The Value of Two-Column Proofs

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.

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/