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### The Value of Two-Column Proofs

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Date: 12/19/2000 at 23:10:26
From: Patrick
Subject: Two-Column Proofs

What is the point of doing two-column proofs? I don't plan on using it
in my field. I think proofs are made to turn our brains into mush.
Geometry is shapes and angles, not writing out two-column and
paragraph proofs.
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Date: 12/20/2000 at 12:29:49
From: Doctor Ian
Subject: Re: Two-Column Proofs

Hi Patrick,

You're half right. Geometry _is_ about shapes and angles (and some
other stuff as well), but the point of geometry is to accumulate
_knowledge_ about shapes and angles. And the difference between
_knowing_ something and 'sort of' knowing it is that you can _prove_
what you know.

For example, you might be playing around with Sketchpad, and notice
that whenever you can inscribe a quadrilateral in a circle, the
opposite angles always add up to 180 degrees. But you don't yet _know_
that this is true. You only suspect it. In order to _know_ that it's
true, in _every_ case, you need to prove it.

Mathematicians care very much about proofs, because they need to be
able to rely on each other's results as starting points for new
investigations. Can you imagine the chaos that would result if
everyone assumed that a particular result was safe to use, and then
turned out to be mistaken?

Proving things is a way of thinking, and ultimately the value of
learning various ways of proving things (not just two-column proofs,
but proof by induction, proof by contradiction, epsilon-delta proofs,
and so on) is that it should help you understand the difference
between knowing something and not really knowing it at all.

In a word, learning how to prove things makes it harder for people to
lie to you. There are plenty of people in the world who are lining up
for the opportunity to exploit whatever chinks they can find in your
cognitive armor. If you think you won't mind letting those people jerk
you around like a trout for the rest of your life, then by all means,
leave this whole proof business to the 'experts'. But if you value
your autonomy, and if you want to be able to rely on your own judgment
instead of trusting society to look out for your welfare, pay
attention.

I hope this helps. Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 12/23/2000 at 16:39:40
From: Doctor Alicia
Subject: Re: Two-Column Proofs

Hi Patrick,

I agree that doing proofs doesn't "seem" practical, at first. The
reason we do proofs is to teach ourselves how to think logically. It's
very easy to see that since these two angles "look" equal that they
must be equal. But it takes practice to be able to explain how you
actually arrived at that conclusion.

Take, for instance, a prosecuting attorney. It doesn't seem that this
person would use math in his everyday dealings with the court system
but he does. He uses logic. It may be obvious to everyone in the
courtroom that Defendant A is guilty. He was found at the crime scene,
he had the murder weapon, and he had a grudge against the victim. But
merely pointing out the obvious isn't enough for the attorney. He must
show logically, step-by-step, how the murder was committed. In
essence, he must construct a two-column proof to show how Defendant A
is guilty of the crime.

Here is the beginning of a typical geometry proof:

Given: Triangle ABC is a right triangle, with right angle 3.
Prove: Angle A and angle B are complementary angles.

Statement                         Reason
---------                         ------
1) Triangle ABC is a right        1) Given
triangle with right angle 3.

Here is the beginning of the logic the attorney must use to convince a
jury:

Given: Victim 1 was found murdered in the back yard with a knife.
Prove: Defendant A is the murderer.

Statement                         Reason
---------                         ------
1) The murder weapon (knife)      1) Defendant A's name was engraved
belonged to Defendant A.          in the knife and his finger-
prints were found on the knife.
There is also a receipt showing
that knife was purchased by
Defendant A and a store clerk
can identify him.

2) Defendant A had a motive       2) Phone records show that Victim 1
called Defendant A's house the
night of the murder. Shortly
after that phone call, computer
records show that Defendant A
made a substantial bid on a
piece of artwork on eBay. It was
later determined that Defendant
A lost a substantial amount of
money when the piece of art was
later determined to be a fraud.

Seems to be an open-and-shut case. But the defense points out that the
prosecution made an error in their case: "It was later determined that
Defendant A lost a substantial amount of money when the piece of art
was later determined to be a fraud." That means that Defendant A
didn't have a motive to kill Victim 1 on the night of the murder. The
prosecution had made an erroneous assumption in his proof. It *seems*
that it would logically follow (as in "obvious" geometric proofs) that
Defendant A had a motive, but the two-column proof clearly shows why
this assumption is wrong.

The defense attorney goes on to use his own "two-column proof" to show
why Defendant A couldn't have killed Victim 1. The defense keeps the
same "prove" statement as the prosecution since a defendant *is*
innocent until proven guilty.

Given: Victim 1 was found murdered in the back yard with a knife.
Prove: Defendant A is the murderer.

Statement                         Reason
---------                         ------
1) Victim 1 was killed by a       1) The medical examiner stated
right-handed person.              that the angle of entry suggests
that the killer was right-handed

2) Defendant A is left-handed     2) It is given that Defendant A is
always seen writing left-handed
and is crippled in his right
hand from a hunting accident

Therefore, with the above argument along with no motive, Defendant A
did not murder Victim 1 and the statement is false.

Although the defense's argument is weak, the fact that his proof
didn't make any false assumptions makes it, overall, a more effective
proof.

So, aside from learning geometry (which, by the way, is very helpful
in and of itself), proofs teach us to use deductive and inductive
reasoning. We apply this reasoning and logic to a variety of jobs,
professions, and everyday happenings. I've just shown you one example
above.

I hope this helps. Write back if you have any further questions.

- Doctor Alicia, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School About Math
High School Geometry

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