Properties and Postulates

Date: 08/04/99 at 11:23:40
From: Kiki Nwasokwa
Subject: Properties and Postulates

When people discover (or create) a property, do they just discover it 
ONCE and then know that from then on it applies to all similar 
situations, or do they just happen to keep discovering the property 
until they decide to call it a property? In other words, how long does 
it take for something to become a property?

COMMUTATIVE PROPERTY:
1. The commutative property of addition seems so intuitive and 
fundamental - (obviously if you have an apple, an orange, and a lemon 
in a box, you could also call them a lemon, an apple, and an orange 
and still be describing the same box) - that it is almost like a 
postulate. What distinguishes postulates from properties such as the 
commutative property of addition?

2. How can people be sure that properties such as the commutative 
property of multiplication, which are not as intuitive as the 
commutative property of addition, work in every case? (I think I found 
a way to prove this property - but I want to know if it is something 
that even NEEDS to be proven.)

Date: 08/20/99 at 12:46:30
From: Doctor Ian
Subject: Re: Properties and Postulates

Hi Kiki,

In theory, once you've discovered a property - that is, proved that 
some theorem is true - then you never need to discover it again. And 
anyone who is playing the same game as you (for example, standard 
number theory) can use your discovery to make more discoveries of his 
own.

But that's in theory. In practice, you would have to publish your 
result, and other people would have to read and verify it.

Gauss made several major discoveries that he wrote in his diary, and 
which only became known decades after his death, after some of them 
had been discovered independently by other mathematicians. 

Similarly, Isaac Newton invented the Calculus in order to prove to his 
own satisfaction that an inverse square law of force would result in 
an elliptical orbit, but he didn't tell anyone until Edmund Halley 
asked him about it many years later. In Germany, Leibniz, not knowing 
what Newton had done, invented it on his own, using a different 
notation.

The Indian mathematician Ramanujan 'discovered' many results by some 
intuitive process that no one understood, but which clearly didn't 
involve the notion of 'proof'. So while he was able to report 
interesting discoveries, some turned out to be wrong, others had to be 
proven by mathematicians who couldn't have dreamed them up, and some 
remain unproven today.

But these are all exceptional cases. Usually, a property becomes known 
when a mathematician either observes or guesses that some kind of 
pattern exists, proves that it does, tells other mathematicians about 
it, and has his proof verified by independent mathematicians. At that 
point, other mathematicians can treat it as if it were 'obviously' 
true.

> The commutative property of addition seems so intuitive and 
> fundamental... that it is almost like a postulate. What distinguishes 
> postulates from properties such as the commutative property of 
> addition?

You're right that many properties seem so basic that it's tempting to 
think of them as postulates. But one of the primary differences 
between postulates and properties is that the number of postulates in 
a given formal system either stays the same or decreases, while the 
number of properties continues to grow.

Properties are knowledge, and knowledge is power, so you want to have 
as many properties as you can find.

But postulates are assumptions, so you want to have as few of them as 
you can get away with.

That's why, for centuries, mathematicians tried to 'prove' Euclid's 
parallel postulate using the other postulates as a starting point. And 
that's why mathematicians find it preferable to prove things like the 
commutative property of addition, even though from a certain point of 
view, proving something so obvious seems like a waste of time.

>How can people be sure that properties such as the commutative 
>property of multiplication, which are not as intuitive as the 
>commutative property of addition, work in every case?

There are two ways to know that something applies in every case. The 
first is to make it a postulate; the second is to prove it as a 
theorem. If you rule out divine inspiration, those are really your 
only choices. Since the commutative properties of addition and 
multiplication aren't postulates, they do need to be proven. 

By the way, I don't agree with you that the commutative property of 
multiplication is any less 'intuitive' than the commutative property 
of addition. Visually, you can represent a sum of two numbers like 
this:

  +--+---+
  |**|***|
  +--+---+

Flip it around, and you get

  +---+--+
  |***|**|
  +---+--+

Since it's the same object, the order of the operation can't matter. 
Similarly, you can represent the product of two numbers like this:

  * * * * * *
  * * * * * *
  * * * * * *

Rotate it 90 degrees, and you get

  * * *
  * * *
  * * *
  * * *
  * * *

Again, since it's the same object, the order of the operation can't 
matter.

But it's important to remember that a picture isn't a proof. A picture 
shows that something is true for one particular case, while a proof 
shows that it is true for all possible cases. When you want to 
convince yourself of something, a picture is often good enough. But 
when you want to prove it to someone else - especially someone who 
might be using it as a starting point for discoveries of his own - you 
have to meet a higher standard.

Also, if you remember how Russell's paradox led to a re-examination of 
the foundations of mathematics, you'll see that we often learn the 
most by trying to really understand the simplest cases, rather than 
the more complicated ones.

I hope this helps. Be sure to write back if you have other questions.

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