
One of the less popular kinds of exercises is a proof:
proofs tend to be very difficult to do right, and since
one already knows the answer (unless the sadistic professor
has assigned a trick question) there seems to be little
point to it.
Why do professors assign proofs?
First of all, just because professor says it's true doesn't
make it true.
Professor might be mistaken, or lying, or sadistic, or trying
to trick or brainwash you.
You don't believe your political science professor just
because he has a Ph.D., do you?
Why should you believe your mathematics professor just
because she has a Ph.D.?
(And don't say, "because it's easier if I just agree with
teacher"; laziness earns you no points in the real world.)
Ultimately, a proof is a convincing argument, an argument
that will convince your kid sibling, even if (s)he is of
the opposite political party.
Which brings us to the critical points:

A legitimate proof will not prove something that is
not true.
That is the point of a proof: to verify that something
is true.

It takes skill and experience to be able to determine
if a proof is legitimate.
And the lazy person who can't be bothered, or the
unthinking one who just goes by what feels right, is
the lawful prey of politicians, advertizers, and
opportunistic office mates.
When Pythagoras introduced proofs to Europe, he introduced
them as a way to settle what he regarded as religious
issues: the stakes were high.
And in applied mathematics, when the issue is whether the
plane will fly or whether the electrical grid will run,
the stakes remain high.
Secondly, proofs are a mathematical version of a kind of
exercise we've all seen: given an issue, take a position
on it and defend that position.
This kind of assignment used be given in rhetoric
courses, which covered valid versus invalid arguments.
The whole point of rhetoric was to learn how to determine
if political, philosophical, social, artistic, scientific,
etc. claims were true or not: people relying on "common
sense" forget Einstein's characterizion of common sense
as "the collection of prejudices acquired by age eighteen."
And looking at the nonsensical results of "common sense,"
and the inability of sensible people to agree on what
"common sense" says, we can see that Einstein wasn't just
talking about general relativity.
Alas, rhetoric courses have gone out of fashion in the last
half century (we live in sad times!), but these kinds of
rhetoric assignments still show up in content courses.
Let's look at an example.
In honor of
Calvin & Hobbes,
we consider the assignment,
"Was Tyrannosaurus rex a predator or a scavenger?"
(Since vultures have been known to kill their prey, and
since coyotes have been known to eat animals found dead,
this probably has an inbetween answer.
Nevertheless, paleontologists are fighting over this
question, so we can, too.)
The point of the exercise is not the answer  as even
leading experts disagree, your teacher will not be
grading on the answer.
The point is the argument supporting the answer.

Calvin & Hobbes fans may recall Calvin's argument:
Tyrannosaurus rex was a predator because that is more
cool.
Actually, many arguments are such direct appeals to
emotion and prejudice (especially in politics) and,
oh yes, common sense.
The problem is that Mother Nature may not share our
prejudices.
If the point is that to persuade a skeptic with different
notions of coolness, or a skeptic aware of Mother
Nature's own indifference to coolness, one needs an
argument based more on how Mother Nature herself
works.
Aha!
We are beginning to see the reasons why teacher assigned
the exercise.

First, to acquaint students with the sad reality that
the universe doesn't necessarily work the way we want
it to.
And in fact, their are many quite intelligent (and even
cool) people out there who actually believe ...
really! ... that Tyrannosaurus rex was a scavenger.
And some people are not sure.
How do you persuade, or at least sway, such people?
You can't just say that, well, it's just logical
(we'll discuss what "logical" really means in a
moment).
You can't just call them names (although this method
is very popular in contemporary politics: "you're
just being a conservative" "face it, you're a liberal"),
especially if you aren't sure what the names mean
(quick: what's a
conservative?
what's a
liberal?).
Or at least, only stupid people are favorably impressed
by namecalling.
So if you want to persuade or at least open minds, you
need real arguments.
That means you have to learn how to construct real
arguments.
And that is the skill that your teacher is trying to
get you to develop when (s)he assigned Tyrannosaurus
rex.

There is another reason.
Take a look at this description of a BBC program on
the Tyrannosaurus rex debate.
Notice that there is a lot of facts, and somewhat
mechanical arguments based on those facts.
What the nasal cavity tells us about the sense of
smell, what the contraction force of muscles tell
us (mathematical computation time!) about how fast
it could run, and so on.
In other words, if you are going to construct an
argument about Tyrannosaurus rex, you are going to
have to not only study, but know a lot about,
Tyrannosaurus rex.
(In fact, Aristotle said that in a controversial
issue, if you cannot construct a plausible argument
opposed to your own position, you probably don't
understand the issue.
This means that no matter what you are going to say
on the paper you turn in, you will have to study
both arguments.)
So the purpose is to learn how to construct arguments,
to learn the subject you are arguing about, and then
(remember this for future reference when you are
planning to argue about welfare or the Mideast or
pollution) to understand the subject you are
arguing about if you are going to make a good
argument.
All this goes for mathematics as well, and the arguments
in mathematics are proofs.
Notice the difference between the words "argument" and
"proof."
The point is that a proof is supposed to be an absolutely
foolproof argument.
It is absolutely mechanical and there's no way out of it.
This makes mathematical proofs different from arguments
about Tyrannosaurus rex; in fact, mathematics is
probably best defined as the subject for which we
have such proofs.
But the purpose is the same: to verify that something
is true (as opposed to looking true), and to study the
issue to understand what is actually going on.
Let's look at the most straightforward kind of proof:
verifying a computation.
For example, one computational exercise is to come up
with a (nice) formula for 1 + 2 + ... + n.
The proof version of the exercise is: verify that
1 + 2 + ... + n = (n + n^{2})/2.
The proof is just the computation, which the reader
just checks, step by step.
For example,
1 + 2 + ... + n 
= 
2 (1 + 2 + ... + n)/2 

= 
[(1 + 2 + ... + (n  1) + n) +
(n + (n  1) + ... + 2 + 1)]/2 

= 
[(1 + n) + (2 + (n  1)) + ...
+ (n + 1)]/2 

= 
[(1 + n) + (1 + n) + ...
+ (1 + n)]/2 

= 
n(1 + n)/2
= (n + n^{2})/2 
To verify, just check each step.
Of course, sometimes we want to prove something that
is not a formula.
Let's look at a very famous mathematical problem: are
there numbers that are not fractions of integers?
Pythagoras had visited Egypt, and returned with a
lot of mathematics, and a firm belief that all numbers
were rational, i.e., fractions of integers.
His precise argument is unknown (nothing he wrote
survives: he led a religious group dedicated to an
identification of arithmetic and religion), but it
was philosophical if not religious, and may have gone
something like this:
  

Since all things arise out of the first thing, all
whole numbers arise out of repeatedly adding one.
From these whole numbers, we obtain fractions that
represent ratios of arbitrary refinement.
Thus for any real number, we must be able to find
a ratio of whole numbers for it.


  
This does not look like a mathematical proof that
you'll find in any textbook, although many "heuristic
arguments" and "plausibility arguments" look equally
mushy.
(NOTE: A heuristic or plausibility argument can only
serve to tell a reader that the assertion is not
silly; but it is not a convincing argument,
as we shall soon see.)
Indeed, someone  no one really knows who  came
up with a complicated geometric construction with
two line segments, L_{1} and
L_{2}, where the ratio between the
lengths was not a rational number.
So here is a plausibility argument that turns out to
be no good: if we are to be certain, we need a proof.
What does a proof look like?
As an example, let's look at one of the nicer proofs that
the square root of 2 is irrational (recall that the square
root of 2 is a number r such that r^{2}
= 2).
This will be a proof by contradiction, which works like
this.
Suppose that you want to prove that A is false.
If you prove that "if A, then NONSENSE is true"
then, since NONSENSE must be false, it cannot be
the case that A is true, so A must be false.
For example, if "A implies that the Moon is made of
green cheese" then, since the Moon is not made of
green cheese, it must be the case that A is false.
One kind of proof by contradiction works like this.
Suppose you want to prove that A is false.
Set up a situation where some statement H is true.
Prove that "if A is true then H is false."
But since we already knew that H was true, if
A is true, then H would be both true and
false, which is nonsense.
The only way to prevent the nonsense is to require that
A is false.
Notice two things, even before we have gotten to the proof:

Everything is mechanical.
There is no handwaving, nothing about what something
seems like like, or what sounds reasonable.
Everything is rigid, and statements are all being clearly
labelled TRUE or FALSE.

Things are getting complicated.
People are used to dealing with complications by relying
on their intuition (common sense!) to get a sense of
whether something looks, sounds, or feels right.
But here, intuition is not allowed: you have to follow
each thread carefully to determine  no fudging! 
whether something works or not.
Now let's look at the proof.
If A is "2 has a rational square root" we will want
to prove that A is false.
Here is the setup.

Suppose, towards contradiction, that A is true.
Then it has a rational square root r.
This means that there are two integers p and
q such that r = p/q.

Now for H.
If r is rational, allowing us to write r =
p/q, then we could divide both the
numerator and denominator of p/q
repeatedly by 2 until either the top or the bottom
(or both) were odd.
So divide top and bottom of p/q by 2
repeatedly, which doesn't change the value of the
fraction (always equalling r), repeatedly to
get r = p_{0}/q_{0}.
Thus we get: H says "either p_{0}
or q_{0} is odd."
We will prove that if A implies that H is
false, i.e., if r is rational then the fraction
p_{0}/q_{0} is a fraction
of two even numbers, even though we had divided both
top and bottom repeatedly until we got a fraction with
an odd numberator or an odd denominator.
Thus by the original setup, H is true, while by
A, H is false, which is nonsense.
And the only way to get rid of the nonsense is to have
A be false, i.e., to have r be irrational,
so it cannot be expressed as a fraction p/q
of integers to begin with.
Here is the proof that A implies that H is
false.
A lets us write r = p/q, which
(dividing top and bottom by 2 repeatedly) gives us r
= p_{0}/q_{0}, which by
H, has either p_{0} odd or
q_{0} odd.
Square both sides of r = p_{0}/
q_{0} to get
2 = r^{2} =
p_{0}^{2}/
q_{0}^{2},
or multiplying both sides by q_{0}^{2},
we get
2 q_{0}^{2} =
p_{0}^{2}.
Now comes the problem.

If p_{0} is odd, then p_{0}^{2}
is odd, and we have "even number = odd number," which is
disallowed NONSENSE.

The other possibility is that p_{0} is even, i.e.,
divisible by 2.
Let s = p_{0}/2, or 2s =
p_{0}, so
2 q_{0}^{2} =
p_{0}^{2} =
4s^{2}.
But then we can divide both sides by 2 to get
q_{0}^{2} =
p_{0}^{2} =
2s^{2}.
so q_{0}^{2} is even, so q_{0}
is even.
But then p_{0} and q_{0} are both
even, contradicting H, which said that at least one of them
is false, which again leads to NONSENSE.
So both "p_{0} is even" and "p_{0} is odd"
lead to nonsense, so nonsense is unavoidable once we let r be
a fraction of integers.
So r cannot be a fraction of integers.
Two observations about this proof.

It was even more complicated than advertized.
The contradicting NONSENSE "H is true" & "H is false"
was what we got in the "p_{0} is even" case; in
the "p_{0} is odd" case, we got a different bit
of nonsense (an even number equaling an odd one).
Nevertheless, if you carefully traced each step, you
would find that as each step worked (save one, which we'll talk
about in a moment), there is no way to refute the proof, so it
must be true.

There was a big leap: notice when H was asserted, I blithely
assumed that I couldn't just keep dividing top and bottom by 2.
Well, why not?
We can keep dividing real numbers by 2.
Why can't we just keep dividing integers by 2?
The fact that we cannot take an integer and divide it by 2 as many
times as we like, and still get an integer, is a fact that we would
have to prove first, before proving that the square root of
2 is irrational.
Once we had proven that, we would then be able to assert, without a
doubt, that the square root of 2 is irrational.
Now, you may be familiar with integers, and so you might want to
say that of course, if you took an integer and kept dividing by
2, eventually you would reach an odd integer which is not divisible
by 2 (just look at 288: 288/2 = 144, 144/2 = 72. 72/2 = 36, 36/2 = 18,
18/2 = 9, and as 9 is odd, 9/2 is not an integer).
But this is just your intuition backed by experience: it is not a
mechanical proof (how do you know that there isn't out there,
somewhere, a nonzero integer that can be divided by 2
repeatedly?).
To claim that any integer can be divided by 2 only finitely many
times, you need a proof.
The Greeks discovered a curious problem with proofs: whenever
you proved something, you found something that you should have
proved something beforehand.
Aristotle proposed a way to deal with this problem:

Start with a collection of hypotheses, which you assume to
be true, for the sake of argument.
Aristotle divided these hypotheses into axioms or
noncontroversial assumptions, and postulates or
controversial assumptions.

Have a universally agreed upon method of verifying assertions
(called propositions) from previously proven propositions:
once you have proven that A, B, and C are
true, if you can use a legitimate method (like a proof by
contradiction) to prove
if (A & B & C) then D,
we can then conclude that D is also true.

Keep on going until you reach the assertion P that you
were trying to prove.
Notice that an argument consisting of a list of hypotheses, and
then a list of these propositions proven one after another by
universally accepted methods, ending with the desired assertion
being the last proposition, then this argument is checkable:
someone skeptical of the desired assertion can check each step
to make sure it is correct.
And this is what is regarded as a convincing argument in
mathematics.

The book that sold this method for valid argument not
only in mathematics, but in philosophy, politics, science,
law, etc., was
Euclid's Elements,
which starts with lists of axioms and postulates, and then
constructs the mathematics of ancient Greece.
This approach is called the axiomatic method, and
is regarded by many philosophers as the primary form of
rational argument.

Aristotle proposed a more rigorous version of this method:
the universally agreed upon method of verifying propositions
from previous assertions must be a mechanical rule of
inference.
The axiomatic method using rules of inference is called
logic, and when someone claims to have a logical
argument, it usually means that they have constructed
an axiomatic argument using rules of inference (often
using Aristotle's own syllogisms, see his
Organon).
Of course, sometimes someone claiming that something is
logical is either deluded, dishonest, or has watched too
much Star Trek.
There are many philosophers who dispute the claims of the
Axiomatic Method, but usually it is because they contend that
the Axiomatic Method cannot decide certain issues (like the
existence of God) one way or the other, or they are
complaining that certain arguments that look like they
are based on the Axiomatic Method actually aren't, or
they disagree with some of the postulates or rules of
inference.
As the Axiomatic Method makes arguments easier to
check, many philosophers still prefer it, if only for
providing a good format (and as the Axiomatic Method
makes arguments easier to check, crooks and demagogues
tend to dislike it).
And mathematics is the place where one can learn the
Axiomatic Method in an environment where all the issues
are most clear (if rather complicated), where even the
limits of the Axiomatic Method can be made precise, and
where the conclusions to be reached are not so gutwrenching
that students are distracted by fears of reaching unsavory
conclusions.
Admittedly, once one has learned the Axiomatic Method in
a mathematics course, there remains the danger that one
could apply it to issues one really cares deeply about,
and reach unpleasant conclusions.
Logic, like life, is not a safe place.
But the nervous individual who will not or can not use
the Axiomatic Method to at least check arguments is the
lawful prey to advertizers, demagogues, and overwheening
relatives.
The choice is up to you.

