What is a Property?

Date: 11/29/2001 at 11:14:54
From: Jim Lyons
Subject: Properties, what exactly are they?

I understand what Undefined Terms and Defined terms are. I also 
understand what Axioms and Theorems are, but what exactly is a 
Property? Is it the same thing as a Theorem? Also, what is a Law? 
The Distribution Property is sometimes referred to as the Distribution 
Law in Math books.

Thanks,
Jim

Date: 11/29/2001 at 12:10:35
From: Doctor Ian
Subject: Re: Properties, what exactly are they?

Hi Jim,

An axiom is a statement that is assumed to be true in some formal 
system. A theorem is a statement that is provable, given a set of 
axioms and proof procedures.  

An axiom or theorem _in_ a given system is often said to be a property 
_of_ that system. For example, in standard number theory, the axiom

  (a + b) + c = a + (b + c)       for any numbers a and b

might be either an axiom or a theorem (depending on which formulation 
you're using); we call it 'associativity of addition'; and we say that 
'associativity of addition' is a property of numbers.  

In math, 'law' is yet another way to say 'property', but implies that 
you're looking at it in a slightly different way. It's more natural to 
think of a property as an 'axiom' or 'theorem' when you're using it to 
prove more theorems. It's more natural to think of it as a 'property' 
when you're trying to use it to solve a particular problem (that is, 
you're trying to figure out what your next step should be). It's more 
natural to think of it as a 'law' when you're trying to check the 
solution to a problem (that is, you don't want to accept a solution 
unless all the relevant 'laws' have been 'obeyed').

But in the end, they're all just axioms and theorems, and the only 
real distinction is between things that you start with (axioms) and 
things you end up with (theorems). 

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/