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/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.