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Undefined Geometry Terms

Date: 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 

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 

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   

   What is a point?   

- Doctor Peterson, The Math Forum   
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High School About Math
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