Why Does Math Need Proofs?
Date: 03/24/2000 at 18:22:41 From: Lesa Subject: Proofs Why does math need to have proofs? I don't understand the importance of them. Please explain. Thank you for your time, Lesa Schultz
Date: 03/25/2000 at 22:55:45 From: Doctor Peterson Subject: Re: Proofs Hi, Lesa. 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. 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. 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. 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. 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. 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. You can find some other answers to your question by looking in our FAQ under Why study math? http://mathforum.org/dr.math/faq/faq.why.math.html and Proofs http://mathforum.org/dr.math/faq/faq.proof.html The latter includes this link, which is worth looking at: Proofs in Mathematics, Bogomolny http://www.cut-the-knot.org/proofs/ If I haven't fully answered your question, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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