Why do we have to learn proofs!?

TARUN GEHLOT

That’s right.
You are going to have to endure
proofs. Like many of my students, per- haps you are asking
yourself (or me), why do I have to learn proofs? Aren’t they just some esoteric, jargon-filled, technical writing
that only
a professional mathematician would care about?

Well, no. And I’d like to offer
a short justification of this claim. My argument is three-fold:
(1) proofs are all around
you, (2) it’s quite possible to get better
at them by practice and by benefiting from the accumulated
knowledge of two thousand years of mathematicians, and (3) this will really help you in “real life,” whether you go into mathematics, carpentry, or child-rearing.

First and foremost, it is important to know what a proof is.
It is not
one of those
horrible
two-column tables of axioms and deductions you saw in 10th
grade geometry. That is a bizarre (though
sometimes
useful)
invention
of mathematics
educators which constitutes a
particular way to
write
down a very special kind of proof in a very narrow area. No mathematician would be caught dead writing
such a thing1.
A proof is not some long sequence
of equations on a chalk board,
nor is it a journal article.
These things are
ways that mathematician communicate proofs, but the truth is,
proof is in your head.

A proof is an argument, a justification,
a reason that something is true. It’s
got to be a particular
kind of reasoning
– logical
– to be called
a proof. (There are certainly plenty of other, equally valid forms of reasoning. And some of them are even used in “doing” mathematics. But they’re not proofs.) A proof is just the answer to the question “Why?”,
when the person asking the question wants
an argument that
is indisputable, in
the sense
that any person of normal
intelligence
who has enough time could be convinced of it.
“Why is the sky blue?” is a question
any answer to which could conceivably be wrong:
perhaps we will all wake up tomorrow, and
find it orange. It’s
possible,
both physically and metaphysically.
(In fact, in New York City,
this
happens quite
frequently.) On the other hand, if one asks, “Why
does

2 + 2 = 2 × 2,” or, “Why is the area of a rectangle the product
of its side lengths?”, I can give you an answer that’s
impossible
to argue with. If we agree on the premises
– the definitions of the terminology being used – and we agree that, if A
implies B, and A is true, then B is true, then we agree on the conclusion. There’s no way around it.2

In fact, you use “proofs”
all the time. When you reason with
yourself
that
it will be cheaper to buy the larger cans of beans,
you are proving something about the respective prices. When you play that terribly
addictive game Minesweeper, and you see that, oh, yeah, there has to be a mine under
this block, well, you’ve proven something to yourself.
When you up the ante because you’ve got three kings, and you know who’s
got the aces, you’re proving something about the possible hands that your

.

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friends hold. When you know you can’t turn
right on red in at a certain intersection,
you’ve constructed a
proof of the illegality of this move. (It goes something
like this: “Normally,
I have to
stop at a
red light. But I’m
in the right lane, so that
rule is superceded by the state law that right-on-red
is okay.
However, there’s
a sign right there that says it’s not
okay at this intersection. So it wouldn’t
be legal to turn.”3 ) Many, many real-life situations require reasoning that is, at least in part, something that would qualify
as a mathematical proof. At the very least,
mathematical-type reasoning
is a powerful
addition to anyone’s critical
thinking toolbox, applicable in a wide variety
of settings.
Mathematics is also a cornerstone
of the sciences, which in
turn provide a profoundly
useful way to understand and interpret the world around us.

Okay, so what?
Why are you, a Drama major, being subjected to the requirement that you write
down a so-called “proof ” in some crazy
mathematical hieroglyphics about
constructs like integrals and binomial
coefficients that you’ll never see again? The reason is that mathematicians have spent
more than two millenia working out what proofs are and distilling them out of the complex and unruly real world. The idea is that studying the concept of proof will make you better at it in all sorts of situations. We could muck it up by giving you problems
involving other types of reasoning, too, by why not
just
boil it down to the mathematics itself ?
(Word problems are sort of a middle-ground: they invite
you to figure out what sorts of argument
you’re going to
need,
and
you’re supposed
to produce a proof and
then
apply
it
to
the ridiculously idealized circumstances
described.) And the reason that the
proofs you are dealing
with involve mathematical
constructs is that mathematical constructs
are things
that
have precise definitions one
can reason about.
Notwithstanding what your mathematics-obsessed
philosophy professor
told you4,
you can’t prove anything about
thoughts, lightning,
or yodelling.
They’re literally impossible to encapsulate in mathematical language, though often it is possible
to “model” them by idealizing the situation first.

Well, that’s the reason
that
practicing proving things is a good idea, but why must you write it
down? No, it’s not because I’m sadistic and proofs are my instrument
of torture. (Not to say that all
math professors aren’t sadistic, as this
is decidedly
not true.)
Here
are a few real reasons. First of all, because otherwise I can’t tell if you know why something is true!
You haven’t learned anything
if you get the final
answer by luck (or by good peripheral
vision). If I could read
your mind, I could check if you thought through the problem “correctly,” i.e.,
how clear your reasoning is.
But I can’t,
so you would you please
just write down your
thinking
so I can see if you get it?
Second, the very act
of writing it down serves as an
extremely
effective way to cut away the chaff,
straighten out your thinking,
and force you to work out the
details – which often are at the heart of the problem – instead of just producing some sort of vague,
hand-wavy, it-could-be-turned-into-a- proof. Third,
a
written representation of
a proof, even if it’s just a sequence of equations,
is a way to check and see if everything
is kosher with your argument. Finally, and this applies

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particularly to people in the
sciences or engineering, you might have to write down proofs or pseudo-proofs in subsequent coursework or even (gasp!)
real life.

Now, given that I’ve convinced you (haven’t I?)
that proofs are useful, let me assure
you that you can do them.
You do them all
the time. I’ve literally never met a student who couldn’t prove things, although I’ve met plenty of students
unwilling to or lacking the time to learn how to. Perhaps you will need some practice when it comes to reasoning about
such abstract objects that come
up in a math class, but that’s
just
a matter of doing your problem
sets. Furthermore there are lots of proof “techniques” that can
guide the way and make the work almost
formulaic once you get the hang of it.
(Induction comes to mind here.) And
getting the hang of it means practice, as annoying as it is. Believe me, I sympathize with your urge to say, “I get it, why do I have to do it fifty times?” But the truth is,
you don’t know if you get it until you’ve done it at least a few times, and you’re unlikely to retain
it for very long
unless you do it a few more. Very often you think
you know what’s going on, until a problem comes up with a twist in it, and you realize
that
you haven’t quite grokked the big picture
yet.

I hope that explains why you’re being tormented so with proofs. Written proofs are a record of your understanding, and a way to communicate
mathematical ideas with others.
“Doing” mathematics is all about finding proofs. And real life has a lot to do with “doing” mathematics, even if it doesn’t look that
way very often.