What is Mathematics?

Date: 09/22/2000 at 04:40:38
From: Erum
Subject: Definition of maths

Hi,

Please tell me the definition of maths. If there is no single 
definition, is there a group of definitions?

Date: 09/22/2000 at 16:15:03
From: Doctor Ian
Subject: Re: Definition of maths

Hi Erum,

Stripped to its barest essence, mathematics is the derivation of 
theorems from axioms.  

So what does that mean? 

It means that mathematics is a collection of extended, collaborative 
games of 'what if', played by mathematicians who make up sets of rules 
(axioms) and then explore the consequences (theorems) of following 
those rules.  

For example, you can start out with a few rules like:

  A point has only location.
  A line has direction and length.
  Two lines interesect at a point. 

and so on, and then you see where that takes you. That's what Euclid 
did, and ended up more or less inventing geometry. And that's what 
other mathematicians have done over the centuries, inventing 
arithmetic, and number theory, and calculus, and group theory, and so 
on. 

It's a little like what you do when you invent a board game like 
chess. You specify that there are such-and-such pieces, and they can 
move in such-and-such ways, and then you let people explore which 
board positions are possible or impossible to achieve.  

The main difference is that in chess, you're trying to win, while in 
math, you're just trying to figure out what kinds of things can - and 
can't - happen. So a 'chessamatician', instead of playing complete 
games, might just sit and think about questions like this:

  If I place a knight (the piece that looks like a horse, and moves
  in an L-shaped jump) on any position, can it reach all other 
  positions?

  What is the minimum number of moves that would be required to get 
  from any position to any other position? 

But they would also think about questions like this:

  What would happen if I changed the shape of the chessboard?

  What would happen if I allowed some pieces ('ghosts') to move 
  through other pieces as if they weren't there?

  What would happen if I made the board three dimensional, 
  or let pieces disappear for specified periods, or made them 
  appear and disappear at regular intervals (for example, if 
  a rook becomes invisible for three moves, then visible for 
  three, then invisible again, and so on)?  

  What would happen if I allowed more than two players, or let players
  take turns in parallel instead of in sequence?

In other words, mathematicians are interested not only in what happens 
when you adopt a particular set of rules, but also in what happens 
when you change the rules. For example, mathematicians in Germany and 
Russia started with Euclid's geometry, but asked: "What if parallel 
lines _could_ intersect each other? How would that change things?"  
And they ended up inventing an entirely new branch of geometry, which 
turned out to be just what Einstein needed for his theory of general 
relativity. 

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

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