Why Proofs? Definitions and Axioms
Date: 09/16/2001 at 18:49:30 From: Allie Subject: Geometry, proofs Why is a proof needed? Why do you think proofs are important in the development of a mathematical system such as geometry? Thank you, Allie
Date: 09/18/2001 at 17:30:49 From: Doctor Achilles Subject: Re: Geometry, proofs Hi Allie, Thanks for writing to Dr. Math. Proofs are the foundation of all mathematical systems. Geometry is perhaps the paradigm (or clearest) case of how to build a mathematical system on proofs. To do this, you start with definitions and axioms. (I should note that the example definitions and axioms I give below are by no means official, they are just samples that I am using to illustrate what a definition and an axiom are.) Definitions are things like "A triangle is a set of three line segments joined at the tips" and the like. Definitions are just a way to create vocabulary. They don't actually say anything meaningful, they just give a word meaning. Basically, it's very cumbersome and difficult to write "a set of three line segments joined at the tips" over and over again every time you want to refer to such an object. So instead you make up a word "triangle" for such an object and save yourself ink. Axioms are things that you claim are true about the world. For example, "Given a line and a point not on the line, there is one and only one parallel line that crosses that point." This is not simply a meaning of a term, it is a substantive sentence about the world: it's a claim that something is true. Axioms are usually chosen to be obvious things that people won't dispute. Once you have a set of definitions and axioms you see what "follows" from it; i.e. you assume that your axioms are true, and then use logic to see what else MUST be true. Basically what mathematics does is show that if we accept certain _basic_ facts, then we must also accept certain other _derived_ facts. So if you accept the axioms of geometry, then you have to accept all the theorems (for example that the area of a triangle is one half its base times its height). Proofs are not just for historic significance. Math grows through new proofs. Mathematicians are constantly using proofs to find out new things. In a way, math is about pushing the limits of what we MUST accept if we choose to accept a few things that seem obvious at first. Without proofs there really is no math. I hope this helps. If you have other questions about this or other math topics, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/
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