Undefined Geometry TermsDate: 09/16/2001 at 21:46:35 From: Jake Subject: The undefined terms of geometry I know that they call point, line, and plane the undefined terms of geometry, but is there a way to give those terms a definition? I've been thinking, and you may not be able to give all them a definition, but a line could a line be defined as the inconclusive conjunction (or joining) of two rays going in separate directions. I've never really thought that anything couldn't have a definition, so is it possible for any of these geometric terms to be defined? Date: 09/17/2001 at 12:52:11 From: Doctor Peterson Subject: Re: The undefined terms of geometry Hi, Jake. Your "definition" would require us to first define "ray" and "direction." Can you do that without reference to "point," "line," and "plane"? Think of it this way: Math is a huge building, in which each part is built by a solid line of inference upon other parts below it. What is the foundation? What is everything else built on? There must be some lowest level that is not based on anything else; otherwise the whole thing is circular, and never really starts anywhere. The "undefined terms" are part of that foundation, along with other things like rules of inference that tell us that logic itself is true. The goal of mathematicians like Euclid has not been to make math entirely self-contained, with no undefined terms, but to minimize the number of them so that we have to accept only a few basics, and from there will find all of math to be absolutely certain. Also, the goal is to make those terms "obvious," so that we have no trouble accepting them, even though we can't formally prove their existence. To put it another way, these terms do have a definition, in human terms; we can easily understand what they mean. They simply don't have a mathematical definition in the sense of depending only on other previously defined terms. I search our archives for the word "undefined" and found these pages that may help you: Unproven Fundamentals of Geometry http://mathforum.org/dr.math/problems/han.05.18.99.html What is a point? http://mathforum.org/dr.math/problems/mhooper.8.26.96.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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