One aspect of mathematics that students often struggle with, particularly in geometry (which traditionally has been where proof is introduced), is writing proofs. Why do we need to learn about proofs? Why are proofs needed in the first place? Here are a few answers we’ve given to these questions.
Why does math need proofs?
First, from 2000:
Why Does Math Need Proofs? Why does math need to have proofs? I don't understand the importance of them. Please explain.
We need proofs in math, first, because we want to be sure that what we do is right. There are enough sources of error in our calculations, from imprecise measurement to misunderstanding of the formulas we should use, that it's important to make sure that our thinking doesn't add more error. Proof just means checking our reasoning. In math, unlike science or any other field, we CAN prove that what we do is absolutely right. That's because math is not dependent on partially known physical laws or unpredictable human behavior, but simply on reason. In math, unlike the real world, we set the rules, so we can know everything we need to know in order to be certain what will happen. For example, we can define what we mean by addition, and then prove that if we add b + a we will always get the same as a + b. Since we can do it, we should take advantage of the possibility. Truth is rare enough to value highly.
Math, that is, is abstract reasoning, which is guaranteed to be true as long as the assumptions we make are true. In arithmetic, we start with basic assumptions about how numbers combine, and reach a conclusion that will be true as long as the things we are counting are actually suitable for counting (as opposed to, say, combining drops of water, where 1 + 1 = 1!).
But the reason we really HAVE to prove things is that we can be easily fooled. Some things that seem perfectly reasonable turn out to be wrong. In fact, even if something is true whenever we try it, that isn't enough to be sure that it always will be. The reason we can be sure that a + b = b + a, for example, is not that we've always seen it work that way, but that we can understand what is happening when we add, and know that this rule is a natural result of the way addition works. Often I see students trying to find a formula for some relation (say, the number of diagonals in a polygon) by making a table and looking for a pattern. Sometimes they find a formula that works for the numbers they have; but if you add a line to the table, the formula will no longer work. What they have to do is go back to the way the table is made and see how a pattern will develop naturally. That's a proof, and when you've done that, you KNOW it's right. You don't need to guess and risk being fooled by a false pattern.
In other words, inductive reasoning (“all my examples fit this rule, so the rule must be right”) can fail when we miss counterexamples. Math is deductive reasoning, where we start from assumptions (definitions and axioms that state how something like addition works) and derive further truths from them.
The Greeks were, as far as I know, the first to develop this love of certainty. They saw that math was not just a tool they could use, but a way to build a world of absolute truth, building one fact on another so that they knew they were right. But they weren't perfect. They originally built large parts of their geometrical thinking on the assumption that any two lines could be compared by finding some unit small enough that both lengths were whole-number multiples of that unit; that is, all lines were assumed to be "commensurable." But it was discovered that the diagonal of a square was incommensurable with the side of the square - that is, the square root of two was irrational. That shook them, and forced them to rethink their proofs, since a lot of what they knew was based on a false assumption. They were able to rebuild their geometry (Euclid's geometry incorporated this rethinking) and as far as I know, nothing turned out to be wrong; but the incident reinforced mathematicians' awareness of the importance of really proving everything.
So the Greeks, for whom math was the epitome of certainty, let themselves be fooled into assuming that all (real) numbers are rational. When this was discovered to be false, it taught them a lesson mathematicians have never forgotten.
On the other hand, we can also be fooled in the other direction: there are some things that are hard to believe without seeing a proof. For example, I think it's hard to believe that the Pythagorean theorem should always be true. I need a proof to convince me that I can always use it and it will always work.
This is one reason people have found hundreds of ways to prove the Pythagorean theorem!
A different kind of proof can be useful in saving effort: the existence proof. Sometimes it can take a lot of work to solve a problem; a mathematician may first be able to prove whether a solution exists, without having to do all the work of finding it. That can either save us from bothering to try it, or allow us to work in confidence, knowing there is an answer.
Knowing that something (like trisecting an angle with compass and straightedge) can’t be done saves us the effort of trying to do it. And knowing that something can be done encourages us to put in the greater effort needed to do it.
Not only mathematicians, but you yourself can benefit from learning to do proofs. The skills you develop in learning to prove mathematical statements are useful in many other areas of life. You learn logic, which lets you recognize when a supposed "proof" (whether in math or life) is flawed and shouldn't be believed. See our FAQ section on False Proofs: http://mathforum.org/dr.math/faq/faq.false.proof.html. You learn how to reason carefully and find links between facts. I myself am a computer programmer, and though I don't prove theorems all the time, I'm often checking whether a program will do what I expect, and using those logical skills. Other people, from lawyers to consumers, need to use logic in all sorts of ways.
We’ll be looking at some of those false proofs next time.
For another answer to a similar question, see
Why Proofs? Definitions and Axioms
Next, let’s look at those logic skills.
Why do I need to learn proofs?
Consider this question from 2002:
Why Learn Geometric Proofs? I am writing an analytic paper for my college writing class on how students are taught things that they will 1) never remember and 2) never have any use for in their lives. I believe learning geometric proofs is one of these topics. So my question is, why are we taught geometric proofs if the vast majority of us will never use them?
Doctor Roy answered, starting with a teaser and some parallel questions:
The fact that you are writing an analytic paper is proof that geometric proofs are useful. But I'll start with other examples. One could argue that teaching English in high schools is equally useless, as few people use obscure grammar rules in their daily lives (standard American spoken English is a good example). So why do we teach it? Why do we teach foreign languages in schools when the vast majority of people will never have a need to speak a foreign language? Why teach history or science or anything else when the vast majority of people never use the subjects?
The same could be said of many things taught in schools.He went on to give examples where school teaches ways of thinking, not just specifically useful facts. He concluded:
But back to the point. The idea of an analytical paper is one of analysis. Analysis implies some ordered process, some type of reasoning based on evidence and logic. The notion of logical reasoning is the basis for teaching geometric proofs. If a person can reason through a geometric proof, then he or she can be expected to learn how to reason logically in areas other than math. Often, this is the only exposure most people have to an orderly thinking process. Logical reasoning certainly isn't taught in any of the other traditional high school subjects. And the ability to reason logically is essential to functioning in society. Of course, we aren't usually as rigorous in everyday life, but the concepts are there to be exploited.
Where will I need proofs?
Here’s one more question, focused on a particular kind of proof in geometry but really applicable to all of math. This is from 2000:
The Value of 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.
We’ve seen above that all of math is built on provable facts, not just blind assumptions. But why learn it when you will not be a mathematician? Two of us replied, starting with Doctor Ian, mentioning facts you might discover by experimentation with the program Geometer’s Sketchpad, similar to the more recent GeoGebra:
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.
This is a big difference, because, as we saw above, what we observe from experience can be very wrong.
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?
The Greeks discovered that; but it’s not just in math that this can happen. What if an engineer designed a new car based on assumptions that turned out to be wrong? Would you want to ride in it? We need to be sure of important facts.
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.
Logic in law
Doctor Alicia added her thoughts:
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.
Of course, they don’t really use “two-column proofs”, which are a particular way to organize a proof meant to help beginners be sure that every step of reasoning is justified.
Here is the beginning of a typical geometry proof: Given: Triangle ABC is a right triangle, with right angle C. Prove: Angle A and angle B are complementary angles. Statement Reason --------- ------ 1) Triangle ABC is a right 1) Given triangle with right angle C.
Step by step, we would use the given information to show, in the end, that angles A and B are complementary. (This is a typical very brief proof meant to teach this format of proof; later examples would be considerably more complex and require more subtle reasoning.)
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.
A proper two-column proof would have more steps, with a single reason given for each; here she has condensed many steps of reasoning into each “reason”. The flaw would be more obvious if it were broken apart that way; do you see it? The key word is “later”.
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.
Doctor Alicia concludes:
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.
For some related thoughts, see
How Do Lawyers Use Math?
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