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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Who can tell
me what a proof is? Any ideas of, what
is a proof, anyway? Any thoughts? Yeah? AUDIENCE: It's a chain of
statements, each logically supported by the
previous ones, that get you from a
set of assumptions to a set of conclusions. PROFESSOR: Very good. I like that. That's very close to what
we're going to do here, yeah. Now, that's a special
kind of proof, though. That's a mathematical proof. And I'm going to write
a definition very close to that in a few minutes. But proofs exist
beyond mathematics. Can anybody think
of a higher level notion of what a proof is? That's correct, what
you said, but there's a higher meta level notion of
what a proof is beyond that. It may have no logical
deductions potentially. It may have no assumptions. Any thoughts about a proof? OK, well, I think
generally, a proof is considered, across
multiple fields, as a method for
ascertaining the truth. And you described one method. Now, by ascertaining, I mean
establishing truth, verifying truth. And there's lots of
ways to ascertain truth in society, and
even within science. What are some examples
of ways that we ascertain truth in society? Yeah? AUDIENCE: Observations, like
seeing that piece of chalk will fall to the ground. PROFESSOR:
Observation, experiment and observation-- excellent. And that's the
bedrock of physics. I mean, who really knows if
there's gravity out there? Well, we observe it. And so we then conclude
that's the truth. There's gravity, and
we have laws about it. That's one good way. What's another way
of ascertaining truth across scientific disciplines,
or beyond science, just in society? How is truth established? What are the ways? Yeah? AUDIENCE: Well,
establishing what's false, you can know what
things aren't true. Then that helps you
narrow down what is true. PROFESSOR: Yes, truth--
yeah, that's great. Truth is the opposite
of falsehood. How do we establish falsehood? What are the ways in doing that? How you decide something is
not true in every day-- yeah? AUDIENCE: Find counterexamples. PROFESSOR: Find counterexamples,
yeah, that's good. So in fact, even a step
more general, sampling. Counterexamples are ways. If you do something, an
experiment, ten times, and every time, it comes out
one way, that's truth, maybe. But there's fields where
that becomes truth. What about other ways? How are we going to
decide if Roger Clemens is guilty of perjury for lying
about steroids to Congress? He did it or he didn't. How are we going to decide that? How is that truth going
to be ascertained? Yeah? AUDIENCE: Would it be by
examining the evidence that we have? PROFESSOR: Examining evidence. And who makes the
conclusion there? AUDIENCE: Juries. PROFESSOR: The jury. Truth is established
by juries or judges. You know, Blago-- I
can never pronounce his name, the Illinois governor,
Blagojevick-- he's guilty. That's the truth of--
not of conspiracy, trying to sell
Obama a senate seat, but of lying to the authorities
about campaign financing. OJ is guilty-- not
of killing his wife, but of breaking into an
apartment to steal back some of his merchandise. So judges and juries
make decisions on truth. What are other truths-- bigger
truths, even, than judges and juries, in society? There's one really big one
that causes a lot of issues. Yeah? AUDIENCE: Religion. PROFESSOR: What is it? AUDIENCE: Religion. PROFESSOR: Religion, the word
of God-- broadly construed here, for religion. Now, that one is really
hard to argue about because you believe it. And especially if you're
not talking to God regularly and somebody is, well, it's
hard to argue about the truth. So you rely on others
to interpret it for you, often-- a priest, or
a minister, a rabbi. And it gets complicated,
because you can end up with conflicting truths based on
who you think you're talking to or who the translator
is for you. Another one is the
word of your boss. Whatever the boss says is right. Often in business, the
customer is always right. That's the truth, whatever
the customer says. With Donald Trump as your
boss, you'd better agree or you're fired. Often in classes, the
professor says it, it is true. Because the authority said it. That's not true here. That will not hold. And one of the nicest things
about math that I like a lot is that the youngest
student can stand up against the most oldest,
most experienced professor, and win an argument
on mathematics. I do get pleasure when a student
comes up and proves me wrong. I loved it when that
student came in, and she showed me
what I said really wasn't right when you
looked at it carefully or in a different light. Now, sometimes if I do it on the
board here and it's in class, well, that's fun. I feel a little
embarrassed afterwards. But it's a good thing
about mathematics, is you can have that
kind of dialogue. OK, another one, which is
related to the word of God sometimes, is inner
conviction-- very popular in computer science, believe
it or not, with the mantra, there are no bugs in my program. I can't tell you how
many times you hear that. Closely related is, I don't
see why not something is true. And that's a good
one, because that transfers the burden of proof to
anybody who disagrees with you. You don't have to prove. You just say, I don't
see why it's not true. All of a sudden, the other
person who's questioning you, it becomes their job to disprove
you, which is not so good. OK, now in mathematics,
there's this higher level. And someone stated it
very clearly up there. Let me write it up here. In mathematics, we have
a mathematical proof is a verification
of a proposition by a chain of logical
deductions from a set of axioms. Now, that's a bit of a mouthful. There's three important
components here-- propositions, logical
deductions, and axioms. And we're going to spend
the rest of the class today talking about
each of these, and then give an
example of a proof. We'll start with propositions. A proposition is a statement
that is either true or false. You may not know which one,
but it's one or the other. A simple example--
2 plus 3 equals 5. Now, that's a true proposition. Here's one that's a
little more interesting. For all n in the set
of natural numbers, n squared plus n plus
41 is a prime number. I know I've used
some notation here. This is-- the upside down
A is the for all symbol. How many people have not
seen that symbol before? A bunch of you. You're going to see a bunch
of symbols here first week. And that means for every
possible choice of n-- and this is in the natural
numbers, which is the set 0, 1, 2, 3, and so forth. It's the natural numbers. It's basically the
integers, but not negative. So we're saying for every
natural number, i.e. for 0, for 1, for 2, and for 3, and
so forth, this expression is a prime. Now, a prime number
is a number that is not divisible by any other
number besides itself and 1. So 1, 3, 5, 7 are prime. 9 is not because it's 3 times 3. Now, this part here is
called the predicate. And a predicate is a
proposition whose truth depends on the value of a
variable-- in this case, n. All right, this is referred to
as the universe of discourse. It's the space of all the
things we're talking about. We're only talking about
natural numbers here. This is called a quantifier. We'll see more
quantifiers later. All right, now to see if
this proposition is true, we need to make sure
that this predicate is true for every natural number n. So let's see if
we can check that. Let's try some values. So we'll try n is 1,
2, 3, and so forth. And we'll compute n
squared plus n plus 41. And then we'll
check, is it prime? So for n equals 0, n squared
plus n plus 41 is 41. Is 41 prime? Yeah, nothing divides
41 but itself and 1. All right, let's try 1. 1 squared plus 1 plus 41 is 43. Is 43 prime? Yes. Let's try 2. We get 4 plus 2 is
6 plus 41 is 47. Is 47 prime? AUDIENCE: Yes. PROFESSOR: Yes. Looking good. 3-- I got 9, 12, 53. Is 53 prime? Yeah. And I could keep on going here. I could go down to 20. I get 420-- 461. In fact, that is a prime. And I could just
keep on going here. Go down to 39. I get 1,601. You can check. That is a prime. The first 40 values of n,
the proposition is true. The predicate is true. It is prime. Now, this is a great example
because in a lot of fields-- physics, for example;
statistics, often-- you checked 40 examples. That's above and beyond
the call of duty. It's always true. So yeah, this must
be true, right? No, wrong. Often, you'll see this in
a lot of scientific fields. It is not true. Can anybody give
me an example of n for which n squared plus
n plus 41 is not prime? Yeah? AUDIENCE: 40. PROFESSOR: 40, good. Let's see about 40. 40 squared plus 40
plus 41 is 1,681. What's that equal? 41 squared. So it is not prime. Somebody give me an obvious
example where it's not prime. AUDIENCE: 41. PROFESSOR: 41--
yeah, 41 squared, we get everything
is divided by 41. But 40 is the first break-point. So the first 40 examples
work, and then it failed. So this proposition is
false, even though it was looking pretty good. There's a reason I'm doing this. In fact, I'm going to
do it some more here. I'm going to beat you
over the head with it. Here's a famous in
mathematics statement. a to the fourth plus b to the
fourth plus c to the fourth equals d to the fourth has no
positive integer solutions. That is a proposition. Now, this proposition was
conjectured to be true by Euler in 1769. Euler's a big honcho in math. We still talk about him
a lot even though he's been dead for centuries. It was unsolved for
over 2 centuries. Mathematicians worked on it. It was finally disapproved
by a very clever fellow named Noam Elkies 218 years
later after it was conjectured. He worked at that other
school down the street. And he came up with this. a equals 95,800. b equals 217,519. c equals 414,560. You don't have to
remember these numbers. We're not going to quiz
you on that-- 422,481. Now, he claims-- I've never
personally checked it, but presumably, people have--
you plug those in here, and you have an equality. So he says. So in fact, the
correct proposition is there does exist a, b,
c, d in the positive natural numbers such that a the fourth
plus b to the fourth plus c to the fourth equals
d to the fourth. I used a new quantifier
here called there exists. Instead of an
upside down A, it's a backwards E. Don't ask me why. That's what it is. The plus means you can't
have 0 or negative numbers. So these are the
positive natural numbers. And here's your predicate, which
of course, the truth of this depends on the values
of a, b, c and d. It took a long time to figure
out that actually, there was a solution here. Obviously, everything they
tried until that time failed. Let me give you another one. 313 x cubed plus y cubed equals
z cubed has no positive integer solutions. This turns out to be false. But the shortest,
smallest counter-example has over 1,000 digits. This one was easy. It only has six digits. So there's no way ever
you'd use a computer to exhaustively search
1,000 digit numbers here to show it's false. Now, of course, some of you are
probably thinking, why on earth would I care if 313 times
x cubed plus y cubed equals z cubed has a solution? And that probably
won't be the last time that thought occurs to
you during the term. And why on earth
would anybody ever try to even find a solution to that? I mean, mathematicians
are sort of a rare breed. Now, actually in
this case, that's really important in practice. This equation is an
example of what's called an elliptic
curve-- elliptic curve. You study these if you're really
a specialist in mathematics in graduate school,
or if you work for certain
three-letter agencies because it's central to
the understanding of how to factor large integers. That means factoring,
showing that-- what was it-- 1,681 is 41 times 41. And I said, OK, who
cares about factoring? Well, factoring is the
way to break cryptosystems like RSA, which are
used for everything that we do electronically today. You have a Paypal account. You buy something online. You're using SSL. They're all using cryptosystems,
almost all of which are based on number theory. And in particular, they're
based on factoring. And if you can find good
solutions to things like this, or solutions to things
like this, all of a sudden, you can get an angle and
a wedge on factoring. And it's because
of that that now RSA uses 1,000 digit
moduluses instead of hundred digit moduluses like
they used to use, because people figured
out how to factor and how to break
the cryptosystem. If you could break
those cryptosystems, well, you can't rule the
world, but it's close. All right, so we'll
talk more about this the week after next when
we do number theory, and we work up to RSA and
how that cryptosystem works, and why factoring
is so important. So yeah, you don't
have to really have to worry about this. But these things are important. And the bigger message is that
you don't just try a few cases, and if it works,
you think it's done. That's not how the game
works in mathematics. You can get an
idea of maybe it's true, but doesn't
tell you the answer. All right, let me
give you another one. This is another very famous
one that probably most of you have heard of. The regions in any map can
be colored in four colors so that adjacent regions
have different colors. Like a map of the
United States-- every state gets a color. If two states share
a border, they have different colors so
you can distinguish them. This is known as the
four color theorem. And it's very famous in
the popular literature. How many people have heard
of this theorem before? Yeah, OK. So you've all heard of it. It has a long history. It was conjectured by somebody
named Guthrie in 1853. He's the first person to say
this ought to be possible. And there were many false
proofs over the ensuing century. One of the most convincing
was a proof using pictures by Kempe in 1879,
26 years later. And this proof was
believed for over a decade. Mathematicians thought
the proof was right until another mathematician
named Heawood found a fatal flaw in the argument. Now, this proof
by Kempe consisted of drawing pictures of what
maps have to look like. So he started by
saying, a map has to look like one of these types. And he would draw
pictures of them. And then he argued that those
types that he drew pictures of, it worked for. Proofs by picture are often
very convincing and very wrong. And I'm going to give you one
to start lecture next time. It'll be a proof by PowerPoint,
which is even worse than proof by picture. And it is compelling. And the point will to be to
show you proofs by picture are generally not a good thing. Because your brain
just locks in-- oh, that's what it has to look like. And you don't think about other
ways that it might look like. Now, the four color
theorem was finally proved by Appel
and Haken in 1977, but they had to use a computer
to check thousands of cases. Now, this was a little
disturbing to mathematicians, because how do they know the
computer did the right thing? Your colleague writes
a proof on the board. You can check it. But how do you know the
computer didn't mess up, or not do some cases? Now, everybody
believes it's true now. But it's unsatisfying. A few years ago, a 12-page
human proof was discovered, but it's not been verified. And people are very
suspicious of it because the proof of
the main lemma says, quote, "details of this
lemma is left to the reader. See figure seven." That's what the main
lemma of the proof is. But people think
that maybe there were some good ideas there,
but very suspicious proof. All right, let's do another
one, another proposition-- also very famous. Every even integer
but 2-- actually, positive integer but 2--
is the sum of two primes. For example, 24 is the sum of
11 and 13, which are prime. Anybody know? Is this true or false,
this proposition? Yeah? AUDIENCE: I wish I knew. PROFESSOR: [LAUGHS]
Yeah, that's right. Me too. Nobody knows if this
is true or false. This is called
Goldbach's conjecture. It was conjectured by
Christian Goldbach in 1742. This is a really
simple proposition. And it's amazing it's not known. In fact, I spent a couple
years working on-- I thought, oh, well, this has
to be easy enough to prove when I was younger,
and didn't get very far. So people still don't
know if it's true. And in fact, it was
listed by the Globe as one of the great
unsolved mysteries. So if you get out this Globe
article here, one of the hand-- does everybody
have this handout? You don't? We'll get it passed out. Somebody missing that handout
up over there and over here? All right, if we get
those passed out. Now, it lists the
three conjectures. Do you see Goldbach's
conjecture there? Now, can anybody
point out something that's a little disturbing
about what the Globe says about Goldbach's conjecture? AUDIENCE: 9 as a prime number. PROFESSOR: Yeah, it
gives the example. Like, instead of
24 is 11 plus 13, it says 20 is the
sum of 9 and 11. Now, if we're allowed to
use things like 9 as primes, Goldbach's conjecture's
pretty easy to prove is true. This won't be the last
time we get examples from the literature. In fact, we're going
to do this a lot, along this theme of, you cannot
believe everything you read. Now, the Globe is easy
pickings, but we'll do some more
interesting ones later. Now, this article lists two
other famous conjectures which most people believe to be
true-- the Riemann hypothesis after an 1859 paper
written by Bernard Riemann suggested that zeros in an
infinite series of numbers known as a zeta function
form along a straight line on that complex plane. The hypothesis has been proved
to 1.5 billion zeros, not far enough to prove it completely. If they did 1.5
trillion zeros, it wouldn't be far enough to
prove it completely, of course. And then the-- no, actually,
the Riemann hypothesis, a couple years ago,
somebody credible claimed to have proved it. Proof turned out
not to be right. Then there's the
Poincare conjecture. Now, this one was finished off. It was proved to be true
in 2003 by a Russian named Grigori Perelman. The conjecture says,
roughly speaking, that 3D objects without
holes, like no a doughnut, are equivalent to the sphere. They can sort of be
deformed into a sphere. This is known to be true in
four dimensions and higher, but nobody could prove
it for three dimensions until Perelman came along. Now, there's a bit of a
controversy around this guy. He had an 80-page proof, but
didn't have all the details. So then other teams
of mathematicians got together and wrote
350 pages of details. And then most people
believe now that it's right, and that his original
proof might not have had all the details, but
he had the right structure of the proof. So he won prizes for this. He won the highest prize in
mathematics, the Fields Medal. And just earlier this year,
he was awarded the $1 million Millennium Prize. And there's about
six problems or so that if you solve one of them,
the Clay Institute gives you a million dollars. And he's the first one to
win the million dollars. Now, the guy's a little strange. He rejected the Fields Medal and
refused to go to the ceremony where he was being honored. And he's recently rejected
the Millennium prize. And anyway, this area's
murky, and we have an expert to explain it all
for us on video, which I thought I'd show. All right, let's do
a simpler one here. For all n in Z, n greater
than or equal to 2 implies n squared is
greater than or equal to 4. Now, Z, we use for the integers. And so that would be 0, 1, minus
1, 2, minus 2, and so forth. And this symbol here is implies. I In fact, one
thing you can notice when you read the text is we
use different notation there as the standard than
I will use in lecture. And there's lots of
ways of doing it. You could have a double
arrow, a single arrow. You could write out
implies every time as it's done in the text. And it doesn't really
matter which one you want to use as
long as you use one of the conventions for implies. And let me define
what implies means. An implication p implies q is
said to be true if p is false or q is true, either one. So we can write this down
in terms of a truth table as follows. You have the values of p and q. And I'll give the
value of p implies q. If p is true and q is true,
what about p implies q? It's true, because q is
true in the definition. If p is true and q is false? AUDIENCE: False. PROFESSOR: False. P is false. Q is true. True. What about false and false? It's true. Even though this is false,
as long as p is false, p implies q is true. So this is important
to remember. False implies anything is true,
which is a little strange. There's a famous expression. If pigs could fly,
I would be king. Is that true? Sort of. In fact, this
statement, pigs fly implies I'm king-- that's
true, because pigs don't fly. Doesn't matter whether or
not I'm king, which I'm not. Since pigs don't fly,
even though that's false, the implication is true. Now, some of you have worked
on these things before. It's second nature. If you haven't,
you want to start getting familiar with that. Let's do another example. What about this proposition? For all integers, n in Z, n
greater than or equal to 2-- this is if and only if--
n squared greater than or equal to 4. Is that true? Is n only bigger
than 2 if and only if n squared is bigger than 4? It's false. What's an example of n
for which that's false? Negative, all right? So it's false. n equals negative 3, all right? Negative 3 squared is
bigger than or equal to 4, but negative 3 is not
bigger than or equal to 2. And in fact this if
and only if means you have to have an
implication both ways. So you have to check
both ways for it. So let's do the truth table--
extend this truth table out here to do the truth table
for p if and only if q. So here are p and q. Is q implies p
true for this row? Does true imply true? Yeah. False implies true? That's true. True does not apply false. That's false. And false implies false. And so now, we can see
where p is if and only if q. If they're both true,
then it's true here. What about here? Is p true if and only if
q is true in this case? No, because p implies q is
false, but q implies p is true. So it's false. False here. I made a mistake there, right? That was true-- oops. And true if and
only if true, OK. They're both true, so we're OK. So p if and only if q is
true when they're both true or both false. And that's it. If they're different,
then it's not true. The key here is to
always check both ways. So if you're asked to
prove an if and only if, you have to prove that
way, and that way. We've just done about
15 propositions. Is every sentence a proposition? Yes? No? AUDIENCE: No. PROFESSOR: No. What's an example of something
that's no a proposition? AUDIENCE: This
statement is false. PROFESSOR: A what? AUDIENCE: This
statement is false. PROFESSOR: This
statement is false. That's true. Well, it's true it's
not a proposition. Because if it were true,
it wouldn't be false. And if was false,
then it'd be true and you'd have a contradiction. So it's neither true nor false. What's a more simple
example of something that's not a proposition? AUDIENCE: This is a tissue. Isn't that a [INAUDIBLE]? PROFESSOR: Ooh. Boy, I would have said
that's true in some world. Because yeah, that's a tissue. So it's a true statement. AUDIENCE: Hello. PROFESSOR: Hello. That's good. That's neither true
or false, yeah. A question. Who are you-- neither
true nor false. So not everything
is a proposition. But in this course,
pretty much everything we deal with will
be a proposition. All right, so that's
it for propositions. Any questions on propositions? Next, we're going to
talk about axioms. Now, the good news is that
axioms are the same thing, really, as propositions. The only difference is that
axioms are propositions that we just assume are true. An axiom is a proposition
that is assumed to be true. There's no proof that
an axiom is true. You just assume it because
you think it's reasonable. In fact, the word
"axiom" comes from Greek. It doesn't mean to be true. It means to think
worthy-- something you think is worthy enough
to be assumed to be true. Now, a lot of times, you'll hear
people say-- sometimes, we'll even say it to you--
don't make assumptions when you're doing math. No, that's not true. You have to make assumptions
when you do math. Otherwise, you can't
do anything because you have to start with some axioms. The key in math is to identify
what your assumptions are so people can see them. And the idea is that
when you do a proof, anybody who agrees with your
assumptions or your axioms can follow your proof. And they have to agree
with your conclusion. Now, they might disagree
with your axioms, in which case, they're not
going to buy your proof. Now, there are lots of
axioms used in math. For example, if a equals b and
b equals c, then a equals c. There is no proof of that. But it seems pretty good. And so we just throw it in the
bucket of axioms and use it. Now, axioms can be contradictory
in different contexts. Here's a good example. In Euclidean geometry,
there's a central axiom that says given a line L
and a point p not on L, there is exactly one
line through p parallel to L. You all saw this in
geometry in middle school, right? You've got a point in a line. There's exactly another
line through the point that's parallel to the line. Now, there's also a field
called spherical geometry. And there, you have an
axiom that contradicts this. It says, given a line L
and a point p not on L, there is no line through p
parallel to L on the sphere. There's a field called
hyperbolic geometry. And there, there's
an axiom that says, given a line L and
a point p not on L, there are infinitely many
lines through p parallel to L. So how can this be? Does that mean one of these
fields is totally bogus, or two of them are? Because they've got
contradictory axes. That's OK. Just whatever field you're
in, state you're axioms. And they do make sense
in their various fields. This is planar geometry. This is on the sphere. And this is on
hyperbolic geometry. They make sense
in those contexts. So you can have more or less
whatever axioms you want. There are sort of two
guiding principles to axioms. Axioms should be-- it's called
consistent-- and complete. Now, a set of
axioms is consistent if no proposition can be proved
to be both true and false. And you can see why
that's important. If you spend three weeks
proving something's true, and the next day, somebody
proves it's also false, I mean, the whole
thing was pointless. So it only makes sense if
your axioms, as a group, are consistent. A set of axioms is
said to be complete if it can be used to
prove every proposition is either true or false. Now, this is desirable
because it means-- well, you can solve every problem. Everything is-- you
can prove it's true, or you can prove it's false. You can get to the end. Now, you'd think
it shouldn't be too hard to get a set of axioms
that satisfies these two basic properties. You're allowed to choose
whatever you want, really. Just, you don't want to be
creating contradictions. And you want a set
that's powerful enough that allows you to prove
everything is true or false, one of the two. Turned out not to be
so easy to do this. And in fact, many logicians
spent their careers-- famous logicians--
trying to find a set of axioms,
just one set, that was consistent and complete. In fact, Russell and
Whitehead are probably the two most famous. They spent their entire
careers doing this, and they never got there. Then one day, this guy
named Kurt Godel showed up. And in the 1930s, he
proved it's not possible that there exists any
set of axioms that are both consistent and complete. Now, this discovery
devastated the field. It was a huge discovery. Imagine poor Russell
and Whitehead. They spent their entire careers
going after this holy grail. Then Kurt shows up
and said, hey, guys. There's no grail. It doesn't exist. And that's a little
depressing-- pretty bad day when that happened. Now, it's an amazing
result, because it says if you want consistency--
and that's a must-- there will be true facts
that you will never be able to prove. We're not going to
prove that here. It's proved in a logic course. For example, maybe
Goldbach's conjecture is true and it is impossible to prove. Now, we're going to
try not to assign any of those problems for homework. And in fact, they do exist. It's complicated. You can state a problem that you
can't prove is true or false. And you may be thinking
that from time to time. Hey, it's one of those. Remember when your parents told
you if you work hard enough, you can do anything? They were wrong. All right, that's
enough for now. And we'll do more
of this next time.
