What Motivates New Definitions?

Date: 02/06/2000 at 02:42:04
From: Ryan
Subject: What motivates new definitions?

After a brief introduction to the idea of groups, I've been wondering 
why so many mathematical objects -- groups, fields, rings, vector 
spaces -- have nearly identical definitions, i.e., they're sets with 
binary operators that have associativity, commutativity, inverses, 
identities, etc. What's so great about these properties that makes 
them apply to so many different objects? Is it just that this mimics 
the way real numbers behave?

In general, if I wanted to define some new mathematical object and I 
can pick and choose, seemingly at random, what properties it's going 
to have, how would I know how to define it so that I end up with 
something interesting and worth studying?

I guess another way of asking this is: when people create new 
mathematical objects, are they usually trying to solve a specific 
problem or do they sometimes just say "let's define something at 
random, play around with it for a while, and see if anything 
interesting can be done with it"?

If they have a specific problem and can just declare that there 
exists an object that happens to solve the problem (like declaring 
that i exists to solve x^2 = -1), that seems kind of like "cheating." 
Does every new object have to undergo some rigorous proof of its 
existence? My algebra professor was trying to prove the existence of 
rationals by representing division of integers as an equivalence 
relation between ordered pairs of integers. I didn't understand at 
all why being an equivalence relation meant that rationals existed. 
Are integers [I'm not picky enough to want to talk about sets right 
now! :-)] the only thing that are taken for granted to exist and 
every new object must be representable in terms of integers in order 
to exist?

Thanks for any replies!

Date: 02/07/2000 at 00:17:55
From: Doctor Schwa
Subject: Re: What motivates new definitions?

>After a brief introduction to the idea of groups, I've been 
>wondering why so many mathematical objects -- groups, fields, rings, 
>vector spaces -- have nearly identical definitions, i.e., they're 
>sets with binary operators that have associativity, commutativity, 
>inverses, identities, etc. What's so great about these properties 
>that makes them apply to so many different objects? Is it just that 
>this mimics the way real numbers behave?

Actually they are all studying almost the same things: properties of 
integers, real numbers, or integers or real numbers in several 
dimensions. So indeed, the fact that numbers have these properties is 
the motivation for the similarity in these definitions. Then people 
got curious: which properties of numbers are special things about 
numbers, and which are just due to the associativity, identity, and 
so on? What other things besides numbers will also have these 
associativity etc.? So the theory evolved to answer these questions.

Note that groups do NOT have to have commutativity; in fact, some of 
the most interesting and often-used groups (symmetry groups, or 
equivalently subgroups of matrices; permutation groups; lots of other 
stuff too) have a non-commutative multiplication rule.

>In general, if I wanted to define some new mathematical object and I 
>can pick and choose, seemingly at random, what properties it's going 
>to have, how would I know how to define it so that I end up with 
>something interesting and worth studying?

Usually it comes up in some context working with another problem, and 
then gets generalized and abstracted to make a general theory that 
applies to other things as well.

>I guess another way of asking this is: when people create new 
>mathematical objects, are they usually trying to solve a specific 
>problem or do they sometimes just say "let's define something at 
>random, play around with it for a while, and see if anything 
>interesting can be done with it"?

Usually people start with a specific problem in mind and then wander 
off in new directions from there as things come up.

For an interesting example of this (the "surreal" numbers), check out 
Donald Knuth's book (_Surreal Numbers_) and John Conway's books 
(particularly _The Book of Numbers_.)

>If they have a specific problem and can just declare that there 
>exists an object that happens to solve the problem (like declaring 
>that i exists to solve x^2 = -1), that seems kind of like "cheating."

Well, as long as everything's consistent in the new theory, then it's 
not really cheating: in math, as long as it follows from the axioms, 
it's okay. And you can make up whatever axioms you want, as long as 
they are either useful or interesting (and ideally they are both).

>Does every new object have to undergo some rigorous proof of its 
>existence?

In the modern tradition, yes. The past two centuries particularly 
have seen a lot of work on putting things on firm footings: building 
up from sets to integers to rationals to reals to complex numbers, as 
in your example below.

>My algebra professor was trying to prove the existence of rationals 
>by representing division of integers as an equivalence relation 
>between ordered pairs of integers. I didn't understand at all why 
>being an equivalence relation meant that rationals existed.

An equivalence relation is a set of sets of ordered pairs of 
integers, so if you believe that the integers exist, and you believe 
you can put them into ordered pairs, and then put those ordered pairs 
into sets, then the rationals exist too.

That is, if the integers exist, then the set of all pairs (a,b) where 
a and b are integers also exists. Then define the rationals to be 
subsets of that set: for instance, the rational "1/2" is the set of 
all pairs (a,b) where b = 2a (except (0,0) of course!)

This constructs the rationals out of ordered pairs of integers.

>Are integers [I'm not picky enough to want to talk about sets right 
>now! :-)] the only thing that are taken for granted to exist and 
>every new object must be representable in terms of integers in order 
>to exist?

When it comes right down to it, people usually want to build 
everything up from sets. But yes, that's the general idea: you 
believe that someone else has taken care of building some stuff out 
of sets, and then you build your stuff out of what they produced for 
you. For instance, in your class it sounds like you're believing that 
someone else has already done the "dirty work" on the integers, and 
you're going to take that and do the rationals (and maybe next, the 
reals...)

Those were some pretty deep questions about philosophy of 
mathematics, but I tried to answer them in as simple and practical a 
way as I could. I hope my answer helps!

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