Math Invented or Discovered?

Date: 10/26/1999 at 09:39:31
From: Tom Demarco
Subject: Origin

Did man create all math? If not, what did? If man created math, then 
all math can be manipulated, and there is no higher 'power' 
controlling math, right? It is whatever we want it to be, right?

Date: 10/26/1999 at 18:02:01
From: Doctor Rick
Subject: Re: Origin

Hi, Tom.

This is related to the age-old question, "Is math discovered or 
invented?" My answer has to be both. We invent new kinds of math by 
setting the "rules of the game." After we have done this, we discover 
all kinds of consequences that follow from these rules.

Thus, we do have a certain freedom in inventing new kinds of math. 
However, once we have done this - set the rules - then only certain 
things are true and other things are false, within the system we have 
set up.

It's like asking, "Since man invented baseball, doesn't anything go? 
Can't I get a fourth strike?" The answer is: "If you want to start 
your own league with different rules (designated hitter, for 
instance), go ahead. But it won't be the same game that I play. As 
long as we're playing in the same game, we have to play by the same 
rules!"

Earlier this year I wrote an answer to another question that might be 
coming from the opposite perspective to yours. My correspondent asked 
whether the existence of unproven and unprovable statements (called 
postulates and undefined terms) in geometry demonstrates that every 
human system must be built upon faith. Take a look at my answer in our 
Archives -- I think it is relevant to your question too.

   Unproven Fundamentals of Geometry
   http://mathforum.org/dr.math/problems/han.05.18.99.html   

You will see from this that as long as we're talking about pure math, 
we are free to try any set of rules and see where they lead. Some 
interesting things can come out of this sort of exploration. But 
freedom ends - and both science and faith begin - when we try to say 
that our brand of math describes the real world.

If you have more to say after you've read my comments, I'd be glad to 
hear it. I haven't really tried to answer your specific questions, 
just to set up a basis for dialog.

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