equal to
514
00:28:18,160 --> 00:28:28,110
two implies N squared greater than equal
515
00:28:22,810 --> 00:28:28,110
to four now Z we use for the integers
516
00:28:28,380 --> 00:28:39,570
and so that would be 0 1 minus 1 2 minus
517
00:28:34,150 --> 00:28:46,720
2 and so forth and this symbol here is
518
00:28:39,570 --> 00:28:47,800
implies alright in fact one thing you're
519
00:28:46,720 --> 00:28:51,670
going to notice when you read the text
520
00:28:47,800 --> 00:28:53,440
is we use different notation there as
521
00:28:51,670 --> 00:28:55,960
the standard that I will use in lecture
522
00:28:53,440 --> 00:28:57,880
and there's lots of ways of doing it you
523
00:28:55,960 --> 00:28:59,860
could have a double arrow a single arrow
524
00:28:57,880 --> 00:29:01,660
you could write out implies every time
525
00:28:59,860 --> 00:29:03,070
is it done in the text and it doesn't
526
00:29:01,660 --> 00:29:04,840
really matter which one you want to use
527
00:29:03,070 --> 00:29:07,930
as long as you use one of the
528
00:29:04,840 --> 00:29:12,450
conventions for implies and let me
529
00:29:07,930 --> 00:29:12,450
define what implies means
530
00:29:25,640 --> 00:29:47,310
an implication P implies Q is said to be
531
00:29:38,220 --> 00:29:50,130
true if P is false or Q is true if
532
00:29:47,310 --> 00:29:57,030
either one so we can write this down in
533
00:29:50,130 --> 00:30:00,000
terms of a truth table as follows we
534
00:29:57,030 --> 00:30:04,440
have the values of P and Q and I'll give
535
00:30:00,000 --> 00:30:06,920
the value of P implies Q if P is Q and Q
536
00:30:04,440 --> 00:30:12,840
is true so I P is true and Q is true
537
00:30:06,920 --> 00:30:17,220
what about P implies Q it's true because
538
00:30:12,840 --> 00:30:24,210
Q is true the definition if P is true
539
00:30:17,220 --> 00:30:33,470
and Q is false false P is false Q is
540
00:30:24,210 --> 00:30:36,240
true what about false and false is true
541
00:30:33,470 --> 00:30:40,950
even though this is false as long as P
542
00:30:36,240 --> 00:30:43,860
is false P implies Q is true alright so
543
00:30:40,950 --> 00:30:48,600
this is important to remember false
544
00:30:43,860 --> 00:30:50,130
implies anything is true which is a
545
00:30:48,600 --> 00:30:53,460
little strange there's a famous
546
00:30:50,130 --> 00:30:58,430
expression if pigs could fly I would be
547
00:30:53,460 --> 00:30:58,430
king is that true
548
00:30:59,390 --> 00:31:16,410
sort of in fact this statement pigs fly
549
00:31:07,100 --> 00:31:19,260
implies I'm King that's true because
550
00:31:16,410 --> 00:31:21,810
pigs don't fly doesn't matter whether or
551
00:31:19,260 --> 00:31:23,910
not I'm King which I'm not since pigs
552
00:31:21,810 --> 00:31:27,960
don't fly even though that's false
553
00:31:23,910 --> 00:31:29,210
the implication is true now some of you
554
00:31:27,960 --> 00:31:30,360
have worked on these things before
555
00:31:29,210 --> 00:31:32,250
second
556
00:31:30,360 --> 00:31:36,990
if you haven't want to start getting
557
00:31:32,250 --> 00:31:49,920
familiar with that alright let's do
558
00:31:36,990 --> 00:31:54,090
another example all right what about
559
00:31:49,920 --> 00:31:57,690
this proposition for all n for all
560
00:31:54,090 --> 00:32:04,770
integers and in Z N greater than equal
561
00:31:57,690 --> 00:32:10,190
to two this is if and only if n squared
562
00:32:04,770 --> 00:32:15,120
greater than equal to four is that true
563
00:32:10,190 --> 00:32:16,410
is n bigger than true if bigger than two
564
00:32:15,120 --> 00:32:20,220
if and only if M Squared is bigger than
565
00:32:16,410 --> 00:32:23,760
four false what's an example of n for
566
00:32:20,220 --> 00:32:29,040
which that's false negative all right so
567
00:32:23,760 --> 00:32:29,900
it's false n equals negative three all
568
00:32:29,040 --> 00:32:31,800
right
569
00:32:29,900 --> 00:32:33,510
negative three squared is bigger than
570
00:32:31,800 --> 00:32:35,210
equal to four but negative three is not
571
00:32:33,510 --> 00:32:40,050
bigger than equal to two and in fact
572
00:32:35,210 --> 00:32:43,920
this if and only if means you have to
573
00:32:40,050 --> 00:32:46,170
have an implication both ways so you got
574
00:32:43,920 --> 00:32:47,610
to check both ways for it so let's do
575
00:32:46,170 --> 00:32:54,480
the truth table extend this truth table
576
00:32:47,610 --> 00:33:05,310
out here to do the truth table for P if
577
00:32:54,480 --> 00:33:11,930
and only if Q all right so here are P
578
00:33:05,310 --> 00:33:18,270
and Q is Q implies P true for this row
579
00:33:11,930 --> 00:33:22,710
does Q imply true imply true yeah false
580
00:33:18,270 --> 00:33:25,440
implies true it's true true does not
581
00:33:22,710 --> 00:33:28,860
imply false that's false and false
582
00:33:25,440 --> 00:33:33,780
implies false and so now we can see
583
00:33:28,860 --> 00:33:38,070
where P is if and only if Q if they're
584
00:33:33,780 --> 00:33:40,170
both true then it's true here what about
585
00:33:38,070 --> 00:33:43,400
here just is P true if and only if Q is
586
00:33:40,170 --> 00:33:43,400
true in this case
587
00:33:43,489 --> 00:33:51,179
no because P implies Q is false but Q
588
00:33:48,179 --> 00:33:56,580
implies P is true so it's false false
589
00:33:51,179 --> 00:33:57,210
here I'm in a mistake there right that
590
00:33:56,580 --> 00:34:01,289
was true
591
00:33:57,210 --> 00:34:05,369
oops yep and true if and only if true
592
00:34:01,289 --> 00:34:07,229
okay they're both true so we're okay so
593
00:34:05,369 --> 00:34:10,409
P if and only if Q is true when they're
594
00:34:07,229 --> 00:34:12,950
both true or both false that's it
595
00:34:10,409 --> 00:34:16,230
if they're different then it's not true
596
00:34:12,950 --> 00:34:18,929
the key here is to always check both
597
00:34:16,230 --> 00:34:21,359
ways so if you're asked to prove an if
598
00:34:18,929 --> 00:34:27,990
and only if you got to prove that way
599
00:34:21,359 --> 00:34:31,349
and that way I we've just done about 15
600
00:34:27,990 --> 00:34:38,609
propositions is every sentence of
601
00:34:31,349 --> 00:34:42,079
proposition yes no no what's an example
602
00:34:38,609 --> 00:34:42,079
something is not a proposition
603
00:34:42,480 --> 00:34:50,399
oh wow this statement is false that's
604
00:34:47,730 --> 00:34:53,490
true well that is true it says he's not
605
00:34:50,399 --> 00:34:54,810
a composition because if it were true it
606
00:34:53,490 --> 00:34:55,649
wouldn't be false and if it was false
607
00:34:54,810 --> 00:34:57,720
and it'd be true you have a
608
00:34:55,649 --> 00:35:00,359
contradiction so it's neither true nor
609
00:34:57,720 --> 00:35:06,079
false what's a more simple example of if
610
00:35:00,359 --> 00:35:08,880
something that's not a proposition hmm
611
00:35:06,079 --> 00:35:12,000
boy I would have said that's true in
612
00:35:08,880 --> 00:35:14,160
some some world you know cuz that yeah
613
00:35:12,000 --> 00:35:18,510
that's a tissue so it's a true statement
614
00:35:14,160 --> 00:35:19,650
um hello
615
00:35:18,510 --> 00:35:23,400
that's good that's neither true nor
616
00:35:19,650 --> 00:35:25,770
false yeah a question who are you
617
00:35:23,400 --> 00:35:27,690
neither true nor false so not everything
618
00:35:25,770 --> 00:35:29,670
is a proposition but this course pretty
619
00:35:27,690 --> 00:35:33,599
much everything we deal with will be a
620
00:35:29,670 --> 00:35:35,010
proposition all right so that's it for
621
00:35:33,599 --> 00:35:38,970
propositions any questions on
622
00:35:35,010 --> 00:35:48,930
propositions all right next we're going
623
00:35:38,970 --> 00:35:50,790
to talk about axioms now the good news
624
00:35:48,930 --> 00:35:53,790
is that axioms are the same thing really
625
00:35:50,790 --> 00:35:56,339
as propositions the only difference is
626
00:35:53,790 --> 00:36:04,230
that axioms are propositions that we
627
00:35:56,339 --> 00:36:18,290
just assume are true an axiom is a
628
00:36:04,230 --> 00:36:18,290
proposition that is assumed to be true
629
00:36:19,160 --> 00:36:24,960
there's no proof that an axiom is true
630
00:36:22,640 --> 00:36:27,990
you just assume it because you think
631
00:36:24,960 --> 00:36:31,560
it's reasonable in fact the word axiom
632
00:36:27,990 --> 00:36:35,190
comes from Greek it doesn't mean to be
633
00:36:31,560 --> 00:36:36,720
true it means to think worthy something
634
00:36:35,190 --> 00:36:38,910
you think is worthy enough to be assumed
635
00:36:36,720 --> 00:36:41,160
to be true now a lot of times you're
636
00:36:38,910 --> 00:36:43,290
Hill people say sometimes we'll even say
637
00:36:41,160 --> 00:36:46,200
it to you you know don't make
638
00:36:43,290 --> 00:36:48,510
assumptions when you're doing math no
639
00:36:46,200 --> 00:36:49,950
that's not true you have to make
640
00:36:48,510 --> 00:36:51,780
assumptions when you do math otherwise
641
00:36:49,950 --> 00:36:54,569
you can't do anything because you got to
642
00:36:51,780 --> 00:36:56,190
start with some axioms the key in math
643
00:36:54,569 --> 00:36:58,050
is to a to
644
00:36:56,190 --> 00:37:01,230
you identify what your assumptions are
645
00:36:58,050 --> 00:37:03,810
so people can see them and the idea is
646
00:37:01,230 --> 00:37:05,310
that when you do a proof anybody who
647
00:37:03,810 --> 00:37:07,710
agrees with your assumptions or your
648
00:37:05,310 --> 00:37:10,050
axioms can follow your proofs and they
649
00:37:07,710 --> 00:37:12,240
have to agree with your conclusion now
650
00:37:10,050 --> 00:37:13,650
they might disagree with your axioms in
651
00:37:12,240 --> 00:37:17,339
which case they're not going to buy your
652
00:37:13,650 --> 00:37:25,099
proof now there are lots of axioms used
653
00:37:17,339 --> 00:37:32,160
in math for example if a equals B and B
654
00:37:25,099 --> 00:37:35,339
equals C then a equals C there is no
655
00:37:32,160 --> 00:37:38,160
proof of that but it seems pretty good
656
00:37:35,339 --> 00:37:43,950
and so we just throw it in the bucket of
657
00:37:38,160 --> 00:37:46,380
axioms and use it now axioms can be
658
00:37:43,950 --> 00:37:50,160
contradictory in different contexts
659
00:37:46,380 --> 00:37:56,069
here's a good example in Euclidean
660
00:37:50,160 --> 00:38:07,319
geometry there's a central axiom that
661
00:37:56,069 --> 00:38:21,540
says given a line L and a point P not on
662
00:38:07,319 --> 00:38:29,490
L there is exactly one line through P
663
00:38:21,540 --> 00:38:32,130
parallel to L you all saw this in
664
00:38:29,490 --> 00:38:33,839
geometry in middle school right got a
665
00:38:32,130 --> 00:38:35,250
point in a line there's exactly one line
666
00:38:33,839 --> 00:38:39,000
another line through the point that's
667
00:38:35,250 --> 00:38:44,119
parallel to the line now there's also a
668
00:38:39,000 --> 00:38:47,520
field called spherical geometry and
669
00:38:44,119 --> 00:38:50,130
there you have an axiom that contradicts
670
00:38:47,520 --> 00:38:57,300
this it says given a line L and a point
671
00:38:50,130 --> 00:39:02,040
P not on L there is no line through P
672
00:38:57,300 --> 00:39:07,880
parallel to L on the sphere there's a
673
00:39:02,040 --> 00:39:10,910
field called hyperbolic geometry
674
00:39:07,880 --> 00:39:13,040
and there there's an action that says
675
00:39:10,910 --> 00:39:23,960
given a line L and a point P not on L
676
00:39:13,040 --> 00:39:27,680
there are infinitely many lines through
677
00:39:23,960 --> 00:39:29,270
P parallel to L so how can this be does
678
00:39:27,680 --> 00:39:31,880
that mean one of these fields is totally
679
00:39:29,270 --> 00:39:34,960
bogus or two of them are because they've
680
00:39:31,880 --> 00:39:37,580
got contradictory axioms that's okay
681
00:39:34,960 --> 00:39:40,910
just whatever field you're in state your
682
00:39:37,580 --> 00:39:43,190
axioms and they do make sense in their
683
00:39:40,910 --> 00:39:44,840
various fields you know this is planar
684
00:39:43,190 --> 00:39:47,930
geometry this is on the sphere and this
685
00:39:44,840 --> 00:39:50,990
is on you know hyperbolic geometry they
686
00:39:47,930 --> 00:39:53,780
make sense in those contexts so you can
687
00:39:50,990 --> 00:39:55,850
have more less whatever-whatever axioms
688
00:39:53,780 --> 00:39:59,110
you want there are sort of two guiding
689
00:39:55,850 --> 00:39:59,110
principles two axioms
690
00:40:14,909 --> 00:40:25,989
axiom should be
691
00:40:16,659 --> 00:40:40,479
it's called consistent and complete now
692
00:40:25,989 --> 00:40:47,380
a set of axioms is consistent if no
693
00:40:40,479 --> 00:40:55,869
proposition can be proved to be both
694
00:40:47,380 --> 00:40:58,869
true and false and you can see why
695
00:40:55,869 --> 00:40:59,469
that's important now if you spend three
696
00:40:58,869 --> 00:41:02,439
weeks
697
00:40:59,469 --> 00:41:04,599
proving something's true and the next
698
00:41:02,439 --> 00:41:07,269
day somebody proves it's also false I
699
00:41:04,599 --> 00:41:08,799
mean the whole thing was pointless so it
700
00:41:07,269 --> 00:41:13,119
only makes sense if your axioms as a
701
00:41:08,799 --> 00:41:15,899
group are consistent a set of axioms is
702
00:41:13,119 --> 00:41:15,899
said to be complete
703
00:41:25,529 --> 00:41:38,579
if it can be used to prove every
704
00:41:34,809 --> 00:41:38,579
proposition is either true or false
705
00:41:46,280 --> 00:41:52,560
all right now this is desirable because
706
00:41:49,860 --> 00:41:55,890
it means well you can solve every
707
00:41:52,560 --> 00:41:57,150
problem all right everything is you can
708
00:41:55,890 --> 00:42:02,120
prove it's true or you can prove it's
709
00:41:57,150 --> 00:42:02,120
false you can get to the to the end now
710
00:42:03,290 --> 00:42:07,920
you'd think it shouldn't be too hard to
711
00:42:05,820 --> 00:42:10,320
get a set of axioms that satisfies these
712
00:42:07,920 --> 00:42:12,840
two basic properties you're allowed to
713
00:42:10,320 --> 00:42:14,700
choose whatever you want really just you
714
00:42:12,840 --> 00:42:16,560
don't want to be creating contradictions
715
00:42:14,700 --> 00:42:17,640
and you want to set that's powerful
716
00:42:16,560 --> 00:42:19,260
enough that allows you to prove
717
00:42:17,640 --> 00:42:22,200
everything is true or false one of the
718
00:42:19,260 --> 00:42:26,070
two turned out not to be so easy to do
719
00:42:22,200 --> 00:42:29,340
this and in fact many logicians spent
720
00:42:26,070 --> 00:42:32,100
their careers famous logicians trying to
721
00:42:29,340 --> 00:42:35,520
find a set of axioms just one set that
722
00:42:32,100 --> 00:42:36,720
was consistent and complete like Russell
723
00:42:35,520 --> 00:42:38,940
and Whitehead are probably the two most
724
00:42:36,720 --> 00:42:41,360
famous they spent their entire careers
725
00:42:38,940 --> 00:42:44,040
doing this and they they never got there
726
00:42:41,360 --> 00:42:48,270
then one day this guy named Kurt girdle
727
00:42:44,040 --> 00:42:50,820
showed up and in the 1930s he proved
728
00:42:48,270 --> 00:42:53,580
it's not possible that there exists any
729
00:42:50,820 --> 00:42:57,300
set of axioms that are both consistent
730
00:42:53,580 --> 00:43:00,360
and complete now this discovery
731
00:42:57,300 --> 00:43:01,160
devastated the field was a huge
732
00:43:00,360 --> 00:43:03,600
discovery
733
00:43:01,160 --> 00:43:05,100
you know imagine poor Russell and
734
00:43:03,600 --> 00:43:08,250
Whitehead they spent their entire
735
00:43:05,100 --> 00:43:09,810
careers going after this holy grail then
736
00:43:08,250 --> 00:43:12,630
Kurt shows up and said hey guys there's
737
00:43:09,810 --> 00:43:14,670
no Grail doesn't exist that's a little
738
00:43:12,630 --> 00:43:16,200
little depressing pretty bad day when
739
00:43:14,670 --> 00:43:18,150
that happened now
740
00:43:16,200 --> 00:43:20,690
it's an amazing result because it says
741
00:43:18,150 --> 00:43:24,090
if you want consistency that's a must
742
00:43:20,690 --> 00:43:26,940
there will be true facts that you will
743
00:43:24,090 --> 00:43:27,870
never be able to prove we're not going
744
00:43:26,940 --> 00:43:29,970
to prove that here it's proved in a
745
00:43:27,870 --> 00:43:33,090
logic course for example maybe
746
00:43:29,970 --> 00:43:36,540
Goldbach's conjecture is true and it is
747
00:43:33,090 --> 00:43:38,400
impossible to prove now we're going to
748
00:43:36,540 --> 00:43:41,220
try not to assign any of those problems
749
00:43:38,400 --> 00:43:43,470
for homework and in fact people they do
750
00:43:41,220 --> 00:43:45,990
exist you can its complicated state a
751
00:43:43,470 --> 00:43:48,810
problem that you can't prove is true or
752
00:43:45,990 --> 00:43:50,340
false and you may be thinking that from
753
00:43:48,810 --> 00:43:53,940
time to time hey it's one of those you
754
00:43:50,340 --> 00:43:57,390
know it's remember what your parents
755
00:43:53,940 --> 00:43:59,220
told you if you work hard enough you can
756
00:43:57,390 --> 00:44:02,700
do anything
757
00:43:59,220 --> 00:44:06,530
they were wrong alright that's enough
758
00:44:02,700 --> 00:44:06,530
for now we'll do more of this next time57443
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